Combinatorial Spaces

Document Type

Conference Proceeding

Publication Date

1-1-2022

Abstract

Combinatoriality—the property that obtains when unions of corresponding subsets within tone rows comprise aggregates—takes various forms, following the canonical operations that relate the constituent rows to one another: transposition, inversion, retrograde, and/or retrograde inversion. The mathematical field of combinatorics presents tools to answer such basic questions as: How many combinatorial sets exist in a space of a given size? To how many equivalence classes do they belong? Such enumeration procedures involve various techniques that have prior connections to music theory. In the process of answering these questions, our results reveal further aspects of combinatorial sets. For instance, no combinatorial n-chords are held invariant by a translation operation with an odd index. The set of I-invariant n-chords that are P-combinatorial is equivalent to the set of those that are I-combinatorial, and this set is precisely the set of all-combinatorial n-chords. Such information sheds new light on these intriguing structures.

Publication Source (Journal or Book title)

Lecture Notes in Computer Science Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics

First Page

48

Last Page

60

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