Two Musical Orderings

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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.

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Most cited references 7

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The geometry of musical chords.

Dmitri Tymoczko (2006)

A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by using short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation. Paradigmatically consonant and dissonant chords possess different near-symmetries and suggest different musical uses.