  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.

General Mathematics