Symmetry relationships

Given that mathematics is most idiomatic for describing symmetry relationships, it is only too easy to get hung up on discussing notions of mathematical determinism and never get to the symmetries themselves. In the wake of the previous article's remarks on reality and indeterminate forms, it is natural to take up the subject of symmetry again. One explicit technique by which some modern composers have hoped to illuminate their deeper philosophical ideas is through particular symmetries. In some cases, especially retrograde melodic inversion, the symmetries can yield a static quality which the composer presumably hoped to achieve. For instance, composers such as Messiaen use this technique extensively, to the point of constructing giant symmetric building blocks for entire compositions. I want to begin by discussing why this method is not very effective.

Although it is certainly possible to hold an entire piece of music in one's mind, and regard it as a full entity as one would a painting, that is not the inherent nature of the medium. Music is laid out in time, and perceived in time. Even if one feels comfortable regarding an entire piece as described, it must first be perceived in time. This is as distinct from the processes of the composer, who might conceive aspects of the whole, and thus be tempted to try to recreate that vision directly. However the process of perception interferes with that idea, because one's memory is likewise time-dependent. Therefore there is no direct way to relate such a unitary perception (except visually, for ideas conducive to that expression), and straightforward symmetry statements can actually serve to blunt any underlying unity in the final perception through the distortion of time-based apprehension. This is also true for prose. To put it bluntly, the audience is bored by repetition. A simple inversion is not sufficient variety, especially because it is laid out on the same time scale. Progressive diminution is the most basic means of restatement, because subsequent statements of equal length will actually seem longer than the original. This fact is perhaps expressed most clearly by the common role, whether consciously or unconsciously, of "golden section" recapitulations. I claim no special preeminence for that placement, but it is one simple form of time relationship which works.

These remarks cannot be fully applied to smaller-scale elements. For instance, the refrain in most song-forms is a direct repetition, yet it is effective indeed. The refrain serves as a point of departure, and remains effective by virtue of its relatively small scale. This is in contrast to attempts to use similar repetition schemes on larger elements. Indeed there are clear practical limits of size to a refrain functioning as such, based essentially on the idea of a "single statement" or a "point" in the Renaissance sense, and similar lengths hold across cultures. The length of the refrain was already taken to its limits in the fifteenth century chanson, and with it the development of those formal schemes. Such a simple factor was as much responsible for the subsequent dramatic change in art song writing as any other. Popular songs continue similar forms and similar length constraints today. Even here, we frequently see such "points" placed in different harmonic or rhythmic guises, especially combinations via syncopation in the more advanced settings. So the applicability of simple symmetry relationships such as repetition is closely related to perceptual scale, and it becomes less easy to indicate something on a larger scale by virtue of such relationships. In short, any vision of reality is necessarily mediated by mundane time when transcribed from its indeterminate form, and music at the highest level cannot hope to make light of this constraint.

Symmetry breaking

Alongside the critique of large-scale symmetry, it is time to turn to the smallest scale. Let us vaguely denote the harmonic series as a symmetry relationship, such that a fully symmetric sound would simply duplicate the entire series unaltered. Of course real objects do not reproduce the full unaltered series, as they can be not only truncated but altered in their frequency ratios. This is basic physics, but the point is that this alteration from a full series reflects a direct transformation of a vacuum into the object itself. One can perhaps return to the topic of dualism and indeterminate forms to realize how it is symmetry-breaking which instantiates the physical world. Returning to music, even the simplest piece (Scelsi aside) makes use of more than a single sound and the corresponding series of its sounding body. Subsequent sounds, forming to either melody or harmony, serve to accentuate certain pitches of this series, thus breaking the symmetry further. In some cases, the geometric relationship between subsequent sounds is so distant that symmetry is reduced essentially to a point. This is the basis of some modern theories, and of course reduction in symmetry opens a greater variety of possibility. Even for most monophonic music, such as Indian classical music, subsequent notes relate in precise ways to previous ones, and specific symmetric (or tetrachordal, if you will) relationships can be prescribed.

In all these cases, the main point is clear. Symmetry relationships are so strongly linked to the act of perception itself that they cannot be used as a straightforward expression. They can work only as mediated by perception, and so only in an imperfect form of a scope which straddles perceptual divides. This is directly related to the fact that only a relatively small range of stimulation densities are apprehensible by ordinary humans, so it should come as no surprise. Can symmetric relationships outside this scale have a subconscious effect? It is a good question, and perhaps unanswerable. Nonetheless, my inclination is to say no, the subconscious produces affects only via repressed stimulation which passes through the higher-level mind. More significantly, I believe that one must also feel something directly in order to produce the sort of cascade effect which actually alters thought processes. In this era of excess stimulation, the last thing one needs is to be immersed in something else which cannot be apprehended. Such a thing brings only befuddlement, whereas the difficult but perceivable brings the twin benefits of apprehension and accomplishment.

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Todd M. McComb