Two Musical Orderings

TL;DR

This paper explores the application of partial orders in music theory, analyzing chord and scale structures and modeling timbral brightness, with practical comparisons and sound design implications.

Contribution

It introduces novel partial order models for musical structures and timbre, providing new insights and computational tools for music analysis and sound design.

Findings

Familiar chords and scales are extremal elements in set inclusion orders.

The brightness order aligns with intuitive perceptions of instrument timbre.

Linear programming can efficiently solve sound design problems based on brightness ordering.

Abstract

We make some general observations about partial orders on quotient spaces, and explore their use in music theory, in two different contexts. In the first, we show that many of the most familiar chord and scale types in Western music appear as extremal elements in the partial order induced by set inclusion on pitch class sets of Tn-type. In the second, we propose a partial order that models the brightness aspect of timbre. We use this order to compare the brightness of six wind instruments, and find that the results conform to intuition. We also use the order to pose sound design problems of a certain type, which can be solved efficiently using linear programming.