Chapter 8. Tutorial 8: Tools for Pitch Analysis

Table of Contents
8.1. Inspecting the Set Class Library
8.2. Searching the Set Class Library for Names, Z-relations, and Super-Sets
8.3. Comparing and Searching Similarity Measures

This tutorial demonstrates tools within athenaCL for analyzing and modeling pitch groups.

8.1. Inspecting the Set Class Library

The Set Class Library consists data on all chord-types, or Tn set classes. For each set, the Forte number (Forte 1973), normal form, Z-relation, Morris Invariance vector (Morris 1987), Forte interval class vector, and all n-class vectors (in both Tn and Tn/I classifications) are available (Straus 1990). Further, contextual data such as common chord-names is also available.

The SCv command provides access to this information. SCv also functions as set translator, easily converting any set, sieve, or Forte-number to its appropriate normal form. Sets can be entered as Forte-numbers, pitch-name sets (with "#" as sharps and "$" as flats), pitch class sets, or pitch space sets. Whenever sets are used in athenaCL, they are treated simultaneously as ordered collections with redundancies, and as unordered collections without redundancies. Either the user or the context determines which form of the set is used. The set below is entered into SCv first as a pitch-name set, then as a Forte number:

Example 8-1. Viewing a set by pitch name or Forte name

[PI()TI()] :: scv  
enter a pitch set, sieve, or set-class: c#, f, e, a, g#, c#, c, a
   SC 6-20 as (C#4,F4,E4,A4,G#4,C#4,C4,A4)? (y, n, or cancel): y
SC(6-20), PCS(1,5,4,9,8,1,0,9), T(0), Z(none), mode(TnI)
Normal Form:                  (0,1,4,5,8,9)
Invariance Vector:            (3,3,3,3,3,3,3,3)
Interval Class Vector:        (3,0,3,6,3,0)
References:
   name                       E all combinatorial (P2, P6, P10, I3, I7, R4, R8,
                              RI1, RI5, RI9), Messiaen's truncated mode 3, Genus
                              tertium, third-order all combinatorial
n-Class Vectors:
3CV(TnI)                                           
          0,0,6,6,0,0,0,0,0,0 - 6,2                
4CV(TnI)                                           
          0,0,0,0,0,0,3,0,0,0 - 0,0,0,0,0,0,3,0,6,3
          0,0,0,0,0,0,0,0,0   -                    
5CV(TnI)                                           
          0,0,0,0,0,0,0,0,0,0 - 0,0,0,0,0,0,0,0,0,0
          6,0,0,0,0,0,0,0,0,0 - 0,0,0,0,0,0,0,0    
6CV(TnI)                                           
          0,0,0,0,0,0,0,0,0,0 - 0,0,0,0,0,0,0,0,0,1
          0,0,0,0,0,0,0,0,0,0 - 0,0,0,0,0,0,0,0,0,0
          0,0,0,0,0,0,0,0,0,0 -  

[PI()TI()] :: scv 6-20
SC(6-20), PCS(0,1,4,5,8,9), T(0), Z(none), mode(TnI)
Normal Form:                  (0,1,4,5,8,9)
Invariance Vector:            (3,3,3,3,3,3,3,3)
Interval Class Vector:        (3,0,3,6,3,0)
References:
   name                       E all combinatorial (P2, P6, P10, I3, I7, R4, R8,
                              RI1, RI5, RI9), Messiaen's truncated mode 3, Genus
                              tertium, third-order all combinatorial
n-Class Vectors:
3CV(TnI)                                           
          0,0,6,6,0,0,0,0,0,0 - 6,2                
4CV(TnI)                                           
          0,0,0,0,0,0,3,0,0,0 - 0,0,0,0,0,0,3,0,6,3
          0,0,0,0,0,0,0,0,0   -                    
5CV(TnI)                                           
          0,0,0,0,0,0,0,0,0,0 - 0,0,0,0,0,0,0,0,0,0
          6,0,0,0,0,0,0,0,0,0 - 0,0,0,0,0,0,0,0    
6CV(TnI)                                           
          0,0,0,0,0,0,0,0,0,0 - 0,0,0,0,0,0,0,0,0,1
          0,0,0,0,0,0,0,0,0,0 - 0,0,0,0,0,0,0,0,0,0
          0,0,0,0,0,0,0,0,0,0 -                    

The first line of the SCv display gives the Forte number, the pitch-class set, the transposition level, the Z-related set (if it exists), and the global Tn/I mode. The following lines give the normal form, Morris invariance vector, and Forte interval class vector. Following the heading "References" is any contextual data known about this set. Here we see that this set is also known as an "E all combinatorial" set, and "Messiaen's truncated mode 3". Following this is a display for all of the set's n-class vectors. N-class vectors display the number and kind of all subsets. Each register corresponds to sub-set. The last vector, "6CV" (for cardinality-6 class vector), shows a value of 1 at the 20th register position. This tells us that only one hexachord is embedded in this set, hexachord 6-20, the set itself. The other vectors tell the same information for pentachords, tetrachords, and trichords.

The number of discrete sets for any cardinality is dependent on the whether or not an inverted chord is counted as unique. When an inversion is counter as unique, the system is said to be in Tn classification. When not, the system is in Tn/I classification (Straus 1990). There are considerably more set-classes under Tn classification, though not every set has an inversion. All set class processing in athenaCL is switchable between Tn and Tn/I classifications. To switch classifications, enter the command SCmode.

Example 8-2. Switching SetClass mode from Tn/I to Tn

[PI()TI()] :: scmode
SC classification set to Tn.

When in Tn mode there are more unique sets then Tn/I mode. Thus, when examining sub-set vectors, there are more registers for each cardinality. In the following example the set used above is displayed again with SCv, though this time in Tn mode. Notice that there are more registers for each of the n-class vectors:

Example 8-3. Viewing Tn subset data

[PI()TI()] :: scv 0, 3, 4, 7, 8, 11
SC(6-20), PCS(0,3,4,7,8,11), T(3), Z(none), mode(Tn)
Normal Form:                  (0,1,4,5,8,9)
Invariance Vector:            (3,3,3,3,3,3,3,3)
Interval Class Vector:        (3,0,3,6,3,0)
References:
   name                       E all combinatorial (P2, P6, P10, I3, I7, R4, R8,
                              RI1, RI5, RI9), Messiaen's truncated mode 3, Genus
                              tertium, third-order all combinatorial
n-Class Vectors:
3CV(Tn)                                            
          0,0,0,3,3,3,3,0,0,0 - 0,0,0,0,0,0,3,3,2  
4CV(Tn)                                            
          0,0,0,0,0,0,0,0,0,3 - 0,0,0,0,0,0,0,0,0,0
          0,0,0,0,0,3,0,0,3,3 - 3,0,0,0,0,0,0,0,0,0
          0,0,0               -                    
5CV(Tn)                                            
          0,0,0,0,0,0,0,0,0,0 - 0,0,0,0,0,0,0,0,0,0
          0,0,0,0,0,0,0,0,0,0 - 0,0,0,0,0,3,3,0,0,0
          0,0,0,0,0,0,0,0,0,0 - 0,0,0,0,0,0,0,0,0,0
          0,0,0,0,0,0         -                    
6CV(Tn)                                            
          0,0,0,0,0,0,0,0,0,0 - 0,0,0,0,0,0,0,0,0,0
          0,0,0,0,0,0,0,0,0,0 - 0,0,1,0,0,0,0,0,0,0
          0,0,0,0,0,0,0,0,0,0 - 0,0,0,0,0,0,0,0,0,0