[microsound] category theory and composition

[Andrew Salch]

On Wed, 23 Dec 2009, Justin Glenn Smith wrote:
>> How do we regard a genre of composition as a category? What are the
>> objects and the morphisms of a genre of composition?
>>
>
> I was imagining that the objects of a genre of composition would be the 
> individual compositions. And the morphisms would be whatever 
> transformations are appropriate to that genre (arrangements, 
> orchestrations, samplings, remixs, covers, each of these having sub 
> variations remix->chop and screw, etc.). Each of these objects would be 
> its own category, with key signatures, samples, instrumentation etc. 
> etc. as its objects.
>
> Thus what I was imagining was a Functor allowing the usage of the 
> material in another genre without losing structure, by remapping to a 
> new category (thus in one mapping of categories for example the point in 
> time in which a sound occurs may be mapped to the timbre in another 
> domain). I was imagining I would allow myself the freedom to potentially 
> invent a new genre (or if I was lucky, happen upon an existing one), by 
> just letting the new genre be whichever one my deterministic mappings 
> created. And reinventing those mappings until I get a result that 
> somehow pleases me.


Let's suppose what you describe is possible, that to every genre G we can 
associate a category C(G) with the following properties:

-the objects of C(G) are the compositions possible within the genre G.

-the morphisms from an object X of C(G) to an object Y of C(G) are the 
possible ways of transforming the composition X into the composition Y by 
a (finite?) sequence of rearrangements which are permitted within that 
genre.

-composition of morphisms is given by composing the rearrangements.


None of this is very precise and it would need to be spelled out in more 
detail. In any case, given genres G and H you could probably work out a 
way of associating a rearrangement in H to every rearrangement in G, and 
get a functor from C(G) to C(H) in this way. A simple example might be to 
let G be the "genre" (speaking loosely) of compositions consisting 
entirely of a single manipulated sine wave, while H could be the "genre" 
of compositions consisting of manipulations of at most two sine waves; 
clearly all compositions and rearrangements possibly in the genre G are 
also possibly in the genre H, so one has a full and faithful functor from 
C(G) to C(H).

Here is another example: let G be the "genre" consisting of all pieces of 
music which can be described in some standard format, say, as a .MID file, 
which the morphisms are given by transposing any number of notes in a .MID 
file; and let H be the "genre" consisting of all 8-bit mono .WAV files, 
with morphisms given by changing the values of any number of bits in the 
.WAV files. Rendering the .MID file as a .WAV file using MIDI player 
software is one functor from C(G) to C(H), but simply appending a .WAV 
header to the .MID file (getting something which, when played, sounds like 
digital noise) is another, very different, functor from C(G) to C(H).

The idea of beginning with established genres G and H, constructing some 
functor from C(G) to C(H), and considering the image of that functor in 
C(H) as a new genre, this might be interesting but only if you can cook up 
some interesting functors, i.e., some interesting ways of transforming 
acceptable rearrangements in one musical genre into acceptable 
rearrangements in another genre.