Cyclical Structures and Linear Voice-Leading in the Music of Ivan Wyschnegradsky
In 1972, La Revue Musicale published an extensive article by Ivan Wyschnegradsky entitled "Ultra-chromatisme et Espaces non-octaviants." In this publication, Wyschnegradsky brought fourth his rational system for composition with micro-tonal division of the pitch spectrum. In this paper, I will show how Wyschnegradsky's methods stem from, and rely on intrinsic properties of the equal-tempered system; i.e. cyclical movements and linear voice leading. Observations and examples will be taken from Wyschnegradsky's Étude sur les Mouvements Rotatoires op. 45, written in 1961 and cited in the Revue Musicale article.
The Cyclical Phenomenon: The Equal-Tempered System
Any one who has ever wondered about the placement of sharps and flats in tonal key signatures has seen the importance of the circle in tonal music. With adoption of equal temperament came the possibility of complete circles of intervals; successions of same intervals that would return cyclically to the original pitch class.
This phenomenon, particularly as it applies to the interval of a perfect fifth, was to shape and govern the evolution of musical language in a drastic way. The system of keys; following rising and falling circles of perfect fifths with C as starting point, the tonic-dominant relationship, applied or secondary dominants, the relative distance of one key to another, are but some of the concepts of tonal music based on the circle of perfect fifths. Through closer examination, we find that these circles can be either complete or incomplete, depending on their cyclical interval. A complete circle goes through all possible pitches of the system before starting the next period. An incomplete circle has a periodicity, smaller than the total number of pitches in the system.
Thus, the semi-tone, with its inversion and extension the major seventh and minor ninth as well as the perfect fifth and its inversion the perfect fourth generate complete circles. All other intervals in the semi-tonal system give rise to incomplete circles (the whole-tone scale, the full-diminished seventh chord, the augmented triad, etc.).
It is important to note that these principals are not invented theories but rather, they are direct consequences of the symmetrical construction of the equal-tempered chromatic scale. These passages from "Ultrachromatisme et espaces non-octaviants" illustrate Wyschnegradsky's acknowledgement of this fact and its importance for his eventual organization of microtonal space.
With the adoption of equal temperament, the tonal continuum was de facto transformed into a sonorous medium. But more than a century was needed for this this transformation to penetrate (our) consciousness and for the equal spacing of the tones to give birth to the sense of their equivalence. Progressive chromatization of the musical language, the more and more extended use of tonal ambiguities arising from the ambiguity of the (equal) temperament gapped the foundation of tonal order until the beginning of the twentieth century accomplished the awareness of the equivalence of (the) twelve tones.
I had spoken of the choice which I had made among all of the possible ultrachromatic mediums, restricting myself to the tonal mediums which are multiples of 12, including 24 quarter tones, 36 sixths of tone and 72 twenfths of a tone. In making this choice, I was guided by the desire to rely on the legacy of a thousand years of perpetual evolution, that is to say, on the sonorous medium of 12 tones and also to limit myself to the most simple and easily realizable. (editor's translation)
It is clear that Wyschnegradsky saw ultrachromaticism as an extension of the equal-tempered system and went about the task of labelling, categorizing, and notating these "extensions". As he demonstrated in the Manuel d'harmonie quarts de ton and later in the "Ultrachromatisme et Espaces non-octaviants" article, all this could be done using cyclical properties of the equal-tempered semi-tonal system as a model or better yet, as a mold.
The Manuel d'Harmonie Quarts de Ton
Nowhere is Wyschnegradsky's adherence to tradition more evident than in the Manuel d'harmonie quarts de ton, written in 1932. Here, the quarter-tone system is presented in the logical point-format, common to all "rudiments" books of the time.
Every aspect of the "new system" is presented as an extension of conventional semi-tonal theory. The notation ( etc.) is sketched on the original alterations. The intervals bear related names; (plus que majeure, moins que mineure, etc.). Concepts of ornamentation, non-structural tones, chords, triads, scales are presented much in the same way as they would in semi-tonal theory.
However, one fundamental concept, much neglected in the teaching of functional harmony in semi-tones, is treated quite extensively here; the concept of cyclical movements. It is here that we may see the most fundamental similarity in the construction of both systems. All complete (semi-tonal and micro-tonal) cycles presented by Wyschnegradsky are symmetrical subdivisions of complete cycles in the semi-tone system. For example, the circle of secondes-tierces is nothing but two circles of perfect fourths, interlocking at the interval of five (5) quarter-tones, thus subdividing the initial fourth in half.
The same is true for all incomplete circles as well. Furthermore, the key to understanding the principals of non-octaviant spaces, which would be shown in 1972, lies in this same trait, and also has roots in the equal-tempered semi-tonal system.
This key lies in the fact that scales can be derived from segments of complete circles of intervals by regrouping the pitch classes in the same register. The C major scale, for example, is obtained by regrouping the pitch classes of an ascending circle of perfect fifths (or descending circle of perfect fourths) starting on the sub-dominant; i.e. FCGDAEB = CDEFGAB.
New scales can hence be constructed by applying the same principal to new, extended circles; eg. the tridecatone-quasi-diatonic scale, which is based on a compression of a descending circle of major fourths (symmetrical middle of the major seventh). cf. Figure 1.
Note that this circle, like all extended circles, is nothing more than two circles of major sevenths, interlocking at the interval of eleven (11) quarter-tones. Hence, half the resultant scale is in conventional pitches and the other half is in quarter-tonal pitches.
Later on, in the "Ultrachromatisme..." article, Wyschnegradsky would refer to this cyclical structure as a binary structure of "regime 11".
I have spoken of the ultrachromatic system as being an extension of the chromatic. This extension, however, is not without structural advantages. Actually, the germ at the base of all Wyschnegradsky's non-octaviant spaces is inherent in a structural difference between chromatic and ultrachromatic systems.
This difference is slight but significant and has to do with symmetrical division of the system's intervals. If we examine the semi-tonal system, we find that the only intervals capable of generating complete circles, (the minor second, the perfect fifth and their inversions), can not be symmetrically divided without introducing ultrachromatic pitches. The only intervals that allow symmetrical subdivision are by their nature non-cyclical; i.e. they generate incomplete circles.
The ultrachromatic system gives rise to symmetrical division of the cyclical intervals. It is through this slight but all-important difference that the quasi-innumerable symmetrical circle-structures of Wyschnegradsky's sound world are born. A quick examination of Wyschnegradsky's "Tableau des structures de base de volume et de densité diverses," with a mind to the continuing or cyclical nature of its segments shows this basic fact.
In regime 11 for example (1st horizontal line), the binary, ternary, and sexanaire structures are symmetrical, that is to say, they show circles of major sevenths interlocking at equidistant intervals. However, Wyschnegradsky's self-imposed limit of the twelfth of tone as the smallest aurally discernable interval brings about the quasi-symmetrical structures (4,8,9,12) which still retain a degree of symmetry in their circles of major sevenths, now interlocking at approximate "center" points. cf. Figure 2 above.
With the notion of perfect (symmetrical) and imperfect (quasi-symmetrical) structures, it is possible to imagine the "intra-polation" of Wyschnegradsky's structures back to the semitonal system. In the case of regime 11, by imposing the semi-tone as the smallest possible interval, all structures become imperfect. cf. Figure 3.
Figure 3: Imposition of Semitone as Smallest Unit in Regime Onze
Looking at these structures as quasi-symmetrical circles shows the basic similarity and obvious relationship between the chromatic and ultra-chromatic systems. cf. Figure 4.
Figure 4: Comparison of Quasi-Symmetric Circles of Chromatic and Ultrachromatic Systems
These cyclical constructions are inherent in the equal tempered system. The ultrachromatic subdivision of the equal-tempered system results in a higher degree of symmetry within these cyclical structures. As we can see in most of Wyschnegradsky's music, the extension of equal temperament permits more inter-play between symmetrical and assymmetrical constructions. Furthermore, the similarities between ultrachromatic and chromatic systems shown above permit interaction between all levels of chromaticism within the total-ultrachromatic.
However, these concepts alone fall short of giving a clear picture of Wyschnegradsky's compositional language. In order to do so, we must examine another concept of traditional composition to which Wyschnegradsky adhered very strongly; the concept of linear voice-leading.
Linear Voice-Leading: The Importance of Oblique Motion
In the history of music, harmony has evolved as the simultaneous sounding of independent melodic lines. From the early polyphonists through Palestrina, Bach, Beethoven, Brahms, Schoenberg, Scriabin, etc...the emphasis has oscillated from diatonic to chromatic back and forth several times, but the basic concept of voice-leading has remained unchanged.
The underlying principle governing the concept of voice-leading is the independence of the voices. This independence of voices brings about change in vertical sonorities in which some voices move while other remain static. This "oblique" motion can be found throughout musical history and transcends concepts of modality, tonality, expanded tonality, atonality, polytonality, etc. It is therefore not surprising to find in Wyschnegradsky's non-octaviant spaces, adherence to the concept of oblique motion.
The Need for Homologous Positions
All the cyclical structures examined so far were shown at the initial position of the system; i.e. non-transposed. Wyschnegradsky explains in his Revue Musicale article, the possibilities of transposing any cyclical construction to "as many positions as there are spatial units within the system"; the spatial unit being the chromatic or ultra chromatic type-interval needed for symmetrical subdivision of the "cyclical" interval.
In other words, these cyclical structures, like all symmetrical structures, are of limited transposition. Each perfect (symmetrical) structure has a finite number of positions, numbered in relation to the conceptual starting point which is Eb. These different positions being identical in construction within one structure, share no pitches and render oblique motion between adjacent positions impossible.
It is now obvious that, inherent in the symmetrical make-up of these structures, is the need for non-symmetrical or quasi-symmetrical, imperfect structures. These structures, known as homologous positions, will enable oblique motion in the voice-leading between perfect structures. This creates the possibility of horizontal movement for these purely vertical constructions.
In effect, the homologous positions of any given structure in any regime arise through staggered movement between perfect, symmetrical positions. cf. Figure 5.
Figure 5: Regime 13, Binary Structure
It is interesting to note that Wyschnegradsky demonstrates oblique motion between imperfect structures in his discussion of different regimes in non-octaviant space but fails to show the most essential oblique movement between perfect structures. It is voice-leading that generates most of Wyschnegradsky's imperfect positions in binary, ternary and quaternary structures. Other types of oblique motion, in ternary and four-part (quaternary) structures, bring about a greater array of imperfect structures. cf. Figure 6.
It becomes clear that the whole maze of Wyschnegradsky's system of interlocking positions is possible thanks to fundamental principals of semi-tonal music. This is not only evident in Wyschnegradsky's theoretical writings but foremost, in his works. The examination of Wyschnegradsky's op. 45 will prove enlightening in this regard.
Figure 6: Regime 11 Ternary Structure
Étude Sur Les Mouvements Rotatoire: Wyschnegradsky's op. 45 and the Inaudible Zone
As I have mentioned before, Wyschnegradsky referred to his op. 45 in his presentation of cyclical structures in Espaces non-octaviants ou cycliques. The title itself, "Étude sur les mouvements rotatoires" suggests that the piece might shed some light on the nature of Wyschnegradsky's musical language.
Before we turn to the music of op. 45, one further concept must be brought forth. Given the nature of cyclical structures, they need a fixed band of the pitch spectrum in which to unfold. Some circles like that of the perfect fourth of fifth unfold over five and seven octaves respectively. This frequency band lies within the threshold of human hearing. However, most structures in Wyschnegradsky's system are based on the major seventh and minor ninth (regime 11 and 13). These circles occupy a band of eleven (11) and thirteen (13) octaves respectively. These go beyond the threshold of hearing and imply an inaudible zone in which the cyclical process unfolds.
In his preface to op. 45, Wyschnegradsky exposes this concept and sets the practical limits of the audible zone to the approximate range of the piano keyboard (A1 to A7) with Eb as the symmetrical middle point. As he points out, the cyclical structure used in op. 45 (like all cyclical structures) brings with it the "'paradox' of the junction of the extreme low to the extreme high registers."
As we will see, in writing the Étude sur les mouvements rotatoires, Wyschnegradsky seems to have given justification to the paradox.
Op. 45: A Guide to Understanding the Cycle
In op. 45, Wyschnegradsky sets the wheel in motion. The work is based on a binary structure in regime thirteen (13). Two (2) such circles of minor ninths interlock at the interval of thirteen (13) quartertones (minor fifth), thus giving a symmetrical structure. This structure is set in motion by the application of a constant pulse; the eighth note. This constant, periodic motion permits the measurement of the exact time interval between the entry, the disappearance into the inaudible zone and the reentry at the lower limit of the audible zone.
Close examination of Figure 7a in relation to 7b reveals eleven (11) eighth note pulses between the end of the first audible part of the circle and the entry of the second. This represents the time needed to "rotate" the inaudible section of the circle. cf. Figure 7a and 7b. Once the regular motion is established throughout the entire audible zone, forward horizontal motion is imposed upon the perfect structure. This is done through oblique voice-leading. Along with this forward motion comes a progressive disintegration of the steady eighth note pulse as well as a gradual decrease in the degree of structural symmetry.
The piece consists of two (2) such integration-disintegration complexes, with a third receding reintegration as a coda. These sections are situated as follows in the score:
1 - 10 - 46 - - 56 - 77 - 84
periodic mvt. disintegration inverted periodic mvt. disintegration re-integration
Figures 8 and 9 are reductions of the structures starting at measures 1 and 46 respectively. Closer examination of these two complexes shows a certain similarity in the forward motion process. Both sections start with the exposition of the perfect structure in its initial position. The first moves up through homologous positions (1b - 2 - 2a - 3 - 3b) to position 4. The second (measure 46) moves up in parallel motion (positions 1 - 2 - 3) and then back down througjh oblique motion to position 12, one full semi-tone below the starting point.
At this point in both complexes, the composer begins to interlock two (2) binary structures at the interval of seven (7) quarter-tones, thus dividing the structural minor fifth asymmetrically. This creates four-part (quaternary) structures which are imperfect, even in their most basic forms. The progression from the perfect binary structures to imperfect four-part structures through oblique motion and interlocking of cyclical structures exemplifies the "decrease in structural symmetry" mentioned above.
It is of note that in the first complex (Figure 8), imperfect (semi-tonal) structures interlock at the neutral third (7 quarter-tones) to create quaternary structures of Type B. In the subsequent section (from measure 46), the four-part structures are the result of the interlocking of perfect (quarter-tonal) binary structures and are classified Type A.
Following this observation , it may be stated that the second disintegration complex retains a higher degree of structural symmetry than the first (at least as pertains to the sections analyzed). The logic here is that two interlocking symmetrical structures, while creating an assymmetrical structure, retain their initial symmetry. However, structures with intrinsic assymmetry remain non-symmetrical within a quaternary configuration.
these differences, both integration-disintegration complexes show similar
construction. A quick glance at figures
8 and 9 is sufficient, in my opinion, to observe the importance of cyclical
structure and oblique linear
integration-disintegration process is evident at many perceptual levels
If one thinks of functional
harmony as vertical aggregate sonorities
Subdividing semi-tonal intervals into symmetrically related parts, Wyschnegradsky has brought a logical extension to the limited pitch spectrum of the "total-chromatic" and has expanded it into a "sonic continuum". Through his adherence to principals inherent in the equal-tempered system, he has arrived at a system that integrates the old and the new. Is this not what Ferrucio Busoni was calling for when he wrote these words at the turn of the century?:
...all signs presage a revolution, and a next step toward that "eternal harmony".
Wyschnegradsky heard this "eternal harmony" and realized that it existed within equal temperament; of this, his works are living, sounding proof.
Lucille Gayden; M.P. Belaieff, Ivan Wyschnegradsky,
Ivan Wyschnegradsky, "Ultrachromatisme et Espaces non-octaviants ou cycliques", La Revue Musicale #290-291, 1972.
Ivan Wyschnegradsky, Manuel d'harmonie en quart de ton, Éditions Max Esching, 1932.
Ferrucio Busoni, Three Classics in the
Aesthetic of Music, Sketch of a New Aesthetic of Music,
Ivan Wyschnegradsky: Biographic Note
Ivan Wyschnegradsky devoted the greater part of his life to the fulfillment of his vision; the sonic continuum.
He was born in
The need for new instruments led
After the second world war, Wyschnegradsky was not attracted by serialism
as were many of his contemporaries. Yet
his language did evolve into a more rigorous system; the system of
non-octaviant spaces described in La Revue Musicale. Towards the end of his life he began to
receive the recognition he had long been denied. Thanks to the efforts of composers such as
Claude Baliff in
Ivan Wyschnegradsky died in September 1979 but his music and his prophetic vision live on.
Ibid. pp.82-83. "Avec l'adoption du tempérament
égal l'échelle sonore s'est de facto transformé en milieu
sonore. Mais il faudra plus d'un siècle pour que cette
 Ibid. p.83. "...J'avais parlé du choix que j'avais fait parmi tous les milieux ultrachromatiques possibles, en m'arrêtant sur les mileux sonores multiples de 12 et, parmi eux celui des 24 quarts de ton, des 36 sixièmes de ton et des 72 douxièmes de ton. En faisant ce choix j'étais guidé par le désir de m'appuyer sur l'héritage de mille années d'évolution perpétuelle, c'est-à-dire sur le mileu sonore 12 sons et aussi de me limiter au plus simple, facilement réalisable.
 The semi-tonal structures of figure 4 would be labelled as "homologue" positions of perfect structures by Wyschnegradsky. The ternary (4,3,4) structure appears as regime 11. Type C (12 9 12), one of the "homologue" positions discussed later in this paper.