Symmetry as a Compositional Determinant

copyright © Larry J. Solomon, 1973, revised 2002


Chapter IX. Some Psychological Considerations

It has been shown that repetition is an important aspect of symmetry. Repetition is probably the most widely used procedure of organization in music. Constant, unchanging repetition, as encountered in primitive chanting or drumming is almost perfectly symmetrical. This kind of music is often capable of inducing a hypnotic effect upon its listeners, putting them to sleep or into a trance.

When we consider that this repetition consists of a regular and unchanging recurrence of tone, melody, rhythm, etc., we can show its relationship to a psychological phenomenon called auditory adaptation, i.e., the ear and mind become insensitive to a constant sound or tone present. An example of this is the humming of fluorescent lamps in a lighted classroom. The occupants are aware of this sound when the lamps are turned on, but thereafter, auditory adaptation may diminish or obliterate awareness of the sound. If the lamps emit irregular sounds or noises, however, they may compete with an instructor for attention.

Auditory adaptation refers to the fact that a sound is loudest when it first reaches our ears. Afterward the hearing sense adapts, and the subjective loudness decreases as time passes even though the intensity of the sound remains fixed. {Bergeijk, 96}

Does symmetry, then, stimulate boredom or insensitivity? In the opposite extreme, very irregular, unpredictable sounds are normally disturbing. Leonard Meyer has proposed that emotional and meaningful responses to music depend upon the inhibition of expectancies, and, therefore, he says, a successful composer will create constant expectations and resolve them in unexpected ways. {Meyer, 195-6} If this is so, symmetry plays an important part in stimulating these emotional and meaningful responses by setting up expectations. Translational symmetry is a prime example. In Rossini's overtures we often hear translations of melody in time and dynamics, repeating phrases over and over and getting louder constantly. This can set up the expectancy of continuation, i.e., more repetition and more crescendo. But after firmly setting up this expectation, the music usually climaxes with different material, or it may not climax but become suddenly soft. This is not a complete surprise since we can expect change, especially when familiar with the style. Unless the audience knows the specific piece, however, they cannot be sure how or when a composer is going to break the sequence.

It may be that symmetry and asymmetry result as basic principles in music because unity and freedom are basic desires. {Richards, 56} The two are inseparable in nature. Translational symmetry defines the leaves of a maple, yet each leaf is different from any other. Perfect symmetry exists only in theory.

Symmetry of diverse elements of composition serves to fuse or relate these elements. The mind can derive satisfaction and pleasure from accomplishment. If it can fuse the diverse elements of music easily, the sense of accomplishment may be small, but if relationships are fused with difficulty, a greater sense of accomplishment could result. (For a similar, yet not identical treatment of fusion, see Moore's theory in: Robert W. Lundin, An Objective Psychology of Music, 1967, pp. 95-96).

If fusion does not result after a difficult attempt, frustration is likely. This could be the reason for such strong opinions about "difficult" modern music. One who is able to fuse the elements and events of a modern work into a comprehensible order may derive satisfaction from that accomplishment, but those who attempt and fail could turn against it. Symmetry serves as a guideline in the search for this fusion.

The theory that symmetry via repetition is important for comprehensibility has been presupposed by many twentieth century composers, but Webern has been unusually clear on this point: "The highest principle in all presentation of an idea is the law of comprehensibility." {Webern,17} Comprehensibility is the direct result of unity, he says, and is something composers of all times have striven for: "... composers' every effort went to produce unity among the various parts, in the interests of comprehensibility." {Webern, 25} Repetition is the means by which this unity, and hence, comprehensibility, is acheived: "What's the easiest way to ensure comprehensibilty? Repetition. All formal construction is built up on it, all musical forms are based on this principle."{Webern, 22}

Webern goes on to explain that the twelve tone compositional method is used in the interests of comprehensibility:

But development is also a kind of repetition.... Beneath it all is the urge to express oneself as comprehensibly as possible.... The basis of our twelve note composition is that a certain sequence of the twelve notes constantly returns: The principle of repetition!.... But why was it interesting to us that the same thing was sung all the time? One tried to create unity, relationships between things, and surely the maximum unity is achieved when everyone sings the same thing all the time-- the maximum unity imaginable! {Webern, 40}

Schoenberg, too, is clearly in agreement on this point:

Form in the arts, and especially in music, aims at comprehensibility. The relaxation which a satisfied listener experiences when he can follow an idea, its development, and the reasons for such development is closely related, psychologically speaking, to a feeling of beauty. Thus, artistic value demands comprehensibility.... Composition with twelve tones has no other aim than comprehensibility.{Schoenberg, 103}

It is ironic that a compositional method which has had the sole aim of comprehensibility is so widely misunderstood. Alban Berg has tried to provide an answer to this dilemma:

The music of that period [the last century] is homophonic almost throughout: the themes are built on two or four measure phrases, the growth and development of which would be unthinkable without sequences, copious repetitions-- mostly of a mechanical type-- and the relative simplicity of harmonic and rhythmic events thereby conditioned. Imbued with such things for decades, the listener of today is incapable of understanding music of a different kind. Deviation from even one of these familiar musical features-- though the rules may well permit it -- is irritating to him. How much more so when, as in Schoenberg's music, there exists a simultaneous combination of these qualities, usually regarded as attributes of good music, but generally found isolated and diffused throughout the various epochs.{Berg, 68}

The values placed upon music are apparently dependent upon comprehension and expectations via repetition and symmetry. These expectations are also important for defining symmetry. One can ask, for instance, how many times a figure need be repeated in order to possess translational symmetry? It has already been stated that translational symmetry at least implies indefinite repetition. What defines "implies?" Is twice enough, or three times? The answer is not so simple; it is dependent upon the psychological perception and context. A composition which is full of many types of repeating figures sets up the expectation of these figures. Twice may be enough to imply reiteration. If, for instance, a motet is dominated by a rhythmic figure such as quarter-eighth, repeat, repeat, etc... and at some point we hear quarter-eighth, quarter-eighth, we probably would expect to hear its continuation; but it could change. In a work with very little repetition, it would take more of a pattern to establish the same expectation.

The dividing line, then, between simple recurrence and translational symmetry cannot be precisely defined. As a guideline, however, we can say that once a pattern is expected to continue it may be regarded as translational. Such a pattern has set up the expectancy of continuation which we do not demand or sometimes even want to be realized.

Reflective symmetry may have its psychological basis in the sense of balance and what Rudolf Arnheim calls "fittingness... the mutual completion obtained by the matching of things that add up to a well-organized whole." {Arnheim, 64-65}

Convexity fits concavity, the key fits the keyhole, and in the fable told by Aristophanes, the male and female yearn to restore the spherical wholeness of the original human body.... For example, in Kohler's experiments, a chimpanzee sees that two hollow bamboo sticks of different diameter fit each other as soon as their position suggests a visual relation. {Arnheim, 65} Such may be the psychological relationship between subject and inversion, forward and cancrizans, question and answer phrases.

Balance defines stability. A constantly rising line, without reference to a tonality, is unbalanced, hence unstable. It may be balanced by a line in a downward direction, giving a sense of stability. Many musical phrases have this contour, sometimes called an arch. Tonic, dominant, tonic is considered more stable than tonic, dominant or even dominant, tonic. The motion is here in reference to a rotary pitch class or tonality. Without this tonality, stability may be manifest by the balancing of registrations, as shown in the analysis of Webern's Variations, Op. 27.

An imbalance may be created by a seven measure phrase following a four measure phrase. This suggests that a psychological pivot is sensed which weighs the two phrases. On a larger scale this sense can apply to whole sections of music,or on a smaller scale, to intervals. This pivotal sense may be in the fore of our perception of music, from the weighing of registrational distributions to the "fittingness" of a cancrizans.

The Composer's Consciousness of Symmetry

Since symmetry is such a prevalent aspect of musical composition, an interesting question insistently arises: How aware are composers of the symmetrical construction in their own music? Do they calculate phrase lengths, sectional correspondences, reflections, etc.? Or do they simply write music in an intuitive, "felt" manner which may later be analyzed to reveal such symmetrical superstructures as those shown in the works of Bartok and Webern?

The answers to these questions cannot be simply yes or no for two important reasons:

1. Different composers work differently. Some have clearly stated their intuitive, "felt" approach to composition; others, such as Webern, have clearly shown their rational and calculated approach.

2. Most composers seem to use a mixture of both intuitive and planned construction in which no clear boundary can be determined. One can stand in awe of the structure of Bartok's Music for String Instruments, Percussion and Celesta, perhaps this awe would be strengthened if we could know that the structure was not calculated. A flower or a snowflake can have no awareness its own structure, and if awe is related somehow to cognition of this lack of calculation, the same awe should be felt upon viewing a flower or snowflake.

This uncalculated structure is not as rare as it may first seem. When children first learn to read they are conscious of the individual letters of words. It is difficult for them to make out the meanings of sentences and paragraphs. But when adults speak or write their native language, they are not conscious of the spelling of the words they speak or the syntax of the sentences they write. A thought is expressed, more or less, by automatic, intuitive utterance, and yet, structure and symmetry are there.

In the same way, although performers may focus their attention upon fingering and the mechanics of playing when initially learning to play a piece of music, they are not conscious of these things, hopefully, when performing a work in public. The responses usually become intuitive and automatic. Self-taught jazz musicians may not be conscious of the theory of scales, harmony, and form which they use habitually. In the same way, skilled composers may have relegated structure, even on a monumental scale, to their intuition.

An account, by Arnold Schoenberg, of his own compositional-process for his Kammersymphonie serves as a striking example of this idea:

From my own experience I know that it [the device of mirroring forms] can also be subconsciously received from the Supreme Commander. After I had completed the [Kammersymphonie] I worried very much about the apparent absence of any relationship between the two themes. Directed only by my sense of form and the stream of ideas, I had not asked such questions while composing; but, as usual with me doubts arose as soon as I had finished. They went so far that I had already raised the sword for the kill, taken the red pencil of the censor to cross out the theme b.

Example 108. Schoenberg's analysis of his themes from the Kammersymphonie, Op. 9

Fortunately, I stood by my inspiration and ignored these mental tortures. About twenty years later I saw the true relationship. It is of such a complicated nature that I doubt whether any composer would have cared deliberately to construct a theme in this way; but our subconscious does it involuntarily. In c [Example 108] the true principal tones of the theme are marked, and d shows that all the intervals ascend. Their correct inversion produces the first phrase f of the theme b. It should be mentioned that the last century considered such a procedure cerebral, and thus inconsistent with the dignity of genius. The very fact that there are classical examples proves the foolishness of such an opinion.... To the imaginative and creative facility, relations in the material sphere are as independent from directions or planes as material objects are,in their sphere, to our perceptive faculties. Just as our mind always recognizes, for instance, a knife, a bottle or a watch, regardless of its position, and can reproduce it in the imagination in every possible position, even so a musical creator's mind can operate subconsciously with a row of tones, regardless of their direction, regardless of the way in which a mirror might show the mutual relations, which remain a given quantity, {Schoenberg, 112-113}

This statement reveals the importance that Schoenberg, as a composer, placed on matters of formal symmetry and his distrust of intuition.

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