Qualified Types and Musical L-Systems

Due: Wed Oct 29 at noon.

Read Chapters 10 and 11 in HSOM.

1. We have worked with several different Music types, such as Music Pitch,  Music (Pitch,Volume), and  Music (Pitch,[NoteAttribute]).  In the end though we usually want our music values to have type Music (Pitch,[NoteAttribute]), and we find ourselves having to write coercion functions to do this.  What would be nicer is the following:
   
   Define a type class called StdMusic having a single method:
   toStdMusic :: Music a -> Music (Pitch, [NoteAttribute])
   such that toStdMusic can be applied to any m :: Music T, as long as T is an instance of StdMusic, in each case yielding a value of type Music (Pitch, [NoteAttribute]).  Then define instances of StdMusic for Pitch, (Pitch, Volume), and (Pitch,[NoteAttribute]).
   
   The nice thing about this design is that if we have n types we no longer need to remember n-1 coercion functions -- we can just use toStdMusic, and in that sense you can view this as an overloading of the name.
   
   Note: to get this to work in GHC you will need to start GHCi with certain type extensions enabled, like this:
   > ghci -XFlexibleInstances -XTypeSynonymInstances
    
2. Now define a type class called PrimEquiv with one method:
   (===) :: a -> a -> Bool
   such that x === y tells us whether or not "primitive musical values" x and y are "equivalent" from a "musical point of view".  For example, we would expect all of the following to be True:
   Cs === Df
   Note qn Cs === Note (1/4) Df
   cs 4 qn === df 4 qn
   Note qn (Cs,[Volume 5]) === Note qn (Df, [Volume (2+3)])
   To get all this to work, you will need to define instances of PrimEquiv for PitchClass, Pitch, Primitive a, (PitchClass,[NoteAttribute]), and Music a, although for the latter it should only work for primitive music values (i.e. an error should occur if non-primitive values are compared).
    
3. Define an L-System grammar to create some "interesting" music using the functions defined in Chapter 10.
    
4. Extend the L-System design in Chapter 10 to include a parallel construction; i.e. add a constructor (:=) to the LSys data type.