Inductive Proofs and an Algebra of Music

Due: Monday Oct 20 at noon.

Read Chapters 8 and 9 of HSOM.

1. Prove that dur (revM m) = dur m.
2. Prove Axiom 9.3.4.
3. Prove Axiom 9.3.7.
4. Haskore permits a lot of flexibility in the interpretation process, specifically through a player.  But the axioms in Chapter 9 are supposedly true for all PMaps and Contexts, in other words for all possible players.  Explain how this can be.  For an extreme case, suppose that a player completely ignores a certain sequence of notes -- how can the axioms still hold, even when other players do not ignore this sequence of notes?  Can you think of an axiom that one might like to hold, but that in fact does not, because of the presence of players?

In addition, grad students should:

1. Prove Lemma 9.1.1.  Note that this relates somewhat to problem (4) above -- you will probably need to make an assumption regarding the way in which players interpret music.  State any such assumptions in your proof.
   Note:  The lemma as stated in the text is wrong!  It should be:
   For all pmap, c, and m:
   perf pmap c m = (perform pmap c m, dur m * cDur c)