A Polychordal Approach to Serial Harmony - Part 1 - Online Version

A Polychordal Approach to Serial Harmony – Part I

Barnaby Hollington

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CONTENTS

Introduction

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Chapter 1

Interval-Class Set Taxonomy

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Chapter 2

Tonal Consonance and Dissonance

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Chapter 3

Four Problematic Terms: Atonality, Post-Tonality, Pantonality, Polytonality

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Chapter 4

The Polychordal Approach

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Chapter 5

The Art of Thinking Clearly: A Polychordal and Serial Analysis

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Chapter 6

Serialism, Polychordally Conceived

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Bibliography

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INTRODUCTION

My research aims to test the effectiveness of a consciously polychordal approach in clarifying so-called ‘atonal’/‘post-tonal’1 harmonic trajectories to the listener. My term for the system I am developing is the Polychordal Approach. I anticipate that my research will demonstrate that this method allows a composer to minimise the ‘cognitive opacity’ 2 that, according to Lerdahl (1988), Hicks (1991), Meyer (1967), Taruskin (1994, 2008) and other commentators, renders much atonally/post-tonally-conceived harmony – particularly serial harmony - inaccessible to most listeners. The Polychordal Approach is founded on the premise that, within the bounds of equal temperament, all pitch-class sets have potential tonal implications – as commentators such as Réti (1958), Parncutt (2009) and Adès (2012) maintain. Most pitch-class sets have multiple latent tonal centres – i.e., they are inherently polychordal. A few pitch-class sets possess only one possible tonal centre – i.e. they are inherently ‘tonal’. Any pitch-class set can therefore be spaced, horizontally and/or vertically, in such a way as to maximise the audibility of its tonal connections – i.e., in all but the simplest cases, spaced as a polychord. In the process, one or more tonal centres within the pitch-class set will be perceived as the focal point(s) or tonal anchor(s) of the pitch-class set. This is the key to meaningful cognition of any conceivable succession of pitch-class sets by the listener; the key to consistently avoiding the ‘cognitive opacity’ that will otherwise continue to prevent atonally/post-tonally-conceived music from ever communicating meaningfully with a wider audience.

1 ‘Post-tonal’ and ‘atonal’ are the generally accepted terms for the type of harmonic territory that my music tends to cover. I hold that neither term satisfactorily describes my music. Moreover, both terms pose certain more general problems: for example, certain composers conceive of their music in ‘atonal’ or ‘post-tonal’ terms, but whether listeners always perceive atonally/post-tonally-conceived harmony in the same terms in practice is another matter. The question will be explored further in Chapter 3. 2 Lerdahl, F.: ‘Cognitive Constraints on Compositional Systems’ in Generative Processes in Music, ed. John A. Sloboda (Oxford: Clarendon Press, 1988), p.231.

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The primary purpose of this paper is to demonstrate the workings of the Polychordal Approach, through harmonic analysis of sections of The Art of Thinking Clearly, for solo piano. This analysis, in Chapter 5, forms the main body of this paper. But before any cogent discussion of the Polychordal Approach can ensue, three sets of terminological and conceptual hurdles must first be overcome: 1. Some well-documented difficulties arising from Forte’s ‘pitch-class set’ system. I have devised a new, more transparent system which addresses those difficulties. 2. Conceptual problems arising from varying definitions of consonance and dissonance. 3. Contradictions arising from the terms ‘atonal’ and ‘post-tonal’, and limitations imposed by the accepted definitions of the terms ‘pantonal’ and ‘polytonal’. Chapters 1-3 deal with each of the three questions outlined above, preparing the ground for an explanation of the rationale informing the Polychordal Approach in Chapter 4. Chapter 5 will analyse the workings of this approach in practice, dissect some of the harmonic phenomena created through employing the Polychordal Approach in conjunction with serialism, and – in the process - refute Lerdahl’s claim (1988) that serialism can only result in harmonic opacity, from the listener’s perspective.3 Chapter 6 concludes by considering some of the broader implications of this Polychordal Approach to serial harmony.

3 I do not dispute Lerdahl’s assertion (1988) that the harmony in serial and other atonally/post-tonally-conceived works – such as Boulez’s Le Marteau Sans Maître (1956) is cognitively opaque. I do contend, however, that serialism can be rendered tonally clear. Schoenberg and Berg both attempted to achieve this, with mixed results. Webern (1933) also held that his own serial harmony could be interpreted in tonal terms. After 1945, the generation of composers who took over serial techniques chose to write as though tonal implications can and should be eradicated from new music altogether, typically resulting in ‘cognitive opacity’. But if previous generations of serial composers have failed to achieve harmonic clarity, from the listener’s perspective, it does not follow that current or future generations of serial composers cannot attain the desired coherence. Indeed, the analysis in Chapter 5 will demonstrate that certain serial techniques can also be used to increase harmonic intelligibility, in themselves, if skilfully handled.

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CHAPTER 1 INTERVAL-CLASS SET TAXONOMY

For the purposes of my research, Interval-Class Set Taxonomy is intended as a substitute for the classification system devised by Allen Forte in The Structure of Atonal Music (1973) – ostensibly a pitch-class set taxonomy - which is currently in widespread use. In seeking to demonstrate the effectiveness of the Polychordal Approach over all conceivable harmonic territory within equal temperament, I intend to use not only all conceivable interval-class sets, but exploit each portion of this expressive territory roughly equally, over my composition portfolio as a whole. Individual pieces will concentrate on different portions of this territory, to different expressive ends. Forte’s convoluted classification system precludes sufficiently concise and accurate presentation of the full range of tonal and expressive implications possible within equal temperament for my purposes. Several commentators have identified flaws in Forte’s classification system and analytical methods, including Taruskin (1987), Perle (1990) and Parncutt (2009). Those points most pertinent to my research are summarised below.

1A. Pitfalls in Forte’s ‘Pitch-Class’ Set Taxonomy Parncutt (2009) establishes that Forte’s ‘pitch-class sets’ are not, in fact, sets of pitchclasses, but sets of interval-classes, defined as though they were sets of pitch-classes. Under Forte’s system, the listed elements of a ‘pitch-class set’ are not pitch-classes, but rather a group of numbers which would correspond to pitch-classes only if the first element listed (always 0) happened to be C, and only then if the set were uninvertible. All we can ascertain from this list of numbers – and then only indirectly, through arithmetic - are the intervalclasses between adjacent pitch-classes (by substraction). An additional ‘interval-class vector’ is appended to each Forteian ‘pitch-class set’, listing all the interval-classes within the set, but this is not enough to distinguish them from one another: there are instances of separate Forteian ‘pitch-class sets’ sharing the same interval-class vectors.

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Furthermore, Forte’s system treats inversionally-related sets – e.g. the minor triad and the major triad - as identical. Evidently, the minor and major triads do not possess the same expressive qualities, and the same can be said of other numerous other inversionally-related sets in both Western and non-Western music4. Nor can the tonal implications of inversionallyrelated sets be considered identical. The Polychordal Approach to post-tonal/pantonal harmony is concerned with these tonal implications. Therefore, any system that does not distinguish between major and minor, nor between any other inversionally-related intervalclass or pitch-class sets, would hinder my research. As a consequence of listing the elements of each set as a numbers indicating neither the pitch-classes, nor (directly) the interval-classes, nor (if the set is invertible) which way up the interval-classes are, Forteian ‘pitch-class set analysis’ is an unnecessarily labyrinthine, tortuous affair. Too often, the system effectively does little more than muddy the water. Perle (1990) has persuasively demonstrated how this has adversely affected Forte’s own analytical work. I have therefore devised a simpler, more elegant and more consistent system, which avoids each of the pitfalls described above: Interval-Class Set Taxonomy.

1B. My proposed solution – Interval-Class Set Taxonomy Under Interval-Class Set Taxonomy, the listed elements of each set are interval-classes. I list only the intervals between adjacent pitch-classes, rather than the full interval-class vector. For example, the pitch-class set [B, C, D, F], is a subset of the interval-class set [1, 2, 3, 6]. In other words, the interval-class between B and C is 1 (a semitone), the interval-class between C and D is 2 (2 semitones), etc. Of course, I could have listed the elements as [2,3,6,1], [3,6,1,2] or [6,1,2,3]. To avoid confusion, the smallest number(s), i.e. interval(s) are always 4 For example, Hindustani Thaats have clearly-defined, distinct expressive qualities. Two pairs of Thaats happen to be inversionally-related – Khamaj (flirtatiousness, sensuality) and Asavari (renunciation, sacrifice, pathos); and Bilawal (devotion and repose) and Bhairavi (devotion and repose, but also compassion and sadness). The interval-classes between the degrees of the Khamaj Thaat are, in ascending order: 2,2,1,2,2,1,2 (= the Mixolydian mode). The corresponding interval-classes for Asavari are 2,1,2,2,1,2,2 (= the Aeolian mode). For Bilawal, 2,2,1,2,2,2,1 (= the Ionian mode); for Bhairavi, 1,2,2,2,1,2,2 (= the Phrygian mode).

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listed first – therefore [1,2,3,6]. Under Interval-Class Set Taxonomy, the name of the intervalclass set [1,2,3,6] is simply 1236. Under this new system, inversions of interval-class sets are listed as separate sets. All that is necessary to discover the inversion of an interval-class set is to reverse its numbers: 1236 becomes 6321. But the 1 is then placed first: so 6321 becomes 1632 (listed as a distinct set from 1236). In other words, to invert an interval-class set under my system, one keeps the first number in place, and flips the rest around: 1236, flipped, becomes 1632, 1227 becomes 1722, 1245 becomes 1542, and so on. The major triad, 354, flipped, becomes the minor triad, 345. 1272, flipped, remains 1272, i.e. 1272 is uninvertible. Thus, unlike Forte’s system, Interval-Class Set Taxonomy allows both inversions and (self-evidently) transpositions of a set to be spotted instantaneously. Interval-Class Set Taxonomy also allows the analyst to list sets of pitch-classes, e.g. [B, C, D, F], without risking ambiguity. I shall henceforth use the phrase ‘sets of pitch-classes’ to denote this kind of list, so as to avoid confusion with Forte’s ‘pitch-class sets’, which are not sets of pitch-classes.

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CHAPTER 2 TONAL CONSONANCE AND DISSONANCE

The next terminological and conceptual minefield requiring clearance, before any meaningful discussion of the Polychordal Approach can ensue, is the question of consonance and dissonance. There are several conflicting definitions. These fall under two broad categories – ‘tonal’ (Plomp and Levelt, 1965)5 or ‘sensory’ (Lerdahl, 1988), versus ‘contextual’ (Hill, 1986) or ‘musical’ (Lerdahl, 1988).6 These describe two very distinct phenomena, which happen to coincide for most of the history of Western art music. Contrary to assertions from Rosen (1975) and other authors, both tonal and contextual dissonance can remain meaningful, and can be coherently and consistently described with reference to atonally/post-tonally-conceived music, in which the two do not coincide - where tonal dissonances have been ‘emancipated’, i.e. no longer necessarily serve as contextual dissonances. Plomp and Levelt (1965) and other commentators employ the terms ‘tonal dissonance’ and ‘tonal consonance’ to describe the sensory difference between intervals, and therefore also between chords. (Lerdahl and others prefer ‘sensory’ consonance/dissonance). According to Plomp and Levelt, a ‘tonally dissonant’ interval produces audible beats between the partials of the two tones. These beats occur when the difference between the two frequencies falls within a critical bandwidth. Plomp and Levelt thus identify the most tonally dissonant interval as the minor 2nd, followed by the major 2nd and the tritone. Through the centuries, there have been many divergent rationalisations of tonal consonance and dissonance. The resultant hierarchies are generally quite similar. I consider Plomp and Levelt’s hierarchy and definition the most persuasive, for the purposes of my research.

5 Plomp, R. and Levelt, W.J.M.: ‘Tonal Consonance and Critical Bandwidth’, in Journal of the Acoustical Society of America 38 (1965), pp.548-560. 6 This is a broad generalisation. Hill (1986) identifies three types of definition. Cazden (1980) identifies fourteen separate definitions, but was unaware of Plomp and Levelt’s work at the time of writing.

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Another group of theorists and composers, including Babbitt (1965), Rosen (1975), Cazden (1980), Kamien (2010/1976), and Anderson, have argued that consonance and dissonance can only properly be defined within the context of a specific piece of music, or, at best, a given set of harmonic rules as defined in a particular era. Rosen (1975)7 writes: ‘[A dissonance is]… any musical sound that must be resolved, i.e. followed by a consonance: a consonance is a musical sound that needs no resolution, can act as the final note, that rounds off a cadence.’

‘Tonal’ or ‘sensory’ consonance/dissonance is crucial to the Polychordal Approach. ‘Contextual’ or ‘musical’ consonance/dissonance is less central to my concerns, although not insignificant. One may measure the relative levels of tonal dissonance between interval-class sets simply by counting the instances of the more tonally dissonant intervals within each set, according to Plomp and Levelt’s hierarchy. On that account, borrowing my terminology from Krenek (1940), I am listing interval-class 1 (the semitone) as a ‘strong dissonance’, and interval-classes 2 (the tone) and 6 (the tritone) as ‘mild dissonances’. The most basic hierarchical strata can be distinguished simply by examining the number of semitones in the interval vector. Under Interval-Class Set Taxonomy, the names of the sets themselves give this information: it is simply a case of counting the number of 1s in the set name. This produces, for tetrachords, the following hierarchy:

Strong Dissonance s 3 semitones

Sets

2 semitones

1128, 1137, 1146, 1155, 1164, 1173, 1182, 1218, 1317, 1416, 1515

1 semitone

1227, 1236, 1245, 1254, 1263, 1272, 1326, 1335, 1344, 1353, 1362, 1425, 1434, 1443, 1452, 1524, 1533, 1542, 1623, 1632, 1722

0 semitones

2226, 2235, 2244, 2253, 2325, 2334, 2343, 2424, 2433, 3333

1119

7 Rosen, C.: Arnold Schoenberg, (New York: Viking Press, 1975), p.33.

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We can then add the second and sixth numbers of Forte’s intervalclass vectors for each set (i.e. the rest of the ‘dissonance vector’), to determine the number of ‘mild dissonances’ – i.e. tones (2) or tritones (6). For tetrachords, 11 hierarchical strata ensue:

Strong Dissonances 3 semitones

2

1

0

Mild Dissonance s 2 tones/tritone s 2 tones/tritone s 1 tone/tritone 0 tones/tritone s 2

1 0 4 3 2 1

Sets

1119

1128, 1146, 1155, 1164, 1182, 1515

1137, 1173, 1218, 1416 1317

1227, 1362, 1722 1254, 1344, 2226, 2244 2253, 2343

1236, 1245, 1263, 1272, 1326, 1425, 1524, 1542, 1623, 1632, 1335, 1452, 1533 1353, 1434, 1443 2424 2325, 2334, 2433, 3333

The following chart shows similar hierarchies for all Interval-Class Sets containing five elements or fewer. Interval-Class Sets (IC Sets) of the same colour contain the same number of strong dissonances. Within each such category, higher rows contain more mild dissonances; lower rows contain fewer of these. Inversionally-related sets will always contain the same levels of tonal dissonance, and are now shown side by side. This chart will suffice for the analytical purposes of this research paper (Chapter 5); later in my research, I will expand this classification to include all 350 IC Sets.

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Of the 350 possible IC Sets (136 of which are listed above), only 8 are tonal consonances: silence, 0 (unison), 39 (minor 3rd/major 6th), 48 (major 3rd/minor 6th), 57 (perfect 4th/5th), 345 (the minor triad), 354 (the major

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triad) and 444 (the augmented triad). Of all the IC Sets serving as contextual consonances throughout the history of Western art music until the late 19th century, there are only 7 types (if one includes silence) – 7 of the 8 possible tonally consonant IC Sets listed above. The exception is 444 (the augmented triad). This obvious shared inherent trait – none of these 7 IC Sets contains tonally dissonant interval-classes; whereas, of the other 343 possible IC Sets, all but one contain tonally dissonant interval-classes – casts doubt over Charles Rosen’s claim that a consonance or dissonance should only be defined contextually: ‘It is not… the human ear or nervous system that decides what is a dissonance, unless we are to assume a physiological change between the thirteenth [century, when only 4ths, 5ths and octaves could serve as contextual consonances] and fifteenth century [when 3rds and 6ths could also serve as contextual consonances, but 4ths could no longer do so].’8

That all contextual consonances in the history of conventional Western tonal grammar are also tonal consonances, by Plomp and Levelt’s criteria – and that, 444 aside, no other tonal consonances are possible - would suggest that the human ear does, in fact, decide ‘what is a dissonance’. Whilst IC Sets possess measurably variable inherent levels of tonal dissonance, in practice these differences do not correspond directly to perceptible levels of tonal dissonance. Rather, IC Sets constitute only the first

and

most

straightforwardly

analysable

of

several

factors

in

determining perceptible tonal dissonance levels. The other relevant factors are spacing, register, dynamics and timbre. My research is not 8 Rosen, C.: ibid., p.33. Moreover, Rosen’s implication that perfect 4 ths never serve as contextual dissonances between the Renaissance and the 20 th century is not strictly accurate. Throughout this period of Western musical history, although a 4th could not serve as a cadential resting-point in itself, 4 ths frequently do appear at cadential resting-points, in the middle or upper voices of larger chords. Considering that in conventional Western tonal music, ‘chords… are [considered] dissonant if they include even a single dissonant interval’ [Hill, C.: ‘Consonance and Dissonance’ in The New Harvard Dictionary of Music (Cambridge, Mass. and London, 1986), p.197], if 4 ths had truly been considered dissonant between the Renaissance and the 20 th century, 4ths would never have appeared at cadential resting-points over that period – but they do.

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primarily aimed at finding failsafe methods of measuring tonal dissonance levels – although later in my period of research, I will be measuring the effect of Polychordal spacing on perceived levels of tonal dissonance, and considering the effect of register. However, the classification of IC Sets by relative tonal dissonance levels enables me to define different expressive territory in broad terms from piece to piece, or from passage to passage 9. Consequently, I can test the effectiveness of the Polychordal Approach across the entire available so-called ‘post-tonal’ harmonic spectrum.

9 For example, The Art of Thinking Clearly, analysed in Chapter 5, uses very tonally dissonant IC Set material - yet in practice, the polychordal spacing of those sets results in a much lower level of perceptible tonal dissonance than the IC content would suggest. I will demonstrate this fully at a later point in my research.

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CHAPTER 3 FOUR PROBLEMATIC TERMS: ATONALITY, POST-TONALITY, PANTONALITY, POLYTONALITY

Several theorists and composers have asserted that all sets of pitch-classes have tonal implications, including Reti (1958), Webern (1933)10, Auhagen (1994), Adès (2012) and, most significantly for the purposes of my research, Parncutt (2009). 11 Parncutt argues that it is impossible for a composer to eliminate tonal implications from their music, unless they also eliminate pitch altogether, as happens in Sciarrino’s …da un Divertimento (1970), Reich’s Clapping Music (1972), some of Xenakis’s Pléïades (1978) and most of Varèse’s Ionisation (1931). The term ‘atonal’, therefore, ought only to apply to music which excludes pitch: ‘Since every interval, sonority and melodic fragment has tonal implications, even the so-called “atonal” music of composers such as Ferneyhough, Ligeti and Nono is full of fleeting tonal references.’12 ‘All… Tn-types of cardinality 3 [i.e. trichords] have tonal implications… For example, … [IC Set 192, e.g. C, D, Eb] may be heard as either the 1st, 2nd and 3rd degrees of a minor scale or the 6th, 7th and 8th degrees of a major scale, suggesting that either the …[C] or the …[Eb] may be heard as a point of reference. The major-third (4-semitone) interval[-class] embedded within … [IC Set 138, e.g. C, Db, E] suggests that its reference pitch is …[C], regardless of whether the pattern is heard as Neapolitan, Arabic or Flamenco…’ 13 ‘...it is surprising that many pc-set [‘pitch-class set’] theorists tacitly consider all pc-sets a priori to be equivalent or value-free, as if they had no tonal implications – or as if tonal implications did not exist. Can the tonal 10 ‘One can certainly take the view that even with us [Schoenberg, Berg, Webern] there is still a tonic present – I certainly think so.’ Webern, A.: The Path to the New Music, W.Reich, ed., L.Black, transl., (Bryn Mawr, Pennsylvania: Theodore Presser, in association with Universal Edition, 1963(1933)), p.39. 11 Parncutt, R.: ‘Tonal Implications of Harmonic and Melodic T n-Types’ in Mathematics and Computing in Music, T.Klouche and T.Noll, eds. (Berlin: Springer, 2009), pp.124-139. 12 Ibid., p.124. 13 Ibid., p.126.

14 implications that we learn from music simply disappear (which is psychologically implausible), or are they arbitrary (which is psychoacoustically and ethnomusicologically implausible)?’ 14 ...in real music heard by real human beings, pc-sets will always have tonal implications.’15

In other words, if a composer conceives of their harmony in so-called ‘atonal’ or ‘posttonal’ terms – that is, denies or ignores the question of tonal implications in their harmony altogether, or considers that tonal associations only appear within certain harmonic territory, and that other territory remains free of such associations - this will not prevent most or perhaps any listeners from hearing, feeling and attempting to contextualise those frequent unintended or unacknowledged ‘fleeting tonal references’ that inevitably become audible. (Of course, experiences vary according to each listener’s prior musical experience, and psychological disposition.) It would be unreasonable and unworkable for a composer to demand that listeners suppress this part of their perception and cognition, particularly considering that identification of tonal implications tends to induce the impression of understanding the harmony on some level. If the term ‘atonal’, as commonly understood, is unsatisfactory, the same can be said of ‘post-tonal’. The premise behind ‘post-tonal’ is that only certain elements of the kind of harmony in question have tonal implications – and that these can only ever represent isolated instances of decontextualized, post-modern references to a bygone system within a wider musical context in which tonal implications are otherwise supposedly either absent or irrelevant, as in ‘atonality’. Parncutt’s article (2009) is restricted to the tonal implications of trichords, and does not address the question of multiple roots or tonics within a single sonority – i.e. polychords. Once these are considered (the Polychordal Approach), it is possible to identify all the possible hypothetical tonal implications of any given set of pitch-classes. It might have been tempting to conclude that all so-called ‘atonal’ or ‘post-tonal’ vocabulary is therefore in fact theoretically ‘polytonal’, albeit to greatly variable levels of intelligibility in practice, depending on how clearly any given vocabulary is presented, in polychordal terms. However, the term ‘polytonal’, as generally understood, implies that each of the multiple tonalities 14 Ibid., p.126. 15 Ibid., p.126-7.

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present in any one sonority is clearly sustained over a reasonably extended period, as in Milhaud or Stravinsky – e.g. one strand of the texture might proceed in G major over 16 bars, with another strand proceeding in C# minor. By contrast, in atonally/post-tonally conceived music, although any given sonority is theoretically analysable in polychordal terms, successive sonorities will normally outline very different tonalities. There might, at best, be an element of, say, G major, somewhere to be found within two or three adjacent sonorities, but the sense of any textural strand remaining ‘in’ G major will generally vanish almost as soon as it has appeared (and provided that the spacing of the IC Sets has allowed it to become relatively unambiguously discernible in the first place, whether by accident or by design). All we are left with, then, are the ‘fleeting tonal references’ that Parncutt refers to. Schoenberg, of course, preferred the term ‘pantonality’ to signify this type of harmony, borrowing the term from Réti, and either ignoring or misreading Réti’s original definition. Schoenberg’s personal definition of pantonality signified ‘the relation of all tones to one another.’16 Certainly, since the Polychordal Approach aims to maximise the audibility of multiple latent tonal centres in any conceivable pitch-class context, the term ‘pantonal’, which could in theory etymologically be interpreted as simply suggesting the hypothetical presence of any tonalities in any possible harmonic context, might ostensibly seem a viable upgrade on problematic terms such as ‘atonal’ or ‘post-tonal’. In that sense, one might be tempted to apply the term ‘pantonal’ to any music composed using any conceivable IC Set vocabulary – particularly, perhaps, my own music, which is explicitly designed to maximise the audibility of tonal implications of all IC Sets. But this would match neither Réti’s original definition nor Schoenberg’s attempted re-definition. There are points in common with Réti, but Réti also asserts that pantonal music requires tonal dissonances to resolve contextually, whereas so-called ‘atonal’ music (which in Réti’s terms would include music written using the Polychordal Approach) does not: ‘… in the specific atonal sphere the [tonal] dissonances appear without being identified as [contextual] dissonances, as though there were no tension, no ‘longing to be resolved’ inherent in them. But in the ‘pantonal’ utterances of our time, which at their face value may appear just as full of [tonal] dissonance … we see quite a different tendency… [Pantonality] is based on constant tensions, i.e. [tonal] dissonances which, however, are constantly resolved – yet whenever a tension seems to abate, a new dissonance is again at work.’17 16 Schoenberg, A: Theory of Harmony, transl. Carter, R., (Berkeley and Los Angeles: University of California Press, 1978 (1911)), pp.432-3.

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Of course, in Schoenberg’s so-called ‘atonal’ music, for which he preferred the term ‘pantonal’, tonal dissonances do not need to resolve. But Schoenberg’s re-definition of ‘pantonality’ also implies that although all tones are related to one another, tonal implications no longer exist: ‘Even a slight reminiscence of the former tonal harmony would be disturbing, because it would create false expectations of consequences and continuations. The use of a tonic is deceiving if it is not based on all the relationship of tonality.’18

Any attempted appropriation of the term ‘pantonal’ to describe the kind of harmony in question would therefore require a third separate definition, cherry-picking certain elements from both Réti and Schoenberg, and discarding other elements. Further confusion would ensue, regarding the very different harmonic practices of composers such as Debussy and Wagner, for whose music Réti’s original definition was tailored. In short, the types of harmony created through the Polychordal Approach cannot be accurately described by any of the terms ‘atonal’, ‘post-tonal’, ‘pantonal’ or ‘polytonal’. Indeed, it could be argued that none of the four terms accurately describe the types of harmony that they purport to describe. (Just as ‘pitch-class sets’ are not sets of pitch-classes, and dissonances are not necessarily dissonances, in tonal and contextual terms respectively.) Much musical terminology is based on assumptions that appear to hold universally, but in truth only prove useful for certain styles or traditions. From time to time, existing assumptions are questioned, rules are updated or broken, and new styles evolve – but in certain cases, the contradictions, inconsistencies and fallacies behind the original definitions of certain terms remain. These can present formidable obstacles to a musician attempting to break new musical ground, and then accurately explain what they are doing, using language that other musicians can understand. Perhaps any musician whose practice happens to genuinely redefine what is musically possible is in a position to suggest that, besides introducing new terms, certain existing terms perhaps ought to be redefined, as and where necessary. In that respect, I have every sympathy for, say, Schoenberg’s attempt to hijack the term ‘pantonal’ for his own ground-breaking purposes, although I cannot agree with his attempted re-definition. 17 Réti, R.: Tonality, Atonality, Pantonality (London: Rockliff, 1958), pp.72-73. 18 Schoenberg, A: Style and Idea (London: Williams and Norgate, 1951), p.108.

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CHAPTER 4 The Polychordal Approach

The Polychordal Approach is founded on the following hypothesis: 1. Within equal temperament, each of the 350 possible Interval-Class Sets, and each of the 4,096 (or 212) possible sets of pitch-classes, without exception, is not only analysable as either polychordal (in most cases) or straightforwardly modal, but also, crucially, perceptible as such. In other words, any set of pitch-classes contains one or more latent tonal centres. 2. Therefore, if a composer presents the tonal centre(s) of a given set of pitch-classes with sufficient clarity, most if not all listeners will sense one or more harmonic anchors for that sonority.19 3. The same holds true for any conceivable succession of sets of pitch-classes. I.e. provided that a composer consistently maximises the audibility of polychordal connections, any conceivable succession of sets of pitch-classes can be rendered harmonically intelligible, and therefore harmonically meaningful to most listeners. To illustrate point 1, let us examine a relatively tonally ambiguous or so-called ‘atonal’ set of pitch-classes - [E, F, F#, G, C, Db], belonging to IC Set 111513. This set of pitchclasses contains numerous clearly diatonic subsets, including:    

Gb 7th [Gb, Db, F] C major [C, E, G] th F# 7 [F#, C#, E] th F9 [F, C, (E), G]

19 There has been considerable debate among musicologists and composers concerning whether one can, in fact, perceive more than one simultaneous tonal centre. Van den Toorn (1983), Babbitt (1949), Hindemith (1942), Baker (1983) and Forte (1955) have argued that only one tonal centre can be perceived at any one time. Tymoczko (2002), Taruskin (1990), Milhaud (1923), Stravinsky, and Thompson and Mor (1992) all claim the contrary. Perhaps the most persuasive cases have been made by Wolpert (2000) and Hamamoto, Botelho and Munger (2010), whose research suggests that some listeners can perceive multiple tonal centres, and others cannot. As a rule, listeners familiar with Western Art Music are more likely to perceive several tonal centres simultaneously.

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 

Db major 7th E minor 9th

[Db, F, C] [E, G, F#]

It is easy enough to present the set of pitch-classes [E, F, F#, G, C, Db] vertically as a polychord – i.e. as a chord in which two or more of the possible diatonic subsets are clearly audible. There are many possible solutions. The first five of the six solutions below involve clear registral separation of the two or three diatonic units. The sixth is perhaps a little more ambiguous, as the registers are interwoven. Nevertheless, the sixth chord is also clearly audible as a polychord. The first two chords occur in The Art of Thinking Clearly, to be analysed in Chapter 5.

Where two or three diatonic subsets are emphasised in this way through spacing, the tonal centres of each of these subsets are then clearly perceptible as rival, simultaneouslysounding tonal centres. Furthermore, of these tonal centres, one often dominates. In the first of the six chords above, it is reasonable to expect that most, if not all listeners will perceive C major more strongly than Gb7 – and while some listeners may only perceive C as the tonic, others may also perceive Gb as a rival, if weaker tonic. 20 In the second chord, due to both the greater registral proximity of the two subsets, and the C major subset shifting to the first inversion, C major will still normally be perceived as the stronger tonal centre, but rather less emphatically than in the first chord. In all cases above, the lowest tonal centre happens to be the strongest – F#7 for the third and fourth chords, F9 for the fifth, C major for the sixth. I maintain, furthermore, that where one pitch is clearly perceived as tonally stronger, all of the other pitches in the sonority can also be heard, to some extent, modally in relation to it. 20 See Hamamoto, Botelho and Munger (2010).

20

For example, the F in the second chord above can be heard by an astute listener not only as the 7th of Gb major or minor, but also simultaneously as the 4 th of C major. The top Db can be heard not only as the 5th of Gb major or minor, but also as the flattened supertonic of C, and the Gb as a blue note in C, as well as a rival tonic in its own right. Indeed, if any one of the constituent elements of any conceivable non-diatonic set of pitch-classes were arbitrarily assigned as a hypothetical tonic, each of the other pitch-classes could be defined in relation to that tonic according to either a Western or non-Western mode, or else a hybrid of two existing modes. In the case of [E, F, F#, G, C, Db], the presence of three consecutive semitones prescribes hybrids in all cases - but whichever arbitrary tonic one assigns, there are numerous hypothetical solutions. Here are solutions for each of the 6 possible tonics within the set. Each solution is a hybrid of a Western scale and a Hindustani Thaat:

21

To give a more extreme example, the set of pitch-classes [C, C#, D, D#, E, F, F#, G] could, for example, be construed as a blend of part of the C major scale [C, D, E, F, G] and C Todi [C, Db, Eb, F#, G]. If a composer were to desire to present that particular set of pitchclasses in those specific terms, and were sufficiently skilled, all pitch-classes within the set could then be rendered audible in terms of their relation to the tonic C. Since all possible IC Sets - and so theoretically all sonorities within equal temperament possess latent tonal centricity, it should therefore be theoretically possible to present sequences including any pitch-class vocabulary in the form of readily identifiable harmonic patterns. Metaphorically, if one clearly perceives a sequence of harmonic locations, one can then infer the direction of movement between those locations, and so follow a harmonic journey. Any such harmony then effectively becomes ‘functional’. I hold that the term

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‘functional’ can potentially can become applicable to other existing or future types of harmony, besides conventional Western tonal grammar – provided that the new types of harmony also appear to ‘function’, i.e. possess the following: 1. Clearly perceptible harmonic centricity throughout a piece of music – metaphorically, as a series of harmonic locations. 2. The presence of harmonic movement between various locations. 3. Discernible patterns governing this movement. The harmonic conventions we hear in operation in the Western tonal tradition create the illusion of purpose, simply because the patterns and the inherent internal logic are so transparent. Shermer (2011) has coined the term ‘agenticity’ to describe ‘the tendency to infuse patterns with meaning, intention and agency.’ 21 If we perceive patterns clearly, they appear to ‘function’, to have a purpose, but this is, strictly speaking, illusory. Traditional Western tonal harmony satisfies the three above criteria more thoroughly than any other harmonic system yet devised, and is therefore generally regarded as the only possible embodiment of ‘functionalism’ in harmony. However, it does not follow that other types of harmonic locations and trajectories cannot be made to seem equally purposeful, and therefore ‘function’ in a distinct, yet equally compelling way. In Chapter 5, I shall analyse both my attempts to clarify harmonic locations via the Polychordal Approach, and my use of serialism to create patterns. Of course, legions of critics from Hindemith (1952) to George Benjamin (quoted in Nieminen and Machart, 1997) have maintained that the kinds of patterns created by serialism will always defy aural comprehension – a straightforward position, which in almost all cases seems to hold true. However, the Polychordal Approach allows serialism to operate in a completely different, far more audible way, as will be demonstrated in the following pages. The resultant music is both centric, and in its own way, ‘functional’.

21 Shermer, Michael: The Believing Brain (New York: Times Books, 2011), p.87.

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CHAPTER 5 THE ART OF THINKING CLEARLY: A POLYCHORDAL AND SERIAL ANALYSIS

Of my recent compositions, The Art of Thinking Clearly (2013) serves the most usefully as an introduction to the Polychordal Approach. Each of my recent works covers distinct harmonic ground, and employs one or more devices not found elsewhere in my work. But the combination of territory and techniques covered in The Art of Thinking Clearly enables a clearer exposition more of the key concepts than would be possible via discussion of Partita (2013), Prosthesis (2014), Two Sketches (2014) or Velvet Revolution (2014).

Example 1: The tone-row used in The Art of Thinking Clearly All of the pitch-class material in The Art of Thinking Clearly derives from a single 12tone row. An analysis of any row’s constituent IC Sets will reveal which portions of the spectrum between consonance and dissonance it employs. The following example shows each of the dyads, trichords, tetrachords and pentachords formed by adjacent elements of row for The Art of Thinking Clearly:

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Of course, one could draw up similar charts for all possible hexachords, heptachords, etc. But from the smaller IC Sets identified above, we can already identify the general trend, in terms of levels of tonal consonance or dissonance, with reference to the chart produced in Chapter 2. The following chart demonstrates that the row for The Art of Thinking Clearly uses solely the most tonally dissonant portion of the available spectrum. The larger the IC Sets in question, the greater the concentration of tonal dissonance. This is readily apparent, visually: in the tables of IC Sets comprising more than 2 elements, the relatively tonally consonant ends of the spectrum simply do not feature. Of the possible trichordal territory, the 7 least tonally dissonant IC Sets - 37% of the available spectrum – never occur. In tetrachordal terms, this grows to 42%. In pentachordal terms, the figure is 58%.

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Naturally, the inversion of this row – frequently used throughout the piece - would produce a similar-looking chart, because any two inversionally-related IC Sets will always lie on the same rung of the tonal consonance-dissonance spectrum. Throughout The Art of Thinking Clearly, therefore, most IC Sets are tonally dissonant – with occasional, notable exceptions deriving from certain conscious choices concerning the development of this central material. Of course, as mentioned in Chapter 2, the level of tonal dissonance of an IC Set is only one of several parameters contributing to the perceptible tonal dissonance of an actual sonority. At a later point in my research, I anticipate that I will be able to demonstrate that the Polychordal Approach tends to result in vertical spacings which tend to generally reduce levels of audible tonal dissonance. But at this stage, since the pitch-class content of The Art of Thinking Clearly is, for the most part, very dissonant, it already follows that – besides certain obvious general expressive consequences - this material will have posed me greater challenges, on the whole, than more consonant material would have done, given that my aim is to test whether the Polychordal Approach can successfully facilitate perception of any conceivable harmonic vocabulary, and trajectory, within equal temperament. If the harmony audibly ‘functions’ in a cognitively

26

transparent manner, consistently throughout The Art of Thinking Clearly, this would suggest that the Polychordal Approach is also likely to succeed in its aims, when applied to more consonant material – given that such material tends to be more tonally explicit.22 We can now analyse the workings of the Polychordal Approach more closely.

Example 2A: Pentachordal Cycle, bars 1-7, polychordally conceived The Art of Thinking Clearly opens with a 36-bar harmonic arch. In quasi-Schenkerian terms, there is a clearly perceptible tonal shift from B (firmly established in bars 1 to 7) to F (a quasi-cadential resting-point in bars 33-36). This shift is achieved through a combination of two factors: polychordal vertical spacing (the Polychordal Approach), and a cyclical serial process, moulding the horizontal harmonic movement. To begin with, let us examine the polychordal spacing in bars 1-7, itself one of several harmonic mini-cycles within the wider, 36-bar harmonic arch. In the example below, I have divided each chord into two or more sub-chords, to illustrate the presence of multiple tonal centres in each sonority. I have shown what I perceive as the strongest tonal centres within each chord in bold, but I am more than happy to acknowledge the probability that some 22 There are, however, two significant exceptions. Of the more consonant IC Sets, 444 (the augmented triad) and 336 (the diminished triad) are problematic in this regard: whilst they are relatively tonally consonant (according to Plomp and Levelt’s criteria), they are also inherently tonally unstable (in contextual terms). As noted in Chapter 2, of the 8 tonally consonant IC Sets, 444 is the only one never to have served as a contextual consonance in the history of Western tonal grammar. The inherent instability of 444 and 336 would appear to explain why, in the course of composing Partita, in which the available 19 trichordal IC Sets were spread evenly over two tone-rows which generated most of the pitch-class material, 444 and 336 became notably under-represented in practice: subconsciously, I was avoiding this harmonic vocabulary to some extent. The underlying harmonic aim of my next work - Prosthesis - was, therefore, to explore the tonal implications of 444 and 336 – in order to test the effectiveness of the Polychordal Approach on this more tonally ambiguous material. Elsewhere, it is remarkable that none of Babbitt’s ostensibly ‘all-trichordal’ rows actually contain either of these trichords – the other 17 trichords (in Forte’s classification, only 10) are always present, but 444 and 336 never feature at all. These so-called ‘all-trichordal’ rows are employed in many of the works from Babbitt’s middle period, including Paraphrases (1979), Dual (1980), Ars Combinatoria (1981), the Piano Concerto (1985), Lagniappe (1985), The Crowded Air (1988) and Consortini (1989). Mead (1994) claims that the omission was made because, unlike other trichordal IC Sets, 444 and 336 ‘cannot be ordered to represent the four classical transformations [original, inversion, retrograde, retrograde inversion] unambiguously, and the latter [336] is the single trichord type that cannot generate an aggregate [of all 12 pitch-classes].’ (p.155-6). Perhaps, consciously or otherwise, the inherent tonal (rather than serial) ambiguity of 444 and 336 may have played a greater part than Babbitt would ever have admitted, or that Mead would ever admit.

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listeners will in some instances perceive other sub-chords more strongly than I do, including sub-chords not shown below. I do not specify 1 st inversions, 2nd inversions, etc, partly because these will be self-evident, and partly to avoid unnecessary clutter.

The polychordal content of this 8-chord extract may be summarised, chord by chord, as follows: 1. The tonal centre of the first chord is emphatically B. Take away the top Eb, and we have a straightforward B minor 9th chord. Respell the top Eb as a D#, remove the D, and this time we have a B major 9th – both sub-chords are anchored by the same pitch-class, B. 2. I perceive the strongest tonal centre in the second chord as an E major 7 th, with the 4th (A) included, but there is an additional, rival tonal centre: the lowest 3 pitches form a Bb major 7th.

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3. The C major triad included in the third chord establishes C as the prevailing tonal centre, with the F and F# perceptible as the 4 th and sharpened 4th within a C major tonality. A faintly perceptible secondary tonal centre is an F# flattened 9th, given that F# is the lowest note of the chord. 4. In the 4th chord, B major is again the overriding tonal centre, except this time in the 2nd inversion (the first chord was in the root position). The C# is perceptible as a 9 th; the C natural as a flattened 9th. I am tempted to hear a secondary sonority, with an Eb, Gb and C over a non-existent Ab – possibly because I have internalised the harmonic cycle enough to anticipate the Ab in the following sonority. Whether any other listeners would hear this possible secondary focus is another matter. Parncutt (2009) and other commentators certainly acknowledge the cognitive possibility of a listener hearing a tonal centre that does not actually sound in the chord. 5. The fifth chord is the most tonally ambiguous, with at least five tonal sub-chords, and at least four plausible tonal centres. The highest four pitches could be heard in terms of either a Bb major 7th (+ 4th) or an Eb flattened 7th (+4th). The low A could also be cognised as a sharpened 4th in Eb, suggesting that Eb is perhaps the strongest tonal centre – but this is far from clear-cut, to my ears. Ab and D are other plausible tonal centres. 6. The sixth chord might be heard in terms of an overriding F minor 9 th – but to some listeners, perhaps the same pitches might appear to be anchored around Bb. One could also sense the lowest four pitches as the first four notes of the Todi Thaat, starting on E. 7. C# major would seem the prevailing tonal centre in the seventh chord, with the B and C (enharmonically B#) perceptible as the 7th and sharpened 7th respectively. 8. The eighth chord is the same as the first. It is audible as a quasi-cadential resting point – a return to a tonic established in the first chord. Another notable feature of this chord sequence is that each chord contains one or two pitch-classes in common with its adjacent chords. These common notes are in most cases registrally fixed, thus acting as pivots between successive sonorities. This voice-leading decision was intended to further clarify the harmonic movement. But besides presenting each successive chord as explicitly as possible, in polychordal terms, and clarifying the voiceleading as much as possible, there is another contributing factor to the harmonic cohesion of this opening chord sequence – one that helps to explain why the return to the first chord in bars 5-7 feels like a perfect cadence.

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Example 2B: Pentachordal Cycle, bars 1-7, serially conceived That factor is the serial mechanism behind the pitch-class content. This claim might seem contentious and improbable, certainly to advocates of Lerdahl’s dictum that serialism causes ‘cognitively opaque’ harmony (and it frequently does exactly that). Furthermore, most serial devices are not directly intelligible to listeners. But this does not render all serialism esthesically meaningless, if the composer is sufficiently skilled. The serial mechanism in operation here governs all harmonic movement from bars 1 to 36. The following table summarises the content of each successive chord until bar 6, in serial terms:

Element Pitch-Class Chord 1 Chord 2 Chord 3 Chord 4 Chord 5 Chord 6 Chord 7 Chord 8

1 C# x x x x

2 B x x x x

3 Eb x x x x

4 D x x x x

5 A x x x

6 Bb

7 Ab

8 E

9 G

10 F

11 F#

12 C

x

x

x x

x

x

x x

x x

x

x

x x

x

x

x x

x x

x

Each successive pentachord shifts along the series, but retains alternatively either one or two pitch-classes from the previous chord. With each new group of pitch-classes comes a corresponding shift in tonal centres; given sufficiently clear polychordal spacing and good voice-leading (including, in this case, pivot notes), these tonal shifts will appear audibly coherent. But besides hearing new harmonic ground with each new chord, one can sense, from Chord 4 onwards, that the sequence of tonal centres heard in Chords 1-3 is now being developed – approached again, but with some changes - as shown by the blue lines and red crosses. The second blue line, through Chords 5, 6 and 7, traces through similar harmonic territory, in a similar sequence to Chords 2, 3 and 4: in pitch-class terms (and therefore tonal terms), Chord 5 is quite similar to Chord 2, Chord 6 is quite similar to Chord 3, and Chord 7 is quite similar to Chord 4. Consequently, following Chord 7, we expect something like Chord 8. The satisfaction of hearing Chord 8 is enhanced further, because:

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1. Chord 8 is identical to the Chord 1; one senses that a cycle has been completed. 2. The tonal centre is unambiguously B. 3. B seems to be recurring at fairly regular intervals: in Chord 1 (root position), then Chord 4 (2nd inversion), and now returning decisively in Chord 8. 4. The lowest two pitches in Chord 8, B and C#, are carried on from Chord 7, boosting their tonal weight. Of course, the success of this type of mechanism depends entirely on whether the tonal centres of each chord can be rendered sufficiently audible in the first place. The following example shows the tonal and serial movement, side by side:

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Example 2C: Full Pentachordal Cycle, bars 1-36 In theory, the serial cycle begun in bars 1-7 would have lasted for 24 chords, consisting of three sub-cycles of the kind demonstrated above, in which the first chord is identical to the eighth. The entire sequence, in terms of elements 1-12 of the tone row, would have proceeded as follows:

Sub-cycle 1 Sub-cycle 2 Sub-cycle 3

Chord 1 1-5 5-9 9-1

Chord 2 4-8 8-12 12-4

Chord 3 8-12 12-4 4-8

Chord 4 11-3 3-7 7-11

Chord 5 3-7 7-11 11-3

Chord 6 6-10 10-2 2-6

Chord 7 10-2 2-6 6-10

Chord 8 1-5 5-9 9-1

In practice, I chose to set up the chord consisting of elements 9-1 of the row as a quasidominant (F being the unambiguous tonal centre in this case), with the first chord – elements 1-5 – serving as a quasi-tonic (anchored on B). I then appended the quasi-dominant to the end of sub-cycle 2, as the next chord in the sequence, repeated sub-cycles 1 and 2 before finally presenting sub-cycle 3, and completing the cycle. In quasi-Schenkerian, quasi-tonal terms, the movement from B to F is thus delayed, with the chord comprising elements 10-2 functioning as a transitional sonority, comprising two tonal centres – B and F. There are brief

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flourishes, in the form of chord-multiplications; otherwise the harmonic sequence simply spins round:

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34

35

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Example 3A: Chord Multiplication in Appalachian Spring Several serial techniques are inherently polychordal. That is: by their very nature, these serial devices can present multiple tonal centres, simultaneously and/or successively, and these tonal centres are readily discernible to listeners if skilfully handled by the composer. One such technique is chord multiplication, developed by Boulez and others. Chord multiplication involves the superimposition of multiple transpositions of an initial chord. If the initial chord to be multiplied is spaced so as to maximise tonal intelligibility, and then transposed at intervals conducive to clear discernment of the multiple tonal centres that result, chord multiplication can often yield vast, highly complex sonorities whilst retaining harmonic intelligibility. The spacing of the transpositions of the original cell is crucial, if the resultant polychordal structures are to remain aurally coherent. The best way of achieving this is to ensure that the transpositions of the initial chordal cell do not overlap registrally – allowing the ear to pick out the multiple presentations of that cell (and tonal entity) with relative ease. The following is a particularly transparent example of chord multiplication, albeit in a tonal, non-serial piece:

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The transpositions of the initial Ab major cell could scarcely have been rendered more aurally transparent. If one were to present these pitches as a block chord in the manner shown above (‘Harmonic Aggregate’), the highest nine pitches would certainly still articulate the three relevant tonal centres in an aurally coherent and satisfying manner. If the low Ab were then moved down an octave, the same could be said of the entire chord. If we consider a hypothetical alternative, in which Copland had allowed the transpositions of the original Ab major cell to overlap registrally, the results would have remained tonally intelligible to the listener, so long as the pitches were presented melodically and not as a chord. The following hypothetical example demonstrates that, with the registral overlap and a block chord presentation, any sense of tonal coherence would disappear, despite the harmonic clarity of the original cell:

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In order to render the block chord shown above tonally intelligible to the listener, the composer would need to separate out the various transpositions of the Ab major cell registrally.

Example 3B: Chord Multiplications in The Art of Thinking Clearly, bars 378-408 If a composer chooses to apply chord multiplication to the broadest possible range of IC Sets, the original chordal cell to be multiplied will normally be rather less tonally straightforward than the simple major triad multiplied in Example 3A. In most such cases, besides ensuring a clear polychordal spacing of the original cell, registral separation of the cell’s transpositions is even more important, if the results are to be harmonically intelligible. In The Art of Thinking Clearly, and in other works, I typically have the highest note of a cell as the lowest note of an adjacent transposition, and vice versa. If the initial cell is judiciously spaced, in polychordal terms, this is often enough to ensure that the multiple tonal centres are presented with sufficient clarity to the listener. Over a series of seven chord multiplications in bars 129-135 of The Art of Thinking Clearly, this device is applied consistently, with no exceptions or changes to spacing: no further work was necessary in this case, to ensure the requisite harmonic cohesion. However, in the sequence of six chord multiplications selected from bars 378-408, shown below, each instance required a certain amount of tinkering, to ensure harmonically satisfactory results. For the purposes of this discussion, bars 378-408 are therefore somewhat more interesting. In the example shown below, besides a breakdown of tonal centres and intervals for each actual harmonic aggregate, I have shown the hypothetical aggregates that would have resulted, had I stuck dogmatically to serial principles throughout, and ignored empirical harmonic reality. Pitches changed from the hypothetical model are shown as diamond shapes, and pitches hypothetically present but omitted in practice are shown as crosses.

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In the 6 chord multiplications shown above, the tonal centres are easy to analyse, given that one original cell-type is then replicated elsewhere – and this is aurally evident. E.g. in the first chord multiplication shown, the lowest ‘Ab 9 + 4’ cell (i.e. Ab, Bb, Db, Eb) is replicated as further ‘9 + 4’ cells higher up. Likewise, the minor 6/5 cells in the second multiplication, the major 9th cells in the third multiplication, the 9ths in the fourth multiplication, and so on. I made three types of change to the hypothetical models. Each of the three kinds of modification originated from the same concern: to maximise the audibility of the multiple tonal centres. The reasoning behind the first type of change is straightforward: low registers normally require wider intervals between pitches, to ensure aural clarity. Therefore, in each case shown above, one of the lowest pitches is moved down an octave. In the fourth and fifth instances, this the second-lowest pitch. In all other cases, the lowest pitch is moved down. I do not always make this type of adjustment: e.g. in the sequence of chord multiplications in bars 129-135 this was never harmonically necessary.

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The second type of change resulted from a desire to avoid octaves, in this instance. Again, this is not always necessary: in those instances, I keep the octaves. For example, over bars 129-135, there are eleven instances of octaves between right and left hand:

In bars 378-408, however, the presence of octaves in the second and fourth chord multiplications would have interfered with clear perception of the multiple tonal centers. In general, where the root notes of these tonal centres are doubled at the octave, as is the case in bars 129-135, the octaves prove aurally satisfying: they reinforce the tonal anchors. In the second and fourth examples shown on pages 36-7, however, the octaves reinforce pitches which otherwise would not have been perceived as tonal centres – consequently, the polychordal spacing would have been rendered more opaque and less aurally convincing, had I allowed the hypothetical models to stand in this particular texture. In serial terms, therefore, I ‘cheated’, and moved certain pitches in the right hand up or down a semitone, as indicated by the diamond-headed notes. Of course, this type of tinkering further increases the tonal complexity of the sonorities in question, but such increases serve merely to add harmonic spice to the chord, rather than upset the balance of tonal centres, as the octaves would have done. The third and final amendment to the hypothetical serial models occurs in the fifth chord multiplication (page 37). In this case, had I kept the highest note of the original cell as the lowest note of its upper transposition, and vice versa, the highest five pitch-classes of the resultant chord multiplication would have matched the lowest five pitch-classes (‘Hypothetical (A)’. This is due to the interval-class distance between the lowest and highest pitch-classes of the original chordal cell, G and Db respectively – evidently, two tritones

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make up an octave. Again, the resultant octaves are not necessarily a problem: in the third chord multiplication (p.34) the two F major 9ths sound perfectly well either side of a B major 9th. In the fifth chord multiplication, the addition of a minor 3 rd between the various transpositions of the original cell simply created a richer harmonic effect. On one hand, these sonorities could only have been conceived through serial thinking: in all cases, the similarities between the actual sonorities and their hypothetical serial models far outweigh the small divergences. However, as soon as the results are not as harmonically convincing as they might be, I opt to break serial exactitude. The overriding concern is with setting up multiple tonal centres, and presenting these audibly and euphoniously. I consistently steer clear of those serial techniques that actively hinder harmonic intelligibility by their very nature. But rather more serial devices can prove helpful in this respect than is commonly thought. Certainly, basic retrogrades, inversions and transpositions can be employed to aid tonal coherence. The same is true of certain types of serial cycle (as shown in Example 2) and chord multiplication (Example 3). Likewise, Messaien’s ‘chords of transposed inversion’ device is essentially serial, and can enhance harmonic clarity, if intelligently used. I have used this technique frequently elsewhere, but it only appears once, fleetingly, in The Art of Thinking Clearly, so discussion of this technique will have to wait until a later point in my research.

Example 4: ‘Krenek Rotations’ in bars 255-265, as perceived tonally The final serial technique used in The Art of Thinking Clearly to be discussed here, likewise, facilitates tonal coherence by its very nature. The term ‘Krenek rotation’ is my own: it was first employed by Krenek in Lamentatio Jeremiae Prophetae (1944). The technique has been widely discussed, particularly with reference to Stravinsky and Boulez, most notably by Wuorinen (1979). One starts with a small melodic series of pitch-classes (often registrally fixed as specific pitches), and rotates the intervals of the series, starting from the same pitch-class on each occasion. Whilst the mechanism itself is well-known, discussions of its tonal implications are uncommon. Julian Anderson’s ‘Harmonic Practices in Oliver Knussen’s Music since 1988: Part I’ (2002) is an exception. Anderson maintains that

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Knussen’s use of Krenek rotations in Flourish with Fireworks (1988) establishes one pitchclass – A – as a tonic:23 ‘Given the importance of the pitch A in… the linear rotations [‘Krenek rotations’] outlined above, this pitch starts to assume the function of a focal point to the harmony, an easily recognizable modal tonic which guides the ear through the many simultaneous complexities of the music’s textures .’

This explanation arguably holds true in Flourish with Fireworks, but does not apply to my use of Krenek rotations in The Art of Thinking Clearly and elsewhere, nor would it apply to other instances of this device in the work of certain other composers, including Stravinsky and Boulez. The mechanism depends on a pivot note, around which the series of intervallic rotations occur. Naturally, since this pitch repeats, it tends to assume a more important harmonic role than most, or all other pitches. But this is not normally enough to establish it as a tonic. The following example shows three Krenek rotations of portions of the retrograde row in bars 255-265 of The Art of Thinking Clearly:

23 Anderson, J: ‘Harmonic Practices in Oliver Knussen’s Music since 1988: Part I’ in Tempo, New Series, no221 (July 2002), p.4. Anderson also holds that Knussen’s use of Messiaen’s ‘Chords of Transposed Inversion’ technique, likewise, contributes to establishing A as a ‘recognizable modal tonic.’ Whilst I agree with Anderson with reference to the use of both of these techniques in Flourish with Fireworks, neither mechanism necessarily always establishes one pivotal pitchclass as a tonic. Far from it! Rather, both devices more commonly establish multiple, rival ‘tonics’ – that is, they are both inherently polychordal devices. I shall demonstrate this with reference to Chords of Transposed Inversion at a later date. In this example from The Art of Thinking Clearly, my focus is solely on Krenek rotations.

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45

Focusing, for the time being, on the three Krenek rotations out of context (‘Serial Mechanisms (2)’), it is easy enough to see how one might succeed in establishing the pivot notes (F#, A and F respectively) as tonal centres, if one so desired. Simply by playing through each of the matrices, spaced as they are under ‘Serial Mechanisms (2)’, and giving a heavy sforzando on each iteration of the pivot note, the pivot note becomes readily discernible as the tonic in all three cases. However, in practice, none of the three pivots in this example function as tonics (except briefly, at the start of Rotation C of the third matrix). Instead, the pivots take on a variety of other clearly identifiable harmonic functions, in three very different ways for each of the three matrices. For this, I am thankful! Were the harmonic implications of Krenek rotations so predictably crude and static in practice, the ear would quickly tire of the device.

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In the first Krenek rotation box in the example above, C functions as a tonic throughout, rather than the pivot note (F#). This happens, despite F# repeating four times, and the C occurring only once. The repeated Gs and Es/Ebs function audibly as the dominant and major/minor mediant respectively; all other pitches, including the F# pivot, are perceived in terms of their relation to C. This is partly due to the structural placing of C within a larger context, partly due to the sforzando, and partly because it lasts longer than the other notes. In the second Krenek rotation box, the perceptible tonal centres shift quickly, with the pivot note, A, assuming a new function each time. Initially, it is heard as the 7 th of B major/minor. In Rotation B, it is heard as the mediant of an F# minor 7 th. In Rotations C and D, it is heard as the mediant of an F major 7 th. Thus, the major/minor 7th sonority of the original melodic cell proves more important than the repeated pivot note in establishing tonal centres. This is a by-product of an intentional spacing decision: were the outline of the initial 7th not so strongly felt, perhaps the repeated As might have begun to sound like tonics – an option I chose not to take in this instance. In the third Krenek rotation box, the upper voices in the right hand radically alter the harmonic perspective of the left hand line. I have only shown some of the more powerful tonal centres, each of which coincides with an occurrence of the pivot-note, F. At the start of Rotation B, the F is heard as the dominant of a Bb major 9 th (with a flattened supertonic). At the start of Rotation C, the F functions briefly as a tonic, as mentioned above. At the end of the example, immediately after Rotation D, the F is heard enharmonically as an E#, operating as the mediant of C# major. This example illustrates that where Krenek rotations succeed in facilitating harmonic clarity, they do so mainly because they audibly transform a distinctive initial harmonic cell – e.g., in the first Krenek rotation box, a 4-note chromatic cluster, and in the second Krenek rotation box, a major/minor 7th. Of course, this can only work if the initial harmonic cell is presented sufficiently clearly. The fact that a pivot note also happens to repeat is normally of secondary importance. To suggest that Krenek rotations establish harmonic cohesion mainly through the banal repetition of a single pitch24 would be to sell the technique short.

24 Anderson does not quite suggest this, but one might easily misinterpret the statement.

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CHAPTER 6 SERIALISM, POLYCHORDALLY CONCEIVED

Serialism was originally conceived as a means of creating order from chaos. In so-called ‘atonal’/‘post-tonal’ music, in theory any sequence of pitches is permissible. The original concept behind serialism was to create patterns, particularly patterns of pitch-classes which hypothetically ought to give the listener a cognitive foothold of some kind, in a context where, otherwise, anything goes. In Schoenberg’s words: ‘Composition with twelve tones has no other aim than comprehensibility.’ 25

But in practice, as Lerdahl (1988) and numerous other commentators have argued, serial patterns are in most cases insufficiently discernible to fulfil this intended function. According to Lerdahl et al., serialism can never achieve the ‘comprehensibility’ that it was originally devised to achieve, particularly in harmonic terms; serialism will always remain ‘cognitively opaque’, as will most, if not all atonally/post-tonally-conceived harmony in general. Meyer (1967) argues much the same thing, with one crucial exception - Meyer acknowledges the possibility of a breakthrough:

‘… the human mind searching for control, through prediction, will discover implicative relationships wherever and whenever a modicum of redundancy exists… if I am correct in contending that it is of the very essence of human behaviour to predict, then it will be no easy matter to invent a series of sounds, no matter how disjunct and disjointed, which cannot be made to serve as the basis for some sort of functionally interpreted inferences. ’26 ‘Although the rules and constraints of the twelve-tone method may help to 25 Schoenberg, A: Style and Idea (London: Williams and Norgate, 1951), p.103. 26 Meyer, L.B.: Music, the Arts and Ideas: Patterns and Predictions in TwentiethCentury Culture (Chicago and London: University of Chicago press, 1967), pp.298-299. [And see my brief discussion of Shermer’s ‘agenticity’ in creating the illusion of functionality, pp.18-19: if we discern any mobile harmonic pattern sufficiently clearly, our brains are wired to interpret the pattern as imbued with ‘function’ and ‘purpose’.]

48 keep the composer from inadvertently writing passages that sound tonal [I partly disagree]… they do not preclude the possibility of functionalism [my italics].’27 ‘… the fact is that the compositional constraints imposed by the rules of serialism are much less comprehensive than those imposed by the traditional constraints involved in the composition of tonal music.’28 ‘The [serial] system is… patently incomplete. That is, it provides no rules for …, …, for choosing pitch register [my italics], …, …, and so on.’ 29

I hold that the Polychordal Approach allows serially-conceived harmony to be heard ‘functionally’. The Polychordal Approach imposes an additional set of rules onto existing serial rules. This new grammar consistently aims at one thing: to clarify whatever harmonic vocabulary might be thrown up by the serial mechanisms. In doing so, the Polychordal Approach allows far greater cognitive transparency than was hitherto possible serially. It arguably completes the rules of serialism – or, more accurately, possibly represents one of several potential means of completion, some of which have yet to be devised. In so doing, the Polychordal Approach is intended to enable serialism to finally achieve Schoenberg’s original aim: ‘comprehensibility’. Evidently, one could also apply the same Polychordal principles to any so-called ‘atonal’ harmony, without using serialism. But in my own experience, serialism and Polychordality complement one another beautifully. If Polychordality seems to complete the rules of serialism, serialism also seems to complete the rules of Polychordality. I am loath to break the symbiotic relationship. Although I have occasionally strayed from serialism in a few passages from works such as Partita (2013), Dectet (2008) and Bagatelle (2007), I generally prefer to keep using both, side by side.

27 Ibid., p.242. 28 Ibid., p.240. 29 Ibid., p.303.

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