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Pitch Class Sets

Paul Nelson - Revised:  5/16/2004

Pitch Class Sets are a method for describing harmonies in 20th century music. These notations and methods can describe and manipulate any type of chord that can be created within a 12-tone (equally tempered) scale. It is an extremely useful technique for composers to help understand and control the harmonies which make up their music.

Pitch class sets are the chemistry of harmonic color. Modern composers will use pitch class sets like chemistry, to mix and create interesting and vibrantly colorful harmonic sounds, which they then use to create works of music.

Table of Contents:

1         Basic Definitions. 1

1.1          Pitches. 1

1.2          Pitch Classes. 2

1.3          Pitch Class Sets. 2

2         Simple Operations on Pitch Classes and Pitch Class Sets. 2

2.1          "Clock" Math or Modulo Math. 2

2.2          Transposing Pitch Class Sets. 3

2.3          Inverting Pitch Class Sets. 3

3         The Prime Form.. 4

3.1          Similar Pitch Class Sets:  Set Classes & Prime Forms. 4

3.2          Uses for The Prime Form.. 4

3.3          Determining the Prime Form:  The Rigorous Method. 4

3.4          Determining the Prime Form:  Easier Methods. 5

4         Interval Vectors. 7

4.1          Pitch Intervals. 7

4.2          Interval Classes. 7

4.3          Interval Vectors. 7

5         The Table of All Prime Forms - Description. 9

5.1          The Columns of Data in the Table. 9

5.2          The Layout of the PC Sets in the Table. 11

5.3          Forte Names. 11

5.4          Z-Related Sets. 11

5.5          Other Comments on the Table. 11

6         Subsets and Supersets. 12

6.1          Definition:  Transpositional Combination of Two Common Subsets. 12

6.2          Definition:  Inversional Combination of Two Common Subsets. 12

7         PC Set Complements. 13

7.1          Definitions. 13

7.2          PC Set Complements and Their Interval Vectors. 13

7.3          6-note complements. 14

7.4          PC Set Complements Used in Twelve Tone Composition. 15

8         More Properties of Pitch Class Sets and Interval Vectors. 16

8.1          Common Tones when Transposed. 16

8.1.1          More Examples and Transpositional Symmetry. 17

8.2          Inversional Symmetry. 17

9         Other PC Set Similarity Relations. 19

9.1          Special Purpose Relations:  Rp, R0, R1, R2 19

9.2          Other techniques for generating related PC Sets. 19

Copyright 2004 by Paul Nelson, all rights reserved.