Symmetry as a Compositional Determinant

copyright © Larry J. Solomon, 1973, revised 2002


Chapter I. Introduction

To understand the very nature of creation one must acknowledge that there was no light before the Lord said: "Let there be Light". And since there was not yet light, the Lord's omniscience embraced a vision of it which only His omnipotence could call forth.... A creator has a vision of something which has not existed before this vision. And a creator has the power to bring this vision to life. . . . {Arnold Schoenberg, "Composition with Twelve Tones" from Style and Idea}

The essential uncertainty of quantum behavior might create the uneasy suspicion that anything goes in this world. But since the Universe has (on all observed scales of length, time, mass and so forth) a very definite structure, it seems extremely unlikely that everything is allowed. There must be some things that are more allowed than others. . . . As it happens, there are many things that are forbidden (in the sense that they are unlikely in the extreme), and it is the forbidding rules that give structure to the world. We will discuss quite a few of these rules, and it is important to to bear in mind that the origin of these is essentially not understood. It is not known why the rules must be as they are; some day we may attain a deeper level of understanding, which will provide insight into the rules that govern the rules, as it were. But we are not at that stage yet, and for the moment all rules that have been discovered seem arbitrary and highly non-obvious. If you were to design your own universe, it might not occur to you to do it in the way our world is, and we know of no reason why you should impose all the experimentally observed rules in order to obtain a working universe. . . .

Symmetry forbids. Forbidding imposes order, but many different things that possess a certain order may derive from the same symmetry.... That is why physicists believe that the underlying symmetry, which forbids whole classes of occurrences at one stroke, is, in a sense, more fundamental than the individual occurrences themselves, and is worth discovering. {Vincent Icke (1995), The Force of Symmetry, 100-103}

The creation of music is called composition, as it is in the other arts. But, what exactly is a "composition"? It is "the organization of forms and colors in a work of art" {The New Columbia Encyclopedia, 617}, "the arrangement into specific proportion or relation, and especially into artisic form." {Merriam Webster's Collegiate Dictionary, 236} Synonyms associated with composition include: structure, form, organization, harmony, proportion, and balance.

A question that naturally follows is: Is there any general principle that is used to compose music, i.e., to organize the parts of music into a whole? The thesis proposed here is that symmetry, as defined, is a predominant principle of such organization found in musical composition, crossing international boundaries of style, history, and ethnicity; i.e., in fact, most of the relationships found in music are based on symmetry. Although the examples used in this study are from Western musical literature, the general musical features cited are found throughout the world.

Symmetry is not a concept that is restricted to music or the arts, but it is now recognized as the very foundation of mathematics and physics. The laws of symmetry are general laws that determine a relationship. Mathematical symmetry is defined as "a relation with the following property: if one thing is in a given relation to another, then the second is necessarily in the given relation to the first." {Karush, 263}. E.g., Cindy is a sibling of Michael, therefore Michael is a sibling of Cindy. Or, if triangle A is congruent to triangle B, then triangle B is necessarily congruent to triangle A. A relation fails to be symmetric if there is at least one instance where the condition fails; e.g., if John is the son of Mary, it does not follow that Mary is the son of John; or, if Jim likes Jane, it does not necessarily follow that Jane likes Jim. Similarly, if Jim is a Christian, it does not necessarily follow that a Christian is Jim.

A mathematical operation or transformation that results in the same figure as the original or its mirror image is called a symmetry operation. These operations include reflection, rotation, and translation, which will be defined. The set of all operations that leave a figure unchanged is called the symmetry group for that figure.

The purpose here is to demonstrate that symmetry is a general determinant of musical composition; a composition being an arrangement of parts to form a whole. Musical symmetry will be mathematically defined as a congruence that results from the operations of reflection, rotation, or translation. These may be applied to any parameter of a musical composition. Stated in another way, the purpose is to demonstrate how postoperative congruences define the arrangement and combination of the parts of musical works.

Symmetry can be recognized in the human body, in snowflakes, in beehives, and in geometric figures. It is an object of scientific study in botany and crystallography. Such study has led to a complex but important system of classifying and relating organisms and structures. How does a flower, a seashell, or a millipede relate to music, if at all? How diverse, related, and pervasive are manifestations of symmetry in existing music? Can they be defined and classified? Is there a method or procedure for recognizing this type of organization? What may be the psychological effects or reasons for employing symmetry in a composition? The answers to these and other questions are posed herein.

The word "symmetry" has a common root of origin with "syndicate;-namely the Greek prefix syn-, meaning together, or from sympiptein, to fall together. One of the first English uses of the word in print occurs in a book on architecture by Shute-in 1563: "Concerning ye proportion and simetry to vse the accustomed terme of the arts of the forenamed columbes. {Murray, J., 366} It was applied then to mean the "mutual relation of the parts of something in respect of magnitude and position." Later, in 1624, Wotton, in his Architecture remarks that, "Man it were the Prototype of all exact Symmetrie,"{Murray, J., 366} and Buliver in Anthopomet of 1650 states: "True and native beauty consists in the just composure and symmetrize of the parts of the body."{Murray, J., 366} Subsequently, the term came to have additional, more specific meanings. "Symmetry is a word used to express...the fact that one half of an animal is usually an exact reversed copy of the other."{Murray, J.,366}

Currently, the term "symmetry" is used in all of the sciences and arts. It has similar meanings in geometry and biological science, namely the arrangement of pairs of parts which when joined by a line can be bisected by a line or a

Figure 1. A drawing by Leonardo da Vinci showing symmetry in the human body

Vitruvius man by Leonardo Da Vinci

plane. This type is called bilateral symmetry and is evidenced in the human form.

The same type of symmetry is found in the morphology of many animals and plants, both living and extinct.

Figure 2. Bilateral symmetry in an oak leak and an extinct trilobite

Radial symmetry also occurs in nature as a form whose features are equidistant from a point, and thereby can be rotated.

Figure 3. Radial symmetry in the pine cone and flower

Another type of symmetry occurs as reiterating parts as those in fern leaves and beehives.

Figure 4. Translational symmetry in leaf arrangement, the millipeded, and a beehive

These forms do not have axes but are translational along a line.

Symmetry occurs not only in the direct creations of nature, but in man's own. It is especially apparent in the visual arts, including architecture. From the Parthenon of Ictinus to Raphael's School of Athens to Le Corbusier's Ronchamp Church, symmetry is a strong organizational feature and has been the object of considerable study in art history. Hermann Weyl's book on Symmetry describes its various manifestations in art, mathematics, and in nature. Weyl's book serves as a model for the extensions into music that are the subject of this study.

In music, the word "symmetry" is most often used in its application to phrases and the dimensions of musical forms. Although Hugo Leichtentritt does not define the term, he implies that equal lengths are symmetrical {Leichtentritt, 225}. Wallace Berry uses it in the following way: "Binary form in which the second part is longer than the first is termed asymmetrical binary as opposed to symmetrical binary form, in which the two parts are of equal length."{Berry, 39} Elie Siegmeister similarly uses the term, without defining it, to mean equal phrase lengths. {Siegmeister, 280-282}. Symmetrical meters are normally defined as those that are either divided into equal parts or are indivisible; e.g., 6/8 is 3 + 3, 9/8 is 3 + 3 + 3. Simple triple is a symmetrical meter because it is not divisible. Asymmetrical meters would include 5/8 (divided into 3 + 2, or 2 + 3), 7/8, etc.

Paul Fontaine offers a very different use of the word: "The word symmetric is arbitrarily used to describe a phrase equal in length to an even number of measures. The reference is to length only and not to melodic balances or imbalances within a phrase." {Fontaine, 17} According to this statement, phrases of equal length would be symmetric only if each contained an even number of measures, and a single phrase would be symmetric if it contained an even number of measures.

Another meaning of "symmetry" is the correspondence of parts or elements across a dividing line or axis. George Perle, Colin Mason, and others have used it in this sense for analyses of special works. The most well known example of reflective symmetry, as it will be called, is the so-called "arch form", such as A B C B A. Colin Mason, Mosco Carner and others have shown the use of this form in Bartok's Fourth String Quartet.

Example 1. Relationship of the movements of Bartok's Fourth String Quartet {Robertson, 140}

       |                                                                                                                      | 
Allegro      Prestissimo non troppo         Lento         Allegretto          Allegro Molto

George Perle has pointed out the use of reflective pitch structures in the same work and other Bartok quartets.

Example 2. Reflective harmonic structure in Bartok's Fourth String Quartet {Perle, 1955, 310}.

Related notions of symmetry occur in the literature. Leichtentritt states that inversion is intended to have the "effect of symmetry in opposite directions." {Leichtentritt, 226} Retrogrades are a type of inversion, he says, "retracing the melody's path from the positive plus direction, geometrically speaking, to the opposite, negative, minus direction," thus, also having a symmetrical effect. He goes on to relate invertible counterpoint to melodic inversion in the same way.

Another meaning for "symmetry" is synonymous with shape or form itself. It is used as such by Stewart MacPherson in his book Form in Music. {MacPherson, 2} He states that "music can never be formless; it must always possess that sense of symmetry and fitness for its purpose without which it must inevitably fail as a work of art."

Still another implication for the word "symmetry" is given by Walter Piston in the text Harmony, concerning sequence: "It is generally agreed that a single transposition of the pattern does not constitute a sequence, the systematic transposition not being established until the third appearance of the initial group. On the other hand, it is remarkable that composers seldom allow the symmetry to extend beyond the third appearance of the pattern without breaking it up by variation or abandoning it altogether." {Piston, 233}Although he does not define symmetry, Piston's use seems closest to our translational variety.

What can we gather from all these statements about symmetry? We can guess that it must be a part of form, that it involves equivalences, that it may occur as mirrorlike correspondences across an axis, that it has something to do with sequence or repetition, and it may involve contrary motion or exchange of parts. A definition of symmetry in music is needed, and this is proposed in Chapter II, along with original definitions (in italics) of the various classes of symmetry and the principles by which they operate.

Chapters III-IV classify, clarify, and demonstrate, in detail, the three principal classes of symmetry with numerous examples of manifestations in the standard repertoire and theory of music. Following this, in Chapter V, a method for discovering musical symmetry is formulated and discussed with examples and a table of operations.

Chapters VI-VII comprise analyses of sample works from the literature. Those in Chapter VI are "unspecialized"; that is, they were not known to possess any special applications or degree of symmetry before the analysis. Those of Chapter VII were known to contain special characteristics of symmetry, but the full extent of the symmetry was not known before analysis was undertaken. These analyses should reveal symmetry as a generally important builder of musical structure.

Chapter VIII demonstrates a possible new extension of the principles of symmetric operations, used therein to generate new thematic variants. This is followed by a speculative chapter on the psychology of musical symmetry (Chapter IX).

One of the most important aspects of this dissertation is the demonstration of a common relationship among musical structures, that of symmetry. All of the following will be shown to have symmetry as a common denominator:

timbre identity
isorhythmic motets
sonata rondo
mirror chords
parallel organum
scale formation
compound rondo
compound ternary
invertible counterpoint
ostinati, chaconne
cantus firmus composition
melodic inversion
meter and pulse
arch forms
Alberti bass
serial operations
canon, rounds
pitch identity
circle of fifths and sequence

In all of these elements and characteristics, therefore, symmetry, as defined herein, will appear as the common generator of structure, perhaps the most significant determinant of musical composition.

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