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# An algebra of music

Despite being able to play only the drumset even sub-semi-competently, I find music theory interesting. It's fun to play around with.

For a while now I've been fiddling around off and on with the idea of an algebra of music, sort of as an extention to the classical idea of "canon", in which an entire piece of music is described by a single written theme and a rule that describes how it repeats and changes. I was inspired by reading about fugues, and saw that the pattern in which a theme recurs, transposed, can be seen as another melody (think of the pattern formed by the first note of each instance of the theme), and that this combination of two melodies (theme and fuguing-melody) is conceptually a lot like multiplication.

My "theory" (I'll call it that, despite the fact that it doesn't seek either to describe existing music or to proscribe future works) is based on the note. This is the most basic unit: an entity with the properties of pitch, duration, and timing. Pitch represents the physical frequency of the sound, which is technically real-valued, but we'll generally limit ourselves to a tuning: a pattern of pitch classes (C# is an example of a pitch class) that repeats at the diapason (usually the 2:1 frequency ratio, a.k.a the octave, but in the case of the Bohlen-Pierce scale the 3:1 frequency ratio). In the case of tunings with a finite number of pitch classes, it can be represented with an integer. Duration is a real number specifying the length of time the note is to be sounded in terms of beats. Timing is a nonnegative real value that represents when the note is played in terms of beats from the beginning of the motive.

A motive is a snippet of written music. It's a space of notes, like a set except that it can contain more than one, or less than zero, instances of a note. Addition is easy enough to define: A + B = C, such that C contains all of the notes in A and B. In other words, it just superimposes A on B. Subtraction just reverses the process: to get the result of A - B, remove every note in A that corresponds to a note in B. Problems come up when B contains notes that don't appear in A: in that case, we say that the result has negative one of that note (yeah, you can't play music with negative notes. Shut up).

Multiplication is a little weirder. First off, let's define multiplication between individual notes: multiply their respective pitches (in terms of frequency; since we'll be working in terms of steps within a tuning, and that's logarithmic, it'll look like addition to us) and durations, and add their timings. We can then define a motive multiplied by a note as the motive produced by multiplying each note in the original motive by the "scalar" note. The product of two motives A and B would be the sum of products of motive A with each note in motive B.

Lots of operations can be defined on these. Horizontal inversion would reverse the order of notes. Vertical inversion would reverse the direction of intervals between notes ("flipping it upside down", if you were looking at written score). Horizontal rotation by duration would add a given amount to the timing of each note in a motive, modulo the length of the motive, such that a note whose duration overlaps the end of the motive would "wrap around". Vertical rotation by interval  would raise the pitch of all notes in a motive by a given amount (usually expressed as a number of tuning steps), modulo the closest multiple of diapasons equal to or greater than the distance between the highest and lowest notes in the motive (bonus points if you can tell why). Horizontal rotation by notes would add the timing of the last note in the motive minus the timing of the penultimate note to all other notes and set the last note's timing to 0 (making it first), a given number of times. Vertical rotation by notes would transpose the highest n notes down by octaves until they were the lowest n notes (note that, if all of the notes have the same timing, this is equivalent to "inversions" of a chord in traditional theory).

The idea is that, given a suitable theory of consonance and dissonance, a composer could find solutions to compositional problems by writing an equation describing the problem and solving for the unknown. For example, given an existing motive , a composer might want to find a shorter motive he can use as an ostinato bassline that, if added to the known motive, would be within a certain range of consonance.

A few open questions:
• How do you define division? Is it possible to define division so that the existing definition of motives is closed under division?
• Is there a better way to define this stuff?
• What other operations would be useful to define?
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