Symmetry as a Compositional Determinant

copyright © Larry J. Solomon, 1973, revised 2002

Chapter III. Reflection

This chapter is devoted to demonstrating how reflective symmetry is variously manifested in musical composition and how these manifestations fall into similar and dissimilar classes. We have seen that the reflection operation involves the turning of a figure about a linear axis which will result in a congruence. Mathematically, the operation has been defined as:

T(x,y)= (-x,y) or

T(x,y)= (x,-y)

Tonal Reflection in the Elements of Music

A tonal axis in music, or more specifically an axis of pitch class, may or may not have temporal dimensions. It is, therefore, appropriate to classify these accordingly. Non-temporal tonal symmetry is especially applicable to unordered pitch class sets, e.g., chords and scales and other elements of music. These are notably independent.

"Mirror chords" occur in both modern and traditional styles. The ordinary diminished triad can be reflected about its central tone to show its symmetry, as in example 13a.

Example 13. Some common chords showing reflective and nonreflective properties

  

The major chord of example l3b, or a minor chord, cannot be reflected and result in a congruence as the diminished chord can. The diminished seventh also possesses tonal symmetry. In example 13c, above, the axis lies between Bb and B. falling outside of the equal tempered scale. Note that the symmetric properties of these chords are independent of durations or other aspects of time. Their symmetry is dependent only upon the arrangement of intervals or tones.

Any pitch set or pitch class set is tonally symmetric if its intervals are all equivalent in size .

Example 14. Symmetric pitch sets.

Further, any pitch set or pitch class set that switches its interval order across an axis of pitch is symmetric.

Example 15. Reflective tonal symmetry in pitch sets that switch interval order across an axis of pitch

On this topic it is interesting to note Howard Hanson's theory of involution. In this theory the intervals of a pitch set are projected in the opposite direction to create a related sonority. That is, for example, a major chord may be involuted to produce a minor chord, e.g., C E G --> C Ab F. Hanson classifies the results of this operation as "simple", "isometric", or "enharmonic".

The first of these results when the operation yields another type of sound, e. g., C E G --> C Ab F. Isometric involution results in "the same kind of sound," e.g., C E G B --> Db F Ab C and the enharmonic variety results in the same tones enharmonically, e.g., C E G# --> a Fb Ab C.

The operation of involution is our reflection operation. It is being used by Hanson to create new sonorities for composition which are related to the original sonority in a logical way. It should be noted, however, that a major chord is not "symmetric to a minor chord"; i.e., a major chord is simply not symmetric. The major and minor chords are the only asymmetric triads of the standard four. The other two, the diminished and augmented are built upon single intervals, and are, therefore, tonally symmetric in either simple or compound forms. The major and minor chords, however, contain mixed intervals and are not symmetric in themselves, but they are symmetric as a combination.

Example 16. Tonal symmetry " resulting from combinations of major and minor chords.

It can be shown that Hanson's isometric and enharmonic involutions can result only from operations upon tonally symmetric structures, the enharmonic involutions being simply special cases of isometric forms. The result after the operation will also necessarily be symmetric. The involution operation can be considered a reversal of interval direction. Therefore, if the interval content of the original set is symmetric about a tonal axis, a reversal of direction (high-low) will result in a congruence having the "same kind of sound" or even the same tones. A pitch class set which is asymmetric, however, will not result in a congruence after the same operation since reversal will change the distribution of intervals.

An automorphism may result from an expansion of the intervals in a set on either side of the tonal axis .

Example 17. Automorphisms in tonally reflective structures.

In these examples, the intervals on one side of the axis tone may be expanded or contracted by an amount that will make them equivalent to those on the opposite side. With the reflective operation, then, a congruence results. Such a set is used in Bartok's Fourth String Quartet, both in an ordered and unordered way. This set is properly divided into two parts labeled x and y.

Example 18. Symmetric pitch class sets from Bartok's Fourth String Quartet (after Perle and others: George Perle, "Symmetric Formations in the String Quartets of Bela Bartok," Music Review, XVI (1955).

                                                                              Score: Copyright 1939 by Boosey and Hawkes, Inc.

The y set is an automorphism of x, and both are symmetric in themselves or in combination, i.e., the total set, A. These sets are transposable.

Scales are also unordered pitch class sets which may have symmetric tonal content. Essential to a scale is the order of tones and the scale's intervallic content. Scale does not necessarily carry the implication of tonality either, e. g., the chromatic scale and the whole tone scale, although some do; e. g., the C major scale. A tonal center may serve as an axis of reflective symmetry, an in the Dorian mode, which has an axis of the pitch class D.

Example 19. Tonal reflection in the Dorian mode.

The "alternating" scale, a synthetic scale, has similar properties, but it has no axis as part of the scale.

Example 20. Tonal reflection in the "alternating" octatonic scale, with an axis outside of the scale.

The axis shown above is only one of the many possible in this scale; that is, an axis may be placed between any two notes of the scale, and it will retain symmetry. The choice of a tonal center here is not a requisite. A scale such as this also possesses a high degree of translational symmetry. The intervallic order repeats after every two tones. Every scale which repeats its intervallic order will have some degree of translational symmetry.

Polymodality can lead to interesting symmetric forms. Although C major is not reflective in itself, if it is combined with Phrygian on C, the result is symmetric. Lydian may be combined with Locrian and Aeolian with Mixolydian to obtain similar results. It is inconsequential that the finals be held in common but are shown as follows for simplicity.

Example 21. Combinations of modes resulting in symmetric pitch class sets.

Tonal Reflection in Composition

Canons in contrary motion exhibit tonal reflection in the highest degree. Examples are numerous, but to name a few, they occur in Bach's Musical Offering, the Goldberg Variations and the Art of Fugue. Some of the greatest tour de forces utilizing symmetric operations are found in the works of J.S. Bach.

Bach was fond of generating entire works through the use of these operations. The following is a spectacular example.

Example 22. "Trias Harmonica" in J.S. Bach's handwriting, an eight part puzzle canon in contrary motion.

This eight part canon in contrary motion is notated in its most concise form in Bach's script, shown in the above example, and it contains all the necessary information, with its description, for its realization.

Example 23. A realization of "Trias Harmonica" by symmetric operations.

The axis of tonal reflection here is E/Eb, outside of the scale, dividing the tonal motion between the two choirs. Translation is also performed at a time interval of two quarters within each choir and at an interval of one quarter at the fifth between choirs. Another example is a canon from The Musical Offering, notated and realized as follows:

Example 24. Puzzle canon as notated in J.S. Bach's handwriting, from The Musical Offering, and its realization (in part). Note the upside down treble clef.

Normally, this type of canon is written out fully, but the above examples illustrate that the full notation is not essential. Most of these shorthand notations directing the performer to carry out symmetric operations have become obscure, but a few remain with us. A repeat sign, for instance, eliminates the need for notating large sections of music repeatedly, and it directs the performer to carry out a large scale temporal translation.

Dallapicolla's Quaderno di Annalibera contains a fully notated canon in contrary motion in the twelve tone style. One of the most straightforward uses of contrary motion as tonal reflection is the concluding statement of the first movement of Bartok's Music for Strings, Percussion and Celesta.

Example 25. Tonal reflection in the concluding statement of the first movement of Bartok's Music for Strings, Percussion and Celesta. Copyright 1939 by Boosey and Hawkes.

The following are from Bartok's Fifth String Quartet.

Example 26. Excerpts from Bartok's Fifth String Quartet showing their axes of tonal reflection. Copyright 1939 by Boosey and Hawkes, Inc.

a. Third movement, measure 41.                                                                                    b. First movement, measure 217.

These examples are limited in that the inversions are stated simultaneously. Melodic inversions of any transposition and temporal placement will contain an axis of tonal symmetry.

Example 27. Temporally displaced inversion, showing tonal reflection, from Bartok's Fourth String Quartet. Copyright 1939 by Boosey and Hawkes, Inc.

All of these examples are restricted in being of fixed registration. It is entirely possible to mirror register changes, too. Therefore,it is necessary to make some distinction between the mirroring of pitches and that of pitch classes.

Chord progressions may also be mirrored tonally. In this case a group of chords centers on a single or multiple axis.

Example 28. Tonally reflective harmonic progressions with single and multiple axes.

Mirror progressions may also be regarded as more complex examples of melodic inversion. It is possible, however, that voices may change their positions in the harmony.

Middleground and background level structures can also exhibit tonal centering. This centering may occur in melodic turning points, climaxes, voice entries, cadences, etc. An entire work may be organized symmetrically in several levels of its structure, including that of registration. An example is the second movement of Webern's Variations for Piano, Op. 27. The following is a background graph by Peter Westergaard showing the structure of the whole movement as tonally reflective on the pitch A4, although, curiously, the graph was not constructed with this purpose in mind, and the central or axis tone, A, is missing. Notice that the graph is consistent with the actual registrations in the music.

This work is analyzed further in Chapter VII, and a score is included at the end of that chapter.

Example 29. Graph showing the background level structure of Webern's Variations for Piano, Op. 27, second movement. From: Peter Westergaard,"Webern and Total Organization...," Perspectives of New Music, I (1963), p. 120.

Rudolph Reti has abstracted large scale key changes and relates them to motivic cells. His analysis of the key structure in the movements of Beethoven's "Pathetique" sonata is apropos..

Example 30. Tonal reflection in the key structure of "parallel" movements of Beethoven's "Pathetique" sonata, after Rudolph Reti, Thematic Patterns in Sonatas of Beethoven (N.Y.: MacMillan, 1967),69.

Spatial Reflection

The various forms of symmetry in music may be classified according to their type of axis and operation. Axes ordinarily occur in time, space, or pitch; they may occur in other dimensions, but the three most common classes of reflective symmetry, defined by their axes, are temporal, tonal (pitch), and spatial. The first two of these have been most important compositionally.

Spatial reflection was exploited early in the antiphonal tradition of Syriac Chant. This took the form of groups of singers in opposite locations. The reflection was temporal as they responded antiphonally to one another. Ambrose apparently developed a special method of composition for this spatial music. The clear division in space is often held as essential to antiphonal chant.

The Venetian polychoral school later continued this tradition. Willaert and the Gabrielis developed compositional procedures utilizing cori spezzati. The principles of imitative counterpoint were abstracted to spatial imitation employing the translation operation. This was continued into the Baroque with the development of the concerto, but the spatial dimension then became less significant.

Many contemporary composers have become interested in spatial composition, foreshadowed by such works as Varese's Deserts and Ives's The Unanswered Question. Henry Brant and Karlheinz Stockhausen have tried to formulate compositional methods for the use of space. Many of these employ symmetric divisions.

Example 31. Spatial plans by contemporary composers having reflective symmetry; a. placement of performers in relationship to an audience by Henry Brant,"Space as an Essential Aspect of Musical Composition," in Contemporary Composers on Contemporary Music (New York: Holt, Rinehart, and Winston,1967),229; b. placement of sound sources to correlate with binaural perception by Karlheinz Stockhausen, "Music in Space," in Die Reihe, V (1959), 77.

Space may also be used to divide music which is polytonal into distinct tonal choirs. Such a division is used in Ives's "Putnam's Camp" from Three places in New England, as well as in The Unanswered Question and other of his orchestral works. A section in "Putnam's Camp" alludes to Ives's childhood recollection of the marching bands of local rival football teams, each playing their own march and trying to outplay the other as they converged on the village green. The two bands were playing in different keys, speeds and rhythms. Space is used in "Putnam's Camp" to separate the simultaneous tonalities of reflected timbres.

The recent interest in the compositional resources of timbre have led to its alliance with spatial divisions and symmetry. Simple examples of this are found early in the history of music, e.g., in responsorial chant. In the nineteenth century Berlioz employed spatial divisions of timbre in his Requiem and other works, and more recent examples are found in Vaughan Williams's Fantasia on a Theme by Thomas Tallis and Bartok's Music for Strings, Percussion and Celesta, the last of which is discussed in detail in a later chapter.

Temporal Reflection

Temporal organization is critical for music, and the principles of temporal organization have been highly developed. Reflective relationships are among the most common. Consider the movement of tones in time.

Example 32. Reflective patterns in time: a. from Stravinsky's Petroushka, second tableau; b. from Chopin's Etude, Op. 10, No. 12.

The second halves of 32a and 32b are retrogrades of their first halves. Note, however, that non-pitch symmetry is preserved after the operation of reflection of 32a only with respect to the order of notes, rather than in real time.

The order of the tones is the principle aspect of symmetry here. Many similar examples may be found throughout the literature. One of the most dramatic occurs at the climax of the third movement of Bartok's Music for Strings, Percussion and Celesta across bar 48.

Example 33. Temporal reflection at the climax of Bartok's Music for Strings, Percussion and Celesta, third movement. Copyright 1939 by Boosey and Hawkes.

This bar also acts as an axis of symmetry on a large scale of the four movement structure. Compared to a simple ABA, the order of symmetry in the above retrograde is higher, namely order 1. The degree of symmetry, however, is the same in both, 100.

Historically, the use of cancrizans, or retrograde, is usually connected with medieval/ early renaissance (14th-15th centuries) styles or with modern music, and almost exclusively so. However, this is somewhat of a misconception since this operation was used throughout the history of music. To illustrate, let us begin with Machaut's Ma fin est mon commencement et mon commencement ma fin. The upper two voices are in retrograded invertible counterpoint with an axis of reflection at bar 21, and the countertenor is reversed from its midpoint at the same bar; these reversals are meant to enhance the meaning of the text. The degree of symmetry for the entire piece is the highest possible, 100, but the order is less than one due to the necessary partial reflection of the upper two voices after the complete reflection about bar 21.

Example 34. Temporal reflection in Machaut's Ma fin est mon commencement

The Diliges Dominum of William Byrd's Cantiones Sacrae (1575) is an eight voice cancrizans. Analyzing this work for any unusual restrictions which may be the result of the imposed cancrizans, we find the following. Despite its use of imitation, the piece is strongly homophonic. Simple triads are used without exception, most of which are in root position, and the number of different simultaneous pitches generally does not exceed four. Root movement is almost exclusively by thirds, fourths and fifths. Note the absence of seconds, with the exception of the III IV movement before the cadence at the center, bar 16. Curiously, there are no non-harmonic tones. From the harmonic point of view, then, the work is quite simple and greatly restricted. The main interest should lie in the counterpoint, but this is obscured by the rich texture and simple harmonic structure.

The precompositional cancrizans is probably most responsible for the severe restrictions on harmonic and rhythmic motion. The lines do not change register after the axis, and, therefore, preserve the position of the bass line. Both this example and the Machaut exhibit harmonic retrograde, i.e., the cancrizans is effected in the harmonic progression as well as in the individual lines.

Example 35. A harmonic reduction of Byrd's Diliges Dominum, central portion, showing temporal reflection across the central axis.

A retrograde canon occurs in J.S. Bach's The Musical Offering, again using a condensed form of notation which requires one of the performers to flip the page and read backwards, a reflection operation. Note the backward facing symbols at the end.

Example 36. The original notation of Bach's retrograde canon from The Musical Offering, requiring a performer to carry out a time reflection.

From the classical era, in Haydn's Sonata No. 4 for Violin and Piano, the Menuetto at rovescio, the entire movement can be played forward or backward with equal results. This is due to the cancrizans' structure, and it is particularly interesting because such devices are not normally associated with the style.

Example 37. Temporal reflection in Haydn's Menuetto al rovescio, from the Sonata No. 4 for Violin and Piano: piano part only, showing two parts which are mirrored. Axis is measure 11.

In Beethoven's "Hammerklavier Sonata", Op. 106, retrograde motion is frequent in the last movement. What is remarkable is the length of the fugue subject and the corresponding length of the retrograde form.

Example 38. Retrograde in Beethoven's "Hammerklavier Sonata", Op. 106, last movement.

Retrograde motion is common in twelve tone music, both in terms of row variation and in compositional construction. However, even before the development of the twelve tone method, Schoenberg used symmetrical constructs compositionally. One of the most striking examples is the four part double canon from Pierre Lunaire, No. 18. The canon is reversed, a cancrizans, from the tenth measure. The complexity of this piece is compounded by the piano part (not a part of the canon).

The classic examples of symmetry in the serial style are found in the late works of Webern.

Example 39. Reflective orderings of pitch class sets in two of Webern's rows as analyzed by Pierre Boulez, Boulez on Music Today (Cambridge, Mass: Harvard,1971),71.

Reflective tonal landmarks are found in most works to some degree. Any work which starts and ends in the same key has some symmetry, but not all do so. Franz Liszt's Hungarian Rhapsodies are asymmetric in this respect. It is remarkable, however, that so many works in the literature have this type of symmetry in common. Some will even retrace the intermediate steps of modulation, for example:

            

In this case, the keys are arranged in retrograde.

A musical ABA is symmetrical in the same way as a cancrizans. The former, however, is a more general construct. Simple ternary forms are common in music. Only a few types need be mentioned here: the da capo song form, minuet and trio, scherzo, and certain sonata-allegros. Chopin's Nocturnes are frequently in simple ternary. Other forms may show several levels of reciprocity:

a. The simple rondo: A B A B A

Example: Beethoven's Rondos

b. The compound rondo:    A     B     A     B     A
                                             aba         aba          aba

Example: Clementi, piano Sonata No. 5, last movement.

c. Compound ternary:    A       B       A
                                        aba    cdc     aba
                                                 or c

Example: Brahms, Symphony No. 3, third movement.

d. Sonata rondo:     A     B     A     C     A     B     A
                   keys:     I      V      I      R      I      I       I

Example: Mozart's Trio in Bb, K. 502, last mvmt.

All of the above forms are examples of symmetrical forms, synonymous with sectional retrogrades. Another of this type is A B C B A, not uncommon, which may be found in Brahm's Rhapsody in Eb, Op. 119, No. 4 and Chopin's Mazurka, Op. 56, No. 2. Slightly more complex is the "arch" found in Chopin's Waltz, Op. 34, No. 1: A B C D C B A.

Robert Schumann analyzed Berlioz's Fantastic Symphony, first movement, as an arch, and Edward T. Cone has later expanded upon this.

Example 40. Edward T. Cone's expanded analysis of Robert Schumann's on Berlioz's Fantastic Symphony, first movement, from: Edward T. Cone, "Schumann Amplified," in Berlioz: Fantastic Symphony (New York: Norton, 1971), 252.

                                                                                        Theme A

                                                                  Episode  - - - - - - - - - - - - Transition

                                Theme B :||  {Development of Theme A&B} - - - - - - - - Dev. and Recap of Theme B

                      Transition with Cadential Phrase - - - - - - - - - - - - - - - - - - - - - - - - - - Recap & (Dev of Th. A)
                                                                                                                                                 of Cad. Phr.
          ||:  Theme A - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Recap of Th. A

Introduction - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Coda

mea: 72        111       150             168        200                234            280     313        331        360    412         477     


J.S. Bach used large scale reflective symmetry to organize many of his multi-movement works.

Example 41. The order of pieces in J.S. Bach's The Musical Offering, showing temporal reflection.

ricercar       5 canons        trio sonata        5 canons        ricercar
                            |_______________________|
       |____________________________________________|

The movements of Bach's cantatas are frequently arranged in a similar order.

Example 42. Reflective arrangement of the movements of Bach's Christ lag in Todesbanden, adapted from Gerhard Hertz, ea., Bach, Cantata No. 4 (New York: Norton, 1967),85.

       Sinfonia           I                II             III               IV             V               VI           VII
      + versus        chorus           duet          solo            quartet         solo            duet        chorus
           c.f.:          S,A,T,B          S,A           T             S,A,T,B          B               S,T        S,A,T,B
        orch.:             full            cor.trb   cont,Vl.1-2        cont.      Vl,Vlas, cont.   cont.         full
                                                                      |__________________|
                                                      |______________________________________|
                               |____________________________________________________________|

Other cantatas of similar structure are Nos. 56, 78, 106, and 140, and in other of Bach's works: the canonic variations on Von Himmel hoch for organ and the motet Jesu, Meine Freude.

In Hindemith's one act opera Hin und Zuruck (There and Back) the action proceeds to the midpoint, where the jealous husband shoots his wife, and then all goes in reverse, not in perfect cancrizans, but, more or less section by section. The music, too, follows this plan. The Fuga tertia in F of Hindemith's Ludus Tonalis is a cancrizans with an axis in measure 32. The Praeludium and Postludium of this work are retrograde inversions of one another, and, on a large scale, the following plan is revealed.

Example 43. A diagram of the order of the parts of Hindemith's Ludus Tonalis showing the overall reflective pattern; from: Ian Kemp, Hindemith (London: Oxford, 1970),49.

                                            Fugue 6 (arch form)     Fugue 7 (arch form)
                                    Fugue 5 (a a b form)                   Fugue 8 (a a b form)
                              Fugue 4 (double fugue)                            Fugue 9 (I+R)
                       Fugue 3 (repeat by retrograde)                               Fugue 10 (repeat I)
                Fugue 2 (Stretto fugue)                                                        Fugue ll(canon)
         Fugue I (Triple fugue)                                                                        Fugue 12 (binary)
 Praeludium                                                                                                         Postludium

That these operations are still being used by contemporary composers is evidenced in the works of such widely different composers as John Cage, Milton Babbitt, Luigi Dallapiccola and others.

Example 44. Outline of the Reflective plan of John Cage's Sonatas and InterZudes, from: David H. Porter, "Reflective Symmetry in Music and Literature," Perspectives of New Music,VIII/2 (1970),418.

                                              Sonatas I-IV
                                                    Interlude I
                                                         Sonatas V-VIII
                                                               Interlude II
                                                                Interlude III
                                                          Sonatas IX-XII
                                                     Interlude IV
                                              Sonatas XIII-XVI

The recurrence of a' section normally carries the connotation of related lengths. However, such a recurrence is not necessarily defined in terms of length. At times this is stretched considerably to include relationships between sections of like content but of very different lengths. A separate category is needed to accomodate temporal length for both sections of related and unrelated content.

The sections of the following example are indicated to be unrelated in content, but the length of the sections are related reflectively in time about the axis section, C. This relationship may be appropriately called durational reciprocity.

Example 45. The reflection of durations.

                              A        B        C        D        E
No.of measures:    5         8       10       8         5

At first glance, the next example may not seem to possess any durational reciprocity.

Example 46. A reflective form with sectional durations specified

.                       A      B      C    |    C      B      A
unit length of time:   15           5           10     |        4          2           6

However, close inspection reveals an automorphism in the proportional lengths of the sections across the axis.

Example 47. Proportional lengths of the form in example 46.

                         A      B      C    |    C      B      A
                               3    :   1   :    2      |      2   :    1    :   3

The third movement of Bartok's Fourth String Quartet is an illustration of such an automorphism. The divisions between sections are clear and may be outlined in numbers of measures, as follows:

Example 48. A diagram of the durations of sections in measures, of the third movement of Bartok's Fourth String Quartet.

              |             A               |             B            |     A'   |     B'  |
              |    
13  |     8    |    13     |    8    |     5    |     8    |       8      |       8    | 

Sections A and B are related by durational reciprocity after an automorphism:

                            13 : 8 : 13 ~ 8 : 5 : 8

A' and B' are obviously related in duration without the need for automorphism.

Music may also have reflective temporal relationships in tempos, such as fast-slow-fast which commonly exist in the movements of a sonata, concerto, etc. Texture is frequently governed reflectively in time when approaching and receding from climaxes, in the same way as dynamics.

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