program mixmidi:  technical comments

To mix midi files, events must be first grouped together.  The approach adopted 
here is data clustering, a topic in statistics.  This is a very general method.  
In the past, researchers have used sequence alignment algorithms instead. (eg. 
Edward Large, Matching Performance to Notation)  The advantage of the present 
approach is that it naturally handles more than two performances.

The main clustering subroutine is named JCLUST.

Modulo arithmetic is used to calculate distances on the circle of fifths.  
A pitch value should be entered twice for each event: as MIDI note number (pitch 
proximity), and as MIDI note number times 7 with modulo 12 set (circle-of-fifths 
proximity).  This is a trick:

 			d = (7 @ (n1 - n0)) % 12
 			
Where d is the circle-of-fifths distance and n1 and n0 are MIDI note numbers.  
This equation is effective for some reason having to do with a fifth being seven 
half-steps.

	hclust 		
Hclust is a data clustering routine based on the method of partial sorting.  
Instead of choosing one value and sorting each element of an array as greater or 
less than it, choose two points to define a partition.  Each other point is 
assigned its place based on which of the two points it is nearer.

	tclust
This is a mixture decomposition scheme based on supposing that the data can
be fit to a Student t distribution.  Each object is assigned to a cluster
in proportion to the probability that it was drawn from that cluster.

	Stable sorting 	
Radix sort RSORTI sorts integers and carries along an index array.  For long 
lists of short numbers, it is moderately quicker than quicksort.  Merge sort 
MSORTR sorts reals and carries along an index array.