alun_clewe raised some interesting points in the
musical algebra thread. So I'm redefining things a bit to account for the flaws he pointed out.
Instead of using pitch, defined as frequency, I'm using tone, defined as number of scale steps above any reference pitch. So a motive doesn't actually define a series of absolute pitches, but a series of relative intervals. The reference pitch must be defined for performance (for example, modern Western tuning uses A above middle C = 440Hz, but this hasn't always been the case) but it's irrelevant for the algebra, where it corresponds to a tone of 0. This is in line with
pitch set theory as well.
I forgot to address timing in multiplication last time too. Here's how multiplication works now: the product of two notes is a note with timing equal to the product of its factors' timings, duration equal to the product of its factors' durations, and tone equal to the
sum of its factors' tones. I believe this works out the way I need it to, so canon can be defined as the product of two motives. Augmentation and diminution can be defined as multiplication by a note of tone 0 and duration of either more than or less than a beat, respectively. Simple transposition can be defined as multiplication by a note with a tone other than 0 and a duration of exactly one beat.
I should note that what I call vertical inversion is typically just called inversion, and what I call horizontal inversion is typically called retrograde.
Another operation is concatenation: following one motive by another. A concat B would be the sum of A, and B times a note with tone 0, duration of 1 beat, and timing that is the sum of the timing and duration of the final note in A.