Richard L. Crocker - Discant, Counterpoint, And Harmony

Discant, Counterpoint, and Harmony Richard L. Crocker Journal of the American Musicological Society, Vol. 15, No. 1. (Spring, 1962), pp. 1-21. Stable URL: http://links.jstor.org/sici?sici=0003-0139%28196221%2915%3A1%3C1%3ADCAH%3E2.0.CO%3B2-L Journal of the American Musicological Society is currently published by University of California Press.

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Discant, Counterpoint, and Harmony BY RICHARD L. CROCKER

HOW

OFTEN ONE READS, in discussions of medieval music, remarks like this: "Here the voices sound a major triad-but, of course, the composer did not think of it that way." A commendable reservation; but one that raises the urgent question: how (didhe think of it? Many feel that the medieval composer did not think of vertical sonority at all; or, if he did, only in abstract, mathematical terms. This view holds that medieval polyphony is "linear," that vertical sonorities are the product of intersecting melodic lines, and that these sonorities are fortuitous. If the medieval composer did pay attention t o vertical sonority, it was only to ensure the use of perfect consonances, that is, unison, fourth, fifth, and octave. This was obviously due (the argument continues) to a mystical trust in number rather than to a musical trust in the judgment of the ear, since these "perfect" intervals sound bad, or at best disembodied. In any case, neither the composer nor the listener is supposed to have listened to the vertical sonority. This is a hard doctrine t o swallow. I t seems to have arisen when modern ears were first confronted with medieval sounds; accustomed to "traditional harmony," the ear found the sound of medieval music meaningless or intolerable. But when viewed as the result of simultaneous melodies, the crudity of the progressions became acceptable, even interesting. In this way medieval music was made accessible to the modern mind, which was willing to attribute philosophic brilliance but not common sense perception to the musical contemporaries of St. Thomas Aquinas. Is such a drastic, merely cerebral solution as this really necessary-or even tenable-any longer? Is it really necessary to deny the evidence of our senses (and theirs) that three melodic lines sung simultaneously do in fact strike the ear with a progression of three-note chords? Must we deny the logic of history, that to a monophonic age the most striking fact of polyphony must have been the presence of three pitches where there should be only one? Finally, must we deny the facts of a polyphonic style that compressed the three "independent" melodies into a single octave and then fused them together with modal rhythm, the most uniform rhythm known to the history of music? Reasonable observers have for some time suggested a more reasonable interpretation. I t requires, I think, only a summing up of these suggestions in order to present an account of the theory of medieval polyphony more in harmony, so to speak, with the facts. There is one reasonable observer who, it seems to me, must be cited

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b y more than a footnote. In 1937, a quarter of a century ago, Prof. Thrasybulos Georgiades presented the whole matter quite clearly in his thesis, "Englische Diskanttraktate aus der ersten Halfte des 15. Jahrhunderts." H e showed that the procedures of medieval polyphony were to be explained as progressions of intervals, avo-note entities. The logic of such progressions is distinct on the one hand from melodic, "linear," logic, and on the other from the logic of triads. Prof. Georgiades's interpretation was made in connection with the problem of discant in England, and the very few who have bothered to pursue his remarks have done so largely in the same connection.] But a recent writer, Sylvia Kenney, has shown that discant in England is in all essentials the same as discant anywhere else;2 this makes it easy to apply the idea of interval progression to all of medieval polyphonic theory. For it is the medieval view that w e want to understand. W e know how we conceive it; what w e need to know is how they conceived it. T o do this, we must take hold of their theory books with both hands and read. If this reading is done in the light of Prof. Georgiades's remarks, one finds that the discant authors from the ~ ~ tot the h 16th centuries provide a clear, consistent, and pertinent account of medieval-and Renaissancepolyphony. For our purposes "discant" means a system of teaching two-part composition, in use from the 13th to the 16th c e n t u r i e ~This . ~ is the most comprehensive definition of the term; it can also refer to a specific musical style or to the upper voice of a composition. W e will not be concerned with these more restricted meanings. Discant, so defined, shows how to 1 E. Apfel, Studien z w Satztechnik der nzittelalterlichen englischen Mzlsik, 2 vols. (Abhandlungen der Heidelberger Akademie der Wissenschaften. Philosophischhistorische Klasse. Jahrg. 1959, j. Abhandlung); also "Der klangliche Satz und der freie Diskantsatz im I j. Jahrhundert," Archiv fur Musikwissenschaft XI1 (I~s!), pp. 297ff. See also G. Schmidt, "Zur Frage des Cantus firmus im 14. und beginnenden 15. Jahrhundert," Archiv fur Musikwissenschaft X V (19j8), pp. 23off. 2 " 'English Discant' and Discant in England," Musical Quarterly XLV ( 1 9 ~ 9 )pp. , 26ff. Most writers on medieval music are forced, in spite of any convictions to the contrary, to acknowledge in some degree the existence of a vertical component; to list all such references would be futile. As special studies one should mention E . Lowinsky, "The Function of Conflicting Signatures in Early Polyphonic hlusic," Musical Quarterly XXXI ( r g l j ) , p. 227; H. E. Bush, "The Recognition of Chordal Formation by Early A4usic Theorists," Musical Quarterly XXXII (19+6), p. 227; G . Reaney, "Fourteenth Century Harmony and the Ballades, Rondeaux, and Virelais of Guillaume de Machaut," A4usica disciplim VII (1953)~P. 129; H. Tischler, "The Evolution of the Harmonic S q l e in the Notre-Dame hlotet," Acta musicologica XXVIII (1956), p. 87; K. v. Fischer, "On the Technique, Origin, and Evolution of Italian Trecento Music," Musical Quarterly XLVII (rg61), pp. 41ff (too late to be considered in the present article). 3Most of the discant treatises are published by C. E. H. de Coussemaker in Scriptorum de musica medii cevi nova series, q. vols. (Paris, 1864-76) (hereafter abbreviated as CS), and in Histoke de l'hamzonze au moyen 6ge (Paris, 1852). These versions are not, of course, completely reliable and will one day have to be replaced; for the present survey, however, they are adequate. Other texts will be cited a s needed.

DISCANT,

3

COUNTERPOINT, AND HARMONY

combine one (and only one) note with each note of a given melodic progression b y the application of two basic principles. T h e first principle deals with the kinds of sonority to be used, the second with the order in which sonorities may appear.' T h e first principle requires discant to consist essentially of concords and only accidentally of discords. W i t h slight adjustments in the definition of concord, this principle governs not only medieval discant and Renaissance counterpoint, but also Baroque thorough-bass and traditional harmony. T h e second principle requires contrary motion between the two parts. This principle is absolutely binding, but has many, many exceptions. A large part of the typical discant treatise is devoted to circumvention of this principle, laying down conditions under which similar or even parallel motion may be used. Here again, reflecting on the nature of thoroughbass or traditional harmony, we can observe: "Plus $a change, plus c'est la mtme chose." In applying the first principle the critical point is, clearly, the definition of concord. Systematic treatment of the concords of discant appears first in the theorist John of G ~ l a n dat , ~a time (mid-I 3th century) when the new, international style of Leonin and Perotin had firmly established the use of these concords. John's formulation, destined to become classic, is itself a clarification of the previous "common doctrine of discant," or Discantus positio uzdgaris, a short exposition found immediately before John's in the compendium of Jerome of M ~ r a v i a .T~h e Positio vulgaris says that some intervals, namely unison, fifth, and octave, are better than others, and some are more dissonant, but "according to greater or lesser degree." Even in this modest treatise there is no hint of an absolute dichotomy between consonance and dissonance; indeed, the notion of a continuum stretching from consonance to dissonance prevails throughout the Middle Ages and Renaissance. John of Garland arranges intervals on the continuum as follows: Perfect unison octave

Middle fifth fourth

Imperfect major third minor third

lmperfect Middle major sixth major second minor seventh minor sixth

Perfect major seventh minor second tritone

These are rephrasings of Miss Kenney's second and third principles of discant.

My friend and colleague gives as a first principle the requirement that discant should consist of only one note against another. I take this to be not a law like the other two, but rather a part of the definition of discant, like the provision that discant concerns only two voices. 5 O r perhaps Anon. VII (CS I, 38z), if the passage in question really antedates Tohn's treatise. 6 S. M. Oerba, Hieronymrs de Moravia O.P. Tractntus de musica (Regensburg, 1935). T h e Discantus positio wuIgaris is on pp. 189-194; John's De musica m m r a b i l i positio on pp. 194-230, his discussion of consonance on pp. z g f f .

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Being a theorist as well as a practical musician, John goes on to draw the inference: "The more the string ratio of an interval approaches equality, the more concordant it sounds." Certainly a justified inference, and one that brings u p the question of the judgment of concord and discord. Medieval writers, from John of Garland on, consistently invoke the judgment of the ear in discussing the degree of concord and discord. Since this observation is in flat contradiction to the opinion commonly held about medieval musicians, it seems prudent to exhibit a quantity of texts sufficient to convince the most sceptical. John of Garland: "A concord is said to exist when two pitches are joined together at the same time in such a way that the sense of hearing tolerates them one with the other. Discord is described c~ntrariwise."~ Anonymous I: "A concord is the harmony (harmonia) of two or more different sounds produced a t the same time, blending together and reaching the ear in sweet uniformity (unifomziter suaviterque veniens ad a ~ d i t u m ) . " ~ Lambert: "Concord is said to exist when two pitches sounding a t the same time blend together so that they render sweet melody (suavem melodiam) in the ear. . ."9 Anonymous 11: "Discant is composed principally of consonances and only incidentally of dissonances, in order that the discant per se may be more beautiful, and that we may be more delighted by the consonances. Consonance is made of diverse sounds mixed together. Dissonance is a rough collision (dura collisio) ."lo ATScontrapunctus secundum Philippunz de Vitriaco: "These concords (unison, fifth, octave) are called perfect because they bring a perfect, pure (integrum) sound to the ears of the listeners."ll Jacob of Li6ge: "Discant is said to be a consonance of different songs: just as consonance requires distinct pitches mixed together a t the same time, so discant requires distinct songs (cantus) sounding simultaneously. Not all sounds, however, can be combined into a mixture that will present itself smoothly and sweetly to the listener; similarly, not all songs when mixed together make discant, but only those that harmonize with one another so that through their concord they make as it were one song. . ."12 Johannes Tinctoris: "A concord, therefore, is a mixture of two pitches rendered sweetly agreeable to the ear by a natural power (naturali virtute)."lS

.

.

Clearly, from these statements, it is false to believe that the Middle Ages relied solely on mathematics and excluded the judgment of the ear in determining the nature of consonance. These authors say, in sum, that the ear takes pleasure in consonance, and the greater the consonance the greater the pleasure; and that for this reason one should use chiefly consonances in composing discant.14 7 Cserba, Hieronymus de Moravia, p. 207. 8CSI,z97. QCSI,260. loCS1,311. lz Cs 11, 387. l3 CS IV, 78.

11CSIII,t7.

14The formulas used by the discant writers to describe consonance go back, of course, all the way t o Boethius, De insrimtione mzcsica I, viii, ed. G. Friedlein (Leipzig, 1867)~p. 195.

DISCANT, COUNTERPOINT, AND H A R M O N Y

5

H o w do we square this with our own experience? For today many listeners would hesitate to describe the interval of a fifth as "sweet." Perhaps we can explain the discrepancy b y noting at least two distinct bases for judgment of consonance. If we consider a fifth (referring of course to one that is in tune) we can hear that the two tones do indeed blend together, almost as well as a unison or octave. Organists, choral technicians, string and wind players will all find within their experience reason to agree with this judgment. On this basis we find ourselves in complete accord with the medieval theorist. T h e second basis is the function of intervals within the development of style; here our judgment differs from that of the medieval musician. H e finds the simplest intervals to be the sweetest, a judgment which one must admit to have been reasonable in the springtime of polyphony. We, withdrawing a little from sonorous reality, find the more complex, less consonant intervals to be sweeter; today the older generation prefers thirds and sixths, while some of us give our vote for pulchritude to the broad scratch of a major second. T h e discant teacher, then, calls the octave a "perfect concord" because its two pitches blend so well; then he recommends the octave for use in discant because its perfection is stylistically appropriate. John of Garland makes a distinction between the "perfect" concords, unison and octave, and the "middle" concords, fifth and fourth, perceiving in the latter intervals a lesser degree of sonorous blend. H e goes on to assign to the major and minor thirds the grade of "imperfect" concord, which-and this is a fact usually neglected-admits these intervals to most of the rights and privileges of concords. For imperfect concord is, to John, not yet discord. Since discant is made essentially of concords, major and minor thirds are included as basic ingredients. (Hence the rule that discant must end with a perfect or middle concord, not an imperfect one; this is the one place where thirds are not recommended.) I t is not true, therefore, that the medieval theorist regarded thirds as discords: at the time of John of Garland, discant specified the essential concords of composition to be unison, octave, fifth, fourth, major and minor third. T h e description of the continuum between consonance and dissonance does not change much through the centuries-indeed it cannot change, being a description of the plain facts of sound. On the other hand, the description of the functions of the various consonances and dissonances changes steadily, since here the theorist must constantly account for new stylistic practices. In general, the changes seem to occur first in practice; they are reflected almost immediately in the writing of the discant teacher, the practical treatise that teaches one how to compose. But between the description in the practical treatise and the explanation in the theoretical, speculative treatise there was apt to be a lag of several decades or more. Thus 13th-century composers treated the perfect fourth as a concord,

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and discant treatises described it as such. In the 14th century, however, the discant teachers characterized the fourth as discordant, on the grounds that it was now treated like a discord even if it did not sound like one. But this reason did not satisfy the speculative theorist, who knew that the ratio of the fourth (4: 3) occurred within the tetrad, the first four numbers, and that it came immediately after the ratio for the perfect fifth ( 3 : ~ ) Furthermore, . the fourth was a basic element in the Pythagorean harmony:

OCTAVE

Last but not least, the fourth sounded like a consonance. Confronted b y all these arguments, a theorist could hardly assign the fourth a status of discord for merely stylistic reasons. In fact the anomaly of the fourth is so deep-seated that according to latest reports the issue is still in doubt. W e should not be upset, therefore, if the medieval theorist is less than conclusive on this point. But the need for formal explanation did not touch the practical teacher of discant. After all, he had already described the major and minor thirds as concords despite their complex ratios (8 I :64 and 32 : 2 7 respectively15). I t cost him little to place the fourth among the discords. This was the only time that a concord was demoted to a discord; the other changes consisted in raising the major and minor sixths to the status of concords-or better, in moving the dividing line between concord and discord further down the continuum to include more complex intervals as concords. Following Prof. Georgiades's suggestion,16 we can construct a tentative genealogy of the anonymous discant authors on the basis of their treatment of these concords. In the 13th century the anonymous authors I, IV, and VII from Coussemaker's first volume maintain with Franco the classification of John of Garland.17 Anonymous II,lB however, includes the major sixth along with the thirds as an imperfect concord. This change reappears in 15 Their proper names in this tuning are "ditone" and "semiditone" respectively; they are so named by the early discant authors. 16 Englische Diskanttraktate aus der ersten Halfte des 15. Jabrbunderts (Schriftenreihe des Musikwissenschaftlichen Seminars der Universitat hlunchen, Vol. 111, Wurzburg, 1 9 3 7 ) ~p. 61. 1 7 CS I , 298, 3j8, 382; Cserba. Hiero~aymusde Moravia, pp. 207, rjo; 0. Strunk, Source Readings in Mzrsic History (New York, 1 9 5 0 ) ~pp. 152f. CS 1, 312.

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DISCANT, COUNTERPOINT, AND HARMONY

Wolf's C ~ m p e n d i m( I ~336), ~ which furthermore drops the fourth from the list of concords, thus reducing their number back to six. T h e number is increased again to seven in the Ars contrapunctus secundum Philippum de Vitriaco20 (late 14th century?) by the addition of the minor sixth; this arrangement is reproduced in the Ars discantus secundum loannem de M ~ t r i s . ~All l this is better shown in the following table: ' 2 ~ 0

1700 (?)

'336

after 1 3 ~ 0(?)

unison octave fifth fourth major third minor third

unison octave fifth fourth major third minor third major sixth

unison octave fifth

unison octave fifth

-

-

major third minor third major sixth

major third minor third major sixth minor sixth

W h y the fourth should be dropped and the sixths added to the list of concords are questions that would detain us too long here. There are, I am certain, very clear stylistic answers, towards which Prof. Georgiades points the way:22 he says that the major sixth was adopted before the minor one because the major was part of a progression from fifth to octave by contrary motion, whereas the minor sixth could proceed only to a fifth and that with one voice stationary. Since this latter progression was less congenial to the style as a whole, the minor sixth could not attain concordant status as easily. Concurrent with these changes in the classification of concords, there was a change in the treatment of intervals larger than the octave. 13thcentury discant viewed these as compounds of those smaller than the octave, hence subject t o similar treatment: a fifth and a twelfth, for example, were handled in the same way. T h e authors speak of the number of intervals as "infinite," envisaging an endless duplication of intervals upwards b y octaves, but all subject to the rules governing those below the octave.23 Lambert (ca. 1260), however, gives the six concords as follows: octave and double octave (perfect); fifth and twelfth (middle); fourth and eleventh (imperfect) .24 In other words, he insists on the Pythagorean consonances (fourth, fifth, and octave) as the only true concords, being almost the only author to do so; but he adapts this doctrine to John's system of six concords b y including the respective octave compounds. This anticipates the later treatises A r s perfecta in ??zusica nzagistri Philippoti 1 9 J. Wolf, "Ein Beitrag zur Diskantlehre des 14. Jahrhunderts," Smmzelbande der Internationalen Musikgesellschaft X V ( 1 9 1 4 ) ,pp. SO& 2 0 CS 111, 27. cs 111, 70. 22 Engliscbe Diskanttraktate, pp. 64f. 2 3 For example, John of Garland: Cserba, Hieronynrus de Mormia, p. 208. 2 4 CS I, 260; the passage is corrupt but its meaning evident.

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de VitriacoZ5and the Libw m s i c a l i m Philippi de Vitriac0,2~which include the tenth and twelfth among the concords. Since these writers ennumerate thirds and sixths without specifying major o r minor, the number of concords stays at seven. Both old and new listings are found in the Ars contrapuncti secundum Ioannem de M ~ r i s , 2the ~ older one first. T h e newer listing: unison, third, fifth, sixth, octave, tenth, and twelfth, remains standard through the Renaissance, altered only by upward extension. T h e specification of intervals larger than the octave also needs more discussion than we can here afford. In passing it is interesting to observe what the Compendiunz discantus,28 (ascribed to Franco but possibly later) says about these larger intervals: And note that when you wish to ascend above the diapason, you will imagine yourself to be in unison with the tenor, and you will discant in the same way you would over the tenor in the lower register, because there is really no difference except that you are higher in pitch. Such elevation [of the discantl can be repeated indefinitely (multiplex est in infinitum). T h e phrase "in infinitum" connects these remarks solidly with the doctrine of John of Garland, but the word "imaginabis," and the procedure designated thereby, is identical with the 15th-century English principle of "sights."29 Perhaps this modest Compendium is the link that connects the doctrine of sights directly to traditional discant. T h e second principle of discant requires contrary motion between the two parts, using the concords already described. This, of course, is the catch: it would be simple to write two parts in contrary motion if all intervals were permitted; it would be equally simple to use the concords if contrary motion were not required. Western part-music, from then until now, depends upon a delicate balance between the demands of vertical sonority and those of voice-leading. Sometimes the balance is threatened b y too much attention to the vertical or the linear dimension, but equilibrium is soon restored with the realization that each dimension is meaningless without the other. T h e linear interpretation of medieval music depends for much of its evidence on the instructions for composing found in the discant treatises. These instructions are intended to produce the desired concords through contrary, or at least oblique motion. T h e y are apt to take the following form: When one part ascends a step, the other, beginning at the octave above, may descend two steps and be at the fifth. It is argued that the stress on contrary motion in such instructions reflects an emphasis on the linear dimension. I t is further argued that such instruc25 2s

28. 26 CS HI, 36. 27 CS 111, 59f. See Kenney, "English Discant," pp. 33ff.

CS 111,

CS 1, 156.

DISCANT, COUNTERPOINT,

AND HARMONY

9

tion implies two simultaneous but independent melodies in the mind of the composer and also in the ear of the listener. T h e discant teacher does, indeed, stress contrary motion, but so does the teacher of traditional four-part harmony, and to the same degree; only here the result is not so apparent because in four-part writing there is necessarily more parallel and similar motion, there being still only two directions in which to go. As a further rebuttal, let me point out that the discant treatise does not describe what the listener hears, any more than does the treatise on traditional harmony. In both cases the teacher tells the student how to proceed; he does not analyze the result as it strikes the ear. T h e typical discant treatise is a collection of practical precepts on how to make music, not a theory of aesthetics. T h e instructions of discant, therefore, do not imply that the listener hears two separate melodies; at most, these instructions imply only that the composer proceeds by combining two melodies. Do the instructions imply even that? I think not, for it seems to me that the assumption that discant taught how to write a second tune over a first is open to question. Consider the typical instruction again: note that while it may be taken to regulate the leading of one voice contingent upon the leading of the other, the same instruction also regulates the progression from the vertical sonority of an octave to one of a fifth. Now this ambiguity arises only in describing two-part progressions; in triadic ones the vertical sonority is identified as something distinct (a "triad") from the intervals (a "fifth" and two "thirds") that describe the location of its constituent tones. In other words, terminology does not permit a distinction between the location of one note an octave away from another, and the interval of an octave that these two notes form. Hence we can speak of a progression of two triads, one on g and one on c, or over a bass that moves down a fifth, or a dominant triad followed b y its tonic, and no one suspects us of describing linear counterpoint. But when we speak of an octave followed by a fifth, with the lower part ascending one step, then we may possibly be describing two melodic progressions; and then again we may not. Assume for the sake of argument that the medieval teacher does mean to describe a progression of vertical sonorities, each consisting of two notes: how else could he describe it but the way he does? Just as 13th-century discant lays down the basic doctrine of concord and discord, subject only to slight modification in the centuries following, so does it present rules of voice-leading that govern both the later Middle Ages and the Renaissance. These rules are discussed and illustrated by 13th-century writers in a bewildering variety of ways, yet as rules they are broad and simple. T h e first rule is, of course, the basic principle of contrary motion. T h e second (the order is mine, not theirs) is that one should begin with a concord and end with a perfect concord. This permits thirds

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at the beginning but not at the end. John of Garland extends this rule t o include the beginning of modal feet,3O while Franco, carrying out the implications of his mensural theory, applies the rule to the beginning of a " p e r f e ~ t i o n ; "both ~ ~ authors are speaking of imperfect as well as perfect concords. A third rule tolerates similar motion, without, however, precise distinction between similar and parallel motion, and without specifying perfect or imperfect concords. Franco says that one can use parallels for beauty, "propter pulchritudinem," which should not cause our eyebrows t o rise even if consecutive fifths are in question. Parallel motion, properly handled, can indeed be beautiful, as will be apparent to all but the most academic observers. I t should also be apparent that music cannot consist entirely of concords, but must also include discords. A fourth rule of 13th-century discant says that discords should be mixed in with the concords at the proper places;32 these places are sometimes described as being before or between concords, but are not otherwise located. Here, more than ever, it is important to remember what discant aims to do: it gives systematic instruction in writing two parts. If a technique cannot be presented with at least a semblance of system, discant does not treat that technique. Concords are the substance of two-part writing and can be treated systematically; discords are the accidents, to speak in Aristotelian terms, and 13th-century discant found it hard to treat them systematically. For this reason, not because the discant teacher disapproves of them, discords are passed over in the typical treatise. There is no gap here between theory and practice, save that imposed b y the needs of rational discourse. During the 14th century the name "discant" was gradually changed There was, however, no change in the basic princito "co~nterpoint.~~33 ples; they were merely applied in a more specific and refined way. This is best illustrated by the 14th-century treatment of parallel motion, of which the first example is in the little-known Compendium of Petrus "dictus palma ociosa" ( I 3 3 6) .z4 An interesting and well-written work, it sets forth near the beginning an informative discussion of discant. Song ( c m t u s ) is an inflection from one pitch to another. Discant is sweet melody made up of different songs, with two or more pitches reaching the ear in combinations governed by modus and tempus. It is called "discant" as if it were diverse songs, because the songs out of which discant is made ought to differ so that when one goes up the other goes down, and conversely. But both can ascend or descend together for the sake of the song's beauty, or because of 31

Cserba, Hieronymus de Moravia, p. 2 1 I . Cserba, Hieronymts de Moravia, p. 254; Strunk, Source Readhgs, p. 155.

32

LOC.cit.

30

See Kenney, "English Discant," p. 43. by J. Wolf; see note 19. The passage translated starts on p. 507. Ascent "in the same way" means by the same interval; "division" means diminution, as in diminished or florid counterpoint. 33

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II

limitation of range or some other necessity. Such ascent or descent should not be made in the same way; rather it should be as elegant and as graceful as possible. I allow, nevertheless, ascents and descents in the same way either by means of some division of the intervals or by imperfect intervals, such as minor third, major third, and major sixth. I do not advise using two or more consonances repeated in the same lines or spaces, either in perfect species of musical intervals or imperfect or middle ones. That finished, let us see briefly of which musical species discant should be composed. Concerning this you should know that all simple discant (which is nothing but punctum against punctum or one note produced by natural instruments placed against another) can be composed and ordered simply with unison, minor and major thirds, fifth, major sixth, and octave. After stating the principle of contrary motion, the author allows similar motion for the sake of beauty, as did Franco. Then the author goes on to admit parallel motion in thirds and major sixths, and this, it seems, is new. Of greatest interest is the way in which the instruction is phrased: its novelty lies not in the banishment of consecutive fifths and octaves, but in the tolerance of consecutive thirds and sixths. Consecutive fifths and octaves had been banned categorically from the moment when contrary motion became obligatory; consecutives of all kinds were tolerated only as exceptions. Now, in the 14th century, the musician seems to reason: "Some kinds of parallel motion are acceptable, other kinds are not. Those intervals which are not perfect concords and yet not discordant seem t o permit parallel motion without upsetting the delicate balance of polyphony, whereas consecutive perfect concords are too striking in their effect." Incidentally, it is interesting that discant authors frequently bannot parallel motion-but consecutive perfect concords, a clear indication of the medieval concern for the progression of vertical sonorities. T h e real importance, however, of the imperfect concords at this time has to do with contrary, not with parallel motion. Tempting as it is to seize upon consecutive thirds and sixths in older music as evidence of progressive tendencies, these consecutives have but little significance for the future development of musical style. Parallel motion does not produce the basic structures of part-music, as any authority on triadic harmony will testify. By an accident of history, consecutive thirds and sixths remind us of traditional harmony, which causes us to apply the labels "harmonic" or "functional;" but in medieval times as well as modern, parallelism is the antithesis of f ~ n c t i o n a l i t y . ~ ~ O n the other hand, the of thirds and sixths within contrary motion leads us to the center of 14th-century discant, and ultimately to the foundations of triadic harmony. A t this time the progressions major sixth to octave, major third to fifth, and minor third to unison take on more and more importance as the building blocks of counterpoint. These 35

Cf. H. Besseler, "Tonalharmonik und Vollklang," Acta mussicologica XXIV pp. '35f.

('952)t

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progressions acquire the force of necessity: their conclusion becomes obligatory. "The sixth seeks out the octave, and this rule always holds," says the author of the Ars contrapunctus secmdum P h i l i g p de Vitria ~ o Viewed . ~ ~ from this angle, a succession of consecutive sixths or thirds is the interruption of expected resolution--of "function," if you will. 14th-century writers specifically allow several ascending or descending consecutive sixths (or thirds) on condition that they are followed b y an octave (or fifth or unison).37 T h e parallelism allowed here is less significant than the resolution required. T h e importance of these progressions is great enough to demand alteration of the written pitches through musica ficta. If a sixth that proceeds to an octave is written as a minor one because of its position in the scale (for example, a-f proceeding to g-g) , then according to 14th-century writers this sixth is to be made major by raising the upper note.38 In the I 3th century musica ficta (falsa) was used chiefly to avoid u-itones and imperfect octaves, that is, to ensure that the intervals of discant would be concords. This use of musica ficta, declared by Lambert and Philippe de Vitry to be not false but neces~ary,3~ is another clear indication of the medieval concern for vertical sonority. But the alteration of sixths and thirds reveals the equally important concern for progression that emerges during the 14th century. Perhaps the strongest argument advanced on behalf of the linear interpretation has been based on the technique of "successive composition." After two-part writing, some discant and counterpoint treatises go on to describe the addition of a third and even a fourth voice.40 I t is argued that since the medieval composer added his voices onto the tenor "successively," rather than conceiving of his vertical sonorities all at once, he is writing linear counterpoint. But if, as I tried to show, the two-part framework was not linear counterpoint in the first place, then the third voice may not be either. Here we must avoid the false dichotomy between linear counterpoint and (triadic) harmony; we must think in terms of those two-note sonorities called concords or accords-terms not far from "chord." If the first step is the composition of a progression of two-note chords, then the third voice is added not as a third melody but as enrichment of those chords. T h e medieval writer says: "When adding the third voice, proceed as in discant," meaning that the third voice will proceed through the proper concords in contrary motion with one of the other two. For while the discant teacher can think in terms of two-note entities, his comCS 111, 27. example, CS 111, 40. 38For example, UTolf, "Ein Beitrag zur Diskantlehre," pp. 513ff, esp. p. 515. 39 CS I, 258; G. Reaney et al., "The 'Ars Nova' of Philippe de Vitry," Musica disciplim X (1956), p. 22. 40For example, John of Garland: Cserba, Hieronymus de Moravia, p. 225. sa

37 For

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prehension seems to stop there; he can explain a three-note progression only as the summation of two-note ones-much as traditional harmony explains polychords as complex or modified triads. But if this is true, then medieval composition is not more successive than our own. T h e really important difference is that the medieval system uses a basic unit consisting of two notes, whereas we use a unit of three notes. And successiveness in both cases is a feature of teaching rather than of listening. Particularly in the 13th century the authors find little new to say about three-part writing, having said it all in connection with normal discant. They sometimes add, with the brevity it deserves, the maxim that the third voice may ascend or descend now with the first voice, now with the second, but not with both at once.41 But the 14th century once again offers more specific instruction: the Quatuor pincipalia ( I 3 5 I ) refers to the principle that the upper voices must concord with the lowest.42 T h e author describes how to discant below the tenor, using the same concords and procedures as in discanting above, then adds the proviso that while improvising below, no one else should discant above unless he knows what tones are being sung below, "because all the upper parts must be in concord with the lowest voice in order to make good consonance." T h e Ars discantus secundum I o m e m de Muris contains a passage on the composition of two counterpoints over one tenor;43 the author advises against two similar concords over the same note at the same time (that is, a fifth and a twelfth, an octave and a fifteenth, a third and a tenth) because in these there is no diversity. H e also cautions against a fifth and a sixth at the same time, but recommends a fifth and a tenth, for if the tenor were to rest, these two counterpoints would still be in concord with each other. Immediately following this passage in the Ars discantus (but not necessarily by the same author), is a description of three-part writing for tenor, camen, and contratenor. For each concord of tenor and c m e n (unison, third, fifth, sixth, octave, tenth, and twelfth) the author gives all possible concords for the contratenor below. Some of these are described as sweeter, some not so sweet. I t should be noted that the procedure is completely vertical: there is no mention of progression from one sonority to the next, progression being treated in the exposition of discant proper. T h e third voice is understood by the author to be an expansion of the vertical sonority. This helps t o explain the curious matter of the Renaissance bass. I t is frequently pointed out, with solid textual support, that the early Renaissance teacher reckoned his bass notes down from the tenor, rather than the other way around. It is concluded from this that the early Renaissance 41For example, Franco: Cserba, Hieronymus de Moravia, p. Readings, p. I 55. 42 CS IV, 292, 294. 43 CS 111, 92f.

254;

Strunk, Source

JOURNAL OF T H E AMERICAN MUSICOLOGICAL SOCIETY '4 bass was "non-functional," which seems reasonable. But, the argument continues, if the bass is non-functional, then everything over it must be linear, and this is where I think the argument runs off the track. T h e medieval sense of function resides, as we saw, in the progression of concords, especially in the progression sixth-to-octave. These progressions are interrupted, obscured, all but obliterated in the Renaissance; yet somehow they continue to function. Example I shows how the sixth-to-octave (in whole notes) can be enriched b y a third voice (in quarter notes), which is below the tenor in the first chord, and either above or below it in the second. Ex. I

In the first case this third voice is clearly non-functional in its progression: it merely enriches the first chord, then the second. I t cannot be said to progress from one to the other, either harmonically or melodically. But the same is true in the second case: here, too, the third voice merely enriches the sonority. It cannot be said to have any function-save from an 18thcentury point of view. T h e functional parts of the second case are still the sixth and octave, even though masked b y the bass. Such masking of a progression, however, is very different from complete independence of voices. I t is also argued that since discant was reckoned up or d o w n from the tenor, the tenor was never a foundation in the same sense that the 17thcentury bass wras. T h e "fundamental bass" is described as being invented, o r discovered, by the late Renaissance and early Baroque. But we already saw in the Qzratuor principalin ( m i d - ~ ~ tcentury) h that concords were reckoned in some sense from the lowest sounding part. Indeed, 14thcentury discant describes primarily the construction of intervals over the tenor. If we were to survey 14th-century music we would find that in motets n j the tenor is usually the lowest part, hence the foundation in every conceivable sense. In motets n 4 (with a contratenor) it sometimes seems as though the contratenor, when below the tenor, is the lower part of a standard discant progression. Perhaps this is what Anonymous XI means when he says: 44 Another general mle: contratenor can well descend with the tenor in imperfect species, ending in a perfect species; and similarly the tenor with the contratenor. And you should know that the contratenor is said to be the tenor when it is lower than the tenor. In other words: in the 14th century the lowest part is the foundation; together with one of the upper parts it forms the basic two-part framework. 44

CS 111, 466.

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'5

In the early Renaissance this framework is masked by other voices, above and below; the lowest part is no longer the foundation. Then in the later Renaissance the lowest part is once again described as the foundation, first of individual sonorities, and-much later-of progressions. Two-part writing remained the basis for instruction straight through the Renaissance. But in response to contemporary practice, teachers began to offer additional instructions for composing in three parts, that is, they described certain stereotyped formulas for masking discant. These solutions to the three-part problem were, in the early Renaissance, necessarily crude and pragmatic: they involved a good deal of parallel motion and sequence, and lacked the economy of classic ~ ~ t h - c e n t u rprogresy sions. But they did manage to produce that full, rich sonority so much in demand at the time. T h e treatise of William the Monk provides us with a catalogue of such instructions, summarized as follows:45 I. 2.

3.

4. 5. 6. 7. 8. 9.

T h e English "modes" (formulas): Fauxbourdon, a 3, Gymel, a 2. Another formula a 3, "non mutatis." Discant (here is the logical beginning of the treatment of composition). Table for finding concords to notes in the c and g hexachords. Fauxbourdon and Gymel (alternate rules). T h e low Contratenor for No. 5 (hence a 4). More rules, and two exceptions. Another formula a 3 . And another.

T h e only formula of real importance for the future is contained in No. 7

(Ex. 2 ) . Ex.

*C

in

2

CS.

T h e strength of this formula seems to lie in its avoidance of parallel motion while producing a series of imperfect concords. Like the "clausulae" of the 16th ~ e n t u r y , 4this ~ formula taps the resources of traditional discant. T h e counterpoint treatises of the 14th and early 15th centuries, when taken seriously and in order, provide a wealth of material and a fascinat4 5 CS 111, 273; ~88ff;Example 2 from p. 296. The treatise had been ably described and analyzed by Brian Trowell, "Faburden and Fauxbourdon," Mzcsica disciplina XI11 ( 1959)~pp. 64ff. 46See B. Meler, "Die Harmonik im cantus firmus-halti en Satz des 15. Jahrhundem," Archiv fiir Musikwissenschaft IX ( ~ p j z ) ,pp. 27f See also A. Schmitz, Oberitalimische Figuralpassionen des 16. Jahrhzmderts (Mainz, 1955), Vo1. I, p. IT*.

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ing variety of detail. It is remarkable that so little attention has been paid to these treatises since the indefatigable Hugo Riemann described them in his Geschichte der Musiktheorie in 1898. T h e condition of the sources being what it was (and still is), Riemann did not get them entirely in order, but he did take them seriously. His seriousness, however, was one that looked forward to the Messianic appearance of the Dual Nature of Harmony in the Major and Minor TRIADS, the glory of whose coming blinded him to the actual meaning of the medieval authors. Nevertheless, many who are scandalized by his speculations could benefit from his knowledge of the sources. T h e more one becomes acquainted with these authors of the 14th and early 15th centuries, the more one sees how dependent the Renaissance authors are upon them. Tinctoris's rules, for example, reveal no basic novelty when compared to earlier sources.47 T h e most important differrence is the insistence on variety, with urgent prohibitions against repetition. This seems to be related to a greater number of imperfect concords, and a relaxation of the procedures governing their use. N o longer does a composer resolve thirds and sixths, but leads them in unending chains of suspended functions. It must be this that gives Renaissance discant its new sound, since, as we saw, the sound of imperfect concords is as old as discant, and a mere increase in their frequency seems a weak basis for a new style. But this variety, being an avoidance of the obvious, can find no expression in general principles. Therefore the old principles are surrounded in Renaissance treatises by an endless number of provisions against the obvious, and an even greater number of examples showing borderline cases of similar motion and ways of exploiting discords. Composition becomes the skill of producing continuous variety while avoiding on the one hand the barbarous and on the other the too familiar. I t is frequently said that in the three-part formulas of 15th-century counterpoint one can hear "a real feeling for functional harmony." While na'ive, this observation is not without foundation-only we must disentangle the meanings of the terms involved. W e use the term "functional harmony" so often that we say "functionalhannony"-one word with the accent on the fourth syllable. W e forget that there are two words with two different meanings; that there might be "non-functional harmony," or even "function" in the absence of "harmony." hTowit seems clear that "function," since Riemann, refers to relationships between triadic chords, relationships that may be actual or implied. Armed with a more comprehensive view of history, we can proceed cautiously to speak of functions between two-note entities instead of between triads; I have tried, through a discussion of discant, to show how this might be done. W e might even 47

CS IV, 147ff. G. Reese, illusic in the Renaissance (New York, 1954)~p. 144.

DISCANT, COUNTERPOINT,

AND HARMONY

I7

speak of functions in monophony, if we could find the appropriate terms -but that is another matter. T h e formulas of the 15th century, then, are indeed functional: they depend upon the two-note progressions of discant. They also sound like the familiar progressions of "functionalhannony," which simply means that triadic functions and progressions develop in unbroken continuity out of discant. T h e difference between discant and "functionalharmony~' has to do not with "function" (although the specific functions are slightly different in the two systems) but with "harmony." T h e search for the meaning of this term takes us into quite another part of the forest, far from the counterpoint teacher and his practical precepts. W e must at long last take on the speculative theorist and his intricate calculations. In compensation for the thorny mathematics the theorist offers us explanations about the very nature of musical sound. T h e term "harmony" is not unknown in the Middle Ages, whose writers got it from the Greeks. In order to understand its use we must remind ourselves that it is an everyday word for the marvellous quality that characterizes a great painting, a successful piece of architecture, a happy family, and that for which the peoples of the world yearn. As musicians we tend to forget this more basic meaning: our books on harmony do not usually tell us why their subject should be so named. T h e Middle Ages as well as the Renaissance approached this marvellous quality by paradox, explaining it as the "concord of discords." This inscription finally turns up on the title page of Gaffurio's De hamzonia msicorunz instrumentorum ( I 5 I 8). N o t that "harmony" meant "polyphonyv-far from it-but in polyphony they saw yet another manifestation of that quality that ran through the whole creation. Indeed, polyphony becomes the most tangible manifestation of harmony: in the Renaissance, the Promethean musicus speculator seizes upon the Idea of harmony and fixes it in the matter of counterpoint. N o t long after the counterpoint teachers tackled the three-part problem-around the end of the 15th century--certain theorists were pondering the same problem from a different point of view. These theorists were not so much concerned with how to produce three-note chords, but rather why some chords sounded better, more harmonious than others. Their attention was focussed upon the chord itself as a vertical sonority; they sought some tool for explaining its nature. T h e tool had been at hand for some time. Every theorist in the West had presumably read Boethius, and thus knew about the several kinds of "mean" or division of a proportion. Boethius described three means: arithmetic, geometric, and harmonic, as follows:4* 4 8 De institutione musica IT, 1 2 (Friedlein ed., p. 241). Eleven kinds of mean were known to antiquity; see P. H. Michel, De Pytl2agore d Euclide (Paris, ~gj-o),pp. 365,

369s.

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Arithmetic Geometric Harmonic

z:3:4

2:4:8

3:4:6

T h e arithmetic mean divides the distance between the extremes into two equal parts; the resulting ratios, however, are not equal (2: 3 # 3 :4). In the case of the geometric mean, the ratios are equal, but the middle term does not divide the distance into two equal parts. In the harmonic division neither the parts nor the ratios are equal, but the ratio of the parts is equal to the ratios of the extremes. This curious affinity of the division to the whole is the special property of the harmonic mean. Its very name, "medietas harmonica," must have caught the fancy of the Renaissance theorist. T h e application to vertical sonority is striking, and the theorist must have pondered the result with excited s a t i s f a ~ t i o n ~ ~ Ex. 3

In this example the numbers to the right of the notes represent string ratios: thus strings in the arithmetic proportion (2: 3:4) sound a fifth with a fourth below; those in the geometric proportion (2:4:8) sound an octave with another octave below; those in harmonic proportion (3:4:6) sound a fourth with a fifth below. I t is characteristic of the arithmetic mean that the larger ratio occurs between the smaller numbers, hence the larger interval (the fifth) comes at the top, between the shorter strings; the harmonic mean, on the other hand, has the special property of placing the larger interval between the longer strings, hence below the fourth. When judged by a 15th-century ear-perhaps by any ear-the three pitches produced by the harmonic division of the octave have a much more balanced, euphonious sonority than the others. Having reached this conclusion, the theorists reserved the term "harmony" for a chord of three pitches; chords of two pitches were concords or discords. And it was the harmonic mean-0 happy coincidence-that produced the truly harmonious division of the octave, that chord which for a long time had been most worthy to end a song. For reasons that the subtle reader may ferret out for himself, the harmonic mean cannot be applied within the "Pythagorean" tuning beyond the division of the octave. But at the moment when theorists were making this application, they were also busily engaged in modifying the system 49 All this and the following is available in Riemann's Gerchichte der Muriktheorie,

2'.

Ch. XII: "Die Revision der mathematischen Akustik," p 318ff; in addition to clarifying the material, my aim is t o put it in a slightly di erent light. The first theorist actually to apply the harmonic mean t o sonority seems t o be Gaffurio (Riemann, p. 324). Walter Odington does not say exactly what Riemann suggests.

DISCANT, COUNTERPOINT, AND HARMONY

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of tuning.60 This dislocation of traditional concepts resulted in considerable controversy; when the dust had settled, the issue seemed decided, at least temporarily, in favor of those advocating a "pure" third. One consequence of this was an endless series of discussions about the "least" intervals, the commas and their kind, which are apparent to the reason but not the ear-this in an age which allegedly trusted in the sensible. Another consequence more germane t o our topic was this: the pure thirds 5: 4 and 6: 5, by replacing the old ratios 8 I : 64 and 3 2: 27, could now take their rightful positions alongside the perfect concords of octave (2: I ) , fifth (3: 2), and fourth (4: 3). Of course the discant teachers had grouped the thirds with the concords ever since John of Garland in the I jth century, but the Renaissance theorist provided the mathematical justification. And now, reasoned Zarlino, the principal concords of counterpoint can be derived from the ratios of the first six numbers, the ena aria.^^ Surely a remarkable demonstration of the rational nature of music! A demonstration as well that the moderns had surpassed the ancients, who had used only the tetrad, or first four numbers. If the major and minor thirds are expressed by these ratios, another application of the harmonic mean is possible. Gaffurio had perhaps already made this application, but in passing; Zarlino takes it up with more decision (Ex. 4) .52 Ex. 4

Here are two more sonorities that are formed from concords, have three pitches, and fill the ear with sweet harmony. In the first a fifth is divided arithmetically, placing the larger interval at the top; in the second the fifth is divided harmonically, placing the larger interval at the bottom. Once again reason coincides with the judgment of the ear in declaring the latter to be more harmonious. Because it consists of three different pitches it is called a "triad;" because it uses the harmonic mean it is the "harmonic triad." For Zarlino and the 16th century this triad, of all sonorities, manifests most clearly that marvellous quality, harmony. It is important to observe that such analyses in no way conflict with discant. In the third part of the Istitutioni barnoniche, Zarlino discusses discant in the traditional fashion, and quite consciously so, for he refers frequently to the ancients, meaning the earlier discant authors, and shows 6oAlthough treated by Riemann and others, this chapter in the history of theory also needs rewriting. J. Murray Barbour's analysis, (Tuning and Temperament [East Lansing, hlich., 19511 ) while authoritative, wants sympathetic insight; according to Barbour, "just intonation" is something that can and should be "confuted." 5 1 lstitutioni hamzoniche (Venice, 1573 ) , I, Cap. xiiiff. 52 Ibid., I, Cap. xxxix, xi; 111, Cap. xxxi; Strunk, Source Readings, p. 242.

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little inclination to depart from their principles. But one should not conclude from this that Zarlino is a dual mentality, half medieval and half modern, unaware of a contradiction between linear counterpoint and functional harmony. As I have tried to show, he discusses neither of these things; the two subjects he does discuss, counterpoint and harmony, are in no way contradictory. His precepts of counterpoint, like those of his predecessors, teach how to get from one concord to the next, and how to expand this progression of concords into three or more parts. His theory of harmony analyzes the nature of three-part sonorities. This theory of harmony does not treat, in principle, the progression from one harmony to the next: the harmonic triads have no systematic relation, and therefore no function, one to another. His theory is about harmony, but not about functional harmony. A word might be said comparing Zarlino's theory of harmony t o Rameau's. Zarlino demonstrated that the principal concords could be derived from the ratios of the first six numbers, and that the principal harmony was formed from these concords arranged according to the harmonic proportion. Rameau showed, in effect, that the principal concords could be derived from the intervals in the natural (that is, physical) series of partials or overtones, and that the principal harmony was formed by taking certain concords in the order in which they actually appeared in this natural series. T h e one demonstration seems just as valid, and no more so, than the other.53 T h e 16th-century theorist believed that if he could find the form of music in the realm of number, he somehow made music more real or more true; the 18th-century theorist believed the same, only he looked for his proof in the realm of physical phenomena. And the Creator in his Wisdom made the universe big enough so that perhaps both are right. Granting that Zarlino's arguments are cogent, it is still difficult to reconcile oneself to his allotment of space among various topics. T h e senaria by no means dominates the scene, while the harmonic division, considering its implications, seems actually slighted. T h e "least intervals," here as in other Renaissance theorists, get the largest share of space, which inclines one to reject the whole tedious discussion of commas as hopeless 5 8 T h e frequency ratios of the ascending partial series of course produce--or are produced by-the series of whole numbers: 1,2,3,4,5,6 . . . Strings arranged so as to yield these sounds will have lengths corresponding t o the series: I, 1/2,. 1/3, I/.+, / 5 , 1/6 . . . Any three consecutive terms of this latter series yield a harmonlc proportion; in fact this series (the reciprocals of the whole numbers) is the only continuous harmonic proportion, since all cases of the harmonic proportion involving whole numbers (e.g. 2:3:6) come to an end. See P. H. Michel, De Pythagore d Euclide, pp. 394f. If string ratios are used, therefore, the harmonic triad must be explained by the harmonic proportion; if frequency ratios, by the whole-number series, that is, arithmetic proportion. Not without logic is the whole-number series called the "harmonic series" when it refers to frequencies.

DISCANT,

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21

pedantry, remote from musical art. Yet perhaps when an art is drawing to an end, when all the ways out have been explored, all limits reachedperhaps in this moment of closure, the investigation of the whole tonal system, all its cracks and crevices, becomes a matter of great importance. Perhaps we will come to understand this in our own time. Yale University