Symmetry, for the purpose of this discussion, refers (1) to a uniform arrangement of notes around a central axis, which may be stated or implied; (2) to a succession of notes which have the same intervallic series in retrograde or inversion; and (3) to a pattern that repeats in one direction, as do the minor 3rd segments in the octatonic scale.¹ Symmetry may be a surface feature or it may be part of the background, like the symmetrical patterns of the modes of limited transposition.² Underlying harmonic symmetry may be obscured by vertical arrangement, as the chromatic voice leading is hidden in Example 4-1. A symmetrical arrangement of notes around an axis can create a harmonic focus on the axial pitch, whether that pitch is stated or implied. Such focusing may be based on traditional tonal functions, as in the case of the French 6th. In the absence of traditional tonality the focus will continue to suggest a tonal center, or “tonic,” but this will be much weaker. Symmetry based on a mirror image where the focal point falls between notes of the scale tends to obscure tonality.

There are many points of contact between symmetry in Russian Modernist music and traditional tertiary-harmonic and folk-song idioms. To take the tertiary aspect first: apart from the focus suggested by a French 6th or dominant 7th with diminished 5th there is also the dominant minor 9th with diminished 5th and added 6th and the dominant major 9th with diminished and augmented 5th. The former is by implication octatonic and the latter whole tone; both can be arranged symmetrically around the associated tonic.³ Aleksandr Tcherepnin has a very expressive example of a symmetrically voiced chord at the beginning of his Pieces Without Title No. 3 (Example 8-1). It can be heard to focus on B-flat (or E) while emphasizing F in the bass, and it resolves to a dominant 7th on F. The phrase hovers between F and B-flat before closing firmly on the former.⁴ This passage contains the major and minor 3rd of the scale of F. Because of the frequent occurrence of the variable 3rd of the scale (major, minor, and neutral) in Russian folk music it is a natural idiom for Russian composers to introduce both the major and minor 3rds in their harmonies; along with the perfect and diminished 5ths and the minor 7th, the two kinds of 3rds produce another symmetrical focal arrangement, one which is again by implication octatonic.⁵ Skriabin’s music contains innumerable examples of the French 6th and dominant 9th type chords mentioned above, and Example 8-2 shows the major/minor 3rd construction in Shillinger, which is here symmetrical around B.

The basic symmetry in Russian folk music is the double 4th, as in Example 8-3, where it is based on the modal centers A, D, and G.⁶ This is reflected in Russian art music by the occasional resolving of the 7th of the dominant direct to the tonic (Examples 4-16 and 4-24) and in cadences where there is emphasis on the 4th of the scale.⁷ Occasionally the gravitational attraction of the modal centers in folk song is felt to be so strong that the scale closely approximates that shown in Example 8-4. This type of intonation, with its augmented 2nds, is especially characteristic of Armenian and Moldavian music and has also influenced songs of the northern Caucasus region (Example 8-5). Symmetrical or quasi-symmetrical progressions frequently occur in the extended-range pentatonic music of the Bashkir people of the Urals area (Examples 8-6a and b). The analysis of a folk song which appears in Lebedinsky’s collection of Bashkir melodies⁸ (Example 8-6c) suggests a number of quasi-symmetrical features in the phrase structures as well as in the melodic shapes. The particular symmetries of the pentatonic scale are compatible with the double-4th focus mentioned above, and the two may occur together in a variety of transpositions, as Examples 8-7a and b, from the Buryat-Mongol people of Siberia, show. Other symmetrical scale patterns sometimes found are illustrated in the song “Deli Yaman” (Examples 8-8a and b), taken from a setting by Lazar Saminsky (1882-1959) of an Armenian song, and in the Chasidic Sabbath Dance (Examples 8-9a and b), also collected by Saminsky.⁹

Many symmetrical and quasi-symmetrical passages occur in works by the Russian Nationalist composers. Examples 8-10 and 8-11 show passages from Rimsky-Korsakov’s Golden Cockerel which repeat a pattern symmetrically in one direction. Example 8-12, also from the Golden Cockerel, is an almost symmetrical ornamentation of a traditional cadence; and Examples 8-13 and 8-14 show the octave divided into a sequence of major and minor 3rds. One of the bird songs from The Snow Maiden (Example 8-15) is reminiscent of Glinka’s Ruslan and Liudmila, where a prolonged suspension of the 3rd in a dominant triad creates a pile-up of alternating 4ths and 2nds (Example 8-16). A clearly focused symmetry is rarer among the Nationalist composers, but a fine demonstration of this technique is found in Musorgsky’s Boris Godunov (Example 8-17).¹⁰ It is very similar to a passage in Aleksandr Tcherepnin (Example 8-32).

In the early years of the twentieth century there was a fascination with the subject of symmetry among certain Russian theorists. In 1900 Boleslav Leopoldovich Iavorsky (1877-1942), then a pupil of Taneev at the Moscow Conservatory, organized a discussion group at which the symmetrical basis of phrase structure was one of the subjects considered.¹¹ In his first essay, The Structure of Musical Speech (of 1908),¹² Iavorsky attempted a universal theory of music in which all progression derives from the unstable nature of the tritone and its need to resolve. He believed that the human ear was in some way “tuned” to the tritone and on the subconscious level was aware of the tritone basis of all music, even when it was not expressly stated.¹³ The composer Sergei Vladimirovich Protopopov, a pupil of Iavorsky and working under his guidance, produced a fuller statement of the theory in 1931 in a thesis called The Elements of Musical Speech.¹⁴

In a chart drawn from Iavorsky’s model (Example 8-18), the chords are arranged after a symmetrical pattern. Iavorsky admitted that the tritone, as contained in the diminished triad or in a chord with major 3rd and minor 7th, could have a relative stability, and Protopopov quotes the song “Prialitsa” as an instance (Example 8-19). Iavorsky referred to his model as a “symmetrical system” (simmetrichnaia sistema), but he thought of the tritone in connection with the modal or tonal system, which, in many respects, is asymmetrical and in which context the tritone is unstable. He does not appear to have noted that in the context of a fully symmetrical system, where tonality is eliminated, the tritone can be completely stable. This can be confirmed by comparing bar 2 of Skriabin’s Op. 74 No. 5 with bars 5-8 (Example 8-20). The bass tritone at the end of bar 2 occurs in a tonal context and has the usual tendency to move to G-flat (F-sharp), but in bars 5-8, where the dominant coloring has been replaced by the symmetrical patterns of the octatonic scale, the bass tritones provide a stable platform for the harmonies above.

A number of symmetrical formations have already been encountered in music examples in earlier chapters. The focus created by the use of chromatic neighboring tones is by far the most common form of symmetry and usually has the effect of emphasizing the focal pitch (Example 4-29). The next strongest focus is the double 4th, as it appears in the left-hand part of bars 3 and 4 of Skriabin’s Op. 58 (Example 2-5). In terms of arrangement the double 4th has much in common with folk music; in terms of gravitational tendency, however, its power in Modernist music is derived from traditional tonality.¹⁵ Other focal arrangements mentioned in the second paragraph of this chapter are relatively weak. At bars 80-86 of the first movement of Mosolov’s Sonata No. 2 the notes in the bass have D as their axis but, since this focus is weaker than the modal centering of the melody, they have the effect of weakening the tonality (Example 5-20). Instances of symmetry underlying the structure of a section of a work are Examples 7-9a and 10-2, bars 7-14, where the octave is divided into major and minor 3rds respectively, in the first case by the modal centers and in the second by the movement of the bass.

The ostinato at the beginning of Deshevov’s Op. 3 No. 7 (Example 5-8) has a distinct sound of F major because the opening chord is symmetrical around the major 3rd F/A and is also suggestive of a dominant 9th on C. This is similar in arrangement to a chord used in the clock scene from Musorgsky’s Boris Godunov (Example 8-21), though it is different in application. The second chord in the Deshevov piece suggests the triad of F major with added 6th. This ostinato well illustrates the common ground between traditional and Modernist tonal harmony. Arrangements such as these tend to modify traditional functions without destroying them entirely and place emphasis on the linear movement. Frequently, as here, they also emphasize the 6th of the scale (see discussion on bimodality in chapter 10). There are many examples of this type of progression in Aleksandr Tcherepnin (Example 10-14).

There are a number of instances in Russian music where one pitch, which is not a mere passing note, is added to the octatonic scale. In such cases the added note automatically becomes an axial pitch of the collection. How clearly this axis emerges as a focal center depends on the density or transparency of the texture and the voicing of the chord. Its function, if any, will also depend on the context. The bass D in Vishnegradsky’s Op. 2 No. 2 is clearly a dominant because of the voicing of the passage (Example 3-2c). On the other hand, the added F in the first bar of Lourié’s Syntheses No. 1 is a tonic (Example 6-39). The symmetry here is slightly broken by the absence of E-flat, which would make the texture too turgid. It would also obscure the dominant sound of the C, which is important in this context, and contradict the tritone and adjacent semitone effect—triads of G-flat, C, and F (mentioned in chapter 6). The double semitone focus (E and G-flat) and the double-4th focus (aided by the prominence of B-flat in the treble) is clear and an important part of the general effect.

Elements of symmetry began to appear in Skriabin’s music before 1910. His Study Op. 42 No. 2 has many instances of major or minor 3rds alternating with semitones in the left-hand accompaniment figures (Example 8-22). The piece called “Enigma” Op. 52 No. 2 has a variety of symmetrical forms: mirror-image chords, minor 3rd bass progressions (Example 8-23a), and augmented 6th arrangements with mirror-image ornamentation (Example 8-23b).

After 1910 symmetrical fragments of melody and decoration appear in the music of both Skriabin and Roslavets (Examples 8-24, 8-25, and 8-26). Various arrangements of tritones also occur in chords. In Skriabin’s Op. 58 most of the left-hand chords consist of two minor 7ths separated by a tritone, except in bar 9 (which in many ways is pivotal), where the chord consists of two tritones separated by a major 3rd (Example 8-33). In No. 2 of Two Poems (1920) Roslavets has two tritones separated by a 5th in bar 7, and by an augmented 6th in bar 11, all in the left hand (Example 8-27). In Poem No. 1 he has two 5ths separated by a minor 3rd (C/G and B-flat/F) in the right-hand part of bar 1 and two tritones joined by three 4ths in bar 10 (Example 8-28). In Skriabin’s Sonata No. 7 there are cases of two tritones below and above a minor 7th in the left hand and of two minor 3rds separated by a 4th in the right hand (Example 8-29). In his Op. 59 No. 1, fourteen of the nineteen bars not involved with the roots B and E-sharp (F) (see discussion in chapter 2) have, as the basis of the left-hand harmony, dominant-structured major 9ths without the 5th, i.e., the implied symmetry of two major 3rds separated by a tritone. This serves as a neutral background against which the tonal “argument” between the roots F and B takes place.

The double 4th is used in special ways by both Skriabin and Roslavets. In bar 3 of the Albumleaf Op. 58 (Example 2-5), Skriabin not only suggests a B to follow in the bass, as the focal point of F-sharp and E, but also focuses on the following D-sharp by approaching it from A-sharp and G-sharp and juxtaposing it with C-double sharp and E. The result is that the D-sharp has greater harmonic value than normally associated with an added 6th. The procedure is repeated a tone higher in bars 7 and 8. In bar 8 the F is not only an added 6th in its immediate context but also a hint of the emerging tonality of F (Example 8-33a). In the first bar of Roslavets’s Poem No. 1 (1920) there is a related procedure (Example 828). Within the context of a complex dominant-structured chord on E-flat, the C and B-flat together with E and the bass G-flat focus on F, which follows in bar 2.

Aleksandr Tcherepnin’s chords are frequently built outward symmetrically from the center (Example 7-12, bar 7). In Pieces Without Title No. 1, Tcherepnin uses a chord common in Russian folk music, consisting of a major 3rd with a minor 3rd above and below. It is closely associated with the scale used in the melody at the conclusion of Saminsky’s arrangement of the song “Deli Yaman” (Example 8-8). It underlies the polyphony of many folk songs, including the one in Example 8-30, which was collected in the district of the Don River.¹⁶ The chord not only occurs clearly in bars 1 and 2 but is the core of much of the song. It is the basic structure of one of the modes variable at the minor 3rd; this can be heard at the beginning and on the third beat of bar 2, where there is a hint of A. At first Tcherepnin uses the chord in both hands to produce an extended symmetrical system; he then employs it in bars 4-8 to accompany a modal and folklike melody centered on G and B-flat (see Example 10-13).

Tcherepnin has another symmetry underlying the passage at bars 9 and 10 of Pieces Without Title No. 7. Two diminished 7ths forming the octatonic scale are heard first and then two notes (B-flat and D-flat) which lie outside the scale but form a symmetry in relation to it (Examples 8-31a and b). These are followed by a chromatically ascending sequence of chords having an underlying nonfocal symmetry (Example 8-31c). No. 8 of these pieces has a short middle section consisting entirely of a passage with D as its axis, in contrast with the E-flat tonality of the outer sections (Example 8-32). It is very similar to the example quoted earlier from Musorgsky’s Boris Godunov (Example 8-17).

Symmetry is employed on a structural level, particularly by Skriabin, Roslavets, Aleksandr Tcherepnin, and Aleksandr Krein. Other Russian composers in the Modernist period do not appear to have taken up Skriabin’s experiments in symmetry to any great extent. Skriabin’s premature death in 1915 meant that his work with the modes of limited transpositions in the Preludes Op. 74 had to await the arrival of Messiaen for further development (see discussion in chapter 14). Albumleaf Op. 58 by Skriabin (Examples 8-33a and 2-5) is tonally structured, but the tonal contrast is now merely suggested rather than explicitly stated. It is based on the symmetrical division of the octave into two tritones supported by a number of other symmetrical devices. In addition to the basic F/B of bars 11 and 18-22 there is a second, less-obvious tritone involved in this work. F-sharp (G-flat) is prominent in the melody throughout and in the bass in bars 3 and 17-22, and C is important in the bass in bar 1 and bars 10-16. The two appear together in the treble and bass in bars 1 and 15-16, at the beginning of the two main sections. This tritone is the common element between the two cadential progressions G-flat, C, F and C, F-sharp, B which underlie the work. All the notes of the first group are present in prominent positions in the treble and bass at the cadence in bars 9-11, and the second group is clearly expressed in the bass at bars 1 and 14-18. The augmented triad centered on F-sharp in the right hand of bar 11 is partly responsible for the weaker effect of this cadence in relation to the final one. Example 8-33b shows that the pitch collection employed at the first cadence in bars 10-12 has the tonic, F, as its axis. The collection at bars 16 and 17, on approaching the B cadence, is not strictly symmetrical in relation to B but it becomes so when the G-sharp gives way to F-double sharp immediately before the B is heard for the first time in the bass at bar 18.

There are other symmetrical patterns underlying the construction of this piece which reveal Skriabin’s attention to detail, even in such a seemingly simple composition. The melodic and bass structures, outlined in Example 8-33c, are based on three phrases of eight, seven, and eight bars respectively. This agrees with the harmonic outline shown in line 2, where the first and last phrases both begin with the bass progression C/F-sharp. However Skriabin’s harmonic schemes seldom exactly match the phrase construction in his late works, and it is also possible to regard the harmonic framework as in line 3, where the first part of the piece consists of a departure of the bass from C/F-sharp to the tritone D/A-flat, followed by a return to C/F-sharp in the second part. A third symmetry lies in the placing of the first entry of the two perfect 5ths, each introducing a passage of seven bars, as in line 4. In the case of the last two patterns the single bar 9, with E in the bass, assumes a pivotal role, returning the harmony to that at the beginning.

In “Etrangeté” Op. 63 No. 2, the various textures are subjected to a complex system of transpositions by various intervals which divide the octave symmetrically (Example 8-34).

Prelude Op. 74 No. 1 (Example 8-35a) is a reworking of the harmonic plan first developed in Op. 58 and illustrates developments in Skriabin’s technique during the four intervening years. The augmented 6th chord emerging in the upbeat to bar 1 corresponds to the opening chord of Op. 58, and the bass notes B-sharp and F-sharp are equivalent to the C and F-sharp of the earlier piece. The most significant change is the move from whole-tone to octatonic coloring. Instead of the two tonics, F and B, which occur in the bass in Op. 58, these pitches become the focal points of the symmetrically based harmonies of bars 1-3 and 9-15, as shown in Example 8-35b. The same two centers of pitch emphasis are also subtly suggested by the added 6ths in bars 3 and 11-15. The chords heard across the bar lines at bars 5-6 and 7-8 are related to the added 6th chords in the outer sections not only by rhythmic placing but by the fact that these too are symmetrical pitch collections (Example 8-35c). The D-natural in the treble of bar 8 and the alto of bar 15 is evidently a harmony note but one which breaks the harmonic symmetry and is outside the octatonic scale. Its sound in bar 15 is particularly singular because it wipes out the effect of the added 6th, which has been important in the first and last sections and draws a curtain of mystery over the ending. The Ds are related in that they come at the harmonic end of their respective sections, follow immediately after the metrical accent, and are the terminations of a rising or falling chromatic line.

The unity of texture and thematic content seen in these pieces was a dominant characteristic of Skriabin’s smaller works throughout his later years. They are usually expressed through structures which divide into two similar and approximately equal parts; occasionally, as in Op. 63 No. 1, with a more or less extended central link. Within this pattern various arrangements of pitch centers are worked out, usually based on the symmetry of the tritone. In Op. 58 each part starts from the same bass note C and ends on a different note of the tritone F/B. In Op. 59 No. 1 both parts commence over the bass B in bars 1 and 13, and the piece ends with the bass progression F/B, the reverse of the initial tritone. Op. 63 No. 1 has a repeat of the opening material at the original pitch (bass C) and concludes on the tritone E-flat/A, after a central linking section. Op. 63 No. 2 and Op. 69 No. 1 are similar in that both parts commence on the same pitch and emphasize this note again at the end. The second part of Op. 71 No. 1 begins with a tritone transposition of bar 1 but ends on the same note as the first section. In Op. 71 No. 2 the second part begins with a transposition of the opening a minor 3rd higher but ends with the reverse of the opening D/G-sharp. The tritone as used in these pieces produces no sense of an outward journey comparable to a move to the dominant followed by a return to the tonic. Instead there is a unified whole with the feeling of viewing the music from its opposite pole.

The two pieces called Garlands Op. 73 demonstrate an extension of the usual two-part form. In each piece the two parts are subdivided, giving an overall AB:AB structure with a symmetrical pattern of bars. In No. 1 the bars may be grouped as follows:

Each of the six sections ends with a four-bar link passage using the same material of descending 3rds. A symmetrical pattern of bass notes also pervades the whole design:

The tritones descend in a pattern of major 3rds and complete the octave cycle twice. The order of pitches in the tritone pairs in bars 7-36 is reversed from bar 37 on, with the evident purpose of causing the piece to end on A, to match the opening. The steps in the cycle of pitches do not coincide with the main phrases listed above. There is, however, a direct connection which is not apparent at first: the pitches change in the midst of each phrase, immediately before the four linking bars. The new pitches are then taken up again in the following phrase.

Skriabin’s major achievement in his search for a new structural principle is undoubtedly his Sonata No. 10, a work which demonstrates his development of symmetry from its smallest details to the total structure.¹⁷ Just as interesting as the symmetry of the overall design is the extraordinary care the composer has taken to join the various sections and so balance symmetry with continuity. It is frequently difficult to define the exact point where one section ends and another begins, and there are places where, in performance, one does not experience any clear break.

The center and chief climax of the work occurs in bars 178-221, the first fourteen bars of which may also be regarded as transition (Example 8-36). The ending of this central section is clear because it is followed immediately by a recapitulation of the second subject, commencing at its original pitch, as in bar 37. The start of the central episode is marked in a similar way: bar 177 is the end of a four-bar semitone sequence, corresponding to bars 51-58, and the end of the recapitulation, based on the first and second subjects, which commences at bar 154. These two recapitulations, which frame the middle section, are each 24 bars in duration.

Surrounding this central block are extended passages drawn entirely from material in the exposition, except for recurrences of the four-bar phrase which appears first at bar 84. The first of these sections commences at bar 59, the second at bar 246. Symmetry of design is again demonstrated in that the first 50 bars of these two sections are identical except for transpositions. The remaining 47 bars of the first commentary continue in very much the same vein, until the first recapitulation commences at bar 154.¹⁸ The remainder of the second commentary, however, takes the form of an increasingly frenzied climax from bar 294 to bar 359, built from ever-more fragmented motifs from the exposition. Although greater in number than the 47 bars of the first commentary-transition, these 66 bars are shorter and occupy slightly less playing time. They begin with a reminder of the central climax from bar 192 and place increasing emphasis on the bass F/C on which the work ends. The whole Sonata is framed by the exposition, bars 1-58, and the brief coda based on the first subject. The work ends with seven bars containing the material of the eight bars with which it began, transposed down a minor 3rd.

Many Russian composers copied some of the more obvious aspects of Skriabin’s harmony, but Aleksandr Krein was one of the few to understand what Skriabin was doing in trying to develop symmetry as an alternative structural principle to tonality. This is demonstrated in Krein’s Piano Sonata Op. 34 (Examples 3-7 to 3-10). (The work will be more fully analyzed in chapter 12; here we are concerned only with the element of symmetry).

A feature of the introduction (Examples 3-7 and 3-8, bars 1-17) which stands out is the palindrome arrangement of the bass of bars 1-15 and the tritone relationship between the central F and the B of bars 1-7 and 13-15. Example 837 is an attempt to note some of the symmetrical and related features of this passage. Line 1 shows the bass notes: the time values do not follow the symmetrical pattern. Line 2 is a close-position arrangement of one of each of the left-hand chords: from this it is apparent not only that some of the chords have an underlying symmetrical structure in themselves but that the voice leading, by step in contrary motion, is also generally symmetrical—though the effect is somewhat obscured in the score by the shape of the bass. Line 3 takes note of the symmetrical scales used in certain places in the melody. Line 4 examines the underlying scales of the harmonic structure and reveals that the first five bars are wholly octatonic, except for the occasional passing note (D-sharp), while the chords on the first beats of the following bars are strongly whole-tone. Line 5 examines the phrase structure of the melody and line 6 the nearly symmetrical layout of chords in the right hand.

It is interesting to note the interplay of symmetrical and nonsymmetrical features. In reference to most aspects of the introduction there is a definite break between bars 1-5 and 6-16. The symmetry of line 1, however, overlaps to include bar 5, and the chords of line 7 frame the whole section. The break between the second and third melodic phrases comes slightly before the center of the harmonic symmetry. The relationship between the chords noted in line 7 is emphasized by the persistent presence of a tritone in the bass in all the intervening chords (except in bars 3-5, where the presence of both the minor and major 3rd also weakens the diatonic effect). The general trend of the passage is to move from an octatonic sound to a more whole-tone one, with the complete section framed by two chords having strongly diatonic associations (line 7).

One other symmetrical feature should be noted: The “Meno Mosso” section starts in bar 133, exactly in the middle of this Sonata of 266 bars. It is followed by five bars from the beginning of the introduction at their original pitch and preceded by five bars which also have B in the bass. Bars 132-34 are marked by repeated emphasis on the reference chord first heard at the beginning of bar 1 (the omission of the accidental G-sharp from the right-hand chord of bar 132 appears to be a misprint). The passage stands at the center of 99 bars (85-183) which form the middle section of the work.

In the pieces quoted from Aleksandr Tcherepnin earlier in this chapter the symmetries are of local importance, but in Pieces Without Title No. 5 there is a structural and particularly subtle play on tonal centers in which symmetry has an important part (Examples 8-38a and b). The ostinato emphasizes D because of its position at the beginning of the bar, but it has G as its axis; while the right-hand melody is centered modally on A, with some prominence given to the 4th, E. At bar 25 the right hand sounds as though it is moving into agreement with the left, with F-sharp, E-flat, and A-flat, all being neighboring tones of D and G (the F-sharp and A-flat also support the focus on G).

In bar 27 there is a comic turn of events as the top notes (B-flat and E) recall the tritone E/B-flat from the melody of the opening seven bars. There is another humorous touch in bar 29, when the alto moves to G while the top notes still resist and (in bars 31-32) go through a little contrary-motion symmetrical pattern of their own before finally breaking away and going to E and A, in agreement with the opening right-hand tune. There follows a coda in which the axes of the symmetrical seven-note groups are alternately A and E while the bass draws A into association with D. Finally there is a symmetrical D chord with major and minor 3rd and a flourish of alternating semitones and minor 3rds with a hint of both D and A major/minor triads. The pitch D prevails at the end.

Shillinger’s “Humming Machines” Op. 12 (Examples 8-39a and b) opens with a nonfocusing symmetry based on a series of alternating semitones and minor 3rds, though the dynamics leave no doubt that it is the G triad which is important. The scale underlying the melody of the next three bars leads symmetrically outward to G (and of course focuses on its tritone C-sharp) while the bass moves down from F-sharp to C, reaching a C major triad in bar 6 (subdominant to the central G). There is also a tritone relationship between the C minor scale of the right hand in bars 3-4 and the 3rd on F-sharp in the left. The pitches of bar 8 focus on G (except for the D, which in any case resolves to C-sharp). The music reduction sums up the symmetries found in this piece.

A system based on symmetry has many features in common with one founded on the cycle of 5ths. Pitches may still be presented in a hierarchic order of importance, which may be changed during the course of a composition (as was seen in chapter 2 in Skriabin’s Op. 63 No. 2). The use of harmonic and melodic integration, involving selected intervals as practiced by Skriabin and Roslavets, tends toward symmetry and leads eventually to the formation of symmetrical scales, whether chromatic, whole-tone, or octatonic. Symmetry allows composers a good deal of flexibility; it does not commit them to an exclusively atonal style. As has been demonstrated, Skriabin and Roslavets were able to make use of tonally derived harmonic coloring, even when they no longer employed the cycle of 5ths as the basis of structure. One new-found freedom offered by symmetry is the possibility of building a section of music on one or more of the intervals which form the basis of the symmetry in a particular work. This aspect of technique will be pursued more fully in chapters 11 and 12. Meanwhile, chapter 9 will examine the scales employed by Russian composers, their frequently symmetrical basis, their historical origins, and their application by Modernist composers.