Struggling Towards an Algebraic Theory of Music

For the last few months, I’ve been trying to come up with a nice, denotational basis for what music is. But I’m running out of steam on the project, so I thought I’d write what I’ve figured out, and what I’ve tried but doesn’t work. Hopefully this will inspire someone to come tell me what I’m being stupid about and help get the whole process unstuck.

What Music Is Not

It’s tempting to gesticulate wildly, saying that music is merely a function from time to wave amplitudes, eg something of the form:

μ Music = Time -> Amplitude

While I think it’s fair to say that this is indeed the underlying denotation of sound, this is clearly not the denotation of music. For example, we can transpose a song up a semitone without changing the speed—something that’s very challenging without a great deal of in the waveform representation. And we can play a musical phrase backwards, which is probably impossible in a waveform for any timbral envelope.

Since we have now two examples of “reasonable to want to do” with musical objects, which cannot be expressed in terms of a function Time -> Amplitude, we must conceed that waveforms-over-time cannot be the denotation of music.

What Music Might Be

Music is obviously temporal, so keeping the “function from time” part seems relevant. But a function from time to what? As a first attempt:

data Note = Note Pitch Timbre Volume

μ Music = Time -> Set (Duration, Note)

which, for a given time, returns a set of notes starting at that time, and how long they ought to be played for. An immediate improvement would be to parameterize the above over notes:

μ (Music a) = Time -> Set (Duration, a)

It’s tempting to try to eliminate more of the structure here with our parametricity, but I was unable to do so. In contrapuntal music, we will want to be able to express two notes starting at the same moment, but ending at different times.

One alluring path here could to write monophonic voices, and combine them together for polyphony:

μ (Voice a) = {- something like -} Time -> (Duration, a)
μ (Music a) = Time -> Set (Voice a)

Such an encoding has many unfavorable traits. First, it just feels yucky. Why are there two layers of Time? Second, now I-as-a-composer need to make a choice of which voice I put each note in, despite the fact that this is merely an encoding quirk. So no, I don’t think this is a viable path forward.

So let’s return to our best contender:

μ (Music a) = Time -> Set (Duration, a)

This definition is trivially a monoid, pointwise over the time structure:

μ (Music a <> Music b) = Music (μ a <> μ b)
μ mempty = Music mempty

If we think about abstract sets here, rather than Data.Set.Set, such an object is clearly a functor. There are many possible applicatives here, but the pointwise zipper seems most compelling to me. Pictorally:

pure a
=
  |----- a ----forever...


liftA2 f
  |---- a ----|-- b --|
  |-- x --|---- y ----|
=
  |- fax -|fay|- fby -|

Such an applicative structure is quite nice! It would allow us to “stamp” a rhythm on top of a pure representation of a melody.

However, the desirability of this instance is a point against μ (Music a) = Time -> Set (Duration, a), since by Conal Elliott’s typeclass morphism rule, the meaning of the applicative here ought to be the applicative of the meaning. Nevertheless, any other applicative structure would be effecitvely useless, since it would require the notes on one side to begin at the same time as the notes on the other. To sketch:

-- bad instance!
liftA2 f (Music a) (Music b) =
    Music (liftA2 (\(d1, m) (d2, n) -> (d1 <> d2, f m n)) a b)

Good luck finding a musically meaningful pure for such a thing!

Ok, so let’s say we commit to the pointwise zippy instance as our applicative instance. Is there a corresponding monad? Such a thing would substitute notes with more music. My first idea of what to do with such a thing would be to replace chords with texture. For example, we could replace chords with broken chords, or with basslines that target the same notes.

Anyway, the answer is yes, there is such a monad. But it’s musically kinda troublesome. Assume we have the following function:

notes :: Music a -> [(Maybe Interval, a)]

which will convert a Music a into its notes and an optional temporal interval (optional because pure goes on forever.) Then, we can write our bind as:

m >>= f = flip foldMap (notes m) \case
  (Nothing, a) -> f a
  (Just (start, duration), a) ->
    offset start $ _ duration $ f a

where offset changes when a piece of music occurs. We are left with a hole of type:

Duration -> Music a -> Music a

whose semantics sure better be that it forces the given Music to fit in the alotted time. There are two reasonable candidates here:

scaleTo  :: Duration -> Music a -> Music a
truncate :: Duration -> Music a -> Music a

where scaleTo changes the local interpretation of time such that the entire musical argument is played within the given duration, and truncate just takes the first Duration’s worth of time. Truncate is too obviously unhelpful here, since the >>= continuation doesn’t know how much time it’s been given, and thus most binds will drop almost all of their resulting music.

Therefore we will go with scaleTo. Which satisfies all of the algebraic (monad) laws, but results in some truly mystifying tunes. The problem here is that this is not an operation which respects musical meter. Each subsequent bind results in a correspondingly smaller share of the pie. Thus by using only bind and mconcat, it’s easy to get a bar full of quarter notes, followed by a bar of sixty-fourth notes, followed by two bars full of of 13-tuplets. If you want to get a steady rhythm out of the whole thing, you need a global view on how many binds deep you’re ever going to go, and you need to ensure locally that you only produce a small powers-of-two number of notes, or else you will accidentally introduce tuplets.

It’s a mess. But algebraically it’s fine.

What Music Seems Like It Should Be

The above foray into monads seems tentatively promising for amateur would-be algorithmic composers (read: people like me.) But I have been reading several books on musical composition lately, and my big takeaway from them is just how damn contextual notes are.

So maybe this means we want more of a comonadic interface. One in which you can extend every note, by taking into account all of the notes in its local vicinity. This feels just as right as the monadic approach does, albeit in a completely different way. Being able to give a comonad instance for Music would require us to somehow reckon with having only a single a at any given time. Which appeals to my functional programmer soul, but again, I don’t know how to do it.

But imagine if we did have a comonadic instance. We could perform voice leading by inspecting what the next note was, and by futzing around with our pitch. We could do some sort of reharmonization by shifting notes around according to what else is happening.

But maybe all of this is just folly.

Music as it’s actually practiced doesn’t seem to have much of the functionaly-compositional properties we like—ie, that we can abstract and encapsulate. But music doesn’t appear to be like that! Instead, a happy melody takes a different character when played on major vs minor chords. Adding a dissonant interval can completely reconceptualize other notes.

It feels like a bit of a bummer to end like this, but I don’t really know where to go from here. I’ve worked something like six completely-different approaches over the last few months, and what’s documented here is the most promising bits and pieces. My next thought is that maybe music actually forms a sheaf, which is to say that it is a global solution that respects many local constraints.

All of this research into music has given me much more thoughts about music qua music which I will try to articulate the next time I have an evening to myself. Until then.

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