4.7. Analyzing and Comparing PathSets

One use of SetMeasures is to compare Multisets (as set classes) of a PathInstance. The PathSet "PS" commands deal with the PathInstance as a series of set classes. The user-entered pitch space or pitch-class representations have no relevance on set-class analysis as performed by these SetMeasures.

There are two methods of set class analysis. The first, PScma, compares each set with the adjacent set. Entering the PScma command will perform such an analysis on the active PathInstance with the active SetMeasure. In the example below, a different SetMeasure is first selected:

Example 4-19. Comparing adjacent sets in a Path

[PI(pathVL)TI()] :: smo rel
SetMeasure REL now active.

[PI(pathVL)TI()] :: PScma
PI: pathVL, Lewin REL analysis
Tn
similarity range:   0(min)         .              |             .         (max)1
          3-11B                                                                 
0.50                ..............................+.............................
          3-7A                                                                  
0.00                +...........................................................
          3-12                                                                  
0.43                .........................+..................................
          3-11B                                                            

athenaCL produces text-based horizontal graphs to demonstrate relative similarities. The final graph compares the last SC to the first. It may seem odd that the comparison of 3-7A and 3-12 produces a minimum value of similarity (0.00). This can be explained by examining the Interval Class Vectors for these two sets, as done below:

Example 4-20. Examining interval class vectors

[PI(pathVL)TI()] :: scv 3-7a
SC(3-7A), Z(none), mode(Tn)
Pitch Space:                  (C4,D4,F4)                     
Pitch Class:                  (0,2,5)                        
Normal Form:                  (0,2,5)                        
Prime Form:                   (0,2,5)                        
Invariance Vector:            (1,0,0,0,5,6,5,5)              
Interval Class Vector:        (0,1,1,0,1,0)                  
References:                                                  
   name                       incomplete minor-seventh chord 
Subset Vectors:
3CV(Tn)                                          
          0,0,0,0,0,0,0,0,0,0 - 1,0,0,0,0,0,0,0,0

[PI(4-3)TI()] :: scv 3-12
SC(3-12), Z(none), mode(Tn)
Pitch Space:                  (C4,E4,G#4)                                   
Pitch Class:                  (0,4,8)                                       
Normal Form:                  (0,4,8)                                       
Prime Form:                   (0,4,8)                                       
Invariance Vector:            (3,3,3,3,9,9,9,9)                             
Interval Class Vector:        (0,0,0,3,0,0)                                 
References:                                                                 
   name                       augmented triad, equal 3-part octave division 
Subset Vectors:
3CV(Tn)                                          
          0,0,0,0,0,0,0,0,0,0 - 0,0,0,0,0,0,0,0,1

The SCv command shows that these two sets have exclusive Interval Class Vectors: the two sets have no vector entries (or intervals) in common, and thus are at maximum set class dissimilarity.

The second method of set class analysis is done by comparing one set to every set in a Path. This method of comparison, PScmb, is performed on the active PathInstance with the active SetMeasure. The user must provide the comparison, or reference, set class. In the example below, the reference set class is 4-11:

Example 4-21. Comparing one set to all sets in a Path

[PI(pathVL)TI()] :: PScmb
enter a SC for comparison to the path...
enter a pitch set, sieve, or set-class: 4-11
   SC 4-11A as (C4,C#4,D#4,F4)? (y, n, or cancel): y
PI: pathVL, Lewin REL analysis
Tn
reference SC 4-11A
similarity range:   0(min)         .              |             .         (max)1
0.47      3-11B     ............................+...............................
0.54      3-7A      ................................+...........................
0.27      3-12      ................+...........................................