After three articles with symmetry in their titles, there remains little sense that the real significance of symmetry per se has been imparted. Whether it is mathematics, the perception of symmetry, or cycles & fashion, there is always a tangent to discuss. This is a place to collect some symmetry-specific thoughts, and perhaps further illuminate what is such a core element of the way both the universe operates and humans perceive. Cycles are a type of symmetry, with one wave following another, and so any remarks on cycles are a subset of those applying to the more general idea of symmetry. Likewise, mathematics is created to describe symmetries, where even counting is based on the very simple symmetry of linear repetition. Finding the symmetries or cycles in complex phenomena such as the weather has been a human undertaking since the beginning of history, an ongoing attempt to make sense of changes and bring them into the realm of human understanding. The seasons have provided a ready starting point, but investigations go further in the differential equations of today, and the quest to predict the weather exactly. Once a symmetry is established, it can be the surest knowledge we have, and it is an understanding of symmetry which is inherent to human creation.
This understanding need not be explicit, of course. Indeed, various postmodern styles have removed the explicit act of composition from the creative process, forcing us to wonder whether there are resulting symmetries to be perceived. As discussed in the "relationships" article, the idea of "note" itself, as well as other aspects of music, is grounded in symmetry relationships. There are always symmetries to perceive, and indeed perception would be impossible without them. One can go on to ask if perceived symmetries in a random composition mean anything, and most may mean something only about the listening mind (not a mean undertaking), but symmetry as the basis for sound does mean something. Perhaps future composers (and "noise" composers already attempt this) will find a way to undo it, creating vibrations so random that no note or overtones can be perceived. Although such experiments can provide lessons and increase our understanding, it is symmetry as consciously utilized which remains the primary concern of most musical exploration. It may not be used consciously as symmetry, but it is in intended structures and correspondences that music finds its greatest glory of expression, and those correspondences have their basis in symmetry, whether conceived at that level or not. Melodies have a symmetry about them, as does harmony. Although not the same, serialism has a very definite sense of symmetry, a more mechanized sense which takes notes in isolation, rather than grouping them by overtone relations or immediate repetition.
Entire musical structures are built on symmetry, not only in serialism, and not only via the remarkably persistent phenomenon of refrain, but through cadential expectations and motivic development themselves. Much as postmodern styles seek to thwart expectations, traditional styles thrive by interacting with listener expectations in concrete ways. The idea of cadence as a stopping point is necessary to any concept of non-infinite music, but the idea of cadential motion as conditioned by specific musical factors is not. It involves an act of communication, however, letting the listener know a stopping point has arrived. As stopping points are regularly marked, they form symmetries, and these symmetries have been among the most basic in the evolution of Western musical style. Even serial technique has not (or at least not often) explored the idea of systematically varying cadential motion. Likewise, as polyphony changed from what was essentially decorated melody, and as literal repetition in refrain seemed too uncreative for some circumstances, the coherence of individual pieces was maintained by varied repetition & interplay of motifs and themes. After numerous earlier prototypes, this sort of abstract symphonic thought was consummated in the Franco-Flemish era. In the work of the Josquin generation, motivic development became the stuff of musical creation, an emphasis which has had its ebb & flow through history, but one decisively reanimated as least as recently as the serialist movement. Development was then the basis for musical apprehension & communication.
Manipulating musical motifs, varying but ultimately repeating them, is about interpreting time itself. It connects moments in the past with moments in the present, perhaps inverted, perhaps in different proportion. It points to different parts of a work, which in turn point elsewhere or to themselves again. It yields a multidimensional geometry of ideas relating to each other, perhaps projecting forward or back to something outside themselves, perhaps not. Determining different ways to combine motifs is an algebraic process, built on symmetry relations, but a process the ultimate success of which depends on something other than mathematics. One must interact with the constraints of human perception, notions of artistry, and ultimately illustrate something other than that one plus one equals two. Indeed, the great composers seem to have intuited most of these relationships, based at least partly on the depth to which symmetry permeates our lives. It is found not only in nature at large, but in the human body, and it is the nature of human thought which conditions not only what our musical notes are, but how large a musical motif can reasonably be or how convoluted its transformations. Although specialists, or even fans, can study a work and determine more intricate motivic transformations, we usually demand a sense of "classical" balance in our music, insisting that it not encompass too much or illustrate too many transformations. One can consider this an argument against a life of excess, and so it is by generally communicating thought processes or ideals that "musical algebra" has reached its stature.
Musical development certainly meets something of its success in what is not done, letting alternate possibilities fill the mind and suggest an array of parallel paths not taken. There is a feedback process at work in the act of apprehension, as one's own thoughts interact with the musical development at hand. Indeed, once a transformation has been attempted, once a particular symmetry is illustrated, it necessarily broadens the pallet of future creation, even if not used. In this sense, there is progress in art, although not the progress of industrialization. It is also a part of the weight contemporary composers feel. Composers of earlier eras often sought to develop their ideas through subtlety, while it was only the modern era which attempted the mechanization of time. Whereas one almost has the sense that modernist works are appraised by a checklist of which symmetries they observe, whether a composition seems to arise spontaneously or not is at least as important a criterion. Organic time and its inherent symmetry, brought as it is to the intellect not by force but by nature, is among the most communicative settings we know. Symmetries are then not only about making sense of, or controlling, the world, but about simply living one's life.