Rex Weyler and Bill Gannon - The Story of Harmony

The Story of Harmony

Rex Weyler and Bill Gannon

Justonic Tuning Inc.

Acknowledgements The Authors wish to thank the following people for their kind help: Gordon Rock, Joel Solomon, Dave Rees, Olwen Rees, Ann Rees, Steve Berry, Debera Barager, Mark Deutschmann, Pierre deTrey, Carol Newell, Norine MacDonald, Bob Anderson, Jeff Haley, Linda Gannon, Oliver Gannon, Peter Gannon, Shivaun Gannon, Patty Hervey, Yasha Spong, Yvonne Kipp, Andrew Davis, Kaye Moss, Cynthia Ann Culkin, Bruce Weyler, Doug Weyler, Joanne Weyler, Lisa Weyler, Juliet Eastmen, Maureen Bracewell, Norman Gibbons, Denise Gibbons, Thomas Langley, Shelly Kantrow, Frank Gigliotti, Anne L. Coulombe, Erwin Liem, Ed Gallagher, Art Lilly, Peter Jensen, Steve Dame, Debbie Dame, Tom Jeffries, Robert Cribbs, Fathi Saleh, Paul Smith, Walter Beebe, Jay Katz, Lome Kellett, Paul Horn, Scott Wilkinson, Robert Barstow, Margaret Taylor, David Darling, Erv Wilson, Ben Johnson, John Chalmers, Robert Rich, John Loffink, Kent Ormiston, Chuck Jonkey, Ronald Wells, Rick Ingrasci, Gordy Ryan, Jack Weyler, Ibolya Weyler, Steve Finlay, Lynn Leboe, Paul Barriscale, Brent Flink, Darren Little, Laura Green, Lionel Wilson, Dave Wade, Jean Gascon, Mark Hasselbach, Helen Bonny, Rick Bockner, John Gibbon, Wayne Silby, Stephen James Taylor, Tom Campbell, Stanley Burke, Bill Weaver, Ron Markley, Dave Nedding, Jennifer Malloy, Haresh Bakshi, David Mastrandrea, and Bruce Jacobson. Many others have helped, some whom we may not even know; to everyone, thank you.

Story of Harmony Contents Chapter One: Showdown in Jena ......................................... 1 Chapter Two: The first musical scale ................................. 17 Chapter Three: Pythagoras: music & numbers ................. 23 Chapter Four: Harmonists east and west ........................... 29 Chapter Five: The stream divides ....................................... 37 Chapter Six: A just diatonic major scale ............................ 43 Chapter Seven: The Arabian contribution ......................... 49 Chapter Eight: Keyboards and polyphony in Europe ........ 53 Chapter Nine: As sharp as the ear will endure .................. 59 Chapter Ten: The art and science of sound ........................ 65 Chapter Eleven: The age of pianos .................................... 73 Chapter Twelve: A paradigm entrenched .......................... 81 Chapter Thirteen: The new Harmonists ............................ 87 Chapter Fourteen: Justonic ............................................... 99 Chapter Fifteen: Perfect tuning: so what? ....................... 109 Appendix A: Harmonic overtone series ........................... 117 Appendix B: Derivation and use of just musical scales ... 119 Index: ................................................................................ 143 Notes: ................................................................................ 153 i

The Story of Harmony

1 Showdown in Jena

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t was on a bright autumn day in 1706 that a great tuning showdown took place in the German town of Jena, in the hill region of Thuringia. Two sets of church organ pipes, two renowned music masters, and two musical paradigms clashed on that day. The themes and issues of this clash had been brewing for four millennia and reverberate even today. This great musical debate involved certain material facts about the nature of sound, and therefore could not be entirely subjective, and yet posed aesthetic questions, so could not be solved with pure rationality or science. Indeed these questions defied systemization then, as they do now. World views come and go, aesthetic tastes move in and out of fashion, and yet the nature of sound holds firm.

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The Story of Harmony

These were essential questions, being debated no doubt by choristers waiting along the choir loft balustrade of St. Michael’s church in Jena, questions as critical to any art or time as to music at the turn of the 18th century, for they touched at the very heart of artistic creation, namely: the material application of nature’s gifts to the artist’s vision. The Gothic church of St. Michael had been completed in 1556, a hallenkirche with side aisles that rose nearly as high as the central nave. The church sat at the center of town, and on the main street, inside the fortified walls of Jena on the river Saale. Perhaps some impatient choristers, waiting for the masters to tune the organ pipes, climbed the white stone church tower, where they could see the silver water curl north on its way to the Elbe, which in turn flows through the vast Mecklenburgh plain to Hamburg, and on to the North Sea. Perhaps they wandered into the dirt streets of Jena, where they would hear the singing pipes echo in the green, rolling Thuringia foothills, and turning would see where the river rose to its source in the Bavarian peaks of Frankenwald and Fichtelgebirge 90 kilometers to the south. This was the “Green Heart” of Germany. Forests and lakes surrounded Jena, and vineyards covered the nearby hills. To the northeast, a faint haze of smoke and dust marked the sky above Halle and Leipzig, seats of regional power and learning. Jena had just weathered a brutal 17th century. One hundred years earlier a great flood known as the Thuringia Deluge had wreaked havoc in the city. Hardly had the citizens collected themselves and their few remaining possessions when they were subjected to repeated outbreaks of bubonic plague. The survivors of flood and Black Death were then greeted by the murderous onslaughts of pillaging soldiers from both the Protestant and Catholic armies of the Thirty Years War (1618-1648). Many buildings in Jena still showed the scars of war. Hardly a family in Jena, or in all of Prussia, had escaped suffering and loss. With peace came the harsh taxes imposed by local princes consolidating their power and financing their private armies, sumptuous palaces, lavish feasts, festivals, and hunting expeditions.

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Nevertheless, those who endured were about to achieve perhaps the greatest German academic and cultural achievements in architecture, philosophy, literature, and music. A great cultural awakening was afoot on the Continent, and the town of Jena, and the people of Thuringia, would play a key role.

The fortified town of Jena in the region of Thuringia in 1652. The Church of St. Michael stands in the center of town, site of the great Bach/Neidhardt tuning contest. The view is from the northwest with the River Saale in background.

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The university at Jena, founded in 1548, was becoming an important center of academic study. Its reputation had been greatly enhanced when the learned Erhard Weigel became Professor of Mathematics in 1653. Weigel considered the study of music an essential part of a liberal education, and likely had a hand in bringing to Jena the renowned organist and choir master Johann Nicholas Bach. Nicholas Bach, elder cousin of the 21-year-old prodigy Johann Sebastian, had arrived in Jena in 1690, and had become cantor, or organist and choirmaster, at St. Michael’s six years later. He possessed a profound musical knowledge, including a deep understanding of musical instrument theory and construction. He had designed and built the unique Lautenklavier, achieving a lute timbre with pianoforte action. He had come to influence both the academic and musical life of Jena, which gradually became a center for progressive musical thought under his guidance. As a master tuner, skilled in all the current tuning systems for choirs, keyboard, and lute, he was one of the participants in the tuning contest on this day in Jena.1 Today, for most western music, we use a single tuning system, equal temperament, in which all semitones are of equal size, and modulation among keys is unrestricted. Some a cappella choral groups and chamber ensembles use forms of pure harmonic tuning, or just intonation, and certain progressive composers use extended just intonation. Although most popular music relies on equal tempered tuning, there exists today a movement among western musicians calling out for more flexibility and precision in tuning. At the beginning of the 18th century the situation was somewhat reversed. Musicians in Europe used a variety of tuning systems, and the equal temperament we use today was only a theory. No truly equal temperament was achieved until the 20th century because of the difficulty of tuning such a system by ear. Lute tuning in 1706 was a quasi-equal temperament because frets demand such a system. Choral groups of the time might sing in pure just intonation when unaccompanied, singing both pure thirds and pure fifths, but such a system required that notes shift pitch based on the context in which they were sung.

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Keyboard tuning at this time was either meantone or well temperament, systems with slightly sharp thirds and flat fifths, but in which each key had a unique character because the sizes of these intervals were different. A master like Nicholas Bach was well acquainted with all these systems, the nuances, advantages, and compromises of each. Choir masters like Nicholas Bach knew the beauty and power of a pure triad, with thirds, fifths and octaves all blending in heavenly perfection. But they knew also that to achieve this perfection on keyboard instruments was nearly impossible, and that to do so seriously limited modulation. The more classical solution at the time was meantone temperament, the many varieties of which used some pure thirds, some slightly sharp thirds and sixths, and flat fifths. Meantone temperaments dated from the 16th century, replacing the earlier so-called “Pythagorean” tunings in which the fifths and fourths were pure, and the thirds quite sharp. The physical problem was a simple law of acoustics: a series of twelve pure harmonic fifths did not land precisely on an octave. Likewise, three pure harmonic thirds did not make an octave. In fact, no series of a single pure harmonic interval will precisely match up at any point with another series of a single pure harmonic interval. A string of thirds will never land on the same point as a string of fifths, or octaves, or wholetones. The discrepancies were known as “commas.” Three pure thirds are flat of an octave by a comma almost a quartertone in size. If fifths were tuned pure, the resulting third would be out by about 1/5 of a semitone, an interval known as the comma of Didymus, a fact known since at least the first century AD. Unaccompanied singers could make the necessary adjustments as they modulated, but to make a free modulation system work with the fixed tones of a keyboard instrument, the major thirds had to be sharpened, and the fifths flattened. Indeed, it was not for the sake of singers, nor the demands of harmony, nor even the demands of modulation that drove the centuries-long experiments with tempering the pure harmonies. Rather, the motivation that drove Nicholas Bach, many before him, and many after, was the convenience offered by the two families of musical instruments with fixed tones, instruments employing frets or digital

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The Story of Harmony

keys, namely lutes and organs. Eighty years earlier, the great French theorist Marin Mersenne had stated that the just intervals were “the easiest to sing and most natural,” and that “if it were as easy to mark the just consonances as the imperfect ones, there is no doubt that the performers would do it.” He pointed out, however, that tempering “is done for the reason of convenience,” and added that “nature has no regard for our convenience.”2 The driving force in music theory of the eighteenth century was simply the diatonic keyboard. At this time, German musicians were unsurpassed in the field of keyboard music, and the church organ was esteemed above all other instruments. The three leading composers for the church organ in the early 18th century were Dietrich Buxtehude (c.1673-1707), the expatriate Dane who had spent most of his working life in Germany as organist and choir master at the Marienkirche in Lübeck; Georg Böhm (1661-1733) from Thuringia, and organist at Lüneburg; and Johann Pachelbel (1653-1706) who had written his now famous Canon in the year J.S. Bach was born, and who had just recently passed away in his birthplace of Nuremberg. This great flowering of keyboard music was centered here in northern Germany where the modern style keyboard itself had been born. About 100 kilometers northwest of Jena was the town of Halberstadt. Here, two and a half centuries earlier, on February 23, 1361, organ builder Nicholas Faber had completed the now famous Halberstadt organ. The organ featured three keyboards for various modal scales and accidentals. The upper of the three keyboards featured a diatonic major scale (Do, Re, Mi...) in C, with five “accidentals” slightly raised and behind the main keys. At the time perhaps nothing much was thought of this arrangement of digitals. There were many modes, many tunings, and many keyboard designs in use. There were keyboards of seven tones per octave, twelve tones, seventeen, nineteen, and many others, all arranged in various modes of various scales, and supporting certain natural harmonics and certain temperaments. At the time of the Halberstadt organ, and later, on the day of the tuning showdown in Jena, there was no unequivocally correct or proper way to cast a

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scale and make music. Naturally, there were advantages and shortcomings to all of this. One of the advantages of these many systems was the use of natural harmonics for singers trained in and sensitive to precise harmony. A disadvantage was that all instruments could not necessarily play with each other. Lutes, for example, were often tuned in an approximately equal temperament required by the frets, keyboards were tuned to varieties of meantone or well temperaments, and a cappella choral groups performed in pure harmonic scales. In the search for a unified and simple system, instrument builders in northern Germany and Prussia had settled on the Halberstadt keyboard with its seven diatonic major digitals and its five accidentals. The major scale in C had now, by custom, been represented by white keys, and the five remaining chromatic notes by black keys, still raised and to the rear. In the streets of Jena, in the fall of 1706, no one knew that this keyboard would dominate musical perception in the West for the next four hundred years. No one knew that the winner of the day’s contest would have the most to lose, and that the loser would, at least for several centuries, win. The singers who were there to judge the tunings had only one purpose, and that was music. They were there to judge the purely musical benefits of each system. “Keyboard temperament” was in a transition phase from meantone to well temperament. Meantone was always tuned to a key, or tonality, usually C. In the earliest traditional manner, the E was tuned as a just third (five-fourths the frequency of C), and D was tuned as the “mean” halfway between C and E. The wholetone, D, was also the “mean” or mid-point between the just Pythagorean wholetone and the flatter “minor” wholetone necessary to achieve good harmonies in certain modulations. From this, meantone temperament later took its name, although in the 16th century it was simply called “keyboard tuning.” The system resulted in two wholetone scales on C and C#, but the thirds varied in size, the fifths were flat, and certain thirds and sixths were unusable “wolf” notes from which musicians stayed away.

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In 1523 Pietro Aaron had described in Thoscanello De La Musica3 the meantone temperament of his time. The major thirds C-E and E-G# were made “sonorous and just,” but the fifths (C-G and G-D) were “a little flat,” and A was tuned to make the fifths (D-A and A-E) equally flat. Then C# and F# were tuned as pure thirds of A and D. This tuning resulted in a variety of major and minor thirds. Some minor thirds were nearly just, and the flatter minor thirds were near what today we would call blues minor thirds, or “septimal” thirds based on the pure seventh harmonic. Aaron’s tuning had eight pure or just major thirds, and four “wolf” thirds almost a quartertone sharp. The fifths were slightly narrow, and the fourths slightly wide. By the time of the tuning contest in Jena, meantone had evolved to the point that the thirds were smoothly and progressively sharpened so that they were near just in the common or “natural” keys, (C, F, G, A, Bb etc.), but were still discords in the “distant” keys (C#, G#, etc.). The fifths were flattened as much as “the ear will bear.”4 The advantage of this tuning was that triads in the more common keys were very good, and some modulation was possible. Each key in this tuning had its own character since the semitone sizes were varied. Therefore, a piece of music was written for a particular key, and transposing to a new key could render the piece unacceptable. Within a piece of music modulation was restricted by the wolf intervals. These restrictions led tuners to experiment with temperaments in an attempt to create a tolerable tuning system that would allow free modulation. The acoustic compromise was to slightly widen the thirds in the common keys. These new tunings were known as “well temperaments,” which Johann Sebastian Bach later made famous with the 48 pieces of his Well Tempered Clavier. In 1691 Andreas Werckmeister, the organist at Quedlinburg, near Halberstadt, published Musicalische Temperatur, a treatise on musical tunings including meantone and well temperament.5 He described a well tempered system that included eight pure fifths, and thirds that were progressively sharper as one moved through the keys. The thirds of “distant” keys were sharp, but “tolerable.”6 Well temperament had

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achieved the goal of being able to support free modulation to all keys. The thirds were slightly wider than meantone thirds in the common keys, and increased in size until the largest thirds (usually in B, F#, and C#) were about 1/5 of a semitone wide, about the size of a Pythagorean third resulting from a series of pure fifths. Nevertheless, the orderly progression in the size of thirds was considered an aesthetic advantage. Since the semitones varied in size, each key still had its own unique character, as with meantone, so as musicians modulated they would take advantage of the unique character of each key. Although it was possible to transpose a piece from one key to another, the results would certainly sound different in the new key. Nevertheless, the meantone wolves had been eliminated, albeit at the cost of more compromised thirds and fifths, and free modulation was possible. It was a variety of this well temperament tuning that choir master Nicholas Bach tuned by ear in the Church of St. Michael in Jena on this day in 1706. On his shoulders he carried the tradition of harmonic singing discovered and enriched by the ancients of every culture on the earth; as refined by the Chinese, the Babylonians, Egyptians, Greeks, and the nomadic Semite tribes; the sacred harmonies chanted in the Gregorian Liturgy, the corporal harmonies sung in Athenian theaters, through to the radical dominant sevenths of Monteverdi’s Italian opera; these harmonies that were not quite achievable on the various keyboard systems without compromising free modulation; these mysterious harmonies that could be sung, but would not yield to any mechanical system; these sweet harmonies that everyone could hear, but no one in 1706 could quite explain. His adversary on this day was one of his brightest students at the university, the young, precocious Johann Georg Neidhardt (1685-1739), whose first book had just been published, Die beste und leichteste Temperature des Monochordi (The Best and Easiest Temperament of the Monochord). Neidhardt, 21 years old, had read all the theorists, from Ptolemy to Mersenne and Werckmeister. He advocated the system of twelve equal semitones, a system that had been a theory for over a thousand years. The system was impossible to tune by ear without knowing and

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listening for the precise number of beats in the mis-tuned intervals, a skill not practiced in the 18th century. However, mathematicians had made discoveries that promised to make such a tuning realizable on keyboard instruments. Neidhardt represented the emerging science of mechanics, the rationalist tradition of Descartes and Kepler, the astounding calculus of Leibniz, and the mechanical laws of Newton. The age of science was in full swing. Only twentytwo years earlier, in 1684, Gottfried Wilhelm Leibniz (1646-1716) had published his system of the infinitesimal calculus in Leipzig, a discovery that would have great impact on music. Newton, who published his version of the calculus in London in 1687, had been working on this new mathematics for a number of years, and it was because of this, as well as the fact that Leibniz was known to have been in communication with certain individuals in Newton’s circle, that the two became embroiled in a rather undignified academic controversy over who was the true discoverer of calculus. In any case, the new mathematics of plotting values on exponential curves had immediate application for instrument makers trying to define equal semitones along the frequency curve of the octave. The theory itself was first given voice in about 330 B .C. by a student of Aristotle, Aristoxenus of Tarentum. Aristoxenus had been a follower of the Pythagorean Harmonist school, but had been so baffled by the Pythagorean comma, by the fact that 12 consecutive pure fifths landed sharp of seven octaves, that he proposed distributing the comma among the 12 fifths to make them equal to an octave. Seven hundred years later Ho Tcheng-Tien (AD 370-447) gave approximate string lengths for a scale of 12 equal semitones. His mathematical derivations do not survive, but his maximum deviation from 20th century equal temperament was less than 1/10 of a semitone, and it is possible that he achieved his results by trial and error.7 In 1596 Chinese prince Chu Tsai-Yü gave string lengths for equal semitone temperament which were correct to nine decimal places. Without the aid of calculus this mathematical feat required extracting the 12th root of numbers containing as many as 108 zeros. French theorist Marin Mersenne had

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given accurate approximations of equal semitone fret positions for the lute in 1636. At Quedlinburg, Andreas Werckmeister, who had written on the current practices of meantone and well temperaments, had also investigated the equal semitone temperament, and had used the new calculus to actually plot out precise string lengths and fret placements, marking the intervals that no ear had precisely found in five millennia of musical investigation. Neidhardt studied Werckmeister, wrote his book, and is credited in history as the co-founder of the formula for the equally tempered semitone: the number, which multiplied by itself twelve times, equaled two. The equal tempered wholetone was the number which multiplied by itself 12 times, equaled four. Instrument makers could thus divide an octave into twelve equal parts, and map those equal parts to the frets of a lute or the digital keys of the organ. Werckmeister and Neidhardt held in their hands the new technology that was going to make playing music more convenient. Everyone knew that these twelve precise semitones compromised the purity of harmony. Over this fact there was no debate. Everyone knew the equal tempered thirds were sharp, the fifths just slightly flat. Singers knew that in natural harmony all semitones are not created equally, that the distance between a major and minor third, for example, is not at all the same as the distance between the major third and the fourth. The singers knew that in modulating by a series of pure fifths, one would never land on the original tonality, and that to do so required the little comma jumps. Twelve equal semitones were the wave of the future, not because of their musical merits, but because of their convenience. Most musicians of the 18th century were dead set against such a system. Scottish theorist Alexander Malcolm, born in 1685, the same year as J.S. Bach, wrote “that tho’ the Octave may be divided into 12 equal Semitones ... ‘tis impossible that such a Scale could express any true Musick.” Malcolm added that the temperaments derived “not from the Nature of the System of Musick itself, but the Accident of limiting it to fixt Sounds.”8

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And thus the terms for the showdown in Jena were set. The traditions of harmonic singing were being challenged by the latest mathematical music technology. At the center of the conflict was the 12-semitone keyboard, with its eminent functionality and its inflexible tones. The musical issue was the translation of harmonic tradition to the fixed tones of the keyboard. Nicholas Bach represented the current practice of maintaining as much natural harmoniousness as possible in the common keys, and retaining the character of individual keys, while still allowing free modulation. Neidhardt represented the final and complete compromise of pure harmony in favor of the convenience of keyboards and frets, the new era of music in which every semitone and every key would be identical. The choristers lingering in the church or outside on the street would have heard Nicholas Bach tuning his set of organ pipes. We do not know what standard pitch he might have used as a starting point. German church organ pitch at that time ranged from A=393 Hz in Strassburg to A=495 Hz in Holstein, a difference of more than four semitones. The tuning fork had not yet been invented, and tuners used their own local standard or perhaps their own best sense of pitch. In any case, Nicholas Bach probably began with a middle C somewhere between 235 and 300 Hz. From there he would have tuned a pure octave below, and from there a fifth, G, that was about 5% of a semitone flat of the pure harmonic. Then he would have tuned another fifth, D above that G, also flat by a similar amount, a pure octave D below, and then another slightly flat fifth, the A above, and again the E above that. The flatness of these fifths he would have judged by ear and experience, knowing that the flatter these fifths were tuned, the more harmonious would be the thirds and sixths. He would then have carefully tested certain intervals such as the sixth between C and A, the tenth between the lower C and the E, the third between middle C and E, and the minor triad A-C-E. In each case he would have checked the color effects of various chords and intervals, and made slight adjustments to bring these intervals into the best possible relationship based on his musical experience and aesthetic judgment. He would have made sure that the diatonic major thirds were close to pure. After tuning the entire keyboard, he would have

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checked the potential wolf notes, such as G# - Eb, to make sure they were tolerable. He would then have tested all the major triads progressing by fourths or fifths to determine if the key colorings changed in a gradual and even manner. The result would have been the highest and most refined version of well temperament. Neidhardt had calculated the precise organ stops using the 12th-root-of-2 formula. His job was much easier, more precise, and entirely objective, requiring no listening to intervals nor aesthetic judgments. Neidhardt himself admitted in his writings that “equal temperament carries with it its comfort and its discomfort.”9 No one doubted that this was the case. The advantages were more practical than aesthetic. The predominate musical advantage was the ability to transpose a musical piece to any key without changing its nature. Modulations were exactly even, without variety and key coloring. The major thirds were all equally sharp, and the fifths all equally flat. This was a tuning that was indeed in step with the emerging scientific paradigm. At last the choir was called in and assembled in the loft. They sang a program of material in several popular keys. They modulated through complex pieces, sang the common harmonies of the more simple hymns, and performed each piece in both tunings. Musicians, students, and the curious denizens of Jena crowded into the church, drawn by the beauty of the music as well as by the promise of a good contest. And the conclusion? On this particular day in 1706 the traditional well temperament of choir master Nicholas Bach won. The choristers claimed to prefer well temperament, with the orderly key colorings, even in the distant keys. Singing a variety of thirds posed no particular problem, and the relatively sweet triads in the common keys were considered a great advantage. In fact, equal temperament would not entirely insinuate itself into most western music for another 200 years, but to some extent the die had been cast. The growing popularity of keyboard and fretted instruments would eventually make equal temperament the most practical tuning for instrument makers, although not the most universally loved among musicians.

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Three years after the tuning showdown in Jena, in 1709, Italian harpsichord maker Bartolommeo Cristofori made four gravicembali col piano e forte, the first pianos. Keyboard manufacturers clearly favored the simplicity of the equal tempered scale. The tuning fork was invented in 1711 by King George’s Royal Trumpeter John Shore, partially in response to the need of piano tuners who found they could not tune the equal tempered scale precisely by ear because of the imperfect intervals. Nevertheless, thirty years after equal temperament was calculated by Werckmeister and Neidhardt, both Handel and Bach continued to play in either just tuning, in meantone, or in well temperament. Handel often played a split-key organ with 17 or 19 tones per octave to allow for alternatives to the wolf notes. Bach would often retune his clavichord between pieces when he changed the key. He was very adept at this, but many musicians using fixed tone instruments found this to be a considerable handicap. There is a misconception that Bach supported equal temperament because of his famous 1722 collection, Das wohltemperierte Klavier (The Well Tempered Clavier). Bach's clavichord was not equally tempered, but well tempered, like the title of the work says. The 48 pieces, two in each major and minor key, were written to show the character of each key in this temperament, the effect being completely lost in equal temperament. One can experience this by tuning a keyboard to well temperament, and then transposing Bach’s Prelude and Fugue in C-major to C#-major, and vice-versa. The results will clearly show what the master was up to, and that the pieces were written for the nuances of each key with its particular coloring. Bach's theoretical framework was for pure intonation, and his music was written to be played in pure intonation, either by altering the tuning of the keyboard for each piece, or by using a flexible temperament that allowed pure tones in most popular keys. As instrument manufacturers pushed equal temperament, they were met with considerable resistance by musicians prior to the 20th century. In 1749 Dr. Robert Smith in England referred to equal temperament as an “inharmonious

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system of 12 semitones, [producing a] harmony extremely coarse and disagreeable.”10 In the 19th century William Pole complained in his book The Philosophy of Music that “the modern practice of tuning all organs to equal temperament has been a fearful detriment to their quality of tone. Under the old tuning an organ made harmonious and attractive music... Now, the harsh thirds, applied to the whole instrument indiscriminately, give it a cacophonous and repulsive effect.” Herman Helmholtz observed that “when I go from my justly-intoned harmonium to a grand pianoforte, every note of the latter sounds false and disturbing” and added that “these are unpleasant symptoms for the further development of art. The mechanism of instruments and attention to their convenience, threaten to lord it over the natural requirements of the ear, and to destroy once more the principle upon which modern musical art is founded.”11 A leading modern authority on historical tunings, Owen H. Jorgensen, has written that “temperament developed because of the mechanical impracticability of constructing instruments with more than thirteen key-levers to the octave which also caused increased performing difficulties. Tempering was easier.” He adds that “the acoustical truth is that in history as one quality was gained, another quality or virtue was sacrificed and lost. The gain for equal temperament is a homogenized neutral gray coloring that is completely dependable ...,” and that “western music now exists under the dictatorship of this one homogenized temperament.”12 The fact is, in Jena in 1706, nothing at all was settled. Today we are faced with the tuning question just as surely as it was faced three hundred or three thousand years ago. The nature of sound is unchanging, and the challenge is always the same: to understand sound, to apply the nature of sound to instruments, and to use those instruments to make music. We are now entering a new era of music because we have a whole new generation of musical tools. Just as keyboards and frets were the new musical technology of the Renaissance, synthesizers and digital processing are the new musical technologies of the future. These technologies do not replace the older

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technologies, they simply add new capabilities. It is worth remembering that keyboards and frets were once the new digital instruments, because they digitized pitch selection at a time when string and wind instruments had complete freedom of pitch selection. Keyboards and frets did not replace the old musical instruments, but they added new capabilities. For three centuries, keyboards have driven western music. The digital revolution will change this to some extent, but we do not yet know how all these changes will occur. Pitch selection, however, is one change that is already well under way. We now have the capability to map a wide selection of pitches to a single key. This is the breakthrough that promises to liberate keyboards and fretted instruments from the limits of their fixed tones. But let us go back again to the very beginning of musical understanding, so that we can see how our conventions were developed. It is helpful to understand how sound works, why instruments were designed to support the nature of pure sound, and how certain musical systems were designed to make the most of the instruments available at the time. The showdown in Jena has not even yet been settled, but has merely passed from one generation of musicians to the next. Now, at the dawn of the 21st century, we are greeted once again by these fundamental musical issues. The following historical survey will give the reader some understanding of and appreciation for the musical tuning dilemma that has faced musicians from ancient China and Greece, to Bach and Mozart, and still to our present day.

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2 The first musical scale

T

he earliest human music was almost certainly singing or chanting and drumming, a part of human culture that predates any archaeological record. The oldest evidence of human music instrumentation is a flute fragment with four very precisely drilled holes in the thigh bone of a bear, discovered in Slovenia, and dating from 82,000 to 43,000 years ago, at the time of the Neanderthals, before modern Homo sapiens. According to Bonnie Blackwell at Queens College in New York, the fragment “looks very similar to the bird bone flutes of much later periods.”13 The hunting bow was developed some 20,000 years ago, and may have been the first string instrument, plucked to vibrate the string, and bent to change the pitch. The earliest wind instrument was probably a form of the didgeridoo from Melanesia, a simple hollow log manipulated by the lips and vocal cavity. Bamboo flutes were common in China 6,000 years ago, and Egyptian musicians accompanied the voice with harps, flutes, lyres, and even an early double clarinet.

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Four thousand years ago there were trumpets in Denmark, and the nomadic Hittite tribes carried lutes and other stringed instruments.

A flute fragment that is 42,000 to 82,000 years old. The flute was made by Neanderthal musicians from the thigh bone of a bear. Music is as old as any human art or craft. Knowledge of pure harmonics is at least 4,500 years old, perhaps much older.

Most early human literature is song, the history of tribes passed from generation to generation. The English Beowulf, the near eastern Gilgamesh, and the Hindu Ramayana are all songs, the words of which have been passed down, the original music lost. Song in this early context was the cultural thread by which moral and practical teachings were communicated. The elements of verse, rhyme, rhythm, and melody were all memory aids that insured the accurate transmission

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of cultural knowledge. The earliest extant sources of the Old Testament of the Bible contain musical notation for the singing of verses. In the old Anglo-Saxon poem Widsith, the Song of the Wandering Minstrel, the poet declares: "When Scilling and I with clear voice raised the song before our victorious lord, loud to the harp the words sounded in harmony..."14 We do not really know how ancient the art of harmony might be. We think of early music as monophonic, consisting of a single melodic line without harmonic embellishment. However, the historical record is incomplete, and it seems likely that over thousands of years of singing, humans experimented with all sorts of harmony, unisons, octaves, fifths, fourths, thirds and sixths long before these relationships were given a name. It is certain, however, that the earliest recorded knowledge of human sound indicates an understanding of a fundamental tone to which other tones were related. We tend to think of harmony evolving after melodic forms, but the historical record indicates that the earliest melodic scales were derived from a recognition of the harmonic qualities of sounds. The earliest known scales all reflect the harmonic relationships of octaves, fifths, and fourths. These early musicians discovered that certain tones, when sounded together, created pleasing sounds, and these relationships became known as harmony. Therefore, in a very real way, melody evolved from harmony, although the process was more likely auditory and instinctual, rather than theoretical. Sound’s inherent harmonies were simply strung out into scales and melodies. Today we know something of the physical structure of sound, and we know that these musical relationships are entirely natural. We know that fifths and octaves and thirds are all supported in the harmonic series (or overtone series) of a fundamental tone. A pure harmonic fifth is as much a part of nature as sound itself. Early singers discovered these relationships, eventually discovered the properties of these relationships, and later gave them names. For a more detailed explanation of how scales evolved from the harmonic series, see the appendix, The Derivation and Use of Harmonic Scales. Every musician ought to be aware of the harmonic series, since it is the foundation of music. It is as simple as 1, 2, 3. A

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The Story of Harmony

tone is sounded because something vibrates, a string, a vocal chord, a column of air in a hollow tube. When a string vibrates at a given frequency, say 100 cycles per second, it also has a secondary vibration at 200 cycles per second, and the series continues at 300, 400, 500, and so forth. The higher harmonics tend to grow weaker, but on a good cello string, for example, one can pick out the first 12 or 14 overtones. In the history of human music, the “octave” is the most ubiquitous of all harmonic relationships. We take it for granted, but in its nature it is no different from other harmonic relationships. An octave is the second harmonic after the fundamental tone, and it vibrates at twice the frequency of the fundamental tone. If the fundamental tone is one, the octave has a relationship of two-to-one, which we write: 2/1 This relationship was discovered by all the earliest cultures who developed musical scales. The next relationship they discovered was what we now call the “fifth,” the third harmonic. This harmonic has a 3-to-1 relationship with the fundamental, and a 3-to-2 relationship with the octave. This 3/2 is a pure fifth, a pure G in the key of C, etc. The earliest known musical scale was derived from these natural harmonics, a five-tone scale from China attributed to Ling Lun, musician in the court of legendary emperor Huang-Ti in 2,700 B.C.15 We cannot know if Ling Lun heard overtones, but even if he did, he had no real understanding of the harmonic series. Nevertheless, he clearly understood the concept of a fundamental tone and knew that the tones that sounded pleasing in relation to that fundamental tone had precise numerical relationships with it. He built his scale on a series of 3/2 harmonics. Ling Lun fashioned a set of bamboo pipes, or “lü” measuring 81, 72, 64, 54, 48, and 40.5 units, recording the earliest human understanding of harmonic sound. The pipes were closed at one end, and a player would sound them

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by blowing across the open end. His pipe of 81 units is the fundamental tone, or tonic, and is exactly 3/2 times the length of the pipe of 54 units. By continuing to multiply by 3 and divide by 2, Ling Lun calculated the lengths of the other pipes, and discovered the series of tones that form this early pentatonic (5-tone) scale. This natural pentatonic scale, based on the harmonic of the fifth (3/2), was also discovered or learned in ancient Java, Sumatra, New Guinea, North America, Japan, Babylon, and Greece, and can still be heard in modern oriental music and Gaelic airs. In our system of notation, in the key of C, this scale is: C, D, E, G, A, C These are, of course, the tonic, a whole-tone, a third, a fifth, a sixth, and the octave. Because the scale relies only on 3/2 for all the calculations, both the E and A, the third and sixth, are slightly sharp of modern just intonation. What we call a pure musical third is based on the fifth harmonic (5-to-1) of the fundamental, and has a 5/4 relationship with the tonic. Ling Lun did not yet know this, and his third is based on a cycle of pure fifths. It is sharp of the pure third by what later became known as the comma of Didymus. The Ling Lun sixth is sharp by the same amount. The numerical relationships of Ling Lun’s scale were as follows: 1/1

9/8

81/64

3/2

27/16

2/1

It is unlikely that the sharp third or sixth were ever actually sung by voices unless they were following accompaniment by an instrument tuned in this manner. Thus we see that the difficulty of creating instruments that could imitate the natural intonation of the voice is as old as instruments themselves. Any instrument that fixes the pitch of the notes, such as Ling Lun’s set of pipes, is going to encounter the difficulty of matching the flexible voice. The natural instinct of the voice is to follow the harmonic series. Ling Lun’s sharp third and sixth do not

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The Story of Harmony

follow the harmonic series, but rather are derived from the series of pure fifths. Therein lies the rub, which we are still dealing with nearly five millennia later. Octaves and fifths were known also to the Assyrian-Babylonian culture, and to the Egyptians. The Babylonians may have used an entire 12-tone chromatic scale by 2000 B.C., based on a series of 3/2 fifths, and may have learned it from the earlier Sumerians.16 We do not know if the Mesopotamians discovered these concepts themselves, or if the information filtered across the Eurasian steppe, but by 750 B.C. the Babylonians were writing cuneiform notation for hymns using 5tone, 7-tone, and 12-tone scales. The Greeks would later name certain scale modes after the nomadic Middle Eastern tribes of this period, the Phrygians and Dorians. How much these peoples actually understood about the fundamentals of music we do not know, but it is possible that some theoretical discourse circulated among the civilizations of the first two millennia B.C. The Greek theorists would borrow from and build upon this body of knowledge, and they would make further advances in the tuning of musical scales.

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3 Pythagoras: music & numbers

B

y 700 B.C. the Greeks had developed a diversity of short song styles and scale modes. Itinerant singers, or “rhapsodes,” traveled the countryside with 5-string and 7-string kitharas or lyras, the strings tuned to various modes, and the tuning based on pure fifths. In 650 B.C. famed kithara player Terpander wrote Compositions for voice and string accompaniment, and is credited as the originator of the Greek Mixolydian mode.17 The Greek scholar Pythagoras (c. 540-510 B.C.) traveled to Babylon, Alexandria, and Chaldea, probably learning the numerical proportions of musical harmony in his travels.18 We have no first-hand account of what Pythagoras learned from others and what he may have discovered himself, but he was clearly a brilliant mathematical innovator, and the leading musical theorist of his time.

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The Story of Harmony

Pythagoras is credited with inventing the monochord, a single string stretched over a moveable bridge. By dividing the string into proportional parts, Pythagoras was able to experiment with two-note harmonies, and determine the

The monochord of Pythagoras (top), and the natural harmonics of a vibrating string: 1/1, 2/1, 3/1, and 4/1.

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numerical relationship between them. By moving the bridge one-third of the way along the string, he divided the string into two parts, one twice as long as the other. This was, of course, the octave, 2/1. Likewise, he could divide the string into a 3/2 fifth, and perhaps his own discovery, a 4/3 fourth. (“Fourth,” and “fifth” are modern terms that Pythagoras never used.) Pythagoras was smitten by the perfection of these numerical relationships. Harmony was not a rational calculation, but was rather a natural observation discovered by the acute ears of musicians. Yet these perfect harmonies were shown to have exact numerical relationships. This was a revelation for Pythagoras. He saw a mystical and universal beauty in the perfect numbers of the musical intervals. Aristotle wrote of this in his Metaphysics 300 years later, speaking of the Pythagoreans: "they say that the attributes and ratios of the musical scales were expressible in numbers . . . [and] they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number." Pythagoras founded an entire school of learning and a cosmological view around the perfection of numbers, a view that spread among the Greek city-states and lived long after he himself passed away. His students were taught to live a pure moral life as well as to seek an understanding of the natural laws of heaven and earth. The perfection of the pure harmonic 3-to-2 musical fifth was at the heart of his philosophy. He postulated that this perfection was a sign of the creator’s hand, and a symbol of the perfection of the universe. Western mathematics starts with Pythagoras, and for Pythagoras mathematics started with the perfect musical fifth. We could say then that western mathematics starts with music, or at the very least was motivated by music at its inception. Music and numbers are therefore eternally linked, not due to convention or theory, but because of the very nature of sound. Pythagoras’ pentatonic scale was identical to Ling Lun’s with the thirds and sixths derived from a series of fifths. However, Pythagoras took this series out even farther, and made his most important musical discovery. Pythagoras

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The Story of Harmony

discovered that a series of twelve pure fifths almost landed on the same tone as did a series of seven octaves, thus creating - almost - a cycle of fifths. Ironically, his great discovery was neither the fifth, nor the cycle, but the “almost.” Pythagoras noticed that the twelve fifths did not land precisely on the octave, and this disturbed him. The tiny discrepancy seemed to shatter the perfection he thought he had perceived in the harmonic proportions. He had fashioned an entire philosophy around that perfection, and suddenly his world view had a kink. He calculated the amount that the series of fifths was sharp of seven octaves, and he arrived at this troubling interval: 531441/524288 This unwieldy fraction, which we now know as the Pythagorean comma, is about a quarter of a semitone, a little interval with huge implications. Poor Pythagoras. He got trapped inside his own paradigm. Had he been able to see that this was simply the natural order of things, that a series based on 3 could never and would never match up with a series based on 2, that is, that a series of fifths would never coincide with a series of octaves, he could have saved himself much aggravation, and perhaps saved future musicians a great deal of trouble. But Pythagoras presumed that the twelve fifths should match up at the seventh octave. The perfection of the universe implied that this should be so. The nagging little quarter of a semitone drove him to distraction, and the assumption that it should be eliminated sent music theory on a millennial quest after a scale that would make a cycle of fifths fit inside an octave. For musicians who may think they don’t understand the math, consider this: imagine a waltz rhythm, 3 beats to a bar. Inside each beat, let’s have triplets, so now you have 9 triplet-beats in the bar. Okay? Now imagine a 4/4 rhythm, 4 beats to a bar, and inside each beat let’s sound the back beat (eighth notes), so we have 8 half-beats in a bar. If you assumed that 9 was supposed to equal 8, you’d have a problem, because it never will. No matter how many times you multiply by 3 you

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will never have a number that is divisible by 2. Never. That’s just the way it is. For this same reason, a series of pure fifths will never match up with a series of pure octaves. The commas also appear when attempting to modulate in a scale of pure intervals. Singers, it was known even in Pythagoras’ time, could change key, and instantly sing the new intervals by ear, but the fixed strings of the kithara, precisely tuned to one key, had to be retuned for the new key, a cumbersome maneuver. In modern terms, the problem is this: when singers are singing pure intervals, and then modulate from the key of C (the fundamental tonic represented by the ratio 1/1) up one step to the key of D (the wholetone, 9/8 of C), some of the new tones in the key of D will be different. For example, the pure F note in the key of C is not the same as the F note in the key of D. In the first case the F is a fourth, 4/3 of C, and in the second case it is a minor third, 6/5 of D, which is 9/8 of C. Therefore, the F in the key of D is: 6/5 × 9/8 = 54/40 = 27/20 This is all fifth grade math today, multiplying fractions. The two F tones are not equal because 4/3 does not equal 27/20. The two tones differ by the small fraction of 81/80, that we already know as the comma of Didymus. Modern equal temperament splits the difference and says all F notes are the same. This compromise, however, is the destroyer of pure harmony, and Pythagoras did not rest well with it. Pythagoras and his followers were perhaps fooled by the smallness of these necessary adjustments. They erroneously assumed that the tones of different keys should be interchangeable, since they were so close. It is quite phenomenal that a well trained musician can make these minute adjustments in tuning by ear, and it has been troubling that fixed-tone instruments cannot. We will do well to remember that the theorists like Pythagoras and Ling Lun only calculated and confirmed the musical ear of the singers. They did not

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invent the pure intervals out of numbers. The numbers were revealed to them when they calculated precise methods of tuning the harmonies that singers discovered naturally. Although the tones used in a particular musical scale may vary from culture to culture, and from style to style, the mathematical ratios of the tones, the perfect intervals - 2/1, 3/2, etc. - remain constant. It is one of the great marvels of music that such diverse cultural or personal expression can arise from these simple relationships. Indian ragas, classical sonatas, and modern jazz are all based on these intervals, regardless of whether or not the tuning of the true intervals has been tempered. In fact, for example, the flat “blues seventh” is a tone that matches the seventh harmonic, quite flat of the equal tempered minor seventh. Musicians just naturally find these harmonics that are reinforced by the harmonic series of the fundamental tone. On the other hand, our modern keyboard fifth was not discovered by any musician’s ear. It is simply a slightly altered pure fifth, and its musical meaning is 3-to-2, the third harmonic of the fundamental tonic. Pythagoras may have died a discouraged perfectionist, but in his memory a school of Harmonists flourished. Greek theater and choral music was at a zenith, with song, poetry and dance all a part of a typical performance.

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4 Harmonists east and west

B

oth Greek music and music theory flourished in the third and fourth century B.C. At the theater, the chorus was accompanied by the stringed kithara and lyra, and also by the aulos, an oboe with a double reed and up to 15 holes. The aulos often had two pipes, one being a drone (the fundamental 1/1) and the second playing the melodies. Thus we see that harmony was indeed a part of the early Greek music that we think of as “monophonic.” It is true that the primary structure of this music was melody, but harmony was understood, sung, and played. Pythagoras had added an eighth string to the kithara to assist in modulation and changing mode. In 457 B.C. Phrynis won the music award at the Greek Panathenaic competition with a kithara to which he had added a ninth string to further facilitate changing mode. However, when Timotheus of Sparta added three more strings to make twelve, he evidently stepped beyond the bounds of good taste, for he was viciously mocked by the poet Pherecrates for introducing “weird music,” and unceremoniously booted out of Sparta.19 From this little incident we

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see that the cultural inertia resisting innovation in musical instruments is nothing new. The “traditionalists” always forget that their tradition was once a novelty. Timotheus later settled in Athens where his innovations were tolerated, and where he received encouragement from another innovator, the great dramatist Euripides. At this time a group of Pythagoreans began to call themselves the “Harmonists.” In Greek mythology Harmonia was the daughter of Venus and Mars. Cupid was her brother. She had a son and four daughters with her husband Cadmus, but her grandmother, Juno, mother of Mars, despised Venus and took it out on poor Harmonia by persecuting her children. Harmonia and Cadmus were so

A Greek kithara. Pythagoras added an eighth string to accommodate various scales and modes, Phrynis added a ninth and won the Panathenaic music award, but when Timotheus added three more strings to make twelve he was kicked out of Sparta for making “weird music.”

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distraught by this that they asked to be sent away to the Elysian fields, resting place of the virtuous. Harmonia personified order, perfection, and peacefulness. The Harmonists took her name to symbolize those qualities in the pure harmonies of music. The greatest scholar among the Harmonists was Archytas from the Greek colony in Tarentum, Italy. He was the first in history to notice that the interval of the third that singers naturally sang was not the sharp Pythagorean / Ling Lun third derived by the series of fifths (81/64) but was rather the pure harmonic tone represented by the relationship 5/4. Archytas was not aware of the harmonic series, but his pure third was the fifth harmonic in that series.20 The singers instinctively sang this harmonic tone unless forced sharp by an instrument tuned to the Pythagorean third. Archytas noticed this, calculated the relative string lengths on a monochord, and recorded for history the pure just intonation third. Archytas did not stop there with his observations and experiments, and later added a tone based on the seventh harmonic, an alternative wholetone about a quarter of a semitone sharp of the Pythagorean 9/8, a tone represented by the ratio 8/7. Thus, he is credited as the first in history to recognize and document the natural tones based on the fifth and seventh harmonics. Greek musical scale theory at this time was based on the concept of the tetrachord, the interval of the fourth, and the division of this interval into three subintervals by the placement of two other tones between the tonic and the fourth. These four tones were called the hypate, parhypate, lichanos, and the mese. The first tone was always the fundamental 1/1, and the mese was always the pure harmonic fourth 4/3. The four tones created three intervals, and these were always represented as ratios of whole numbers since the Greek Harmonists understood this to be the true expression of a harmonic interval. The two internal tones gave the tetrachord its particular quality. When the three intervals were two wholetones and a semitone the tetrachord was called diatonic. When the three intervals were a minor third and two semitones the tetrachord was chromatic. A major third and two quartertones created an enharmonic tetrachord. There were many different

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The Story of Harmony

sizes for a “wholetone,” for a “semitone,” or “quartertone,” and also for either a “major third” or a “minor third.” These terms were general, not specific. For example, a wholetone could be 9/8, the sharper 8/7, or the flatter 10/9. A typical diatonic tetrachord might be composed of the three intervals: 16/15

10/9

9/8

These are a semitone, a small wholetone, and a Pythagorean wholetone. These three intervals would create a tetrachord scale with the following four tones: 1/1

16/15

6/5

4/3

These are the tonic, a semitone, a minor third, and the fourth. Musicians unfamiliar with seeing intervals and musical tones represented this way may find the fractions (or ratios) confusing, but it is worth attempting to understand these harmonic expressions, because they are the very foundation of the musical arts. In a more familiar language, the four note scale above, in the key of C is: C

C#

Eb

F

The intervals are then clearly recognized as a semitone and two tones. In our modern equal temperament all the semitones and tones are the same size, but this is not the case in just intonation, or pure harmonic music, and was not the case with the early Greek Harmonists. The tetrachord above (with various sizes of tones and semitones) is found in Babylonian, Greek, Balinese, East Indian, Arabic, and African music. For musicians interested in deepening their understanding of this fundamental scale theory, or expanding their use of scale options, see Divisions of the Tetrachord by John Chalmers.21 The Harmonist Archytas was the first to define tetrachords in all three classical genera, the diatonic, chromatic, and enharmonic. Octave scales were

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constructed by sticking two tetrachords together with a disjunctive tone between them. For example, if we took the tetrachord above, added it to itself with a wholetone in the middle, we would have: C

C#

Eb

F - G

Ab

Bb

C

This is the Greek Dorian (Ecclesiastical Phrygian), and also the East Indian Hanumat Todi, and the Arabic Ishartum. In modern, western style these tones would all be equal tempered tones and semitones, but in the other traditions the precise harmonic intervals can vary. In any case, this is the fundamental form of scale construction upon which our entire western musical tradition was founded. In terms of just intonation harmonic intervals, the above scale could look like this: 1/1

16/15

6/5

4/3

3/2

8/5

7/4

2/1

The Greek modes were created by shifting the tonic within a given scale. In the Greek Dorian mode in C above, if we shift the tonic to the Ab, we have the Greek Lydian, our modern diatonic major scale in Ab. Archytas was also the first to describe the difference between the arithmetic and harmonic division, or “mean,” in the construction of scales. The arithmetic mean was simply the equal division of a string as performed by Pythagoras. This division is spatially equal but not aurally equal. By dividing the wholetone harmonically, Archytas introduced the semitone 16/15 which has stood the test of time as a popular just intonation semitone of choice.22 This harmonic semitone is about a quarter of a semitone sharper than the Pythagorean version derived from a series of fifths. Archytas and the Harmonists can be credited with liberating harmonic music from the Pythagorean series of fifths. They recognized that the semitone, the thirds, and the sixths were harmonic tones in their own right. By instinctively hearing and recognizing the harmonies of the higher harmonics, they set pure

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harmonic music on its way. Archytas himself must have had an extraordinary ear. He literally picked these pure harmonic tones out of the air without any tradition or aid to guide him, and some of his enharmonic tetrachords suggest a keen ear that was able to discriminate among a variety of tiny intervals. Composer Harry Partch has commented that “in a healthy culture differing musical philosophies would be coexistent, not mutually exclusive,” and certainly the era of the Greek Harmonists witnessed this sort of musical openness and curiosity. Both Plato and Aristotle extrapolated harmonic theory and attempted to apply the sense of order and perfection to human morality and governance. Perhaps they carried the harmonic concepts too far, but in any case we can see how powerful was the influence of early music theory on mathematics, philosophy, and political theory. Meanwhile, in the far east, the harmonic tradition of Ling Lun had evolved along lines very similar to the developments in Greece. Chinese musicians had, like Archytas, recognized the tones based on the fifth and seventh harmonics. During the fourth century B.C. a bronze kin (small koto), called the “scholar’s lute,” was tuned as follows: 1/1

8/7

6/5

5/4

4/3

3/2

5/3

2/1

In our nomenclature, in the key of C, this scale would be: C D Eb E F G A C The harmonically interesting feature here is that the intervals all appear in Greek music at this time, although the scale style is entirely distinct from anything one would see in Greek music. This shows that the harmonic fundamentals are universal, but that the musical application is cultural. The distinguishing feature in the scholar’s lute scale is that the tetrachord, the interval of the fourth (4/3), is divided into four intervals rather than the Greek three. This allows for both the

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major and minor third in the same scale. The wholetone (8/7) is supported by the seventh harmonic of the fundamental, and is slightly sharp of the wholetone of Ling Lun and Pythagoras (9/8) which is based on the third harmonic, the pure fifth (3/2). These harmonic discoveries were made almost simultaneously in both China and Greece, and it is safe to say they were made independently. Thus we see that the ears of the musicians in both the east and west were in complete agreement as to the precise harmonic tones that were considered pleasing. Without knowledge of the harmonic series or other physical qualities of sound that would be discovered later, these musicians independently found by ear the just intonation ratios of pure harmony. However, the practical matter of building fixed-tone instruments that allowed free modulation was about to raise its head, and everything would be thrown into question. The Pythagorean notion that a cycle of fifths should fit into an octave would not go away.

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5 The stream divides

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ristoxenus was the son of musician Spintharos in the city of Tarentum, home of Archytas a generation earlier. As a student of Aristotle, he learned the theories of Pythagoras, Archytas, and other Harmonists. However, Aristoxenus became obsessed with the Pythagorean comma, the amount by which a series of 12 fifths was sharp of seven octaves. Aristoxenus subscribed to the notion that this series of fifths should be a cycle, without having to make the comma adjustment, and that this cycle should fit neatly into the octave. He proposed distributing the Pythagorean comma among the 12 fifths to make them fit. His proposed solution, however, put all the fifths out of tune, hardly acceptable to the singers of Greek choral music, the fifth being the most prominent of all musical intervals. The Harmonists were appalled, and as Harry Partch noted two millennia later, “the war was on.” Aristoxenus set out to fashion a theory that would explain and support his position. He was a formidable scholar, author of some 453 published works, including biographies of Pythagoras and Archytas, and his Elements of Harmony

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is probably the earliest extant treatise on Greek music theory.23 In laying the foundation for his theory of tempering the fifths, Aristoxenus claimed that the tones of the scale had a narrow range of acceptability that the ear would tolerate. This, of course, is entirely subjective since the acuteness of the ear varies widely among musicians, and the ear might grow accustomed to anything. On the other hand, the Harmonists argued, a pure harmonic interval is either in tune or it isn’t. A cycle of fifths that fit into an octave, however, required that these intervals be compromised, so Aristoxenus held firm to his conviction that the ear would accept such a compromise. He gave new definitions for fourths and thirds and wholetones. He defined a wholetone as the interval between his tempered fourth and tempered fifth, or simply one-sixth of an octave. A semitone was a twelfth of an octave, and a fourth was a semitone and two wholetones. It almost seems that he was describing equal temperament, although he did not use any such term. Nevertheless, he abandoned harmonic ratios altogether, and described his scales in units that were 1/12 of a wholetone. Therefore his tetrachords were simply divided into 30 parts, and those parts subdivided into intervals. For example, two of his chromatic tetrachords were 4, 4, and 22 parts, or 6, 6, and 18 parts. He claimed that any interval smaller than a fourth was a dissonance, and therefore could be of any size. He allowed the wholetone to be divided “as melody admits of half-tones, thirds of tones and quartertones, while undeniably rejecting any interval less than these.”24 Thus he set a lower limit on the size of musically usable intervals, and a limit on how finely a wholetone could be divided. Aristoxenus clearly understood the music of his time, and some of his tetrachords are fair approximations of those of Archytas. His scholarship was thorough. Still, his logic seems forced, somewhat circular, and his rationalizations are sometimes even contradictory. For example, Aristoxenus maintained that the ear, not numbers, should determine proper tuning. He faulted the Harmonists for relying on numbers rather than on the ear. On this point he clearly misinterpreted the history of harmonic tuning. The intervals of the Harmonists did not evolve

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from mathematics. On the contrary, the intervals were recognized by ear, and the numerical relationships were discovered when theorists from Ling Lun to Archytas applied these intervals to pipes or strings. The ear clearly favors the simple harmonic ratios (2/1, 3/2, etc.), the octave being the simplest and most obvious of these. Aristoxenus also says that “the subject of our study is the question, in melody of every kind, what are the natural laws according to which the voice in ascending or descending places the intervals.”25 Left to itself, the voice places the intervals at the pure harmonics since this is the easiest and most natural. We are again reminded by Aristoxenus’ comment that Greek music was primarily monophonic, with choirs perhaps doubling at the octave. However, we would be mistaken to assume, therefore, that the Greek musicians and theorists did not understand harmony as two tones being sounded simultaneously, for surely they did. That is what the monochord of Pythagoras and Archytas was all about. The earliest notation is melodic, so we can only speculate what singers may have achieved harmonically. It is hard to imagine that with all of the choral singing going on, no one ever sang a fifth or third harmony part, or droned a tonic while the melody was sung, if not by design, then at least by experiment or even by accident. It is equally hard to imagine with all those strings being added to the kithara that no one ever plucked more than one at the same time or sang and played different parts. Almost certainly they did, and in any case, they knew what a simultaneous fifth or fourth or third sounded like. We tend to think of the evolution of knowledge, in this case musical knowledge, as being linear monophonic, homophonic, polyphonic, harmonic - but in fact the evolution of knowledge is never quite so neat. Clearly, early Greek music was monophonic in execution, perhaps with the octave doubling (which is itself a harmony, albeit the simplest possible harmony), but it is equally clear that these early Greek musicians fully understood more complex simultaneous harmony, learned the musical intervals from such harmony on their monochords, and built their melodic scales from these natural harmonic principles.26

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Finally, in some cases, Aristoxenus is just simply wrong. He claimed that the Harmonists “fabricated rational principles, asserting that height and depth of pitch consist in certain numerical ratios and relative rates of vibration - a theory utterly extraneous to the subject and quite at variance with the phenomena.”27 Here, Aristoxenus goes too far. He attempts to throw out the discoveries of Ling Lun, Pythagoras, Archytas, and others. His attempt to temper the fifths to fit a cycle of twelve into an octave is not in itself wrong; it is a subjective approach to music, and has some practical value, but to base this approach on the assertion that the harmonic ratios are “extraneous” and “at variance with the phenomena” of acoustics is wrong. All tempered music is a sacrifice of harmony for a gain in simplicity. Furthermore, to base everything on the octave, 2/1, as Aristoxenus does, and then reject all other harmonic ratios is entirely inconsistent. Meanwhile in third century China, musical theorists were struggling with these same issues. Rather than temper the fifths, King Fang (“King” is a name, not a title) extended Ling Lun’s series of 3/2 fifths looking for the point at which it would match an octave (an impossibility we know will never happen). He calculated the lengths for a series of 60 lü, bamboo pipes like those of Ling Lun. He observed that the fifty-fourth lü was only a tiny bit sharp of a higher octave of the fundamental. The comma between the two was infinitesimal, about 36thousandths of a semitone. Western music theorists did not discover this until Nicolas Mercator proposed a similar system in the 17th century, and later the 53notes-per-octave scale was used by some composers as a good approximation of just intonation. In China, however, the notion of tempering the fifths, although considered, was not practiced. The practice may have even been forbidden by some Chinese rulers who considered such a compromise to be risking the wrath of the gods. There was a belief that altering the harmonies violated natural principles and would lead to social decay. There are stories of emperors traveling the country, requiring local musicians to play for them, to ascertain the stability of the regions.

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In Greece, the tempering theories of Aristoxenus met an immediate challenge by Euclid (c. 300 B.C.), the founder of western geometry. Euclid exposed Aristoxenus’ most obvious errors, and asserted correctly that the harmonic ratios were natural, not theoretical. He demonstrated that six wholetones (six 9/8s) were sharp of an octave (Theory 9 in his Section of the Canon), and therefore showed that describing a wholetone as “one-sixth of an octave” was arbitrary. Euclid showed how to divide an interval geometrically, and demonstrated that the Aristoxenean ideal of dividing a wholetone into “halves,” “thirds,” and “quartertones,” was impossible, since 9/8 cannot be divided into aurally equal parts by ratios of rational numbers. Indeed the war was on. In fairness to Aristoxenus, and in the spirit of co-existing musical systems, his proposed tempering of the fifths was an inevitable theory as later systems of temperament attest. Aristoxenus erred in discounting the pure harmonic ratios as extraneous. Had he been more precise in his evaluation perhaps he would have viewed the tempering for what it was, a practical compromise for the benefit of fixed-tone instruments. It was during this same time, in Alexandria, that another musical innovation was made that would influence music even until our time. Ktesibios of Alexandria designed and built the first keyboard instrument, a pipe organ called a hydraulos. The wind for the pipes was supplied by hydraulic pressure from water enclosed in a container that was fixed with a hand pump. Today, music can be said to be divided into two separate streams. On the one hand there is keyboard and fret based music which has traditionally required temperament, most notably equal temperament. On the other hand, there still exists today pure harmonic music as performed by a cappella choirs, barbershop quartets, certain chamber groups, string and brass ensembles, and as written by many modern composers. These two streams of music are isolated from each other because they use different fundamental tuning systems. The third century B.C. can be seen as the point at which the streams began to diverge. The theories of Aristoxenus, and the invention of the keyboard set music on two separate courses,

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and although many musicians since have temporarily succeeded in merging the two streams, they remain asunder. The 21st century may see these streams merge once and for all.

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6 A just diatonic major scale

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he last three centuries B.C. experienced an information explosion, which from the perspective of the time was quite dramatic. The silk road became the conduit of learning among the Chinese, Indian, Persian, Arabian, Babylonian, Egyptian and Greek cultures. Science and the arts benefited and flourished. Geographers and astronomers shared knowledge and theories. Forms of writing and the recording of those ideas also flourished. An obscure astronomer in Samos, Aristarchus (c.270 B.C.), actually hypothesized correctly that the earth moved around the sun, although he found no believers, and his theories were ignored for 1,800 years until the time of Copernicus, another example of how knowledge is not at all a linear progression. Music and music theory benefited from this intellectual commerce, and the center of western music began to drift across the eastern Mediterranean from Athens to Alexandria, Syria, Byzantium, and Persia. Eratosthenes (276-194 B.C.), director of the Library of Alexandria, known as “pentathlos,” an all-around, versatile scholar, devised a method of extracting prime numbers (1, 2, 3, 5, 7, 11,

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13, 17 ...), wrote history and literary criticism, measured the circumference of the earth with great accuracy, and advanced the musical theories of Archytas. He contributed to the understanding of how acoustics suggest the prime numbers and the harmonic ratios. His contribution to scale theory was to substitute the harmonic minor third (6/5) for the flatter and less singable Pythagorean version (32/27) derived from the series of fourths. Musical notation was now widely used. The oldest extant manuscripts of the Old Testament, dating from around 165 B.C., contain 25 notational signs for the melodic phrases and cadences of Jewish ritual chants. This notation may have evolved from earlier Egyptian demotic characters. Similar musical notation had been developed in Greece, suggesting a common origin in Egypt. Musical notation had also been developed in Han China, and from there had traveled the silk road into the Middle East. In the first century AD the directorship at the Library of Alexandria passed to the scholar Didymus, who advanced the Harmonist tradition. He suggested a chromatic tetrachord (C, C#, D, F) that was more harmonious than earlier versions: 1/1

16/15

10/9

4/3

The interval between the wholetone (10/9) and the fourth is a pure harmonic minor third. Didymus also constructed enharmonic tetrachords and scales in which intervals smaller that a semitone were used. However the actual use of these enharmonic scales was going out of fashion by this time, in favor of diatonic and chromatic scales. Harmonic theory continued to revolve around the harmonic ratios, and the theories of Aristoxenus had claimed no champions. The name of Didymus is forever linked to the comma 81/80, which he noted was the difference between the two wholetones that were used in scales, the Pythagorean 9/8 and the flatter 10/9 used since the time of Archytas. The comma of Didymus, is also the

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distance between the Pythagorean third and the pure harmonic third of 5/4 which singers naturally used. Claudius Ptolemaeus, or Ptolemy, (A.D. 90-168) was a geographer, astronomer and musician in Alexandria a generation after Didymus. He was known among the Greek intelligentsia as megiste, the greatest, and among the Arabic scholars as Almagest, the Great One. He gave the positions of over a thousand stars, devised a system of longitude and latitude for geographic positions, and made maps of the world which were used by Columbus thirteen centuries later. He catalogued all of the tuning systems known at his time, including the tetrachords of Archytas, Aristoxenus, Didymus, Eratosthenes, Babylonian, Arabic, other obscure versions, and his own. He attacked Aristoxenus openly for discounting the harmonic ratios, and for his arbitrary definitions of tones, semitones and other intervals. He was considered by Marin Mersenne (1635) as “the most knowledgeable of all those who have taught us Greek music.”28 Ptolemy’s Harmonics provides us with a very clear picture of the state of harmonic theory in his day. He states flatly that “the aim of the musician consists in keeping the law of canon, or of the harmonic rule, which does not repel the senses at all and which are accepted by the greater part of mankind.”29 Countering Aristoxenus, he states that the natural harmonic ratios display a “beautiful order, and that it is not by chance.” He also faults some of the Pythagoreans for being too fixated on numbers, and says that the ear and the rational ratios are in agreement. He gives one of the earliest classifications of the consonances, placing the octaves in the first category, fifths and fourths next, followed by wholetones and thirds. This classification of levels of consonance is necessarily subjective, and the debate is to continue even until the 20th century. Ptolemy advanced the tradition of Greek scales and modes, and eventually arrived at our modern diatonic scale. The Greek diatonic tetrachord is constructed from a semitone and two wholetones (C, C#, Eb, F), regardless of the precise tuning of these intervals, and there were many versions of the tuning. Ptolemy’s

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version used 16/15 for the semitone, 9/8 for the first wholetone, and 10/9 for the second. These harmonic ratios share the characteristic that the higher number is larger than the smaller number by 1. This feature of harmonic ratios (called “superparticular ratios”) was considered by Ptolemy to be characteristic of the most pleasing scales. The tetrachord resulting from these interval ratios would have been tuned as: 1/1

16/15

6/5

4/3

Adding a second tetrachord of the same construction with disjunctive tone in between yields a version of the Greek Dorian in pure harmonic intervals. Transposing to the Lydian mode of this Dorian scale gives a scale that includes the sixth (5/3) and the major seventh (15/8) which were Ptolemy’s innovation: 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1. This scale is the diatonic major scale that we know as do, re, mi, fa, sol, la, ti, do, the white keys of the piano. Our modern tuning is tempered, although Ptolemy's tuning was pure harmonic. This ratio tuning of the major scale, the “Ptolemaic Sequence,” is the first in recorded history to reduce all the intervals to their lowest natural ratios, achieving perfect just intonation. Another innovation of Ptolemy’s appears in an alternative version of the diatonic tetrachord. Rather than the 16/15 semitone, he used the 12/11 tone which is about a quartertone sharper. Ptolemy may have heard a similar three-quarter tone in Arabic music, but his specific tuning introduces the eleventh harmonic of the fundamental, and thereby implies the seventh harmonic as well. These higher prime number harmonics add new tones to the palette. The non-prime higher harmonics do not add new tones since, for example, the fourth harmonic is simply an octave of the second harmonic, and the sixth harmonic is an octave of the third harmonic. (See the table of harmonics in the appendix).

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Ptolemy was the first to record the observation that when two strings of differing thickness are stretched to the same tension their pitches have the same ratio as their two circumferences. To bring the pitches into unison, the fatter string must be stretched tighter by the same proportion. These observations were helpful in the development of stringed instruments, and were later expanded upon by Mersenne in the seventeenth century. Ptolemy was a great scholar, but not infallible. He disregarded Aristarchus, maintaining that the earth was the center of the universe, and he underestimated the earth’s circumference. This latter error helped convince Columbus that the ocean passage to India was achievable, and led to Europe’s arrival in the western hemisphere. Still, in the end, Ptolemy made significant advances in musical scale theory, settling the just intonation diatonic scales, establishing the fact that the musician’s ear and the harmonic ratios were in agreement, and expanding the resources to the higher harmonics. His work greatly influenced Arabic and European music of the Middle Ages.

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7 The Arabian contribution

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s with early Chinese and Greek musicians, Arabic music was based on series of 3/2 fifths and 4/3 fourths; the fifths and fourths being inversions of

each other. As early as the Sassanides Persian dynasty (226 A.D.), Arabic musicians were aware of the commas between the tones of this system and other pure harmonic tones. They had read and translated Aristotle, Ptolemy, and other Greek scholars, and were therefore up to date with Greek music theory. Their octave scale construction was, like that of the Greeks, based on tetrachords linked together with a disjunctive tone. However, Arabic music took its own course in the Middle Ages, creating stylistic distinctions that persist to the present. Whereas European music drifted toward diatonic scales of tones and semitones, Arabic music embraced the enharmonic or quartertone intervals. Although communication existed both east and west of the Middle East, we do not know how much theory the Arabic musicians developed themselves, and what may have come from Greece and

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China. We have already seen that the fundamental understanding of harmonic intervals was universal. In China, during the fourth century A.D., musician and theorist Ho TchengTien (c.370-447) calculated history’s first numerical approximation of equal temperament. He gave string lengths for 12-tone equal temperament in three base pitches, and his maximum deviation from exact equal temperament was about a tenth of a semitone. His mathematical derivations do not survive, and it is possible that he achieved his result by ear and by trial and error.30 In practice, the Chinese declined to use this equal temperament, and if knowledge of the system traveled to the Middle East, Arabic musicians also declined. Early in the eighth century Zalzal the Lutist revised the Pythagorean system as applied to frets for the lute. Zalzal was a composer and performer more so than a theorist, and he made his adjustments by ear.31 Lute frets were, at this time, made from chord tied about the neck of the instrument, and Zalzal moved his frets to suit his ear. Like Ptolemy, he introduced the eleventh harmonic intervals, but he carried this much farther, using the three-quarter tone 21/11, the quartertonediminished fifth 16/11, and the quartertone-diminished sixth 18/11. All three of these tones are still used in 20th century Arabic music. However his most celebrated contribution was the neutral third, a quartertone between the major and minor thirds, 27/22, known as the Wosta of Zalzal in Arabic tradition. This quartertone third is still used today, and is a distinctive feature of Arabic music. Some Arabic compositions use all three thirds - major, minor, and neutral - in a single piece. In the tenth century Al-Farabi gave directions for a 22-tone scale. The first seventeen tones were calculated on a series of fourths, and he ended up with several unwieldy ratios, such as a slightly flat major sixth of 32768/19683 and a major seventh of 4096/2187, slightly flat of the more harmonic 15/8. It is unlikely any singers would sing such tones unless guided by instruments tuned in this manner. He did construct some theoretical tetrachords that employed the seventh harmonic tones such as the large wholetone, 8/7. He included Zalzal’s neutral third and

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other eleventh harmonic tones in his 22-note scale, and added a very sharp major seventh, 64/33. He then advanced to the next prime harmonic, the 17th harmonic, and included the semitone 18/17. This interval is almost an equal-tempered semitone, about one percent of a semitone flat, and was therefore sometimes used as the interval for lute frets; a series of 12 of these semitones would almost make an octave, and with the slightest of fudging would do so. Al-Farabi also calculated a forty-tone arithmetic division of the string known as the Tanbur of Baghdad. In his book Al Musika Al Kabir, he defined the four main Arabic regular tetrachord types as Agam, or major (tone - tone - semitone); Nahawand, or minor (tone - semitone - tone); Rast (tone - 3/4-tone - 3/4-tone); and Higaz (semitone minor third - semitone).32 Any two of these tetrachords (in any particular tuning) can be joined by a disjunctive tone to create a scale. From any such scale, all modes or transpositions of the tonic are possible, creating a wealth of scale options still used in Arabic music today. As polyphonic music was developing in Europe, Arabic musicians continued in a homophonic tradition, and built up this sophisticated system of modes. In the thirteenth century Safiyu-d-Din devised tables of tetrachords and their inversions. He also gave divisions of pentachords (the interval of the fifth), which suggests origins independent of the Greeks, possibly the Babylonians or Chinese. His Zirafkend Bouzourk pentachord is: 1/1

14/13

7/6

6/5

27/20

3/2

In modern nomenclature, in the key of C, this would be: C

C#

Ebb

Eb

F

G

The Ebb is what we might call the “blues third,” a very flat minor third, but not quite a quartertone, formed on the seventh harmonic. The C# would sound a

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bit sharp to modern ears; it is a linear division of the Ebb in the manner of Didymus. The F would sound as a disturbingly sharp fourth, more of a theoretically derived tone, than a practical vocal tone, but the other tones are all quite singable, and the scale clearly has an Arabic feel. In the fourteenth century, Mahmud Shirazi and Abdul Kadir formalized the Maqamat, or set of modes, as still used today. The four regular tetrachords patched together create 84 unique Maqams. The modern tuning is consistent with that set out by Al Farabi. Some of the modes use the leading tone, the major seventh leading to the octave, a feature that was also becoming popular in European music. Arabic composers also learned to employ irregular tetrachords such as enharmonic versions with two quartertones and a major third, and developed the practice of changing scale or mode within a piece of music. What Arabic music lacked in polyphonic development, it gained in scale variety. Arabic lute tunings, particularly the 18/17 near-equal semitone, would later appear in Europe.

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8 Keyboards and polyphony in Europe

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hroughout the Middle Ages in Europe, as polyphonic music inexorably matured, musicians and theorists struggled to sort out the harmonic and acoustic issues that they were only beginning to understand. Harry Partch pointed out that the four central harmonic ideas that would dominate discussion for the next thousand years had already been articulated by the time of Ptolemy, and they were: (1) the harmonic proportion, (2) the arithmetic proportion, (3) the Pythagorean series of fifths, and (4) the Aristoxenean tempering.33 Not until the late ninth century (Regino, Abbot of Prüm, De harmonica institutione) and early tenth century (Hucbald, De harmonica) do we have clear reference to simultaneous harmony as a feature of musical performance. As discussed earlier, we know that musicians and theorists had heard simultaneous harmony much earlier, on the monochord and other instruments, and that melodic scales were constructed from the tones revealed by such harmony, but prior to the development of the early parallel organum in the ninth and tenth centuries we have no record of harmonic musical performance other than perhaps octave doubling.

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In the 11th century, Guido d’Arezzo introduced solmization for the eight notes of the diatonic major scale, that is: “ut, re, mi, fa, sol, la, ti, do,” although his own hexachord for singers did not include either the major seventh nor the octave. At that time, only three intervals were considered consonant, the octaves, fifths, and fourths plus the octaves of these (11ths, 12ths, and double octaves). Guido wrote that “these three intervals blend in organum congenially and smoothly ... hence they are called ‘symphonies,’ that is, compatible unions of notes ...”34 The exclusion of the major third could be explained by the possibility that the tuning was still Pythagorean, and therefore the thirds were sharp of the harmonic. The sharp Pythagorean third will beat - we know now - not only with the fundamental’s fifth harmonic, but with the combination tones created by these simultaneous tones. Therefore, if the thirds were being sung or played in Pythagorean tuning the sonorous effect would certainly be more rough than the pure 3/2 fifths in resonance with the harmonic and the subsequent combination tones. All of this reasoning was unknown in the 11th century since the discovery of the harmonic series and combination tones was still seven centuries away. However, it is also possible (and even likely since they had been known for well over a thousand years) that some singers were singing pure 5/4 thirds, and that these dyads were just simply not recognized yet as consonances. However, by the close of the 11th century thirds were appearing in the harmony parts of free and contrary organum. The keyboard instruments were also beginning to have an impact on musical styles. By the end of the tenth century the Winchester Monastery in England had an organ with 400 pipes. It was also about this time that a technological advance in balanced levers linking keys to hammers made keyboard instruments more practical. By 1250 the portatio, a small, portable organ, had been designed and built. The new and better keyboards allowed for rapid polyphonic playing. Hence by the mid twelfth century melismatic organum could feature several notes in the harmony part against a single melody line. In the 12th century John of Garland, Franco of Cologne, and other writers were admitting both the major and minor thirds as “imperfect consonances.” These were

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most likely pure harmonic major (5/4) and minor (6/5) thirds. Franco of Cologne also mentions the major and minor sixths as “imperfect dissonances,” and the semitone, tritone, and sevenths as “perfect dissonances,”35 showing an acute awareness of the harmonic qualities of these intervals. We get confirmation that singers were singing pure thirds, rather than Pythagorean thirds, from the English Monk Walter Odington (c. 1240-1280). He says that singers in the faux bourdon vocal tradition intuitively used the pure ratio intervals and not the Pythagorean intervals.36 This would require altering some melodic intervals to accommodate the vertical harmony, and indicates that singers were making these melodic adjustments in favor of pure simultaneous harmonies. Odington includes the major sixths as “discordant concords” and also mentions the major chord, the earliest record of this triad being mentioned. Throughout the Middle Ages, there was a secular music tradition as well as ecclesiastical and scholarly ones, and among these there was a cross influence in song styles during the development of polyphonic music in Europe. Celtic, Germanic and English minstrels; French troubadours; German minnesingers and Meistersingers; and the trouvère poets of Charlemagne’s court all likely contributed to the rise of polyphonic music through stylistic experimentation. There may well have been more experimentation with polyphony among some minstrels than among certain church musicians since Pope John XXII was still forbidding counterpoint in church music as late as the mid-fourteenth century. The text of his Papal decree of 1322 makes it clear that secular music was influencing polyphonic development. In his tirade he bemoans the fact that “certain disciples of the new school... truncate the melodies with hoquets, they deprave them with discants, sometimes even they stuff them with upper parts made out of secular songs.”37 The pope forbids taking such liberties, but allows “occasionally, and especially upon feast days,” the use of octaves, fifths and fourths, so long as they do not distract from the melodies. Thirds and sixths were not to be sung in the church.

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Chinese musicians had, like the Europeans, discovered the pure thirds. The Tang Dynasty court orchestra in 619 A.D. included over a hundred instruments (flutes, gongs, bells, kotos, and other string instruments) tuned to pure harmonic ratios, including the 5/4 harmonic third. Chinese opera began at this time, predating European opera by a thousand years. By the ninth century, the Persians and Chinese had both developed bowed instruments, again predating Europe by some six centuries. The erhu, developed during the Sung Dynasty (960-1279), with two strings, the bow clasped between them, and a sound box covered in snake skin, is still used today. In Europe, by the 14th century, as counterpoint styles were developing, both the major and minor sixths were considered as “imperfect consonances.” In Ars nova, Philippe de Vitry (1291-1361) includes the sixths and states that both the major third and sixth are more consonant that the minors. Four part canons and motets were already well established, and Guillaume de Machaut wrote a four part Mass. In singing these pieces, however, musicians came face to face with the challenges of pure harmony. Singers knew that when they modulated through a piece of music, notes of the melodic scale would have to change pitch to balance the harmony. This changing of the musical key was not a problem when the voice was unaccompanied. However, when keyboards or fretted instruments accompanied the voice problems did arise because of the fixed tones. The fixed-tone instruments could only be tuned to play in one key. Some keyboard makers inserted the extra notes needed between the keys of a single scale. These "split-key" keyboards would sometimes contain 17, 19, or even 31 tones per octave instead of 12. However, this was only a partial solution, allowing musicians to play pure tones in some, but not all, keys. These systems were also cumbersome to build, and complicated to play, and this put pressure on musicians to compromise pure harmony for the sake of these instruments. Then in 1361, Nicholas Faber completed the Halberstadt organ with its diatonic major scale on the lower digitals and the chromatic accidentals on the

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raised digitals of the upper keyboard. This humble little keyboard would come to dominate western musical thought for the next six hundred years, right up until our time. The tuning of this instrument was likely Pythagorean, and possibly featured some tempering of the fifths.

The first modern style keyboard, built by Nicholas Faber in 1361, one of three keyboards for the organ in Halberstadt, with the diatonic

scale in front and raised accidentals behind.

(From Michael Praetorius' Syntagma musicum, 1618).

Organ players had already experimented with tempering a justly tuned instrument by simply lopping off some pipes and extending others. Once thirds and fifths were routinely played together in triads, and once fixed-tone 12-notesper-octave keyboards were established, some method of tempering seemed

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inevitable. However, the equal tempered tuning that would, in the 19th and 20th centuries, be wedded to this diatonic major keyboard had not yet been realized. The dream of Aristoxenus, of fitting twelve fifths into an octave, was still only a theory. The stage was thus set for the 18th century showdown in Jena which was really just a single skirmish in a protracted battle between acoustic perfection and practical convenience. The particulars of the musical and acoustic issues may have been complicated, but the musical choices were simple. Musicians playing polyphonic and harmonic music from the Renaissance to the 20th century could choose among the following: (1) play pure harmonic music, but don’t use fixed-tone instruments; (2) play pure harmonic music, but add extra digitals to the fixed-tone instruments to allow for enharmonic shifts of some notes; (3) temper the notes in favor of relatively pure triads in the common keys, and accept some unusable wolf intervals in the distant keys; or (4) equally temper all the notes, and accept the compromise of all harmonic intervals, except the octaves. It was not until the end of the 20th century that a new option would present itself, as instruments were able to take advantage of electronic and computer technology: (5) build keyed or fretted instruments in which the tones of all notes can change as needed. Debates over the first four of these options would dominate intonation theory and music instrument design for the next 500 years following the completion of the organ at Halberstadt.

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9 As sharp as the ear will endure

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he growing complexity of music from the Renaissance motets and hymns of John Dunstable and Guillaume Dufay, to the Italian operas of Monteverdi, culminating in the brilliance of J.S. Bach, coupled with the growing popularity of keyboards, seemed to make some sort of temperament inevitable. The 15th through the 17th centuries witnessed a long running debate on the relative merits of pure harmony and tempered harmony, and what form such temperament should take. All of this, as we have seen in Chapter 1, led to the tuning showdown of 1706 between Bach’s cousin and the young man credited with the co-discovery of the equal tempered solution, Johann Georg Neidhardt. We know from Franchino Gafori’s Practica musica in 1496 that organists commonly tempered the fifths and thirds. The organ at St. Martin’s cathedral in Lucca had split keys with separate digitals for D# and Eb, and for G# and Ab. We learn from Bartolomeus Ramis de Pareja’s Musica practica in 1482 that the break had been made with Pythagorean thirds that Ramis claims were “tiresome for

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singers.”38 And just as the singers were bringing the thirds into harmonic tune, the keyboards were forcing them back out. The first certain voice of meantone temperament was that of Arnolt Schlick (1455-1525), the blind organist for Count Palatine at Heidelberg. In his Spiegel der Orgelmacher und Organisten in 1511 he gave a tuning formula that was an early form of meantone. He also mentioned the practicality of split-key organs as a solution. Twelve years later Pietro Aaron in Venice described his meantone temperament with “sonorous and just” thirds, and fifths “a little flat.” Then, in Brescia, Italy, Giovanni Maria Lanfranco gave tuning rules that approached equal temperament in which the fifths are tuned so flat “that the ear is not well pleased with them,” and the thirds “as sharp as can be endured.”39 This description can hardly be interpreted as an endorsement for the system, but it does give us some idea of how sensitive ears first reacted to the tempered intervals. In the meantime, the performing musicians were charging ahead with stylistic advances, variations, ornamentations, diminution, augmentation, modulation, inversions and retrograde motion, all made possible by the relative simplicity of keyboards. It was about this time, mid-16th century, that Henricus Glareanus misnamed all the Greek modes when assigning names to the 12 ecclesiastic modes. It was also the age when violins, previously considered crude folk music instruments, were becoming accepted among serious composers and performers. In 1555 Don Nicola Vincentino built a harpsichord-like instrument that he called an Archicembalo with 31 notes per octave arranged on six ranks of keys. The first rank consisted of the 7 white keys of the diatonic major scale, on the second rank were the five black keys, and above those seven alternative black keys, forming the 19-tone octave: C, C#, Db, D, D#, Eb, E, E#, F, F#, Gb, G, G#, Ab, A, A#, Bb, B, B#, C

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Ranks four and five were “enharmonic” tones with the purpose of reviving ancient Greek scales employing quartertones. The final rank repeated the diatonic scale of the first. Vincentino failed to interest his contemporaries in bringing back the Greek enharmonic genera, but the 19-tone scale gained some support, and was revived a generation later by Michael Praetorius in Germany. Franco de Salinas (1513-1590), another blind organist, thoroughly examined the growing tuning problem, and investigated, by ear obviously, all the known or hypothesized solutions for 12-fixed-tone instruments which, it is interesting to note, he calls “artificial instruments.”40 He set out the rules for meantone which had already been formulated by Schlick and Aaron before him. He noted that for the thirds and sixths to be “made sweeter,” the fifths had to be flattened. For fretted instruments he proposed equal temperament, at least in theory, but not by giving precise formulation. He proposed for the placing of frets on viols that “the octave must be divided into twelve parts equally proportional.” And finally, he proposed a 24-tone enharmonic matrix based on just intonation which allowed some, but not all, pure major and minor triads.41 The theoretical battles of this era are best witnessed, however, in the great clash between Gioseffo Zarlino, choir master at the Basilica San Marco in Venice, and Vincenzo Galilei of Florence, a former student of Zarlino, a lute player, and the father of the more famous Galileo Galilei. Zarlino maintained that the natural voices of the singers demonstrate the true intervals, and that those intervals for the diatonic major scale are represented by the Ptolemaic Sequence (1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1). He reconfirms that singers did not sing Pythagorean thirds. He pointed out that arithmetic and harmonic proportions could coexist in the same musical schema, and he showed how the number series 1-2-3-4-5-6 created both the major and minor tonalities. Zarlino designed keyboards of 17 and 19 tones per octave, but also accepted temperament for keyboards and lutes, with the understanding that such temperament compromised the consonance of intervals and triads. He described meantone temperament, and the Euclidean geometric construction used to calculate the mean proportional.

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Zarlino pointed out that “every composition, counterpoint, or harmony is composed principally of consonances. Nevertheless, for greater beauty and charm dissonances are used,” and that “the ear not only endures them but derives great pleasure and delight from them ...”42 Here Zarlino was referring to pure harmonic

The lute, with its fixed frets, limited the flexibility of the player, and demanded some form of temperament.

(From Marin Mersenne's Book of Instruments, 1636).

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intervals, not tempered intervals. He was a choir master with a keen ear, and although he investigated all the options, he clearly favored the pure harmonies that the voices intuitively and naturally found. Vincenzo Galilei attacked the theories of his former teacher in his 1580 Dialogo della musica antica et della moderna. He claimed that the voice does not teach the true intervals, but rather the voice is taught correct intervals by singing to instruments properly tuned by correct theory. Vincenzo Galilei’s version of correct theory was that of Aristoxenus, a series of tempered fifths that would fit into an octave. He maintained that art does not simply follow nature, but improves upon nature, and he devised a system for tuning the lute that he claimed created twelve equal semitones. To begin, he divides the string length into 18 parts, and the first part from the nut he marks as the first fret. (This is the arithmetic proportion, and gives an 18/17 semitone, flat of the Ptolemaic 16/15.) The length from this first fret to the bridge he again divides into 18 equal parts and marks the first part as the second fret. This procedure he continues until he has marked off 12 semitones of proportionally equal size (18/17). “This brings me to the midpoint of the whole string; the first and lower octave thereof I find I have divided into twelve equal semitones and six tones, as said by Aristoxenus.”43 Vincenzo must have been overwhelmed with excitement at his discovery. Unfortunately, his math did not quite add up, and his “midpoint” was flat of the true midpoint or octave of the string. His semitone is the same 18/17 semitone used by Arabic lutists, but it is slightly flat of a truly equal semitone, and this error is compounded over the twelve divisions. Vincenzo’s octave is flat by about one-eighth of a semitone. Zarlino, clearly the better mathematician in addition to his keen ear, wasted no time in publicly replying to Vincenzo and pointing out his error. He published a “Musical Supplement” to his earlier Dialogo, setting the record straight, and reasserting his position that the natural voice and the true harmonic ratios are in agreement. Commenting on this controversy in the nineteenth century, German scientist, musician, and authority on acoustics Hermann Helmholtz gave Zarlino

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credit for reintroducing "the correct intonation," and added that "singers were then practiced with a degree of care of which we have at present no conception. We can even now see from the Italian music of the fifteenth and sixteenth centuries that they were calculated for most perfect intonation of the chords, and that their whole effect is destroyed as soon as this intonation is executed with insufficient precision."44 Their argument over art, nature, and the perfect tuning system raged on until Zarlino’s death in 1590, and thereafter was taken up by others. A generation later, Johannes Kepler, who inadvertently discovered the laws of planetary motion while trying to show that the distances of the planets conformed to the harmonic series, corrected Vincenzo’s semitone and calculated the correct string lengths to make 12 equal semitones equal one octave. His calculations were published in his Harmonices mundi of 1619. Everyone was getting into the act. Galileo, son of Vincenzo, who had already seen the moons of Jupiter through his telescope, began investigating the laws governing the vibration of strings. And French mathematician and philosopher René Descartes gets credit for being the first voice in history to acknowledge the presence of the harmonic overtones. Descartes wrote his only work on music, Compendium musicae, at the age of 22, shortly after graduating from law school and joining the French army so he could travel and think. He stated that “we never hear any sound without its upper octave.” The work was only circulated in manuscript during Descartes’ life, and was published in 1650, fire year he died. In his Abrégé de musique, Descartes observes “of the two terms required to form a consonance, the lower ... in some way includes the other. This is manifest on the strings of the lute. When one of these is plucked, those an octave or a fifth higher vibrate and sound by themselves.” Knowledge of the overtone series, or harmonic series, would change everything. It gave 18th century Harmonists a physical foundation for the harmonic intervals.

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10 The art and science of sound

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he greatest irony of the entire history of harmonic theory is that just as science was beginning to glimpse the physical foundations of pure harmony, instrument makers were giving up on it. The mechanical limitations of fixed-tone instruments seemed, in the 17th and 18th centuries, to preclude pure harmonic intervals other than octaves, for keyboards and fretted instruments. There was often a conflict between the keyboards using meantone or well temperament, the fretted instruments using approximately equal temperament, and vocal music written for pure harmony. The keyboards and guitars clashed, and neither could play with the justly tuned vocal music. The musicians, however, had not given up. Handel, Bach, and Mozart all wrote music with the intention of pure intonation, particularly vocal music. Their instrumental music was intended for intonation as pure as the instruments could achieve, Bach’s Well Tempered Clavier being an example. Handel, as we know, performed on a split-key instrument, and Bach would often retune the clavichord

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between pieces. It is clear that musicians of the 17th and 18th centuries still had an ear for pure harmony. In the 17th century, Marin Mersenne heard the harmonic overtone series in the sound of the trumpet and realized that he was hearing the major triad in the

One of the many keyboards designed by French theorist Marin Mersenne, who insisted that pure harmony should not be sacrificed to convenience.

(From Mersenne's Harmonics, 1648).

first five harmonics: a tonic, the octave, the fifth, another octave, and the third. He realized that the harmonic series continued, and that this harmonic triad was followed by another fifth and then the seventh harmonic which formed a natural dominant seventh harmony. Because of this awareness of the seventh harmonic,

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Mersenne considered the 7/6 minor third (a flat, or sub-minor third, which we now call a “blues third”) to be a consonant interval. Although Mersenne gave instructions for the main systems of temperament for his time, including quasi-equal temperament, he concluded that "it does not follow that the system of Aristoxenus, in which the fifth contains seven-twelfths of the octave, is more perfect than that in which it is pure." He designed several keyboards to address the problem of keyed instruments. In his Harmonie universelle in 1636 he faults both Zarlino and Vincenzo for arguing over the hierarchy of the voice versus theory, reminding his readers that the voice, the ear, and harmonic theory agree. In reference to the Pythagorean third, which Francisco de Salinas had said was acceptable in keyboard tuning, Mersenne says “actually, the true ratio of the major third, which is 5 to 4, is much easier and much sweeter.” In reference to the pure harmonic fifth (3/2), fourth (4/3), and third (5/4) he says that “there will be no person but does not acknowledge these consonances to be very exact, and that the ratios correspond perfectly to experience.”45 Mersenne also hung weights from brass strings and determined that the frequency is not only inversely proportional to string length (known earlier), but that frequency is also proportional to the square root of the tension, and inversely proportional to the square root of the string thickness. Instrument makers and musicians continued to experiment, bolstered by the wave of new knowledge that was becoming available. In 1630 Giovanni Battista Doni built a double lyra after ancient Greek models, but with added strings to achieve extended harmonic scales. He also built a three-manual keyboard which gave the Dorian, Phrygian and Lydian Greek modes in just intonation intervals. In 1640 Nicolaus Ramarinus constructed a keyboard based on the cycle of 53 fifths, first suggested by King Fang two thousand years earlier. In 1670 Nicolas Mercator advocated the same system. This 53-notes-to-the-octave system of pure fifths gives very close approximations to just intonation, but naturally presents some manufacturing and performance challenges. The system lived a long and healthy life, well into the 19th century.

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Descartes died in 1650, and his Compendium musicae was published, revealing to a wider audience his introduction of harmonic overtones. Antonio Stradivari built his first violin in 1666. Violins could now achieve a volume sufficient to take a major role in symphony hall orchestration. Since intonation is flexible on the violin, it could successfully accompany pure harmonic singing. In 1685, the year before Bach was born, Gottfried Leibniz published his papers on calculus (followed by Newton two years later), providing a means of calculating more precise string lengths for equal temperament. The first church organ tuned to equal temperament was probably that designed and built by Art Schnitger at the Church of St. Jacobi in Hamburg, in 1688.46 Three years later, Andreas Werckmeister published his formula for calculating equal temperament in his Musikalische Temperatur. The seven white, five black Halberstadt keyboard had become the standard. J.S. Bach was three years old, standing on his tiptoes to peck out diatonic improvisations on the family clavichord. After Descartes and Mersenne, soon everyone was hearing overtones. It was becoming common knowledge among the musical elite that sound was a vibration, and that the major triad and the dominant seventh chord were supported by the first seven harmonic overtones. In 1700 Joseph Sauveur published Principes d’acoustique et de musique, measuring and explaining the vibrations of musical tones, and providing experimental evidence of the overtone series. And here, at the dawn of the 18th century, our historical irony is in full flower as precise equal temperament and the laws of harmonics were being simultaneously discovered and understood at exactly the same time. In the first decade of the 18th century, Bach wrote his first cantata, Denn Du wirst meine Seele, Neidhardt and Johann Nicholas Bach had their great tuning showdown in Jena, and in Italy harpsichord maker Bartolommeo Cristofori made a new, heftier keyboard instrument that he called a “gravicembalo col piano e forte,” the first piano. The knowledge of the harmonic overtones lead musicians to listen ever more acutely to the sounds they were playing, and this lead to a completely new discovery about the nature of acoustics. In Italy Giuseppe Tartini, solo violinist in

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Padua, and chamber musician to Count Kinsky at Prague, began to hear additional tones that were not in the overtone series. He called them “terzi suoni,” or third tones, and these were later called Tartini’s tones. These mysterious tones are formed by the difference between two other tones, and today we call them “differential tones.” A high tone and a low tone sounded loudly and continuously together create a third tone in the middle. This acoustic phenomenon was independently discovered by German organist Sorge, and later in 1863 German acoustician Hermann Helmholtz discovered “summational tones,” which are third tones created by the addition of two tones, such that two lower tones will create a third higher tone. Helmholtz gave the name “combinational tones” to this whole family of phenomena. Tartini was also one of the first musicians to recognize that the pure harmonic seventh (harmonic proportion 7/4, quite flat of the equal tempered version), was the true minor-seventh based on the harmonic series, and that “this harmonic seventh is not dissonant, but consonant.” He adds that this pure minorseventh “has no need either of preparation or of resolution: it may equally well ascend or descend, provided that its intonation be true.”47 The discovery of harmonic overtones and combination tones gave musicians a partial understanding of why pure harmony sounded so good. In a pure harmonic triad, the overtone series of the three fundamental tones match up and blend in the higher registers, and the combination tones created not only by the fundamentals, but by the natural overtones, also tend to match up and blend. On the other hand, in a tempered triad, all the upper harmonics clash, and the combination tones also clash. The natural combination tones formed the pure fourth (4/3) and the pure minor third (6/5). Inversions of natural harmonics gave the major and minor sixths. Every tone in the just chromatic scale is accounted for in these natural tones. The great composers found that their carefully worked out harmony parts were imitating the pure harmonies of nature. Later, this understanding of why pure harmony sounds good was filled in by Hermann Helmholtz, Harry Partch, and others.

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Jean-Philippe Rameau, whose Treatise on Harmony (Traité de l'Harmonie Réduite à ses Principes naturels) introduced our modern concept of chord progressions and guided several generations of composers, built his theories around the collective acoustic knowledge that he learned from reading Zarlino, Descartes, Mersenne and others. When he first published in 1722 he had not yet read Sauveur, and did not understand the harmonic series, but he updated and corrected his work in his 1726 “Nouveau Systême.” He gives voice to the notion of tonality when he states that “melody arises from harmony,” and that “the source of harmony ... [is the] harmonic center to which all the other sounds should be related.” He adds that “whether this fundamental sound be implied, inverted, supposed, or borrowed, reason and the ear are in such good agreement on this point that no exception can be found.”48 When Rameau spoke of harmony, he intended pure harmony, pure 3/2 fifths, 5/4 thirds, and so forth. These perfect harmonic proportions were, to Rameau, “simple, familiar, precise, true, and accurate.” He found a foundation for minor tonality in the harmonic overtones 10, 12, and 15. J.S. Bach‘s Well Tempered Clavier was written in the same year, 1722, that Rameau’s first book was published, and it is clear that musicians and theorists of this era tended to think of harmony as pure harmony to whatever extent was possible on the instruments they played. Handel introduced the idea that ending on a minor triad was acceptable, but his intention was a pure harmonic minor triad for voices. The tempered minor thirds of meantone, well temperament, or equal temperament, sound harsh compared to the pure minor third, and Handel’s acceptance of the minor in a final cadence was predicated on the harmonic proportion 6/5. The keyboard minor thirds caused a problem, and this is one reason Handel played split-key keyboards. The popular opinion of equal temperament at the time was summed up by Dr. Robert Smith of England in his Harmonics, 1749, when he described the temperament as “that inharmonious system of 12 semitones [producing a] harmony extremely coarse and disagreeable.”49

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Science continued to advance the understanding of acoustics and the nature of harmony. Swiss mathematician Leonard Euler, Swiss physicist Daniel Bernoulli, and French mathematician Joseph Louis Lagrange worked out the equations for sound waves in horns. Bernoulli and Euler worked out theories to explain the acoustics of vibrating strings, and Bernoulli described how the larynx acts as a vibrational source of sound waves. The next generation of scientists, Michael Faraday, Jean Fourier, and Georg Ohm documented the nature of harmonic motion, complex waveforms, and resonance. Returning for a moment to the illustrious Bach clan, it is not likely that the young Johann Sebastian Bach was in Jena on the day of the tuning contest between his cousin Nicholas and the student Neidhardt. At that time the 21-yearold prodigy was more likely in Arnstadt, 40 kilometers west of Jena, where he had been hired in 1704 to put the new church organ through its paces. The young Bach’s reputation must already have been exceptional, because considerable technical knowledge would have been required for such an assignment as testing all the parameters of a new church organ. He so impressed the Arnstadt elders that they transferred their organist and choir master to another church and hired Johann Sebastian. Although his technical skills were beyond question, his ability to manage and discipline the choristers was lacking, since he was nearly their age, and given to some insubordination himself. One clash with his charges deteriorated into a knife fight in which western music could have suffered an immense loss. Furthermore, his tendency toward improvisation with the chorale standards, not to mention his radical “Easter cantata No. 15,” had infuriated the Arnstadt establishment. He was soon thereafter granted a four week leave. He reportedly walked (likely hitching rides on carts or wagons) the 350 kilometers north to visit Dietrich Buxtehude, Danish composer and renowned organist at the Marienkirche in the Baltic port of Lübeck. There he was exposed to Buxtehude’s “free” choral and orchestral works, an experience that would influence his own creativity and impact upon western music for centuries to come.

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The one month leave became two, and three, and more. The authorities at Arnstadt had had enough, and in any case, Johann Sebastian’s interests had drifted elsewhere. His artistic interests tended toward manifesting his own creative passions, improvising, writing and performing music. His personal passions had turned to his cousin Maria Barbara Bach. On the fateful day in Jena, Bach was likely courting Maria, whom he shortly thereafter married. Although this young prodigy was known locally as a skilled organist, the crowds in Jena in 1706 had no idea that the absent young cousin of the choir master Nicholas was to become almost synonymous with western music, one of Europe’s greatest composers, far overshadowing all but a few others, all of his contemporaries, his family’s other musical geniuses, and the renowned Johann Nicholas who was painstakingly tuning the organ pipes in the St. Michael’s church in Jena. As the eighteenth century closed, the piano was gaining in prominence, and beginning to replace the violin as the driving force in orchestral music, and even in some chamber music. For a century the violin had been the leading instrument of both orchestral and chamber music, and the dominance of pianos changed the intonation landscape. Hermann Helmholtz mentions in The Sensations of Tone that "when quartets are played by finely-cultivated artists, it is impossible to detect any false consonances ... practiced violinists with a delicate sense of harmony, know how to stop the tones they want to hear, and hence do not submit to the rules of an imperfect school." The piano began to insinuate “keyboard” temperament on all the other instruments because the tempered intervals sound even worse when heard with pure intervals. Since the piano player could not alter his or her tones, the pure tone players were forced to succumb to the new temperaments. True equal temperament was still a century away, but quasi-equal temperament began to make inroads into music. Helmholtz later commented that Mozart “is master of the sweetest possible harmoniousness, where he desires it, but he is almost the last of such masters.” The two musical streams - pure harmonic music on the one hand, and keyboard based music on the other - had now clearly drifted apart.

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11 The age of pianos

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he keyboard, its particular temperaments, and the standardization of the diatonic and equal-semitone chromatic scales gave something very valuable to western music, namely simplicity and ease of play. Music is boundless, so by organizing some intervals, making them all comfortably accessible, and making modulation unrestricted, the modern keyboard helped musicians sort out the vast potential of music into a manageable system. Beethoven brazenly exploited the power of this 12-tone system. His music is filled with speed, spirit, energy, and drama. The roughness of equal temperament is somewhat masked with this style of music, particularly with fast moving chords, and Beethoven made the most of it, producing effects no one before him had ever attempted. However, Owen Jorgensen, authority on historical tunings, points out that “for every gain in temperament history, there has been a corresponding loss.”50 Temperament is a compromise of harmonic purity, sacrificing the full power and delicacy of the pure harmonic proportions for simplicity in construction and play of keyboard and fretted instruments. Musicians of the 19th century knew this.

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Today, most musicians grow up and are trained from inside the equal tempered paradigm, and may be only marginally aware that there is such a thing as a pure harmonic interval that is distinct from a tempered interval, or that there are infinite scale choices beyond our diatonic and chromatic default choices. Jorgensen points out that pure equal temperament was not practiced on pianos before 1885. However, quasi equal temperament was used, and the intention was to achieve equal temperament within the limits of available technology. In fact, since tuning equal temperament by ear, by counting the beats of the mistuned intervals, had not yet been developed, tuners of equal temperament in the 19th century relied on devices such as the tuning fork tonometer invented by Johann Heinrich Scheibler in 1835. When speaking of equal temperament in the 19th century, we are speaking of quasi-equal temperament. The J. G. Pleyel piano factory was established in Paris in 1808. Alfred James Hipkins introduced equal temperament to the James Broadwood piano factory in England in 1846, although James Broadwood himself had introduced the idea as early as 1811.51 Broadwood introduced the importance of the striking place on the piano string, a parameter that influences the tone since the striking point may dampen certain harmonics of the string. Broadwood’s solution was to strike the string somewhere between one-seventh and one-ninth of the length. This had the effect of dampening the seventh harmonic which clashes with the minor sevenths of equal temperament. In 1852 the Exeter Hall organ was tuned to equal temperament, and two years later English organ makers Gray & Davidson and Walker & Willis made equal tempered organs. The Henry Steinway & Sons piano factory was established in New York in 1853, and Baldwin Piano & Organ company was founded a decade later. François Fétis introduced the term “tonality” in 1835 to describe the principle of musical intervals having a relationship to a fundamental, and he noted that the music of different cultures and scale styles placed different emphasis on this tonality. Fétis also pointed out that the ear can grow accustomed to almost any

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tuning system, and he tells a story that emphasizes how far habitual hearing can lead one away from the harmonic proportional intervals. Fétis was the teacher of the young composer Nicolas Jacques Lemmens. Lemmens had been born in the village of Zoerle-Parwijs, the district of Campine, in the Belgian province of Limbourg, on a frozen January morning in 1823. As a youth he learned music on a harpsichord that was badly out of tune due to the harsh weather of the region and the fact that no tuner lived in this rural district. Nearby was the abbey of Everbode, and there the organ was also mistuned and broken. An organ builder was summoned to repair the Everbode organ, and by chance this man stopped to visit the Lemmens family. Young Nicolas performed for the guest, and when the tuner heard the dreadful state of the instrument, he offered to tune it. Later in life Lemmens recounted the story for his teacher Fétis. He told how, when the instrument was first tuned, he experienced “the most disagreeable sensations.”52 Growing up with no other musical reference, his ear had adapted to the arbitrary intervals of the mistuned harpsichord, and it took him a long time to feel comfortable with the intervals of the tuned instrument. Nicolas Lemmens later became one of the great organ masters of his time, a teacher at the Brussels Conservatory, and composer of symphonies, choral works, and keyboard pieces. He was certainly a musician of considerable skill, with a good ear, and yet as a youth his ear had completely adapted to an entirely arbitrary tuning. The story reminds us that our certainty about what tuning system we prefer can be biased simply by our habitual way of hearing, a good reminder for 20th century musicians who have grown up with equal temperament. Many music theorists and musicians of the 19th century contributed to the counter-current in the evolution of tuning. Equal temperament was establishing itself, but by no means exclusively. In 1812 the Glover sisters in Norwich, England devised the “movable doh” system for teaching singing, using “doh” as the keytone, and teaching the diatonic major relationships to this fundamental as “re, mi, fa, sol, la, and ti.” They were so successful in teaching young children to

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sing that their system was later introduced into the English schools. Thirty years later the minister John Curwen visited the school where Sarah Glover was teaching the system, then known as the “sol-fa system of singing.” He was so impressed that he joined with the Glover sisters to form the Society of Tonic Sol-faists, and he later published the singing system as Singing for Schools and Congregations. By 1862 there were some 150,000 sol-faists in England. The movement also spread to the continent. One advantage to young singers using the Sol-fa system is that all songs in all keys use the same symbol to express a given interval. That is, the interval of a third from the tonic is “mi.” Since this never changes, a student more easily makes the association between the sound of the harmonic or melodic interval and the symbol. Curwen advanced the system as a purely just intonation system with the help of instrument builder Perronet Thompson who, in 1864, built an “enharmonic organ” with 40 tones to the octave on three ranks of digitals. Advances were also made in the understanding of acoustics. Michael Faraday advanced the theory of resonance, the phenomenon of one vibrating body setting a second vibrating body in motion. Jean Baptiste Fourier described complex wave forms as additions of simpler waves. Today, the “Fourier Transform” allows additive synthesizers to create complex musical timbres from sets of simple oscillations. French scientist Jules Lissajous (1822-1880), hooked a large pendulum to his ceiling. Vincenzo Galilei’s son Galileo had earlier determined that a swinging pendulum keeps perfect time. Lissajous surmised that sound was some kind of a waveform in the air, and he imagined that sound had regular beats like those of the swinging pendulum. To the pendulum he attached a bag in which he made a small hole, in the bag he placed sand, and under the pendulum he placed a long scroll of paper. He set the pendulum swinging in one direction, and pulled the paper across its path in a perpendicular direction. The pendulum, he thought, represented the vibration of sound, and the moving paper represented its movement through the air. Through the small hole in the bag the sand spilled back and forth across the

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paper, but as the paper moved the sand formed a wavy line, a line we now call a “sine wave.” Lissajous predicted that this was the visual image of a single frequency of sound. He was correct, of course, but his theory was not confirmed until the invention of the oscilloscope in the early 20th century. More impressively, however, Lissajous conducted a more elaborate experiment entirely in his mind. He imagined that two sound waves would interfere with each other, and would therefore create more unusual wave forms. He imagined that two imaginary pendulums swinging at different rates could swing across each other’s path, and he imagined the resulting images they might make with sand on the paper below. With this thought experiment, Lissajous predicted that two tones in perfect harmony would create certain well-defined patterns. Again he was correct, and later proven so by oscilloscope images. Each harmonic interval has its own unique signature interference pattern, and these we now call “Lissajous curves.” Just as the wave of equal temperament was about to wash over the music world, German scientist Hermann Helmholtz, began his studies of acoustics. Helmholtz was a great general scientist, unfettered by specialization. Between 1845 and 1894 he wrote over a hundred definitive papers and books on the conservation of energy, propagation of nerve impulses, the theory of complex colors, a review of Goethe's scientific work, optics, muscle movement, weather, electrodynamics, magnetism, the sensation of hearing, the physics of sound, and aesthetics of music. Music was his passion, he was an accomplished musician, and his instrument of choice was a harmonium tuned to just intonation. He began his study of acoustics in 1852 and wrote On the Nature of Human sense perception. He built a "vibrating microscope" using a piece of glass that oscillated in one direction and a tuning fork that oscillated in the perpendicular direction, connected both to sources of pure tone, and found the geometric patterns of sound predicted by Lissajous. He published On combination tones in 1856, followed by On Musical temperament, and On the motion of the strings of a violin. He reviewed the work of every music theorist from Pythagoras

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to Neidhardt and Lissajous, and was arguably the most knowledgeable person in human history on the complete physics, physiology and aesthetics of sound. His magnum opus on the subject, On the Sensations of Tone, was published in 1862, is still in print, and has been translated into every modern language.

The design for Henry Poole's 100-tone per octave organ was published in 1867, but was never built. Similar designs by other researchers, including Helmholtz, proved too cumbersome to be practical for musicians. (From

Helmholtz/Ellis, 1885)

After exhaustive accounts of the physics and physiology of sound, and the mathematics of harmony, Helmholtz at last engaged the historical dialogue of intonation theory and aesthetics. "There is nothing in the nature of music itself to determine the pitch of the tonic of any composition," states Helmholtz. "[It is]

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necessary for musicians to have free command over the pitch of the tonic. For singers these transpositions offer no difficulties ... But the matter becomes much more difficult for musical instruments ... [that] only possess tones of certain definite degrees of pitch,"53 the keyboards and fretted instruments. "The justly-intoned chords," said Helmholtz, "... possess a full and saturated harmoniousness; they flow on, with a full stream, calm and smooth, without tremor or beat. Equally-tempered or Pythagorean chords sound beside them rough, dull, trembling, restless. The difference is so marked that every one, whether he is musically cultivated or not, observes it at once."54 He pointed out that chord inversions, modulations, the contrast between consonance and dissonance, and contrast between major and minor chords "are much more decided and conspicuous" in just tuning. Therefore the effect of these musical techniques becomes "much more expressive." On the other hand "when the intonation of consonant chords ceased to be perfect ... the differences between their various inversions and positions were, as a consequence, nearly obliterated." "In a consonant triad every tone is equally sensitive to false intonation," Helmholtz observed, "... and the bad effect of the tempered triads depends especially on the imperfect Thirds." He outlined the four species of beats heard in a tempered major triad due to impure intervals, and remarked that they were "always quite audible," and that they "strike the ear as a marked roughness ... The beats arising from the Thirds ... are decidedly disturbing in the middle positions, even in quick time, and essentially injure the calmness of the triad." Helmholtz, the scientist and musician, at once appreciated both the technical and aesthetic nature of pure sound. "When I go from my justly-intoned harmonium," he observed, "to a grand pianoforte, every note of the latter sounds false and disturbing." Helmholtz lamented that "these are unpleasant symptoms for the further development of art. The mechanism of instruments and attention to their convenience, threaten to lord it over the natural requirements of the ear, and to destroy once more the principle upon which modern musical art is founded." In conclusion he said that "after all, I do not know that it was so necessary to

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sacrifice correctness of intonation to the convenience of musical instruments. As soon as violinists have resolved to play every scale in just intonation, which can scarcely occasion any difficulty, the other orchestral instruments will have to suit themselves to the correct intonation of the violins. Horns and trumpets have already naturally just intonation." A solution to the problem, in other words, was not out of reach. Helmholtz and others in the late 19th century proposed solutions. Henry Poole built a just intonation pipe organ, and designed (but never built) a keyboard of 100 tones per octave. Perronet Thompson’s 40-notes-per-octave organ included a device for correcting the changes caused by temperature on intonation. He claimed to have taught a blind organist to play the instrument in six days “thereby settling the question of the practicability of just intonation on keyed instruments.” Colin Brown at Andersonian University in Glasgow designed a 40-tones-per-octave voice harmonium, and built three such instruments. One of these was later played by 20th century composer Harry Partch who found it “easy to play and its intervals and triads a delight to the ear.”55 R.H.M. Bosanquet applied the 53-tone division to an enharmonic organ, and founded a company with T.A. Jennings to offer custom built organs with up to 84 tones per octave. These inventions were not complete solutions, and they posed problems of construction and play, but they held promise. Nevertheless, when Helmholtz died in 1894 the general acceptance of equal temperament slowed further research. The secrets of harmony had become visible, but the science of building instruments to play pure harmony faltered as equal temperament dominated western music.

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12 A paradigm entrenched

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he two streams of music - harmonic and tempered - had now wandered along quite distinct paths. Not only keyboards, but guitars had become popular. The C.F. Martin company, founded in 1833, and Gibson Guitar Company, founded in 1896, were turning out high quality acoustic guitars, and the instruments were popular. Keyboards and frets sent the tempered stream along its own path. Certain a cappella choirs and chamber groups continued to perform harmonic music, but from the perspective of 1900, the future of music, at least in the west, looked like equal temperament. By 1937 the Encyclopedia of Music and Musicians confirmed the current paradigm by claiming that tuning by pure intervals was “strictly impossible,” that “the mechanical obstacles to pure tuning in keyed instruments are insurmountable, and the question of some compromise is hence necessary.”56 In 1950, Llewellyn Lloyd wrote in Intervals, Scales, and Temperaments that “the present supremacy of equal temperament will remain unassailable until someone invents a really practicable means of playing ... alternative notes ... by a single key.”57 The main currents of musical theory and composition of this century have

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accepted the compromise deemed inevitable. The parallel chords, wholetone scales, and serial techniques of composers like Debussy, Ravel, Stravinsky, and Schönberg carried Beethoven's instrumental music into ever new subtleties of expression. The chords of Schönberg and Scriabin built on fourths, took advantage of equal temperament, and moved music toward an atonal aesthetic. These composers fully embraced tempered harmony, in part because their music was a rejection of traditional harmony and tonality all together, but equally so because that was the intonation available on modern instruments. In much modern music dissonance is featured, and a sense of tonal center is obliterated. In Problems of Harmony Schönberg introduced twelve-tone and serial techniques. His “emancipation of the dissonance,” was a movement to put all intervals of the chromatic scale on equal aesthetic footing. Schönberg was well aware of pure harmonics, and saw the major scale as the addition of the tones of the tonic, dominant and subdominant major triads. He wrote that “we actually to some extent hear and to some extent feel this relationship in every sounding tone.” He equated the other chromatic tones also with higher harmonics, saying “if we note the more distant overtones (up to the 13th) ... we find the chromatic scale.”58 In fact, as Henry Cowell later pointed out, this is not exactly correct. For example, Schönberg states that the 13th overtone of G is Eb, but the 13th overtone is a 13/8 interval that is almost a quartertone sharp of Eb, a “neutral sixth” that has been used in certain Arabic and other scales. This tone has no western chromatic counterpart. Cowell knew this, and felt that the natural evolution of chord development was not 12-tone chromatic, but rather would follow the harmonic series, “from the seventh overtone upwards.” He adds that “there seems to be need of such a system to further the understanding of contemporary material, which has had no adequate theoretical coordination.”59 Paul Hindemith faced the same contradiction. He attempted to explain the western chromatic scale by the overtone series that did not entirely support it, yet he understood that tonality itself was a natural phenomenon. He wrote in A

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Composer’s World that “in music we cannot escape ... tonality. The intervals which constitute the building material of melodies and harmonies fall into tonal groupments, necessitated by their own physical structure and without our consent.” He calls tonality “a very subtle form of gravitation,” and adds that “some composers who have the ambition to eliminate tonality, succeed to a certain degree in depriving the listener of the benefits of gravitation. To be sure they do not, contrary to their conviction, eliminate tonality.”60 Hindemith may have come up with the best definition of equal temperament: “a compromise which is presented to us by the keyboard as an aid in mastering the tonal world, and then pretends to be that world itself.”61 The aleatory (chance) music of John Cage and Karlheinz Stockhausen took this revolution against tonality to ever further limits. Cage’s experiments with random sound, prepared piano, and even complete silence in 4’33” had nothing at all to do with harmonic proportions. It was sound Dada. He was not only liberating dissonance, but liberating noise and even silence, as musical elements. Blues and jazz forms were developed in the U.S., from the early 19th century, and although this music evolved within the equal tempered stream, it produced some interesting intonation experiments. Equal temperament gave jazz players the freedom to roam unrestricted among tonal centers, and the music tended toward chord sequences with a widening definition of chromatic chordal resources. Musicians like Louis Armstrong, Charlie Parker, Sarah Vaughan, and others were delivering music into new realms of sound, speed, and rhythm. Later Miles Davis, John Coltrane, and a new generation of jazz musicians advanced the form, playing up and down through fast chordal cycles, leaving the melody for spontaneous inspiration, integrating Afro-Caribbean polyrhythms and LatinOriental off-beat melody lines. All of this was helped along by the ease of equal temperament. Horn players like Davis and Coltrane, however, instinctively shaped the intonation to fit the mood of a passage. Later, in the 1970s, flutist Paul Horn recorded over his own echo inside the Pyramids and the Taj Mahal, working

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with the natural resonances of these sites as well as the natural harmonics of the flute tones. String bass players, like singers and horn players have the freedom to place the intonation where they want to because they aren’t limited by frets. However, a bass player with a good natural ear for harmony must be careful not to clash with the lower notes of the piano. This is because the string bass player may instinctively want to stop the note at the natural harmonic, something the piano cannot do. Jerry Coker, a former saxophonist with Woody Herman, Stan Kenton, and others, mentions this problem in his book on jazz improvisation. “If the pianist’s left-hand base note coincides with the root played by the bass player, there is an intonation problem.” In this case, the bass player either has to stay away from this note, or play the piano intonation. Most modern musicians, of course, learn to play the piano intonation since the piano cannot change. Blues musicians also wandered instinctively from equal temperament. The “blue notes” heard in some of this music tend toward the seventh harmonic which is obliterated in equal temperament. In “Septimal Harmony for the Blues,” Dudley Duncan analyzes W. C. Handy’s St. Louis Blues, in G, and concludes that “...the singer or horn player uses inflection of the melody or countermelody for color (in ways probably beyond the scope of any simple theory), [but] for simple harmony the blue notes are merely the just intervals 7/6, 7/5, and 7/4.”62 These are the blue minor third, augmented fourth, and minor seventh. The blue Eb in the key of G, nominally an augmented fifth, Duncan suggests is, in the blues treatment, the quartertone 14/9 which forms a blues third above the fourth. Guitar players can find the pure seventh harmonic of a guitar string by lightly touching the string about halfway between the second and third frets as the string is sounded. On the bass E string this is a D note. Compare this note to the high octave D at the twelfth fret of the D string tuned two fretted fourths from E. These two “D’s” are clearly not in tune with each other. The harmonic D is much flatter, and this is the blues seventh. The tempered flat-seven, found with the frets, is the worst of all the tempered intervals when compared to its natural harmonic.

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The blues singers and horn players simply used their instinct and good ears to find the pure harmonics that were found by the Harmonists of ancient times. Guitar players know that when they tune perfect harmonics from the bass E-string to the treble E-string, the two E's are not in tune with each other. By matching the fourth harmonic of the bass E string to the third harmonic of the A string, etc., and by matching the fifth harmonic of the G string to the fourth harmonic of the B string, a guitarist ascends by four pure fourths and a pure third, but this lands flat of two octaves. The interval by which this is flat is the comma of Didymus that the reader may remember from the first five chapters, a discrepancy that has been known by musicians for two thousand years. It is about 1/5 of a semitone. The ear hears this, and with a flexible instrument, the ear can easily guide the necessary intonation changes. Frets however, demand equal temperament, which disguises these natural discrepancies by spreading them out across the octave and polluting every interval. Noticing the intonation and placement of these harmonics on a guitar is a good way to hear, and even see, the problem with tempered intervals. The fifth harmonic of the G string is a pure major third of G, a B note. When you play the B harmonic on the G string, you are dividing the string into five vibrating parts by touching the string at one-fifth its length, just slightly flat of the fourth fret. Play the B harmonic on the G string of a guitar. Now press down and hear the fretted B on the G string. These two B's are noticeably out of tune with each other. You can see how far behind the fret your finger is when you get the ringing B harmonic. On a Martin D-35 the fourth fret is more than 1/8-inch sharp of the harmonic. They say in framing a barn that 1/8 inch tolerance is acceptable. To the delicate art of harmony, it is a problem. Modern popular music, particularly electric rock guitar playing, is based on fast and colorful chord changes, rather than on the subtleties of pure harmony. Helmholtz predicted this evolution of instrumental music a century ago when he acknowledged that "in rapid passages ... the evils of tempered intonation are but little apparent," and added, "we might, indeed, raise the question whether

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instrumental music had not rather been forced into rapidity of movement by this very tempered intonation, which did not allow us to feel the full harmoniousness of slow chords to the same extent as is possible from well-trained singers, and instruments had consequently been forced to renounce this branch of music."63 Fast, chord-based music is by no means aesthetically inferior, and indeed great music has been created by artists in the blues, jazz, and rock genres. The issue for musicians is one of musical options. The simplicity of equal temperament created new options, but also foreclosed certain other options. As musicians, do we merely accept those limitations, or do we chose to investigate all the musical options that could enhance our ability to make expressive music? The question will have different answers for different artists, but asking the question is the issue, and seeing the options is the opportunity for modern musicians to expand their creative resources. Composer, producer, and music innovator Brian Eno mentioned this limitation of the 12-tone keyboard in a March, 1995 interview in Keyboard magazine. “The keyboard,” said Eno, “gives you distinct islands rather than a continuous set of pitch possibilities. That's a disadvantage for keyboards.”64 Steve O’Keefe, editor of the Piano newsletter was even more blunt in saying “the piano is one of the most frustrating instruments on God’s green planet. ... If you want to play the piano you are locked into that vision, unable to use anything outside of what is provided.”65 The guitar, of course, has the same problem. “Frets are slightly out of tune,” wrote John Schneider, in Acoustic Guitar magazine in 1994, “enough to produce a warbling effect on every chord you play and to sabotage you every time you try to tune.”66 These musicians are addressing the aesthetic limitations of their instruments, a healthy move for any artist.

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13 The new Harmonists

T

he second stream of modern music, pure harmonic-based music, seemed a trickle at the dawn of the 20th century. Some choral groups and string ensembles kept the 18th century traditions alive, but the seeds of Hermann Helmholtz, Perronet Thompson, and other innovative instrument designers of the 19th century fell on fairly dry ground. As we will see, however, these seeds would survive and take root once the weather changed. One modern Harmonist link from the 19th century was Julian Carillo (1875-1965) of Mexico, a solo violinist who experimented with 24-tone, 48-tone, and 96-tone equal temperaments in an effort to find an equal tempered solution that also served the harmonic proportions. He called his system “el Sonido Trece,” the 13th Sound, and formed the “13th Sound Ensemble” which performed his music such as Preludio á Cristobal Colon for harp-zither, octavina, cello, trumpet, and soprano voice. He modified a guitar to achieve quartertones, and built many of his own instruments. Near the end of his life, in 1962, he completed Missa de la restauracion for Pope John XXIII, written in quartertones.

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Ferruccio Busoni (1866-1924) was another link. He transcribed Bach and Liszt; edited the Well Tempered Clavier; wrote numerous operas, choral pieces, violin concertos and chamber pieces including Fantasia contrappuntistica; and experimented with 36-tone equal temperament. He wrote “we have divided the octave into twelve equidistant degrees, because we had to manage somehow, and have constructed our instruments in such a way that we can never get in or above or below or between them. Keyboard instruments, in particular, have so thoroughly schooled our ears that we are no longer capable of hearing anything else - incapable of hearing except through this impure medium. Yet nature created an infinite gradation - infinite!”67 English musicologist Kathleen Schlesinger made a great contribution to the Harmonist school with her 1939 work The Greek Aulos. Schlesinger maintains that the evenly spaced holes in the Greek flute suggest that Greek music was based on the arithmetic progression, divisions of strings (or flutes) into 11, 12, 13, 14, 15, etc. equal parts (not equal temperament). Whether or not she is correct about Greek musical performance, her system of equal divisions gives harmonic proportions and allows her system to achieve versions of all the diatonic Greek modes. For example, the Mixolydian scale uses a 14-division of the string, the Dorian an 11-division, and so forth. The Greek enharmonic and chromatic scales are achieved in the Schlesinger system by doubling the number of divisions. Schlesinger advanced the 11th and 13th harmonics which generate natural harmonic quartertones, and her work inspired American Harry Partch who visited her in London in 1935. Henry Cowell, who corrected some of Schönberg’s approximations with regard to the harmonic series (see chapter 12), also inspired the American Harmonist movement. He encouraged experimental composers of all persuasions by publishing their work through his New Music publishing company. He also invented a keyboard percussion instrument, the Rhythmicon, with innovative instrument designer Leon Theremin. His work with Theremin was early evidence

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of a third stream in 20th century music, a stream that eventually would help reunite keyboard music and pure harmonic music: electronic musical instruments. The “radio valve,” or diode tube had been invented by John Ambrose Fleming in 1901, and five years later the first primitive electronic instrument, the Telharmonium, was built by Thaddeus Cahill. The Telharmonium was a 200-tone machine in which a rotating cam generated the tones of the scale as selected by a keyboard. The signal could be sent over the telephone line, and thus the name. The machine itself, however, was a monster which Cahill moved to New York City on seven railway flatcars. At about this same time the newly invented tape recorder led to experiments in “musique concrete” by George Antheil and Edgard Varèse. Varèse also used a “theremin” invented by Leon Theremin in 1927. The theremin is a vacuum tube that sends a signal up an antenna. The instrument is played by a musician who moves both hands along the antenna thereby generating various frequencies. The instrument was still in use in the 1970’s by groups such as the Beach Boys (Good Vibrations), and Led Zeppelin (Whole Lotta Love). By 1934 Laurens Hammond had invented and built the first fully integrated electronic keyboard, a refined Telharmonium utilizing a synchronous electric clock motor, toothed gears, two upper keyboards and a foot-pedal keyboard, the first Hammond organ. Near the gears were magnets wound with wire which caused the magnetic field to fluctuate as the teeth passed. Each coil output a sine wave, and these were added together to adjust the timbre using manual drawbars. This was early additive synthesis. The tuning gears were set to the ratio 196/185, an approximation of the equal tempered semitone. Equal temperament had been turned into a harmonic proportion, albeit a rather cumbersome one. This event foreshadows the ability of electronic instruments to achieve variable tuning, but in 1934 Laurens Hammond had not yet imagined this potential. It was at this same time that the 17-year-old Lou Harrison became a student of Henry Cowell. The Harmonist stream was swelling. Cowell introduced Harrison to international cultural music as well as to great composers like Varèse and Schönberg, and taught him counterpoint and composition. In 1946 he

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conducted Charles Ives’ Third Symphony at Carnegie Hall. Harrison developed a lifelong passion for cultural tuning systems and pure harmonic intonation. Harrison’s famous line, “just intonation is the best intonation” later became the slogan of the Just Intonation Network in San Francisco. In 1987 the editor of 1/1, the Network’s newsletter, David Doty, interviewed Harrison who said “if unpolluted by other instruments, string orchestras, left to their own devices, play quite well in just intonation.”68 By the time of this interview, the Harmonist stream had swollen to a steady flow, and Lou Harrison was one of the mentors. “Just those tiny, tiny differences, sometimes between a just and a tempered interval, in terms musical, makes all the difference,” he said. “The real intervals of music are very beautiful, there’s no doubt about it, and moving through them is beautiful ... when you can have something real, why mess everything up?” Jazz pianist Keith Jarrett recorded Harrison’s Piano Concerto for well tempered piano, and violinist Lucy Stoltzman joined him on Harrison’s Suite for Violin, Piano, and Small Orchestra. The recording was released by New World Records in 1989. An influence on Harrison, and on the entire revival of Harmonist thinking in the 20th century was composer Harry Partch (1901-1974), a pioneer who set out on his own at the age of 22 to learn how to make the music that his ear heard and his mind conceived. He educated himself in the Harmonist tradition, devised a scale of 43 tones per octave, conceived a notation system to communicate his ideas, wrote music for this set of harmonic intervals, built his own just intonation instruments to achieve the intervals, and performed the pieces himself and with his friends and musical associates. As if this was not enough for one lifetime, Partch set out his theories in a book that became one of the guiding lights of 20th century pure harmonic music. He began the book in 1925, completed a draft three years later, and published the final version, Genesis of a Music, in 1948. Partch called his system “monophony” to emphasize the relationship of all tones to a fundamental consonance represented by the number “1” or the proportion 1/1. “A tone, in music,” Partch explains, “is not a hermit, divorced

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from the society of its fellows. It is always a relation to another tone, heard or implied.” Partch outlines four basic fundamental concepts of pure harmonic music: (1) The scale of musical intervals begins with 1/1, absolute consonance, and progresses through more complex proportions - 2/1, 3/2, 4/3, 5/4, etc. which become more dissonant as the proportional numbers get larger. (2) Every harmonic proportion has a dual identity, as a higher harmonic identity in relation to its tonic, and as a tonic itself. (3) These tonalities represent the nature of sound and an immutable faculty of the human ear to perceive its qualities. (4) Human use of consonance in music has proceeded from the unison, to the simple harmonic intervals, to the more complex, toward the “infinitude of dissonance.” Partch’s 43 tones per octave are based on all the tones generated by the first 11 harmonics. He defines the “3” identities - the fifth and the fourth as “power” intervals; the thirds and sixths (5/4, 8/5, 6/5, 5/3, 11/9, 18/11, 9/7, etc.) which include approximate quartertones and other intervals not found in a 12-tone chromatic scale, are “emotional” intervals; the seconds (8/7, 9/8, 10/9, 81/80, etc.) and the sevenths (15/8, 9/5, 7/4, etc.) are “approach” intervals, and the intervals between the fourth and the fifth, the quasi-tritone intervals are “suspense.” On some of his instruments he lays out these intervals in a “tonality diamond” that recognizes each interval’s dual identity. “The ear,” says Partch “does not budge for an instant from its demand for a modicum of consonance in harmonic music nor enjoy being bilked by near consonances which it is told to hear as consonances.” Music to Partch is “corporeal” rather than “abstract,” arising from spoken cadences, song, recited or intoned poems, chants, dance, movement, and drama. Partch takes up the theme of Zarlino that the human voice naturally sings pure

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intervals, and he defends singers who struggle with the intonational ambiguity of modern music. “Much of the oft-heard railing against the intonation of singers is scandalously lacking in candor,” he says. “As composers and educators we give them an accompanying instrument - the piano - which is continually at odds with

An octave of Harry Partch’s Chromelodeon II keyboard showing the placement of his 43 harmonic tones.

their instincts. After they have mastered this incongruity we pose them in an a cappella choir or before an orchestra, where they are at the mercy of each intonational whim of concertmasters and conductors, and proceed to criticize them for their ‘bad’ intonation.” He adds finally that “the great need for a better instrument than the piano in the training of singers and for accompaniment of songs is too self-evident to be labored.”69 Partch also pointed out that the pure sound of harmony was due to more than just the harmonic series and the combination tones. The ear, he suggested,

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recognizes the simple harmonic proportions - 3/2, 5/4, etc. - because when musical tones are perfect ratios of each other the sine waves of the tones match up exactly at certain points. For example, if the pure third (5/4) and the pure fifth (3/2) are played together, five cycles (wave lengths) of the first tone fit exactly into six cycles (wave lengths) of the second tone, thus creating a perfect and pleasing harmony. The wave length is the inverse of the frequency, so the mathematics of that simple harmony can be written as: 5 × 4/5 = 20/5 = 4 6 × 2/3 = 12/3 = 4 That is, 5 cycles of the 5:4 tone are precisely equal to 6 cycles of the 3:2 tone, and they fit into exactly 4 cycles of the root tone. These three tones played together form, of course, the major triad. This perfect fit of sine waves is harmony. On the other hand, the tones of equal temperament do not make any perfect harmonies (except octaves) because the tempered tones are not perfect fractions, they are what mathematicians call irrational numbers. The 12th root of two, for example, is 1.05946... a decimal that never ends. So, when the tempered third (1.25992...) and the tempered fifth (1.49831...) are played together, the sine waves of the two tones never match up exactly, and the harmony is imprecise. Genesis of a Music was an underground musical classic, read by many young students, and inspiring them to open their hearing to the delicacies of pure harmony. One of these young students was composer Ben Johnston at the Cincinnati Conservatory. Johnston had been influenced by Scriabin and Debussy, but also by Ives, and Carrillo. In high school he had taught a fellow student to play a Bach piece in just intonation, entirely by instinct and without training. He later recalled “I could hear very early that the piano was out of tune and she was in tune. That made me very aware.”70 Partch’s approach to music resonated with Johnston who traveled to California to work with him. Johnston had a great ear, and was able to intone Partch’s intervals, singing on demand a 16/15 interval or a

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16/11 interval. When Partch’s instruments were slightly out of tune, Johnston would notice by comparing his voice intonation. Partch was impressed, and he accepted Johnston as a student. Johnston, now Professor Emeritus of Music at the University of Illinois, credits Partch with opening the floodgates of extended just intonation in the west, but Johnston himself has been equally influential. “Western music has based itself on an acoustical lie,” is a now famous remark Johnston made in the notes to the Fine Arts Quartet’s performance of his String Quartet No. 4, on April 28, 1974. He explained this remark in a paper that appeared in the Proceedings of the American Society of University Composers. “The acceptance of keyboard temperaments in the sixteenth and seventeenth centuries ... impoverished the logic of harmony and tonality by weakening the perceptibility of consonance and dissonance. ... it took only a little over a century for the exhaustion of the fresh possibilities of the system to become a major aesthetic problem.”71 Johnston compares equal tempered music to a movie that is slightly out of focus. The audience, he suggests, may not be consciously aware, but they are straining to see until the film is brought into focus and everyone notices that they were straining. The same phenomenon happens with music that is brought into pure harmonic tune. “Music,” says Johnston, “is not simply an intellectual exercise; it is a physical response to sound,” a point made by Lou Harrison as well as by Harry Partch. In the same year that Partch’s Genesis of a Music was published, 1948, the Radiodiffusion studio was opened in Paris by Pierre Schaeffer and Pierre Henry, the Radio Cologne studio was opened in Germany by Friedrich Enke, and Radio Corporation of America engineers Herbert Belar and Harry Olsen built the RCA Mark sound synthesizer based on the ENIAC computer. A decade later Max Mathews at Bell Labs predicted “the computer will become the ultimate musical instrument.” In 1964 Cornell University doctoral student Robert Moog introduced his first Moog synthesizer to the public. The sound synthesizer used a voltagecontrolled oscillator connected by plug-in telephone jack patch cords. Electronic

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music was a new fact of musical life, and this change took place just as a renaissance of pure harmony was flowering. One of the first to bring these two movements together was Wendy Carlos. Carlos’ original compositions have been performed by the London Philharmonic and Boston Symphony Orchestras, and by the Kronos Quartet. In 1971 she used synthesizers to create the music for Stanley Kubrick’s film A Clockwork Orange. She used a new Moog synthesizer to record Switched on Bach in 1968, one of the largest selling classical recordings ever made. Although this early Moog was tuned to equal temperament, and even suffered from pitch drift at that, Carlos saw the potential for developing electronic music in the direction of pure harmonics. In Just Imaginings, a composition from the recording Beauty in the Beast, Carlos used a Hewlett-Packard computer to access the tuning tables of synthesizers to achieve pure harmonics throughout. In Poem for Bali, she combines Indonesian and western scales, sounds, and styles. She has also experimented with meantone and other historical temperaments, and believes these temperaments that favor certain keys may evolve into new versions with the aid of digital synthesizer technology. In “Tuning: At the Crossroads,” in the Computer Music Journal in 1986, Carlos notes that “computer-controlled synthesis ... has inherent need to respect these heretofore inescapable limitations [of equal temperament].” Among the many scales she uses, she describes a scale built from the first 27 harmonic overtones, mapped to the twelve tones of the keyboard. This scale is: 1/1 17/16 9/8 19/16 5/4 21/16 11/8 3/2 13/8 27/16 7/4 15/8

2/1

These harmonic tones are mapped to the 12-tone chromatic keyboard as: C

Db D

Eb E

F

F# G

Ab A

Bb B

C

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In this natural harmonic scale the Eb, F, F#, Ab, and A are all quartertones that have no direct counterpart in 12-tone equal temperament. When modulating, Carlos uses an external computer to control her precise tuning. “There is scarcely a more worthwhile venture to pursue as soon as possible,” Carlos says, “than adopting a standard for and then manufacturing at least a limited edition of these new keyboards” that allow musicians tuning flexibility. In 1964, La Monte Young, also influenced by Partch and Lou Harrison, wrote and performed The Well-Tuned Piano, a 7-hour piece for justly tuned piano, emphasizing the seventh harmonic minor third 6/7, and major third 9/7, and their inversions as the sixths. His The Second Dream of the High-Tension Line Stepdown Transformer is based on C, F, F#, and G built on the harmonics 12, 16, 17, and 18. Here the “F” is the tonic 1/1, the F# is 17/16 (the same semitone used by Wendy Carlos in the scale above), the G is the classic just wholetone 9/8, and the C is a pure harmonic fifth of 3/2. La Monte Young influenced Glenn Branca who, in 1983, performed his Symphony No. 3 in New York, a large ensemble, rock influenced piece based on high harmonics, and their combination tones, “the first 127 intervals of the harmonic series,” according to Branca, achieving a result he describes as a “field of sound.” In 1978 Dave Smith and Sequential Corp. built the first microprocessorcontrolled, programmable keyboard, the Profit-5. At the National Association of Music Merchants trade show in June of 1981, Smith introduced the idea of a Musical Instrument Digital Interface (MIDI) protocol. The following year the new protocol was adopted, a visionary move by the industry because it allowed all digital electronic instruments to communicate. By this time, the pure harmonic movement was once again a wide, flowing river with many contributors from around the world, including Terry Riley and Pandit Pran Nath; Arthur H. Benade, whose Fundamentals of Musical Acoustics had been published in 1976; James Tenny; Pauline Oliveros; Joseph Yasser who experimented with 19-tone equal temperament; Franz Herf with 72-tone, and

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Adriaan Fokker with 31-tone systems, David Rayna and Harold Waage, who both developed computer-enhanced systems; William Alves who wrote on theory; Carter Scholz and Robert Rich who helped establish a tuning standard for the MIDI protocol; and Douglas Keislar, George Kirck, and Larry Polansky who advanced computer applications for pure harmonic music. John Chalmers, who published the exhaustive study of historic harmonic scales, Divisions of the Tetrachord, published the journal Xenharmonikon to help expose musicians to the work of theorists and composers like Erv Wilson who has advanced the understanding and application of historical and international harmonic structures in music. In 1985 David Doty, Henry Rosenthal, Larry Polansky and others founded the Just Intonation Network in San Francisco, and began publishing 1/1, the Network newsletter featuring the work of the composers, musicians and writers in the Harmonist tradition. In 1988, Electronic Musician magazine writer, and experimental musician, Scott Wilkinson wrote Tuning In, Microtonality in Electronic Music, further exposing electronic musicians to the potential of pure harmonic music. Clearly the Harmonist seeds had taken root in the digital, electronic age, bringing thousands of years of acoustic knowledge and harmonic tradition together with advanced musical instruments. “Lack of access to suitable instruments has always been an albatross around the neck of microtonal music,” wrote Douglas Keislar in Computer Music Journal. Harry Partch had earlier written that “to produce music in Just Intonation we must have instruments, and instruments are no small problem.” The time had come for that problem to be solved. Keyboards gave music simplicity. Pure harmonics give music its fundamental nature. The question we might now raise is whether we can have both, whether we can restore pure intonation to music without sacrificing the ease and simplicity that modern music demands. The technology available for instrument making has evolved several orders of magnitude since the days of Beethoven and Helmholtz. It is possible that the nineteenth century acceptance of a

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sixteenth century compromise to the two thousand year old tuning problem is not the end of the story of musical intonation. Music evolves. Music grows with the spirit of the times, with technology, and with the skill and knowledge of musicians. Early musicians discovered natural harmonics by ear, and from these patterns of harmonic resonance, scales and modes evolved on every continent, in every culture, in a virtually infinite number of variations. Most musicians are aware that the harmonics of their voice or instrument strings do not match the tuning of the piano nor the frets of a guitar. Musicians have been led to believe that the problem was insurmountable, and that the compromise was therefore inevitable. They have been misled. With today's computer technology we are presented with an historic opportunity to restore pure harmony to music while retaining all the advantages of temperament. We can now open a great door to long ignored musical possibilities. The two streams of music may once again merge and flow as one.

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14 Justonic

I

n 1979, when the first microprocessor-controlled musical instruments became available, and the early personal computers were available, Bill Gannon was a self employed accountant. He was also a former professional bass player from a musical family. Gannon had gained a working knowledge of computers, and had an interest in the new synthesizers. He noticed, however, that the synthesizers were using the tempered scale, and he thought that this was a failure to take advantage of the technology. Surely, he surmised, there must be an electronic/computer solution to the problem of free modulation and pure harmony. Gannon did some research, investigated historic scales, and shared his work with his friend and associate Rex Weyler, an amateur musician with some background in mathematics and engineering. By way of introduction, these are us, the authors of this book. In 1991 we proposed a solution that would take advantage of current technology, restore pure harmony to music, and still retain every expediency of temperament, that is, the advantages of ease and speed, simple key modulation and chord changes. In 1993 we founded Justonic Tuning

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Inc. in Vancouver, British Columbia. We built a prototype of our invention and wrote a patent with the intention of marketing our solution to the music industry. Our patent attorney Jeffrey Haley, a musician himself who helped us revise and file our first patent, was optimistic that our solution would work and that our patents could be protected. We found a few visionary investors and set to work on building a commercial version, software to drive the system, and a synthesizer with the tuning resolution to make the most of it. It seemed to us that most music could be enhanced by pure harmonic tuning, pending the availability of a workable just intonation system. For some modern music, based on dissonance and distortion, the change to pure intonation will hardly matter. However, for music based on melody and harmony, the difference is significant. A pure chord and a tempered chord are audibly different. A pure chord rings with unmistakable internal resonance. A tempered chord wobbles and beats, and there is no way around it, except to smother the effect in ever more chaos. A chord is either in perfect tune or it is not, and no convention or special sound effect can change that natural fact. The acceptance of the tempered scale comes from the erroneous belief that the problem is insurmountable. But is it? In the nineteenth century theorists tried to solve the problem with multiple keyboards, with foot pedals, and other devices. These mechanical solutions were cumbersome. Yet, it is important to remember that the problem is not with the music, the problem arises with the limitations of fixed-tone instruments. In fact, keyboard and fretted instruments were the first “digital” instruments, because they digitized pitch selection. Piano keys are even called “digitals,” and indeed on a piano keyboard the player has the choice of this key or that, but cannot access the infinite tones between the two. The compromise solution, the tempered scale, throwing every tone out, putting all harmonies out of tune, seems dubious now that we have the technology to overcome the mechanical limitations of earlier instruments. We believed that with modern digital signal processing, coprocessing technology, and with our

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Justonic system, there was a real solution. The problem, it turns out, was not "insurmountable," only tedious, and too complicated for pre-electronic technology. The first generation of electronic, digital musical instruments had the tempered tones fixed into their circuitry. Later versions included microtuning capabilities, but even so, the ability to tune a single scale to just intonation, without the ability to modulate keys, is of little use in practical music. Our vision for the Justonic system was to create an instrument that was easy for musicians to use, since musicians want to make music, not program computers and tinker with gear. For a complete just intonation system to work for musicians it must be unrestricted as to pitch calibration and key, modulation among keys, scale variations, and other parameters, and it must not encumber the process of making music. Through Haley, our patent attorney, we met another music instrument innovator, Steve Dame, founder of Virtual DSP Corp. Dame was working on pitch detection technology for a guitar-to-MIDI interface, no small problem. Keyboards are easy to interface with digital sound because each key can send out its own note. A guitar interface, on the other hand, must read the pitch of the string and convert it to a MIDI note. To do this in real time, accurately, has been a problem, and Steve Dame was onto an innovative solution. He also had the know-how to build our synthesizer with the tuning resolution we wanted. By January 1996 we had our first working model of a commercial instrument. We asked Gannon’s brother Oliver, a world class jazz guitarist, to test this instrument. He was skeptical at first, saying, “yeah, it’s a good idea, but you guys are going to be swimming up Niagara Falls.” However, after a few sessions with the first prototype, he reported that "I'd like to tell you than I am very impressed. I'm well aware of the intonation trade-offs inherent in the equal tempered system. I rigorously tested every type of chord from C major to F#13 (b9 b5) ... all chords sound better in the Justonic system." We began to introduce the prototype to other working musicians. Jazz singer Patty Hervey said, "before the recent presentation by Justonic I had no idea

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that there could be a solution to the inherent problems I have often encountered in using the familiar tempered scale. The Justonic system ... produces correct, pleasing and in tune chords." Classical and jazz violinist Margaret Taylor commented, "this is a venue to really play in tune ... I was astonished at the pure sound of the simple A7 chord ... In turn, the more sophisticated chords are even juicer." Flutist Paul Horn told us, "the Justonic tuning is more mellow, because of the overtones, definitely." We felt we had something that worked for musicians, and we began to complete the user interface for the software. Musicians suggested to us that the software should work with other microtunable synthesizers, so we set to work to interface with these machines. Unfortunately the microtuning systems of the commercial synthesizers have not yet been standardized under the MIDI protocol although Carter Scholz and Robert Rich have written a tuning standard that was adopted. As a result, we have had to create the unique interface for each microtunable synthesizer or sound card. Two other problems limit the current state of the art in these instruments: tuning resolution, and speed. The tuning resolution is not fine enough in some instruments to achieve beatless, smooth chords. The speed at which the messages are relayed in some instruments makes real time tuning changes difficult. Nevertheless, it is possible to interface with microtunable synthesizers, and the Justonic software does so. Naturally, these instruments will improve their resolution and speed over time. The Kurzweil K2500 operating system, for example, now allows notes to change pitch while they are sounding, one option of the Justonic method. With the Justonic system, the musician plays in the familiar chromatic context although the system is using more than 150 distinct tones within each octave. The Justonic software makes the same tuning adjustments a well trained chamber music player makes based on the parameters of reference pitch, scale, key, and harmonic structure. The system has been designed to be as flexible as possible, so the musician can achieve the tuning effect he or she wants.

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Although the parameters of reference frequency and chromatic scale are fixed in most western music, and the musical key is chosen by the composer or musician, precise just intonation requires that the frequency of each note varies with the harmonic structure. The reference pitch, the scale, and the musical key

The scale editor from the Justonic Pitch Palette software. The user can select from a menu of scales or enter any harmonic scale, mapping the tones to the 12-tone keyboard.

can all change. Within a key, as the music moves through different tonal centers, the tuning adjustments must be made to each note. The Justonic method refers to this parameter as the "tuning root." The tuning root can be sequenced or selected by the Justonic software, but the discerning musician may also want manual control over this parameter. Musicians who use the Justonic tuning method soon discover not only the natural

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beauty of properly tuned chords, but the exquisite power and subtle nuances of tuning root variations. The default reference frequency of A-440 can be changed at the discretion of the musician. A chromatic scale is selected, allowing harmonic experiments with world music scales. The scale can also be changed in real time as one plays. This allows musicians to work with music that requires more than twelve degrees to a scale. An example would be some Arabic music that may require a major third, a minor third, and a neutral or quartertone third. These different thirds can be mapped to the same key by changing scale. The musical key may be preselected, and may change at any time during performance. The choice of a tuning root sets the dynamic intonation of the instrument. Notes are selected by the player in the traditional manner using keyboard, guitar, or other controller. During performance, the scale, key, and tuning root may be changed in real time by several means: Sequenced: key and tuning root instructions on a MIDI sequence track for any pre-determined musical piece or program. Software: For free improvisation Justonic has written software that recognizes harmonic structure on the fly, and assigns a tuning root. However, ambiguous chords such as Cmaj6 and Am7 may be tuned differently. Since these chords are meant to perform different musical roles, the fact that they can sound different is musically advantageous. In this way, just intonation restores the subtlety of tonal techniques obscured by temperament. Musicians will discover that this is a point of artistic choice, and will want manual control over the tuning root. Therefore: Manual: Key and tuning root may be selected manually in real time with foot pedals, a secondary keyboard, or other switches. The switches may control all tuning root selections in a piece of music, or may simply override the sequenced or software selection at the discretion of the musician. The software automatic

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selection option can be turned off entirely. A single "root player" could select the tuning root for all networked instruments in an ensemble. There is an art to selecting the tuning root. If one tunes a standard synthesizer to a just scale in the key of A, and then plays a Bm chord, the intervals of this chord are too narrow, and either the B must be flattened or the D and F# must be sharpened. The choice is an aesthetic one. If the melodic priority is to keep a pure whole tone between an A note in an A chord followed by a B note in the B chord, then the tuning root for the second chord is B, and the other notes are sharpened. If, however, the melodic priority is to keep a pure fourth between a C# in the first chord and an F# in the second, then the tuning root of the second chord would be F#, and the B would be flattened. There are other harmonic considerations as well, and a correct system must allow for a manual selection of tuning root on the fly. Musicians will find that the tuning root parameter is an important and expressive musical element. By ignoring this fundamental feature of music, temperament allowed music to be played without the musician having to listen to and adjust the intonation of each note. Musicians will discover new sources of creativity once this intonation awareness is reclaimed. There are an infinite number of scale choices. Another of the serious aesthetic limitations of tempered music is that we have locked ourselves into one chromatic scale. It's as if painters were forced to throw out all but a few simple colors from their palette, or if dancers hobbled themselves with leg irons. Music hobbled itself to the limitations of 17th century mechanical technology. We have virtually ignored the harmonic seventh, the neutral thirds and sevenths of middle eastern music, the alternative minor thirds, and a variety of wholetones. All of this we have relinquished for simplicity. Modern technology allows us to reclaim these lost musical scale resources. The Justonic system allows the composer or musician to use any cultural, historical, or newly created just scale. This opens the whole world of tonal possibilities to modern musicians.

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Pure harmonic blues scales, quartertone scales, Raga scales, and hundreds of other choices are available to us as musical resources. One difficulty musicians have had in the past with these pure harmonic, or “just” scales is that they require flexibility with each note if one is to play polyphonically. Arabic music, for example, has developed with very little harmonic embellishment because of this problem. Justonic is working with musicians around the world to solve this problem for any harmonic or just scale. Western musicians looking for something new in music will be delighted with the tonal possibilities available simply by having more scale choices that are easily available and simple to apply to familiar harmonic forms. Familiar modes blues, be-bop, minor, etc. - all sound different in various harmonic scales. For choral or chamber ensembles who strive for pure harmony, the Justonic system will be helpful since the chords may be played in precise just intonation, and each player can hear his or her intonation in harmonic context. Musicians may want to detune an interval up or down to achieve any number of musical effects. The Justonic system allows for these additional pitch shifts, and when the default intonation is pure, the musician can more easily hear the desired detuning effect. Musicians deserve access to the full palette of tonal resources. They have been denied this access by our reliance on an old technology. Justonic points back to a fundamental purity of musical harmonics, but at the same time, points ahead to the whole, vast potential of modern music. The system does not force any style or technique upon the musician, but rather makes pure tuning available. When the composer or musician chooses dissonance or atonal forms, the system supports those choices, but when the composer or musician wants pure harmony, they get it. Effects such as modulation, chord inversions, and simple harmonic parts are much more expressive and vibrant with just intonation. There is no way around it: for music to modulate freely, and for all intervals to remain in perfect harmonic relationship, the actual frequency, or pitch, of all notes must be flexible. A good choral group can do it. When we hear that

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sweet harmony our ears perk up, our souls lighten, something resonates inside. There is no cheap substitute. When singing groups like Sweet Honey and the Rock or the Nylons sing a cappella on stage, anyone can hear it. Where does that sound come from? It comes from real harmony. "Music stands in a much closer connection with pure sensation than any other art," Helmholtz noted in the last century, because "... in music, the sensations of tone are the material of the art." He added prophetically that "the theory of hearing is destined to play a much more important part in musical aesthetics ... music has to look for the foundation of its structure." The Justonic system is based on that very foundation. The authors feel that this breakthrough will perhaps help usher in the new era of pure harmony in music. Eventually, we might guess, all serious music will return to the pure and natural harmonics known to musicians for thousands of years, and the era of all instruments being out of tune will be over.

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15 Perfect tuning: so what?

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ntonation theory is as much a fundamental part of music as color theory is a part of the visual arts, as grammar is to literature, or as ethics is to law. For four thousand years musicians have investigated the harmonic nature of sound and applied their discoveries to the artistic making of music. The design of musical instruments is also fundamental to music. Instrument design has always followed from harmonic theory, but also must answer technical demands. Herein lies the challenge to music. As with any craft, skill, or profession, there is a constant struggle to apply fundamental knowledge to actual results. The earliest bamboo pipes of Ling Lun and stretched strings of Byzantine, Egyptian and Greek musicians attempted to mimic the natural harmonic structures of sound, and were successful within limits. In the medieval period, keyboards and fretted instruments introduced to music a valuable simplification, namely the ability to strike a key or fret and achieve a tone of fixed intonation. This simplicity has benefited music for a thousand years. But there was a price: the nagging compromise of tempering harmonic intonation.

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Now, at the dawn of the 21st century, we have the technological ability to retain the simplicity of keyboards and frets while gaining back the perfect harmonic purity recognized by musicians throughout history. Some musicians react to this with enthusiasm and curiosity. However, one might also ask: so what? For about the last hundred years most western music has been played in equal temperament. We’ve been told in our music theory texts that the problem was “insurmountable,” and we have adapted. Our ears suffer the bad musical thirds, and perhaps don’t miss the altogether lost seventh harmonic. Our triads sound roughly in tune, especially if the music moves quickly. For those who want pure harmonics, there are a cappella choirs, string ensembles, barbershop quartets. Why do we need pianos that can play in pure harmonics? Music is doing fine as it is. Silent movies were doing fine before the talkies came along. In fact, many learned commentators claimed that audiences would not care about talking movies. “If they want talking they can go to the legitimate theater.” Radio was doing fine before television. In fact, learned commentators claimed that “people don’t want to sit down and watch something from over the airwaves. People are happy with their radios. They can listen as they work or play.” Writers were certainly doing fine with typewriters, and did not need word processors to create great literature. Some people believed CD’s would never make it. Who wants to spend twenty bucks on a CD when you can get a record at less than half the cost, and it works just fine on the turntable that you already have. On the other hand, a few people throughout history have always advanced art and culture by imagining that “doing fine” does not preclude improvement. Aside from making fortunes, the visionaries also have given tools of great value to the culture. Sound cinema, television, computers, and CD’s are not the half of it. Every tool we have was, at one time, an innovation. Let’s face it: Neanderthals were doing fine before hammers.

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But what about music? What advantages to music can be derived from enabling keyboards and guitars to play pure harmonics, and to modulate in real time while maintaining pure harmonics? The first and most obvious advantage is that music, as we have learned in this long history of harmonic tuning, is divided along two parallel paths. Pure harmonic music is isolated from keyboard-and-guitar-based music. The two can not join forces because the basic intonation systems are mutually exclusive. By giving the keyboard/fretted instruments the option of playing in pure harmonics we simply allow these two strains of modern music to reach each other. There is no artistic down-side in this; there is only creative opportunity. Allowing more musicians to work together has to be a positive step. Some will take advantage, some won’t. To begin with, a cappella choral groups and chamber ensembles who are devoted to pure harmony now have a source of pure harmony for rehearsal purposes. Previously, there was no accurate source for correct intonation. Choral rehearsals have been guided by the excellent ears of a few top directors. “Wean singers early from the piano,“ says Harvard choral director Jameson Marvin, because “when the piano plays, the conductor cannot hear or listen acutely for problems of intonation”72 Dr. Robert Barstow, chair of the Department of Music at the State University of New York at Oneonta wrote to Justonic that “as a choral director, it is a constant hassle to teach my students to sing in tune when in fact they most often sing with an out-of-tune instrument, viz., the piano.” Now, by giving keyboards the ability to achieve perfect harmonics with free modulation, the choral director, chamber group, or barbershop quartet, has a new tool. The director can play the piece, and the singers can hear their precise parts. Just as writers gravitated to word processors because of the added ease and speed of editing, not because it made them write better, these musicians will embrace the new keyboard technology because of the ease and speed of rehearsal. The advantage is simple: save time, save work, and get on with the creativity.

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Furthermore, these pure harmony musicians can now include pianos and guitars in actual performance if they wish. This is simply a new artistic option, certainly not required, but clearly valuable. The arts evolve, and the advancement of artistic tools is a major contributor to this evolution. Bach and Mozart wrote music to be sung and played in precise tune, and they did the best they could with the instruments they had. We can now achieve what they could only approximate. If a choral ensemble wants to do a pure harmonic Mozart piece with piano accompaniment, we can now achieve that. We thereby gain a new advantage. We gain no advantage by limiting ourselves to seventeenth century technology in music. There are many serious modern composers who write for pure harmonic music. These include American composers Ben Johnston, Lou Harrison, Wendy Carlos, La Monte Young, Terry Riley and others. These composers have been frustrated by the lack of adequate instruments. “My own music has had to solve the problem of how to get the intonations I need from instruments ... not inclined to play them,” says composer Larry Polansky.73 Emmy nominee film composer Stephen James Taylor has used pure harmonic pieces in his work including The Lion King’s Timon and Pumbaa on the CBS Saturday show. He comments that “what has kept many of my colleagues from exploring the microtonal area has been the lack of user-friendly interfaces.”74 He adds that “I have been waiting for quite some time for someone to come along and do what you are doing.” The introduction of these new instruments will not only satisfy these artists who have already experimented with pure harmonics, but will introduce this musical option to the broader community of musicians. What is now a flourishing niche technique in music among musicians with the enthusiasm and patience to meet the challenge on their own, will become, with the introduction of the appropriate user-friendly tool, a standard technique. San Francisco Symphony Director Michael Tilson Thomas has featured Lou Harrison and other pure harmonic composers in his programs, earning a reputation for making new music accessible to modern audiences. Thomas is a

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keyboard musician dedicated to classical as well as modern symphonic music. “Who knows,” Thomas asks, “maybe now we can start to develop keyboard instruments that will make acoustically correct adjustments on themselves as you are playing them.”75 This dream is exactly what Justonic Tuning Inc. has achieved with its Pitch Palette software. The pure harmonic keyboard thus opens all music to a valuable option, not just a new gimmick or digital toy for synthesizers, but a priceless and historic musical technique based on the very foundations of sound. Will musicians working the popular genres - rock, country, jazz - want this option? Every professional popular musician works to achieve a unique sound. Having this new option will certainly appeal to some musicians. No single style or technique is going to claim 100 percent of the artists, nor should it. However, any serious musician will, at the very least, want to experiment with pure tuning, and many will adopt it in their work. “The effect of music is heightened by being in tune,” says composer Terry Riley, “what happens when a note is correctly tuned is that it has detail and a landscape that is very vibrant.”76 Musicians know the problem with equal temperament, and they know what true harmony sounds like. One final advantage to pure harmonic music is this: most western music is derived from one single chromatic scale. That this scale is out of tune with natural harmonics is one drawback, but a further limitation is the exclusion of all other perfectly useful historic and international scales. Computers not only give us the opportunity to solve the harmonic problem, but once this problem is solved the computer opens the door to a whole world of harmonic scales, Javanese scales, Arabic scales, East Indian scales, historic scales, Greek, Persian, Romanesque, Gothic ... the list is literally endless. These musical options will also appeal to certain musicians and will fit with the growing internationalism of music. For example, Egyptian musician Fathi Saleh, Egypt’s Cultural Attaché to France, is the creator of the “Musiclabe,” an instrument for the generation of the Arabic musical modes or Maqamat. He wrote to Justonic Tuning Inc. that “I am delighted that you are working on an idea I have dreamt of for years ... one day somebody

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making a machine that can tune to any scale and play in real time.”77 A greater flexibility in scale choice will lead to a greater awareness among musicians of the richness of international, historical, and cultural scale options. This can only be good for the advancement of the musical arts. Musicians know a purely tuned chord from a tempered chord. Pure harmony is the essence of music. Do musicians want it? Well, some do and some may not. This is what freedom of artistic choice is all about. In 1947 American composer Harry Parch said “perhaps the most hallowed of traditions among artists of creative vigor is this: traditions in the creative arts are per se suspect.” He adds that “the extent to which an individual can resist being blindly led by tradition is a good measure of his vitality.” On the subject of keyboard tuning flexibility, he predicted “it is quite conceivable that an instrument could be built that would be capable of an automatic change of pitch throughout its entire range ... the problem of transposition may be considered minor, one for which a solution will inevitably be found,” and added that “few persons will know or care what the means is, what its nature is, or how it came into being, but the rewards ... will be great.”78 It is both this “creative vitality” among musicians, and need for a better instrument, that inspired the Justonic system. It ought to be the job of the innovative music instrument designer to provide choices for artists, not dictate how music should be played. The purpose of new musical instrument design is not to replace one limited paradigm with another, but rather to expand the musical resources. Artists need tools, and a changing art needs changing tools. By expanding the palette of sounds available, by giving musicians new options, the inventors (Gannon & Weyler) of the Justonic system have attempted to add tuning precision and flexibility to the musician’s tool kit. The future of music is destined to be vastly influenced by digital technologies. This change is already in full swing. The piano gave us simplicity but compromised pure harmony. Digital technology allows us to retain the first advantage and restore the purity that was lost in the original compromise.

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It is ironic that computer power and digital signal processing return to music some of its most fundamental qualities, but the long story of harmony is rich with such irony. Harmonic proportions are the essence of musical sound, and this has been known for thousands of years. Musical traditions, musical instruments come and go, but the fundamental nature of sound remains. From the bone flute of the Neanderthal, to the bamboo pipes of Ling Lun, to the kithara of ancient Greece, to the keyboard of Bach, and to our present day the nature of sound has not changed. What does change are the creative instincts of artists and the tools we build to manifest those artistic visions. Shakuhachi master Masayuki Koga reminds us in his 1989 The Japanese Bamboo Flute:79 “The wondrousness of the human mind is too great to be transferred into music only by 7 or 12 elements of tone steps in one octave. There are millions of steps of microtones, and none is to be thrown away, just like nothing is to be wasted in this world.”

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Appendix A The Harmonic Overtone Series With comparative chromatic scale degrees or approximations Frequency Notation in "C" in "C" (or approx.) Hz

Harmonic series

Frequency Relation to tonic

To closest lower octave

1st octave 1

1/1

1/1

1.00000

0.00

528.00

C

Tonic

2nd octave 2 3

2/1 3/1

1/1 3/2

1.00000 1.50000

0.00 701.96

528.00 792.00

C G

Octave Fifth

3rd octave 4 5 6 7

4/1 5/1 6/1 7/1

1/1 5/4 3/2 7/4

1.00000 1.25000 1.50000 1.75000

0.00 386.31 701.96 968.83

528.00 660.00 792.00 924.00

C E G Bb

Octave Major third Fifth Blue minor seventh

4th octave 8 9 10 11 12 13 14 15

8/1 9/1 10/1 11/1 12/1 13/1 14/1 15/1

1/1 9/8 5/4 11/8 3/2 13/8 7/4 15/8

0.00 1.00000 1.12500 203.91 1.25000 386.31 1.37500 551.32 1.50000 701.96 1.62500 840.53 1.75000 968.83 1.87500 1088.27

528.00 594.00 660.00 726.00 792.00 858.00 924.00 990.00

C D E F#b G Ab# Bb B

Octave Major second Major third 1/4-tone flat diminished fifth Fifth 1/4-tone flat sixth Blue minor seventh Major seventh

16/1 17/1 18/1 19/1 20/1 21/1 22/1 23/1 24/1 25/1 26/1 27/1 28/1 29/1 30/1 31/1

1/1 17/16 9/8 19/16 5/4 21/16 11/8 23/16 3/2 25/16 13/8 27/16 7/4 29/16 15/8 31/16

1.00000 0.00 1.06250 104.96 1.12500 203.91 1.18750 297.51 1.25000 386.31 1.31250 470.78 1.37500 551.32 1.43750 628.27 1.50000 701.96 1.56250 772.63 1.62500 840.53 1.68750 905.87 1.75000 968.83 1.81250 1029.58 1.87500 1088.27 1.93750 1145.04

528.00 561.00 594.00 627.00 660.00 693.00 726.00 759.00 792.00 825.00 858.00 891.00 924.00 957.00 990.00 1023.00

C (C#-) D (Eb-) E

(A+) Bb (Bb+) B B#

Octave Minor second (7 cents flat of 16/15) Major second Small minor third (between 6/5 and 7/6) Major third 1/8-tone flat fourth 1/4-tone flat diminished Fifth 1/8-tone sharp diminished fifth Fifth 1/8-tone flat minor sixth 1/4-tone flat sixth 1/8-tone sharp sixth (21.5 cents sharp of 5/3) Blue minor seventh Sharp minor seventh (1/4-tone sharp of 7/4) Major seventh 1/4-tone sharp major seventh

32/1 33/1

1/1 33/32

1.00000 1.03125

528.00 544.50

C C#b

5th octave 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 6th octave 32 33 etc...

Decimal

Log. Cents

0.00 53.27

(F-) F#b (F#+) G (Ab-) Ab#

Western chromatic scale degree (or approximation)

Octave Quartertone, flat minor second

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Appendix B The Derivation and Use of Just Musical Scales

T

he phenomenon of harmony - or resonance, or sympathetic vibration - is not a human construct, but is rather a natural relationship between vibrating bodies in the real world. We make music by arranging audible vibrations into patterns, and traditionally we have been guided by natural harmonics. The relationships among vibrational frequencies are what we call musical intervals. We measure these intervals in vibratory cycles per unit of time. A stretched string or piece of wood or metal may vibrate back and forth once every millisecond. Another string or piece of wood or metal may vibrate twice in a millisecond. The relationship between these two vibrations is "two-toone," written as the ratio 2:1 or as the fraction 2/1, or inversely as the fraction 1/2. Frequencies that have whole number relationships are called "harmonic." A pure harmonic musical scale, or "just" scale, is defined as a series of audible frequencies determined by applying a series of whole number ratios to any fundamental base frequency. Just ratios are made up of whole numbers (1, 2, 3, 4, etc.). These numbers represent whole, complete cycles of vibration. Examples of whole number ratios, or just ratios, are as follows: 2:1, 3:2, 5:4 These ratios may also be written as fractions:

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2/1, 3/2, 5/4 Harmonic relationships were discovered by human ear at least 4,500 years ago, and probably much earlier. What we call an "octave" is a precise harmonic ratio of 2:1. This was probably the first harmonic relationship discovered, and it is the most universal of all musical relationships. Most scales in most cultures repeat a pattern of tones after every 2:1 interval. However, this is a convention, not a demand of nature. It would be perfectly reasonable, as an artistic exercise, to repeat a scale after every 3:1 or 5:1 interval, or at any other point, or not at all. However, the musical ears of most cultures have settled on the 2:1 interval as the basic unit of musical scales. An "octave" may be defined as any doubling or halving of any frequency (multiplying by any power of 2). The western word "octave" refers, of course, to the convention of the eighth diatonic tone being the double of the tonic. This is a convention, not a law of nature. However, the convention is not arbitrary, as Gerald Eskelin points out in Lies My Music Teacher Told Me. As Eskelin, director of the L.A. Jazz Choir, points out, the major/minor diatonic scale degrees are derived from the most natural and useful harmonic relationships. Nevertheless, a just musical scale may have any number of tones within an "octave." The eight notes of the diatonic major scale include both the tonic (1:1) and the octave (2:1). When we speak of a “12-tone” scale, these 12 tones do not include the octave tone. It is best, when analyzing just scales, to remember the simple harmonic relationships behind the conventions. In this discussion we will refer to scales as having a certain number of tones, and in every case the number of tones does not include the 2:1 "octave" tone. Also, please be aware that "tone" in this context refers to any just ratio, and not to a western "wholetone step." Serious problems do arise from our musical conventions, namely the predominant use in western music of a single, mistuned scale, and the exclusion of pure harmonics and countless other scales.

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The equal tempered scale is a mathematical construct created by seventeenth century mathematicians to simplify the construction of keyboard instruments. The equal tempered major thirds and sixths are sharp of their harmonic counterparts, minor thirds and sixths are flat. The fourths are sharp, the fifths flat. These deviations can be seen in the table.

Equal Tempered Deviations from Just Intonation

Deviations in cycles-per-second (Hz), Key of A=440 Hz. The tempered flat-7 is 14 Hz sharp of the natural harmonic, the minor third is 5 Hz flat, the major third is 4 Hz sharp, and so forth.

To use a pure harmonic scale for music, one must alter the frequency of each note depending on key modulation and harmonic structure. This dynamic alteration of pitch is the challenge that choral singers or string ensembles face when they attempt to execute a piece of music in pure harmony. A well-trained

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horn or string player knows that the "D" in the key of E is slightly sharp of the "D" in the key of A.80 Furthermore, melodic considerations may suggest additional changes in pitch. We know, from analysis of recordings, that string players like Heifetz, Casals, and others have used up to 40 tones per octave in performance.81 To play western chromatic music in any and all keys, while maintaining just intonation, requires hundreds of such microtonal, or enharmonic, adjustments. The authors have devised a system that overcomes this problem by analyzing the modulations and harmonic structures of music in real time, and by making the necessary changes in the intonation of each tone. There have been previous attempts to solve this problem with both mechanical and electronic means.82 Mechanical solutions involving extended keyboards have proved too cumbersome to build and play, limiting their popular appeal. Previous electronic solutions have been limited in their ability to respond in real-time, as the musician plays. The purpose of the authors' Justonic software retuning system is to allow the musician to play the traditional western keyboard (or fretted instrument) in the traditional manner, without having to step through computer menu selections, and yet also achieve pure harmonic intonation for all harmonic structures in all key modulations. The history of musical scale development has been encumbered with attempts to create just scales that eliminate the problem. Such attempts have failed for quite natural and logical reasons. It is simply not possible to build a just scale, based on harmonic ratios, that can freely modulate to any scale degree and play the same scale from that point without retuning any of the tones. There is no way to avoid the fact that in true harmonic music the frequency of each scale degree must be flexible. The equal tempered scale does not solve the problem, but rather chooses free modulation over harmonic purity by abandoning harmonic intervals altogether. Harmonic ratios are derived from the harmonic series. These harmonics are also referred to as overtones or as partials. The authors use the common convention which defines these terms in the following manner: A harmonic is a

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pure harmonic, a whole number multiple (2x, 3x, 4x, etc.) of any base frequency. An overtone is not necessarily harmonic. For example, bell, drum, and piano string overtones are not purely harmonic. Flute and horn harmonics tend to be pure harmonics. The series of partials are the same as the series of harmonics in that they include the fundamental tone (1:1) itself. Long before human theorists knew anything about harmonics, combination tones, or any other acoustic qualities of sound, human ears used these qualities to discover the natural harmonies. When ancient vocalists heard themselves sing a perfect unison (1:1), or a perfect octave (2:1), they knew it sounded pleasing. The ancient Chinese discovered another pleasing relationship, what we now call the perfect fifth, and they also discovered that the ratio between these two tones was 3:2. What they didn't know was that the fifth was the third partial of a tone one octave below the tonic. They did not need to know this. Their ears put all this information together and decided that this interval sounded alluring and radiant. The Greek Harmonist Archytas is credited with first documenting what we nowadays call the pure harmonic major third, a 5:4 relationship of tones. Musicians naturally began to string these pleasant sounding tones together into formal scales. The purpose of this paper is to introduce certain scale choices, and to show how one might organize the process of selecting tones for harmonic, or just, scales. We will look at the harmonic logic behind the creation of a just version of the western chromatic scale, and we will then investigate how we might begin to organize optional scale choices into a system of scales that may be exploited by composers and musicians. Deriving a just western chromatic scale A pure harmonic, or just, version of the western 12-tone chromatic scale can be generated through an analysis of the natural harmonic series, the combination tones that arise from these harmonics, and inversions of these tones.

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It took human musicians several thousand years to come up with this knowledge, but it all derives from quite simple and natural acoustic phenomena. The harmonic series for a fundamental frequency of 110 cycles per second (Hz) is: 110, 220, 330, 440, 550, 660, 770, ... etc. The second partial is the octave of our fundamental. The third partial has a relationship of "three-to-one," (330 Hz to 110 Hz) or 3:1. We reduce this tone to the lower octave by dividing by two, making it "three-to-two," written as 3:2. This, we already know is the just perfect fifth in the diatonic scale. Continuing with this process, we see that the fourth partial is another octave, the fifth partial can be reduced to a just third (5:4), and so on. The first fifteen partials of the natural harmonic series give us the following set of unique harmonic ratios (in order of appearance, and reduced to a single octave): 1:1, 3:2, 5:4, 7:4, 9:8, 11:8, 13:8, 15:8 These ratios are a subset of the theoretically infinite Harmonic Set. This subset, above, has a prime limit of 13. We call this set of ratios the Fundamental Chromatic Harmonic Set, because these ratios are the basis of the western chromatic scale of 12 scale degrees in semitone steps. Six of these eight tones fall at scale degrees of the western 12-tone scale. The 11:8 and 13:8 fell approximately at quarter-tone points between the semi-tone scale degrees. We will return to these two harmonic intervals when we discuss extended scales. Withholding these quartertone ratios, we are left with six harmonic ratios: 1:1, 3:2, 5:4, 7:4, 9:8, 15:8

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This is the Chromatic Harmonic Set. Re-arranging the ratios in ascending order, we have: 1:1, 9:8, 5:4, 3:2, 7:4, 15:8 Or, in familiar notation in the key of C: C, D, E, G, Bb, B This hexatonic scale is the most natural harmonic scale within the prime limit of 7, generated directly from the harmonic series. In our modern diatonic nomenclature, these ratios represent the tonic (1:1), second (9:8), third (5:4), fifth (3:2), minor-seventh (7:4), and the major-seventh (15:8). It may not be immediately clear to musicians unaccustomed to seeing intervals written as ratios that a 5:4 is a wholetone larger than 9:8. In any case, musicians ought to be aware of these ratio relationships since they are the very foundation of the musical arts. It is important to hear these intervals and to understand how they differ from tempered intervals. The reader can use the Justonic Pitch Palette software to access these intervals and hear them. String players can find these tones by ear, simply by playing the harmonics of a base tone. Notice that the positions of these harmonic tones do not match the tempered positions. Play a triad with these just intervals, and witness the beatless, smooth quality. Add the minor or major seventh. These chords will sound different from the tempered variety. Both sevenths sound flat to the tempered-trained ear, but notice that they create very sweet harmonies. Inversions of any interval are also quite natural. For example, when the ancient Chinese or Greek musicians discovered the perfect fifth, 3:2, they also discovered its inversion, the fourth, 4:3. Why? Because there is nothing in the nature of these tones to say which is the tonic and which is the interval. The octave

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of the original tonic is the fourth of the fifth. That is, the ratio series, tonic-fifthoctave: 1:1, 3:2, 2:1 (the notes C, G, C) can simply be transposed to make the G the tonic: 2:3, 1:1, 4:3 These are the same tones defined differently, as a tonic in the middle with a fourth above and a fifth below. We say, therefore, that the fourth is the inversion of the fifth. Inversions of the six tones of the Chromatic Harmonic Set give us six new ratios. Excluding the 2:1 octave, we have five new ratios to consider: 16:9, 8:5, 4:3, 8:7, 16:15 Bb,

Ab,

F,

D,

C#

The 8:7 is slightly sharp (approximately an eighth-tone) of our more common second, 9:8. We will save this harmonic ratio for scales extended beyond 12 steps, or with sub-semitone steps. The 16:9 falls between the minor and major sevenths. We will use this ratio as an alternative minor-seventh in certain variations. In the meantime, withholding these two sub-semitone steps, we have three new ratios which fall at western semitone steps, and which we add to our scale: 8:5, 4:3, 16:15 Placing these ratios in their proper ascending order, we now have a harmonic nonatonic (9-tone) scale, the Justonic Inversion Harmonic Scale. In

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diatonic terms, we have added to our scale the minor-second (16:15), the fourth (4:3) and the minor-sixth (8:5). Our scale now looks like this: 1:1, 16:15, 9:8, 5:4, 4:3, 3:2, 8:5, 7:4, 15:8 C,

C#,

D,

E,

F,

G,

Ab,

Bb,

B

This scale has two semitone steps, followed by alternating tone and semitone steps, and a final semitone step to the octave. In addition to natural harmonics, tones played together also create combination tones, the addition and subtraction of two tonal values. For example, if two tones are sounded together, one at 440 cycles per second, and another at 660 cycles per second, these will create two combination tones, one at 220 cycles per second, and another at 1100 cycles per second. That is, a pure A and E create a sub-octave A and a higher C#. These combination tones add significantly to our perception of pure harmony.83 Likewise, if we simultaneously sound the tonic and the minor-sixth (8:5) of our harmonic scale, above, a combination tone is created that has the precise ratio of 6:5, the minor third.84 This tone is created by the difference between the frequency of the tonic and the frequency of the minor sixth. Its inversion is the major sixth, 5:3. Adding these two harmonic tones, we now have an 11-tone scale: 1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 3:2, 8:5, 5:3, 7:4, 15:8 C,

C#,

D,

Eb,

E,

F,

G,

Ab,

A,

Bb,

B

All the steps of this natural scale, the Differential Harmonic Scale, are of semitone magnitude except for the wholetone step between the fourth and the fifth (4:3 to 3:2). The "missing" semitone is the only tone in our 12-tone western scale that is not directly generated by the first 15 partials plus inversions and

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combination tones of the harmonic series; it is also the theoretical center of the octave, and in these characteristics it is unique. The most harmonic choice for this tone, the simplest ratio, is 7:5. We may derive this ratio by taking a minor sixth of a minor seventh and reducing it to our octave. (7/4 x 8/5 x 1/2 = 7/5). There are other choices, such as 45/32, which we may use as variations, and we will discuss these later. Using 7:5, we now have a natural harmonic 12-tone scale in semitone steps, and with a prime limit of 7: 1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 7:5, 3:2, 8:5, 5:3, 7:4, 15:8 C,

C#,

D,

Eb,

E,

F,

F#,

G,

Ab,

A,

Bb,

B

This classic just scale is the default western chromatic scale in the Justonic system, the Justonic Classic Harmonic Chromatic Scale. The equal tempered version of this scale has equal semitone steps. This just scale has semitone steps, but they are not all equal in size. There are five different semitone steps here, ranging in size from the small step between the thirds and between the sixths (25:24) to the large semitone between the tritone and the fifth and between the major and minor sevenths (15:14). Likewise, the wholetone steps vary in size. This variety gives each harmonic scale a distinct feel. As we have already seen, the chromatic scale has several just variations. The most significant variant is in the eleventh chromatic position, the "minorseventh." The choices for this scale degree are: 1. 7:4, the natural harmonic, or seventh partial, of the tonic: This harmonic ratio is the natural choice as a minor seventh. It is flat of the equal tempered variety, and is sometimes called the "blues seventh." This minor-seventh is quite expressive and sounds excellent in major chords or with the septimal (7:6) minor third (see scale g, below). This just tone is the most foreign-sounding to

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musicians who are used to the much sharper tempered minor seventh, but there is no denying its exquisite harmonic feel. 2. seventh.

16:9, the inversion of the 9:8 second: Closest to the tempered minor-

3. 9:5, the perfect fifth of the 6:5 minor third: The most sharp of the three choices, this tone is the correct interval when used with a 6:5 minor third. However, since the same result can be achieved in the Justonic system by "rooting" the minor chord on its own minor-third or fifth, using this ratio as the scale degree minor-seventh is not necessary, and it is quite rough when heard in a major chord. It is, however, a viable artistic choice. By substituting these three options for the minor-seventh we have three alternative chromatic scales, differing only in their eleventh scale degree, and named as follows in the Justonic system: (a) The Justonic Classic Harmonic Chromatic Scale: Using the natural harmonic minor-seventh: 1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 7:5, 3:2, 8:5, 5:3, 7:4, 15:8 (b) The Justonic Inversion Chromatic Scale: Using the inversion of the second for the minor-seventh: 1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 7:5, 3:2, 8:5, 5:3, 16:9, 15:8 (c) The Justonic Acute Minor-Seventh Chromatic Scale: Using the natural fifth of the minor-third for the minor-seventh: 1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 7:5, 3:2, 8:5, 5:3, 9:5, 15:8

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The next most significant scale variant is in the seventh chromatic position, the tritone, or diminished fifth of the diatonic scale. As we have already determined, there are no direct, natural harmonic choices for this unique degree of the chromatic scale which theoretically falls in the middle of the octave. The tempered square root of 2 is the mathematical center of the octave, but since this tempered tone makes no harmonic relationships with any other tones, it is not appropriate in a just intonation context. The two most obvious harmonic choices for this scale degree are: 1. 7:5, the simplest ratio, is the most harmonic choice, and appears in scales a-c above, and g below. 2. 45:32 the ratio derived by taking a third of the second; this tone is slightly sharp of the 7:5 tone. There are other choices, such as the Pythagorean tritone (729:512), the nearly-equal tritone (140:99), and the inversion of 45:32 (64:45). The avid experimenter might want to listen to these tritone alternatives. For our purposes we will substitute 45:32 for 7:5 in the above scales. We have three new scales, named as follows in the Justonic system: (d) The Justonic Derived Harmonic Chromatic Scale: Using the harmonic minor-seventh and the derived diminished fifth: 1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 45:32, 3:2, 8:5, 5:3, 7:4, 15:8 (e) The Justonic Derived Inversion Chromatic Scale: Using the inversion of the second for the minor-seventh with the derived diminished fifth:

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1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 45:32, 3:2, 8:5, 5:3, 16:9, 15:8 (f) The Justonic Derived Acute Minor Seventh Chromatic Scale: Using the sharp, 9:5 minor seventh and the derived diminished fifth: 1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 45:32, 3:2, 8:5, 5:3, 9:5, 15:8 We will look at one more variation. The minor third can be moved lower almost a quartertone, from 6:5 to 7:6. This more flat minor third is sometimes called the "blues minor third."85 It has a unique and interesting feel, and the 7:4 minor seventh forms a perfect fifth with this minor third. By substituting this interval into scale "a", above, we get the following chromatic scale, featuring the seventh harmonic, the harmonic most obscured by equal temperament. (g) The Justonic Septimal Chromatic Scale: Using the seventh harmonic as a basis for the minor third, tritone, and minor seventh: 1:1, 16:15, 9:8, 7:6, 5:4, 4:3, 7:5, 3:2, 8:5, 5:3, 7:4, 15:8 Above we have set out seven versions of the western chromatic scale, each slightly unique, but each performing essentially the same role of a chromatic 12tone scale in semitone steps. Western musicians and composers will find that the scalar resources outlined here have creative applications, and that the harmonic integrity speaks for itself. Chords created with these scales have a harmonic purity unheard in tempered music. Subsets, or modes, of these chromatic scales - such as our diatonic major scale, pentatonic, Gypsy, whole-tone, etc. - can of course be selected at the discretion of the artist. In the Justonic system we call these subsets "modes" of the chromatic scale. This use of the term mode is different than the Greek or ecclesiastical mode which indicates a new, nominal tonic for a given set of

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intervals. For our purposes, a mode is any subset of a scale in any configuration. For example, the Pentatonic scale 1:1, 9:8, 5:4, 3:2, 5:3 is a selection, or mode, of the fundamental chromatic harmonic scale, as is our diatonic major scale of Do, Re, Mi ... etc. A just version of our diatonic major scale is: 1:1, 9:8, 5:4, 4:3, 3:2, 5:3, 15:8 All of the classical diatonic and chromatic Greek and ecclesiastical modes in western music are subsets of the chromatic scale. Every scale is a mode of all larger scales containing its tones. Theoretically, there exists an infinite number of scales, harmonic or otherwise, and therefore every scale is a mode of some other scale. Likewise any mode can be considered to be a scale. However, for the purposes of common usage within the Justonic system, once a fundamental harmonic scale is selected and identified as such, all subsets of that scale are called modes. When modulating key, or changing harmonic structure within a key, we want to keep the ratios pure, and this necessitates the retuning of the actual frequencies. This extra requirement is the price we pay for playing in tune. The discerning artist may find this price to be a bargain. The Justonic system helps by making the tuning adjustments automatically. The seven scales described above only scratch the surface of scale building using harmonic ratios. What we will call extended scales are harmonic scales using sub-semitone steps, or more than 12 steps per octave. Extended Scales The chromatic scale of semitone steps is only one way to organize tones into scales. Throughout history, in many cultures today, and even for some modern western composers, scales take on a variety of patterns, each with its own

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limitations and merits. For the musical artist, this rich variety of scales represents a bounty of tonal resources available for the creation of music. The enharmonic scales, scales using sub-semitone steps - Greek, Arabic, East Indian and other - are not subsets of the chromatic scale. Nevertheless, any such scale that can be described by a set of rational ratios can be considered a just scale. For example, one such scale would be a version of the Greek Enharmonic Dorian, based on an enharmonic tetrachord86 of Didymus, in which the just ratios are: 1:1, 32:31, 16:15, 4:3, 3:2, 48:31, 8:5 Here we have two quarter-tone steps followed by a two-tone step, wholetone step, two more quarter-tone steps, and a final two-tone step to the octave (2:1). Arabic Rast-Rast scales also uses two quarter-tone steps. These scales employ a "neutral" third, and a "neutral" seventh. A typical Rast scale or mode has seven or eight tones per octave. As with the chromatic western scales above, the Justonic system uses several varieties of 12-tone "chromatic Rast" fundamental scales, each of which allows for a number of Arabic Rast modes of seven or eight tones. One example, using traditional Arabic intervals for all the steps, is the following: 1:1, 18:17, 9:8, 32:27, 27:22, 4:3, 24:17, 3:2, 128:81, 27:16, 16:9, 81:44 The fourth and eleventh steps of this 12-tone scale are quartertone steps; the fifth and twelfth steps (the twelfth step is from 81/44 to the octave 2:1), are threequarter-tone steps. The scale has a prime limit of 17, and uses several ratios derived from early attempts to mitigate the modulation problem, with an emphasis on modulations to the dominant and subdominant. Arabic musicians have remained loyal to traditional just ratios. However, the lack of instruments that

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could freely modulate and make microtonal adjustments has limited Arabic music to primarily melodic, modal lines with very little harmonic embellishment. Arabic composers find this frustrating, but are also hesitant to relinquish their pure harmonic scales. Whereas European music sacrificed harmonic purity for ease of modulation, Arabic music has sacrificed polyphonic development for harmonic purity. The Justonic system addresses the needs of both traditions. For the Arabic scales, the system retains traditional harmonic ratios, but adds free modulation and extended harmony by making the necessary microtonal shifts. There are many historic just scales using other than semitone steps. In the context of a just scale, "semitone" and "quartertone" are variable, not precise intervals. We loosely refer to a "semitone" as 1/12 of an octave, but in just intonation scales a semitone may be an interval of 16:15, the slightly more flat 18:17 as in the Arabic scale above, and in other cases may be 21:20, 135:128, or some other ratio of similar size as in the just western chromatic scales above. It is only a convention, after all, to call 1/6 of the octave a "wholetone," and it is only in the equal tempered system that all tones and all semitones are of equal size. We may now return to the harmonic intervals which we gleaned from the harmonic series but did not use in our just versions of the western chromatic scale. Some of these we left out because they formed quartertone intervals, and some were alternatives to certain scale degrees. These harmonic scale ratios are: 11:8, 13:8, 8:7, 16:9, 9:5 Adding these tones to our chromatic scale, we have a 17-tone scale with a prime limit of 13, and with nine sub-semitone steps created by the inclusion of the new ratios: 1:1, 16:15, 9:8, 8:7, 6:5, 5:4, 4:3, 11:8, 7:5, 3:2, 8:5, 13:8, 5:3, 7:4, 16:9, 9:5, 15:8

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We may continue this process by adding ratios such as 10:9, 7:6, and 11:6, giving us a 20-tone scale: 1:1, 16:15, 10:9, 9:8, 8:7, 7:6, 6:5, 5:4, 4:3, 11:8, 7:5, 3:2, 8:5, 13:8, 5:3, 7:4, 16:9, 9:5, 11:6, 15:8 The process of scale-building is theoretically infinite, and indeed there have been hundreds of scales used throughout history by human cultures. Most of these scales are derived from (in order of historic discovery and significance) the octave (2:1), the fifth (3:2), the fourth (4:3), and the major third (5:4). These harmonic intervals are all heard in the first five partials of an audible frequency. They truly represent the harmonic heart of world music.

Cataloging the parameters of scales The authors predict that since computer and signal processing power now makes free harmonic modulation possible, and since the Justonic system makes it simple, western music will naturally move toward a greater variety of scale choices. The 300-year era of most music being written and played in a single outof-tune chromatic scale is over. Composers and musicians will naturally expand their tonal vocabularies as these scales are made accessible. And as this phenomenon unfolds, we might be wise to consider how best to organize and catalog the infinite variety of harmonic scale choices. The history of western music has left us with a tangle of contradictory conventions, as discussed above. It is an irony of western music that we have so limited ourselves to one simple scale, and yet so confounded our young students with a notation system

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that begs its own twisted internal logic. It is quite possible that we could solve both problems, expanding our scale choices, while rationalizing and simplifying (and internationalizing) our notation system. In any case, to begin, we must identify the parameters of scales, and then attempt to organize those parameters into a system that simplifies selection of the scales. The authors suggest the scale parameters in table 3, and invite composers and musicians to suggest other parameters, or changes to those in table 3, that might suit the needs of modern music making. Table 3: Scale parameters I. Intonation Family: A. Harmonic: based on whole number harmonic series B. Overtone: based on enharmonic overtone series 1. Drums 2. Bells 3. Stretched piano 4. etc. C. Tempered: non harmonic 1. meantone 2. equal 3. etc. D. Free Continuum: No restriction, using the entire range of audible tones at the discretion of the composer.

II. Intonation Genus: A. Harmonic: All tones directly taken from the harmonic series. B. Inversion: Harmonics plus inversions

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C. Derived: Harmonics, inversions, plus tones derived by taking intervals of existing tones. D. Free: No restriction on tone selection III. Prime limit: All harmonic scales have a prime limit, the largest prime number harmonic necessary to create the scale. Prime numbers 1 and 2 are assumed, so this category would start with 3, then 5, 7, 11, 13, 17, etc. IV. Scale step construction (Tetrachord family, after Greek tradition, E.M. Wilson, 1986-7, and J. Chalmers, 1993):87 A. Diatonic: wholetones and single semitones 1. soft 2. intense 3. equable 4. etc. B. Chromatic: consecutive semitones and 3/2-tones (minor third), with wholetone disjunction between tetrachords. C. Enharmonic: with sub-semitone steps. D. Hyperenharmonic: with steps larger than a major third and smaller than a quartertone (smaller than 36/35, and in the range of the 81:80 syntonic comma, the "eighth-tone"). E. Reduplicated: with two exactly repeating intervals in each tetrachord, Arabic and others. F. Mean: Calculated mean divisions. 1. Arithmetic 2. Geometric 3. Harmonic 4. Logarithmic 5. etc. G. Tempered: divisions of a tempered fourth

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1. Equal 2. Non-equal H. Non-tetrachordal: with scale divisions based on other than tetrachordal design. 1. Pentachordal 2. Hexachordal 3. etc. I. Mixed: mixed tetrachord construction in a single octave scale. J. Miscellaneous: Chalmers' catch-all family which includes unusual historic (Archytas, Al Farabi, et. al.) and modern (I. Xenakis, H. Partch, J. Carrillo, et. al.) tetrachords. K. Free: no tetrachordal rule limiting scale construction. V. Characteristic Interval (per Chalmers): A sub-classification of the above, by largest interval in the tetrachord, or in the scale. VI. Repetition pattern: A. 2x: traditional harmonic octave B. 2x enharmonic: stretched piano octave, etc. C. Non-2x: 1. 3x, 5x, etc. 2. Variable 3. Non-repeating 4. Mixed 5. Free VII. Number of Tones: Because of the prevalence of 12-tone systems (in western music, and in world musical instrument construction), 12-tone scales might best receive their own category, with extended and subtended scales as the other two categories, with appropriate subcategories.

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A. 12-tone B. Extended (more than 12-tones per repetition unit) C. Subtended (less than 12-tones per repetition unit) VIII. Cultural origin: It may be useful to composers and musicians to be able to catalog and retrieve scales by musical tradition or cultural origin, with subcategories identifying individual originators or documenters of scales, such as Ling Lun, Archytas, Ptolemy, Al-Farabi... Harry Partch, Kathleen Schlesinger, Murray Barbour, John Chalmers, etc. IX. There may well be scalar parameters overlooked by the authors that are important to composers and musicians. As this cataloging of scales is a work in progress, we shall let the categories remain open. There are other important characteristics of scale use, such as notation system, modal system, note change parameters, pitch bend, tuning resolution, and modulation system (spiral, closed, etc.) that also require a close look, and a logical cataloging. These qualities may be considered independently of scale choice, and may best be included in a set of operational parameters. From an artistic point of view, there is nothing in nature to preclude any tone or interval from being used to make music. The historic rules governing scale choices reflect cultural taste and aesthetic habit, but do not constitute laws of nature. Nevertheless, human musical scales are predominantly influenced by the natural ratios of the harmonic series, the basis of harmony. Much of the historic tinkering with the harmonic intervals has been an attempt to mitigate the modulation problem, and avoid the necessary retuning with each change of key or tonal center. There are modern composers who have experimented with scale choice (Harry Partch, Julian Carrillo, Wendy Carlos, La Monte Young, Lou Harrison, Robert Rich, Ben Johnston, Erv Wilson and many others). However, for 700 years

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most western music has restricted itself to one chromatic scale in semitone steps. This limitation is a serious artistic compromise. The trade-off is for simplicity, but musicians might well ask themselves if such a trade-off serves their art. Would painters accept a restriction to a small, fixed set of colors because it was easier to build paint sets that way? Would poets accept the limitation of rhyming couplets because printers found it easier to set type that way? No? Consider that this is precisely what we have accepted in our choice of a single scale for the last 300 years! Musicians have been somewhat misled by keyboard manufacturers and theorists since the days of Bach. Keyboard manufacturers found that the mechanical challenge of creating a just intonation, free modulation, fixed-key instrument was beyond the limit of their technological resources. In short, they gave up. Theorists too often ventured down the road of searching for an acceptable compromise rather than precisely defining and solving the problem. Teachers were left to tell their students that the equal tempered intervals were the best compromise available, and that we all had to suffer the rough, beating chords. Young string players and horn players have been taught to hear and play a pure major third when playing with string and horn ensembles, then taught to play a tempered third when performing with a piano. This has caused untold confusion among young players who never entirely understood the problem and yet could not resolve the conflict so plain to their ears. Choirs learned to sing those beautiful, resonant harmonic triads that send chills down the spine and suggest the presence of angels above. Then, when singing with pianos and guitars they are forced into tempered triads that wobble and beat. It is true that the keyboard, and the system of equal temperament, have contributed to the development of our music. From the dramatic modulations and power chords of Beethoven to the chromatic fluidity of modern jazz, western music has taken full advantage of equal temperament. However, the loss of pure harmonics and scale variety was the price paid to gain this simplicity. Now, with the aid of computer power, we can solve the problem that 17th century keyboard

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manufacturers could not solve. We can also enrich our music with the great spectrum of scale choices that are available. The scales outlined above are little more than a starting place in this investigation. Please see table 4 for some additional scale choices. It is the belief of the authors that it is the responsibility of the music theorist and music instrument maker to serve music first. It is our hope that we might contribute in some small way to a new flowering of western and world music by making these scale choices available, and by making dynamic just intonation with these scales not only possible, but simple. For the modern composer and musician this expanded palette of tones can only contribute to greater variety, subtlety, and power of expression. May the muse be with you. ================

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Index

—1—

—9— 96-tone equal temperament, 95

12th-root-of-2, 13

—A—

—2— 24-tone equal temperament, 95

A cappella, 4, 7, 44, 117, 120, 121 Aaron, Pietro, 8, 63

—3— 36-tone equal temperament, 96

Acoustic Guitar magazine, 94 Acoustics, 5, 99 Agam, 53

—4— Aleatory, 90 48-tone equal temperament, 95

Al-Farabi, 52 Alves, William, 106

—7— Antheil, George, 97 72-tone multiple division equal temperament, 106

Arabic music, 51, 114 Archicembalo, 63

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144

Archytas, 31, 32, 33

Blues third, 51, 67

Aristotle, 25, 34, 37

Bosanquet, R.H.M., 80

Aristoxenus, 10, 37, 38, 40

Bowed instruments, 56

Arithmetic proportion, 33, 53, 88

Branca, Glenn, 96

Armstrong, Louis, 83

Brown, Colin, 80

Atonal, 82

Busoni, Ferruccio, 88

Aulos, 29, 88

Buxtehude, Dietrich, 6, 71

Automatic tuning, 104

—C— —B—

Cage, John, 83

Babylonians, 22

Cahill, Thaddeus, 89

Bach, J.S., 8, 14, 65, 68, 71, 72

Carillo, Julian, 87

Bach, Nicholas, 3, 4, 5, 12, 13, 68

Carlos, Wendy, 95, 112

Bamboo flutes, 17

Chalmers, John, 32, 97

Bamboo pipes, 20

Chanting, 17

Barstow, Dr. Robert, 111

China, 17, 50, 56

Beach Boys, 89

Chord inversions, 106

Beethoven, Ludwig Van, 73

Chromatic, 31, 88

Benade, Arthur H., 96

Chromatic scale, 82

Bernoulli, Daniel, 71

Chromelodeon, 92

Blue minor third, 84

Coker, Jerry, 84

Blue notes, 84

Coltrane, John, 83

Blues, 83, 84, 86

Combination tones, 69

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Comma of Didymus, 21, 27, 44, 85

Digital, 16

Commas, 5, 27

Digitized pitch, 16, 100

Common keys, 8

Disjunctive tone, 33, 46, 49

Computer, 94, 98

Dissonance, 38, 55, 79, 82, 100

Computer Music Journal, 95

Distant keys, 8, 58

Consonance, 54, 56, 79, 91

Doni, Giovanni Battista, 67

Counterpoint, 56

Dorian, 33, 46, 88

Cowell, Henry, 82, 88, 89

Doty, David, 90, 97

Cristofori, Bartolommeo, 14, 68

Drumming, 17

Curwen, John, 76

Dual identity, 91

Cycle of 53 fifths, 67

Duncan, Dudley, 84

Cycle of fifths, 26, 37

—E— —D—

Ecclesiastical, 33

d’Arezzo, Guido, 54

Egyptians, 17, 22

Dame, Steve, 101

Electronic Musician, 97

Davis, Miles, 83

Emancipation of the dissonance, 82

Debussy, Claude, 82, 93

Enharmonic, 31, 49, 61, 88

Descartes, Rene, 10, 64, 68

Enharmonic organ, 76

Diatonic, 31, 45, 54

Eno, Brian, 86

Didgeridoo, 17

Equal semitones, 63

Didymus, 5, 44

Equal temperament, 4, 13, 14, 32, 38, 50,

Differential tones. See combination tones

58, 60, 65, 67, 68, 70, 81, 83, 87, 95

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Equal tempered thirds, 11

Galileo, 64, 76

Eratosthenes, 43

Gannon, Bill, 99, 114

Erhu, 56

Gannon, Oliver, 101

Euler, Leonard, 71

Glarean, Henricus, 60 Glover sisters, 75

—F— Glover, Sarah, 75, 76 Fang, King, 40

Guitar interface, 101

Faux bourbon, 55

Guitars, 87, 112

Fétis, François, 74

—H—

Fifth harmonic, 21, 34 Fifths, 5, 19, 20

Halberstadt keyboard, 56, 57

Fixed-tone, 27, 35, 41, 56, 58, 62, 65

Halberstadt organ, 6

Fixed-tone instruments, 65, 100

Haley, Jeffrey, 100

Flutes, 17, 18

Hammond organ, 89

Fourier, Jean Baptiste, 76

Handel, Georg Friedrich, 14, 65, 70

Fourths, 5, 19, 31, 34

Handy, W.C., 84

Franco of Cologne, 55

Harmonia, 30

Frets, 15, 16, 50, 56, 62, 81, 84, 85, 98

Harmonic division, 33

Fundamental tone, 19, 27, 91, 93

Harmonic fifth, 19, 67 Harmonic fourth, 67

—G— Gafori, Franchino, 59 Galilei, Vincenzo, 61, 63

Harmonic proportion, 38, 53, 61, 70, 73, 87, 91, 93

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—I—

Harmonic ratios, 46, 47, 63. See harmonic proportion Harmonic series, 19, 64, 66, 82

Inversions, 79, 106 Ives, Charles, 90

Harmonic seventh, 105

—J—

Harmonic third, 45, 54, 55, 56, 67 Harmonic triad, 66, 79

James Broadwood piano factory, 74

Harmonics, 24, 39, 97, 98, 106

Jarrett, Keith, 90

Harmonists, 30, 31, 33, 37, 38, 85, 87

Jazz, 83, 84, 86

Harmonium, 80

Jena, Germany, 1, 3, 4, 12, 72

Harmony, 19, 29, 39, 56, 58, 70

Johnston, Ben, 93, 94, 112

Harrison, Lou, 89, 90, 112

Jorgensen, Owen H., 15, 73

Helmholtz, Herman, 15, 63, 69, 72, 77, 85,

Just intonation, 4, 32, 100

107 Herf, Franz, 96

Just Intonation Network, 90, 97 Justonic, 101, 103, 104, 105, 107, 114

Hervey, Patty, 101

—K—

Hexachord, 54 Hindemith, Paul, 82

Kadir, Abdul, 52

Historical tunings, 73

Keislar, Douglas, 97

Homophonic, 39

Kepler, Johannes, 10, 64

Horn, Paul, 102

Key, 104

Hydraulos, 41

Keyboard temperament, 7

Hymns, 59

Keyboards, 15, 16, 54, 56, 67, 73, 81, 83, 88

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Kirck, George, 97

—M—

Kithara, 29, 30, 39

Major, 51, 79

Koga, Masayuki, 115 Koto, 34

Major thirds, 32, 54 Malcolm, Alexander, 11

Kronos Quartet, 95

Manual tuning, 104

Kurzweil K2500, 102

—L—

Maqamat, 52 Martin D-35, 85 Marvin, Jameson, 111

Lagrange, Joseph Louis, 71 Meantone, 5, 8, 60, 61, 65, 70, 95 Lanfranco, Giovanni Maria, 60 Meistersingers, 55 Led Zeppelin, 89 Melodic scales, 19, 53, 56 Leibniz, Gottfried, 10, 68 Melody, 18, 19, 70 Leipzig, 2 Mercator, Nicolas, 40, 67 Lemmens, Nicolas Jacques, 75 Mersenne, Marin, 6, 10 Lissajous, Jules, 76, 77 MIDI, 96, 97, 101, 104 Lloyd, Llewellyn, 81 Minnesingers, 55 Lü, 20 Minor, 79 Lun, Ling, 20, 25 Minor semitone, 51 Lute, 4, 52, 62 Minor thirds, 32, 44, 54, 67, 70 Lydian, 46 Minor wholetone, 51 Lyra, 29, 67 Minstrels, 55 Mixolydian mode, 23, 88 Modes, 52

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Modulation, 79, 106

New World Records, 90

Monochord, 24

Notation, 19, 21, 22, 39, 44, 125, 135, 139

Monophonic, 19, 29, 39

—O—

Monophony, 90 Monteverdi, Claudio, 59

O’Keefe, Steve, 86

Moog synthesizer, 95

Octave, 14, 20, 41

Moog, Robert, 94

Old Testament, 19

Motets, 59

Oliveros, Pauline, 96

Movable doh, 75

Organ, 54, 57, 68, 78

Mozart, W.A., 65, 72

Organs, 74

Music theory, 43, 47

Organum, 53, 54

musical key, 103, 104

Overtone series. See harmonic series Overtones. See harmonic series

—N— —P—

Nahawand, 51 Nath, Pandit Pran, 96

Parker, Charlie, 83

National Association of Music Merchants

Partch, Harry, 34, 37, 53, 80, 88, 90, 91,

(NAMM), 96

93, 97, 114

Natural keys, 8

Pentachord, 51

Nature of sound, 91

Pentatonic scale, 21, 25

Neanderthals, 17, 18

Phrynis, 29

Neidhardt, Johann Georg, 3, 9, 12, 59

Piano, 14, 68, 73, 74, 92, 98, 112

Neutral thirds, 50, 105

Pitch Palette, 103

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Plato, 34

—Q—

Polansky, Larry, 97, 112

Quartertone, 32, 49, 87, 91, 96

Pole, William, 15

—R—

Polyphony, 39, 54, 58 Poole, Henry, 78, 80

Radio Cologne studio, 94

Prime harmonic, 51

Radio Corporation of America, 94

Profit-5, 96

Radiodiffusion studio, 94

Ptolemaeus, Claudius. See Ptolemy

Rameau, Jean-Philippe, 70

Ptolemaic Sequence, 61

Ramis, Bartolomeus, 59

Ptolemy, 45, 47

Rast, 51

Pure chord, 100

Ratios. 32

Pure fifths, 8, 70

Ravel, J. Maurice, 82

Pure fourths, 70

Rayna, David, 97

Pure harmonic, 32

RCA Mark sound synthesizer, 94

Pure harmony, 70

Reference frequency, 103, 104

Pure thirds, 54, 70

Rhapsodes, 23

Pythagoras, 23, 25, 26

Rich, Robert, 97, 102

Pythagorean, 54

Riley, Terry, 96, 112, 113

Pythagorean comma, 26, 37

Rock music, 85, 86

Pythagorean thirds, 45, 54, 55, 59

Root tone. See fundamental tone

Pythagorean tunings, 5

Rosenthal, Henry, 97

Pythagorean wholetone, 32

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—S—

Singing, 17

Safiyu-d-Din, 51

Sixths, 33, 55, 56

Salinas, Franco de, 61

Smith, Dave, 96

Standard pitch, 12

Smith, Dr. Robert, 14, 70

Sauveur, Joseph, 68

Sol-faists, 76

Scales, 33, 103, 105, 106

Sound, 15

Schlesinger, Kathleen, 88

Sound waves, 71

Schlick, Arnold, 60

Split-key keyboards, 56, 59, 70

Schneider, John, 86

Stockhausen, Karlheinz, 83

Scholar’s lute, 34

Stoltzman, Lucy, 90

Scholz, Carter, 97, 102

Stradivari, Antonio, 68

Schönberg, Arnold, 82

Stravinsky, Igor, 82

Scriabin, A.N., 82, 93

Sumerians, 22

Selecting the tuning root, 105

Summation tones. See combination tones

Semite tribes, 9

Sweet Honey and the Rock, 107

Semitone, 32, 63

—T— Sequence track, 104 Serial techniques, 82

Tanbur of Baghdad, 51

Series of fifths, 33, 37, 49

Tartini’s tones. See combination tones

Seventh harmonic, 28, 31, 34, 74, 84, 110

Tartini, Giuseppe, 68

Shirazi, Mahmud, 52

Taylor, Margaret, 102

Silk road, 43

Taylor, Stephen James, 112

Simultaneous harmony, 53

Tcheng-tien, Ho, 10, 50

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152

Telharmonium, 89

Tuning fork tonometer, 74

Temperament, 6, 15, 57, 73, 100

Tuning resolution, 100

Tempered chord, 100

Tuning root, 103, 104, 105

Tempered harmony, 59

Tuning root, automatic, 104

Tenny, James, 96

Tuning root, manual, 104, 105

Terpander, 23

Twelve-tone, 82

Tetrachord, 31, 32, 45, 46, 49, 51

Two streams of music, 41, 81, 98, 111

The Nylons, 107

—V—

Theremin, Leon, 88 Third. See pure third. See harmonic third

Varese, Edgard, 89

Thirds, 33, 110

Vaughan, Sarah, 83

Thomas, Michael Tilson, 112

Vincentino, Don Nicola, 60

Thompson, Perronet, 76, 80

Violins, 60, 68

Three-quarter tone, 50

Virtual DSP Corp., 101

Timotheus, 29

Vocal music, 65

Tonality, 70, 74, 82, 83

Voice, 39, 63, 91, 98

Tonic, 27

—W— Tonic Sol-faists. See Sol-faists Troubadours, 55

Waage, Harold, 97

Trouvère poets, 55

Wave lengths, 93

Tsai-yu, Chu, 10

Weigel, Erhard, 4

Tuning compromise. See

Well temperament, 5, 8, 13, 65, 70

Tuning fork, 12, 14

Well Tempered Clavier, 8, 14, 65, 70

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Werckmeister, Andreas, 8, 11

—X—

Weyler, Rex, 99, 114 Xenharmonikon, 97 Wholetone, 27, 32, 35, 38, 41

—Y—

Wholetone scales, 82 Wholetones, 105

Yasser, Joseph, 96

Wilkinson, Scott, 97

Young, La Monte, 96, 112

Wolf notes, 7, 8, 13, 58

—Z—

Wosta of Zalzal, 50 Zalzal, 50

Zarlino, Gioseffo, 61, 63, 91

Notes Chapter 1 1

An account of this tuning contest appears in Johann Sebastian Bach, by Philipp Spitta, translated by Clara Bell and J.A. Fuller-Maitland, London, 1884; vol. I, 137 f. Also see Equal Temperament, by J. Murray Barbour, Michigan State College Press, 1951, pp. 85-87.

2

Mersenne, Marin F.; Harmonie Universelle. Paris; Sebastien Cramoisy, 1636. Translation; Roger E. Chapman, The Hague, Martinus Nijhoff, 1957, pp. 89-90.

3

Aaron, Pietro; Thoscanello De La Musica. Vinegia; Impressa per Bernardino et Mattheo de Uitali, 1523. Reprint; Broude Brothers Limited, New York, 1969.

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4

Holder, William; A Treatise of the Natural Grounds, and Principles of Harmony. London; J. Heptinstall, 1694.

5

Werckmeister, Andreas; Musicalische Temperatur. Quedlinburg, Theodori Philippi Calvisii, 1691; reprint: Utrecht, The Diapason Press, 1983.

6

Jorgensen, Owen H.; Tuning. East Lansing; Michigan State University Press, 1991. The definitive text on tuning systems from the 17th to the 20th century, with instructions for tuning these historical temperaments by ear. 7 Barbour, James Murray; Tuning and Temperament. Michigan State University Press, 1951; reprint: Da Capo Press, 1972. 8 Jorgensen, Owen H.; Tuning; op. cit.; pp. 63. Jorgensen cites Alexander Malcolm’s A Treatise of Musick, Speculative, Practical, and Historical; Edinburgh, 1721. 9

Partch, Harry; op. cit., 259.

10

Smith, Robert; Harmonics, Or The Philosophy of Musical Sounds. Cambridge; W. Thurlbourn and T. Merrill; 1949; reprint: New York; Da Capo Press, 1966.

11

Helmholtz, Hermann; On the Sensations of Tone as a Physiological Basis for the Theory of Music. Fourth German edition, 1877; translated, revised, corrected, with notes and additional appendix by Alexander J. Ellis. Reprint: New York, Dover Publications, Inc., 1954. 12 Jorgensen, Owen H.; Tuning; op. cit.; pp. 4, 6.

Chapter 2 13

Folger, Tim; and Shanti Menon, Discover; December, 1996. 14 Gordon, R.K., trans. Anglo-Saxon Poetry. London, J.M. Dent & Sons Ltd; 1926. Widsith, possibly the oldest poem in the English language (c. 7th C.) is a fictional account of a wandering minstrel recounting heroic lore. The quoted passage makes clear that instruments and voices sang harmony. The work itself makes clear that song was the primary form of historical record. 15

Yasser, Joseph; A Theory of Evolving Tonality. New York; American Library of Musicology, 1932. Yasser cites Chinese historian Sze Ma-chi’en, a contemporary of Ptolemy, second century A.D.

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16

Chalmers, John H.; Divisions of the Tetrachord. Hanover; Frog Peak Music, 1993. Chalmers cites M. Duchesne-Guillemin, Revue de Musicology, 49:3-17, 1963; and 55:3-11, 1969. Chapter 3

17

When we speak of the Greek modes, we are faced with the poor scholarship of Henricus Glareanus, who in 1547 published the 12 ecclesiastical modes, gave them Greek names, but got all the Greek names wrong. Glareanus’ “Mixolydian,” the dominant seventh mode, is the Greek Ionic, and his “Ionic” is the Greek Lydian, etc. The Catholic Church adopted Glareanus’ terminology, and this has caused confusion ever since. Modern usage usually conforms to Glareanus, not Greek terminology, except when discussing actual Greek modes. This confusion could be settled once and for all if scholars simply adopted the original Greek names. After all, why should we honor Glareanus’ poor scholarship by adopting his erroneous system. In this history we will use the Greek names, preceded by the word “Greek” to make this clear. When we are referring specifically to the ecclesiastical modes, we will identify them as such.

18

Hope, Robert C.; Medieval Music: An Historical Sketch. London; Elliot Stock, 1894. Chapter 4

19

Partch, Harry; op. cit., 238. The nomenclature here can lead to some confusion. The musical “third” is based on the “fifth harmonic,” and the musical “fifth” is based on the “third harmonic.” The numbering of the harmonics is natural and simple enough. The fundamental is the first harmonic, and subsequent harmonics are numbered “second,” “third,” and so forth. The musical “third” is so called because it is the third note in a diatonic major scale, and likewise, the “fifth” is the fifth note. The fact that the third harmonic creates the musical fifth, and vice versa, is simply an accident of our diatonic convention. 21 Chalmers, John H.; Divisions of the Tetrachord. Hanover; Frog Peak Music, 1993. The definitive treatise on scale construction based on tetrachords. For musicians 20

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looking to expand their scale resources, this book will provide you with enough scale options to fill a lifetime of investigation. Chalmers catalogs over 700 tetrachords including the diatonic, chromatic, and enharmonic genera, a section on hyperenharmonic tetrachords (with sub-quartertone intervals), and other unusual constructions. 22

For a discussion on the mathematics of these and other proportional means, see Chalmers (Op. Cit.) pp. 29-31, and Partch (Op. Cit.) pp. 104-106. Chapter 5

23

Smith, Sir William; Dictionary of Greek and Roman Biography and Mythology. Boston, Little & Brown, 1849. The biographies of Pythagoras and Archytas were mentioned by later writers, but have never been found.

24

Schlesinger, Kathleen; “Further Notes on Aristoxenus and Musical Intervals,” The Classic Quarterly, 27:88-96; April, 1993. Also see Chalmers (Op. Cit.) pp. 17-23.

25

Aristoxenus; The Harmonics. Edited and translated by Henry S. Macran. Oxford, 1902; reprint: Hildesheim, Georg Olms, 1974, pp. 188.

26

This point of view is logical in relation to the use of the monochord, but admittedly speculative as far as Greek choral singing goes. For an alternative view see A History of ‘Consonance’ And ‘Dissonance’ by James Tenny, Section I, “The pre-polyphonic era.” 27 Ibid, pp. 188-189. Chapter 6 28

Mersenne, Marin F.; Harmonie Universelle, (Op. Cit.), pp. 84.

29

Ibid, pp. 84. This and the following are taken from Mersenne's paraphrase in which he explains all of Ptolemy’s musical theories.

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Chapter 7 30

Barbour, James Murray; Tuning and Temperament. (Op. Cit.), pp. 155. Barbour cites Maurice Courant, Chine et Coree, Encyclopedie de la musique et dictionnaire du conservatoire, Paris, 1913.

31

Partch, Harry; Genesis of a Music, (Op. Cit.), pp. 244.

32

Saleh, Dr. Fathi; The Musiclabe, An Instrument for the Generation of Arabic Music Maqamat. Paper for the Arabic Music Conference, Cairo University, 1996. Dr. Saleh has designed an ingenious instrument for representing the complex system of Arabic scales and modes, or Maqamat. Chapter 8

33

Partch, Harry; Genesis of a Music, (Op. Cit.), pp. 245.

34

Tenny, James; A History of ‘Consonance’ and ‘Dissonance.’ New York, Excelsior Music Publishing Company, 1988. p. 20. Tenny cites Hucbald, 10th C., as the earliest unmistakable reference to simultaneous “consonance.”

35

Helmholtz, Hermann. Sensations of Tone; Op. Cit.; p. 196.

36

Odington, Walter; De Speculatione Musicae; edited by Frederick F. Hammond. Stuttgart; American Institute of Musicology, 1970. 37 Woldridge, H.E.; The Oxford History of Music. London, Oxford University Press, 1929; reprint, New York, Cooper Square Publishers, 1973; pp. 294-6. Chapter 9

38

Barbour, James Murray; Tuning and Temperament. Op. Cit. 39 Ibid. p. 45. 40

Ellis, Alexander, in the translator’s additions of Herman Helmholtz; Sensations of Tone (Op. Cit.), p. 547.

41

Partch, Harry; Op. Cit.; p. 250. Partch cites James Barbour, Equal Temperament from Ramis (1482) to Rameau (1737).

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42

Zarlino, Gioseffo; Le lnstitutioni harmoniche. Venice, 1558. (Part Three) The Art of Counterpoint Translated by Guy Marco and Claude Palisca; New Haven, Yale University Press, 1968.

43

Partch, Harry; Op.Cit.; p. 254. Quoted from James Barbour’s, Equal Temperament. Helmholtz, Hermann. The Sensations of Tone; Op. Cit., p. 326.

44

Chapter 10 45

Mersenne, Marin F.; Harmonie Universelle, (Op. Cit.), p. 29.

46

Jeans, Sir James; Science and Music. New York, Macmillan, 1937; p. 175.

47

Perrett, Wilfrid; Some Questions of Musical Theory. Cambridge, W. Heffer and Sons, Ltd.; 1926; p. 62.

48

Rameau, Jean-Philippe; Treatise on Harmony. Translated, with introduction and notes by Philip Gosset. New York, Dover Publications, Inc., 1971; pp. 141. 142, 152.

49

Ellis, Alexander, translator’s notes; in Hermann Helmholtz’s The Sensations of Tone; p. 548. Ellis cites Dr. Smith’s Harmonics, or the Philosophy of Musical Sounds, 2nd edition, pp. 166-7.

Chapter 11 50

Jorgensen, Owen H., Tuning; (Op. Cit.); p.1.

51

According to Alexander Ellis in his “Additions by the Translator in Hermann Helmholtz‘s The Sensations of Tone (Op. Cit.), pp. 548-9.

52

Helmholtz, Hermann. The Sensations of Tone, (Op. Cit.). This story is recounted by Alexander Ellis, as told to him by Fétis, in footnote, p. 280.

53

Helmholtz, Hermann. The Sensations of Tone, (Op. Cit.); p. 310. The following quotes are taken from the final three chapters, beginning on this page. 54 Ibid; p. 319. 55

Partch, Harry; Op.Cit.; p. 267.

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Chapter 12 56

Parkhurst, W. and de Bekker, L.; The Encyclopedia of Music and Musicians. New York; Crown Publishers; 1937; p. 274, p. 588.

57

Lloyd, Llewellyn, S.; Intervals, Scales, and Temperaments, 1950. Reprint: St. Martin’s Press, 1979. 58 Schönberg, Arnold; “Problems of Harmony,” Modern Music, 11:167-187; May/June, 1934.

59

Cowell, Henry; New Musical Resources. New York, Alfred A Knopf, 1930.

60

Hindemith, Paul; A Composer's World. Cambridge; Harvard University Press, 1952; pp. 55-6.

61

Hindemith, Paul; The Craft of Musical Composition, V.1. New York, Associated Music Publishers, Inc., 1942; p. 155. 62 Duncan, Dudley; “Septimal Harmony for the Blues,” 1/1, The Quarterly Journal of the Just Intonation Network; vol. 5, n.2, Spring, 1989. 63

Helmholtz, Hermann. The Sensations of Tone, (Op. Cit.); p. 323.

64

Eno, Brian; Keyboard, March, 1995.

65

O’Keefe, Steve; Piano newsletter, Port Townsend, Washington; 1995. Schneider, John; “Fine Tuning,” Acoustic Guitar, May/June, 1994.

66

Chapter 13 67

Busoni, Ferruccio; Sketch of a New Esthetic of Music. New York; Schirmer Books, 1911; p. 24.

68

Harrison, Lou; interview by David Doty, 1/1 (Op. Cit.); V.3, N.2; Spring, 1987.

69

Partch, Harry, Genesis of a Music; (Op. Cit.); pp. 225-226.

70

Duckworth, William; Talking Music. New York; Schirmer Books; 1995; p. 146. Duckworth interviews Ben Johnston, Lou Harrison, John Cage, La Monte Young, Lauri Anderson, and other “experimental composers.”

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Johnston, Ben; “Rational Structure in Music;” 1/1; (Op. Cit.); V.2, N.3, Summer, 1968. This piece was reprinted from the original in the Proceedings of the American Society of University Composers. Chapter 15

72

Marvin, Jameson; “Choral Singing in Tune,” Choral Journal, 32 (5), 1991.

73

Polansky, Larry; 1/1, (Op. Cit.); Winter, 1985

74

Stephen James Taylor, in correspondence with the authors and Justonic Tuning Inc., October 5, 1996.

75

“Michael Tilson Thomas on Contemporary Music, Notation vs. Interpretation, and the Keyboard as the Conduit for Musical Thought”. Keyboard, July, 1996.

76

Riley, Terry; interview with William Duckworth; Talking Music. (Op. Cit.).

77

Fathi Saleh, in correspondence with Justonic Tuning Inc., October 15, 1995.

78

Harry Partch, Genesis of a Music, The University of Wisconsin Press, 1949; p. 136-7.

79

Masayuki Koga, The Japanese Bamboo Flute, p. 123. Appendix B

80

Holland, Jack; "An Introduction to 'In Tune' The Scale of Just Intonation," Brass Bulletin, n.40, pp. 58-64, 1982. Holland describes a simplified "3 positions" system. 81 Podnos, Theodor H.; Intonation for Strings, Winds and Singers. (Scarecrow Press, Inc. 1981). 82 See: Keislar, Douglas; "History and Principles of Microtonal keyboards," Computer Music Journal, vol. 11, n.1, 1987. And: Robert Rich; "Features of Tuneable MIDI Synthesizers," 1/1, The Quarterly Journal of the Just Intonation Network, vol.8, n.1, 1993. 83

Carlos, Wendy; "Tuning: At the Crossroads", Computer Music Journal, 11:29-43, n1, 1987; this is a good account of how harmonic partials contribute to our sense of harmony.

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resulting differential tone will be 264 Hz, a C below the tonic A. We bring this C into our A440 octave by multiplying by 2, giving us C528. This C of 528 Hz is exactly 6/5 times 440 Hz. For a detailed treatment of combination tones see Arthur H. Benade, Fundamentals of Musical Acoustics (Dover, 1976, 1990); and Herman Helmholtz, On the Sensations of Tones, (Op. Cit.) 85 Duncan, Dudley; "Septimal Harmony for the Blues," 1/1, The Quarterly Journal of the Just Intonation Network, vol.5, n.2, 1989. 86 Chalmers, John H., Divisions of the Tetrachord. (Op. Cit.) 87

Wilson, Ervin M., "The Marwa permutations," Xenharmonikon 9, 1986; "The Purvi modulations," Xenharmonikon 10. Also see Chalmers, Op. Cit.