Abstractions Galore

[ChrisEPhifer]

I think there are a few places where we might benefit from writing a type class. I have some comments on this in #12, in particular about a proposed Invertible type class for objects admitting a sensible notion of inversion (which is... a lot of stuff.)

In particular, I would propose that the following type class be defined among our abstractions:

class Invertible a where
  -- invert must be involutive. That is, `(invert . invert) x == x` for every `x :: a`.
  invert :: a -> a

We can get a whole bunch of leverage out of this by defining some instances over data structures we care about, too, eg:

instance (Functor f, Invertible a) => Invertible (f a) where
  invert = fmap invert

instance Invertible a => Invertible (Set a) where
  invert = Data.Set.map invert

It would be worth thinking about whether or not there are other opportunities for ad hoc polymorphism; this ticket can be used for discussion about that.

[ChrisEPhifer]

Per the discussion in #19, I'm going to work on this (plus a couple of other refactors/abstraction efforts) in a new branch, typeclass-refactors.

[edwisdom]

Sounds good! Let me know if there's any instances we're missing of functions that you'd like me to write, e.g. invert for chords or transpose for intervals, etc.

[ChrisEPhifer]

Closing since we've merged the other work / are aware of some of the other opportunities for ad hoc abstraction.