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Counterpoint is, perhaps, the oldest tradition of systemized mathematics and computation in music. Moreover, contrapuntal devices such as augmentation, imitation, inversion, etc., are both products and remnants of mathematical consciousness in musical composition. These techniques are, in many ways, the musical equivalent of simple algebraic functions applied to the prime form of a melody. It should be noted that traditional forms of counterpoint, such as the round and the fugue, are concerned with independent statements of the subject and the permutations of that subject. However, another type of canonic counterpoint called Homosynchrono counterpoint is the simultaneous occurrence of a prime subject/theme/melody with one or more mathematical permutation of itself. Homosynchrono retrograde would be defined as the simultaneous occurrence of a melody and its retrograde. Similarly, Homosynchrono inversion is the simultaneous occurrence of a melody and its inversion (melodic inversion as harmonic inversion results in double counterpoint), and so forth. Furthermore, there is always the possibility of Homosynchrono compounds (i.e. Homosynchrono retrograde inversion).
While all contrapuntal functions can be simplified as basic algebraic functions, the mathematical representation is of very little consequence aside from a means to generate new material. In Homosynchrono counterpoint, however, these functions are necessary to understand and compose the music; we must discover the proper algebraic functions in order to write Homosynchrono counterpoint that is in agreement with itself. Each Homosynchrono technique has its own system of functions and compositional processes, which can be combined to formulate an “algorithm”: a series of steps and mathematical processes through which a Homosynchrono melody/theme/subject can be composed. Furthermore, canonic counterpoint is “pure” enough (i.e. strictly obeys one function) as to be represented by algebraic formulae.
In systemizing the tonal and modal composition of Homosynchrono counterpoint the need for a system of transliteration between melody and abstract algebraic structures became obvious. While musical set theory is able to categorize unordered pitch series, no adequate system for set notation of melody has been develop. In counterpoint the melody is the seed from which all other musical events sprout. It is necessary in function/formula driven counterpoint that the objects of manipulation (the sets) retain the temporal quality of the melody. I propose that, “Melodies can be represented using a combination of musical set notation, time series analysis notation, abstract algebraic structures (namely, Zermelo-Fraenkel set theory), and, moreover, discrete or finite mathematics in general.” We shall call this Discrete Melodic Set Notation, and the sets of concern will be referred to hereafter as Discrete Melodic Sets (DMS).
Additionally, there are a few deviations from both standardized set and time series theory. Aside from utilizing the natural numbers from 0-11 to notate intervals, representation of melody relies more on discrete mathematic and algebraic principals than the set theory codified by Allen Forte. In fact, almost all that we assume to be true about musical sets must be forgotten, reworked, rethought. The largest deviations from musical set theory and standard mathematical axiomatic set theory have to do with the members of the sets, their progression through time (ordinality), and, more broadly, how the sets are notated and interact. To some these discrepancies may seem to be quite a large leap of faith given the practices of set theory. But I assure that these differences are a necessary part of applying a set mentality to the composition of mathematically derived contrapuntal music. Furthermore, the deviations from set theory arise out of the nature of tonal music. In fact, there are many aspects of music that set theory is incapable of analyzing, such as linear melody. I realize that I may be met with some opposition for taking this position, but I simply mean to say that current set theory deals only with unordered composites, and it does not concern itself with the real world application of counterpoint.
At this point I would like to qualify the vantage point from which this work has been conducted. The ultimate goal of discrete melodic set is to provide the composer with a foundation for composing advanced forms of counterpoint. The purpose of my work is not to systemize counterpoint to the point where there is no longer any human creativity or artistic choice. After all, the entire theory rests upon the assumption that we already have a melody/subject/theme that we would like to morph. The initial melody, and the selection process, which occurs in the output stage of the compositional process, are all driven by the composer’s interest, preference, purpose, and taste. Furthermore, the adaptation of mathematical concepts has been performed with sensitivity to musical perception. ZFC set theory has not been arbitrarily thrown out the window, rather, axioms have been rethought and new ones introduced to form a musically geared system of abstract algebra.
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