Topic: A post-tonal method of music composition
If you compose some music or would like to compose some music, you might be interested in the following article "A post-tonal method of music composition" that I wrote recently. It is a synthesis and generalization of three articles that I published in the Journal of Mathematics and Music. The text is intended for readers who have minimal knowledge in abstract algebra or music theory. It is freely available at:
https://hal.science/hal-05425426v1
Title: A post-tonal method of music composition.
Abstract: This monograph describes a method to design chord progressions and structure compositions from the intrinsic properties of artificial scales. A concise introduction to the theory of pitch class sets (pc sets) in any equal temperament is provided. Subsequently, the concepts, results and algorithms are developed with abstract sets. The concepts of transformation (transposition or inversion) and associated equivalence between pc sets were extended to any permutation group on a finite set. Four relations between sets of cardinality n (n-sets) were considered: equivalence through transformation, parsimony (quasi-identity), mildness (equivalence or parsimony) and fuzziness (quasi-equivalence). The results showed that these relations are symmetrical and compatible with any transformation. Therefore, a parsimonious, mild or fuzzy sequence of n-sets exists in four main forms that keep its properties: direct, retrograde, inverted, and retrograde inverted. Circular non-redundant exhaustive parsimonious sequences of the n-subsets from a p-set can be derived for any finite values of p and n ≤ p. If a p-set is invariant by an involution (e.g. an inversion), its n-subsets can be partitioned into two non-redundant parsimonious sequences that are related by the involution and have only the invariant n-subsets in common. The n-subsets and their sequences can be represented in a two-dimensional table of the (n – 1)-subsets of the p-set versus the set classes (for the relation of equivalence) of its n-subsets. Numerous examples of application to the n-chords of a p-scale are given in the 12-tone and 24-tone equal temperaments (TET). The practical consequences for music composition are extensively discussed.
