Will Peterson -- Countin' on the 2 and 4

  This is a very interesting and well-written paper, though it's still rather 
  speculative and the mathematical parts are rough and ``in progress.''  I agree 
  that the combinatorics of rhythms may be an interesting way to think about 
  improvised music (and even some non-improvised music, after the fact of 
  composition).  I think a lot of the mental processes involved in improvisation 
  are probably analogous to forms of counting and "fitting in" units into an overall 
  structure, so a sophisticated understanding of the possibilities is probably part 
  of the toolkit of skilled improvisers (at least on an intuitive level).  

  The Homer seminar I sat in on with Prof. Ebbott in Classics this semester convinces
  me that something like this is going on because of the way Greek dactylic hexameter
  (the meter of the Homeric epics) works.  Most people think these poems were composed 
  orally in performance by a somewhat similar process before they were written down.  

  I think it's great that you devised your own topic this way; although I think
  it turned out to be more complicated than either of us saw at first, I think you
  are "onto" something here.

  Specific Comments:

  0)  Pages 1 and 2:  Agreed that Schenkerian analysis cannot be the whole story.
      It's too much tied to the notions of the Western tonal harmony system 
      and it aims for "big picture" understanding of a piece of music rather 
      than understanding of details (it's a question of what scale you want
      to look at).  

  1)  Page 8:  "For example, a half note in position four opens up a set of six sixteenth 
      notes on the left and six sixteenth notes on the right."  

      > If the half note begins on the fourth sixteenth note of the 4/4 measure, 
        doesn't that mean there are *three* sixteenth note slots on the left and
        *five* on the right?  (The "left-over" part of the measure has to add 
        up to two quarter notes = eight sixteenth notes.)  This is corrected later
        on for the equation (3.1) so I think it's an editing problem(!)
        I can see what equation (3.1) is doing, but it would be good to 
        explain a bit more about what the notation q_3, q_5, etc. means.
        I didn't understand that at first, although I do now :)

  2) Page 9: "If you look at the spaces created by the nine locations of the half 
     note, it becomes pretty obvious that they produce the series of partitions of eight ... "

     > Yes, but only the partitions with exactly two parts, i.e. p(8,2) in 
       Beeler's notation.

  3) Page 10:  The formula (3.3) is exactly right for counting the number of
     "full" measures for each solution of (2.3) with equality.  However, it's
     not correct to call those "derangements" because that word has a
     different meaning.  A derangement is a permutation of [n] (all elements 
     distinct) that has no fixed points.  You are looking at permutations with 
     repeated elements, so the definition of a derangement does not really apply.  

  4) Page 11:  I think your method of counting is essentially OK, but I'm not
     convinced about the total of 166 in equation (3.7).  I think you might 
     be missing some measures.  I tried to think of this a different way by
     exploiting what you were saying before about the "full" measures.  You
     can make measures "full" by including rests in your computations on 
     the same footing as the notes(!) and that simplifies the counting quite
     a bit as long as you are careful to include all the possible combinations
     of rests.  Below, notes are capital letters and rests are 
     lower case, with e = eighth rest, s = sixteenth rest.   So
     for instance the three note measures including a half and a quarter 
     include:

     HQQ -- 3!/2! = 3 of these
     HQEe -- 4! = 24 of these
     HQEss -- 5!/2! = 60 of these  (but maybe you don't want to consider
                                    ss as different from e if they are 
                                    consecutive -- the "ties" issue in a 
                                    different form!)
     HQSes -- 5! = 120 of these

     (I think) this would give 207 different measures if ss is different from 
     e and 207 - 24 = 183 if not(?)


   Final Project Grade Computation 
 
   Bibliography:  10/10  

   Paper:         56/60

   Presentation:  26/30

   Total:         92/100