ComposerTools.com / Theory / Pitch Class Sets / 1. Basic Definitions

1 Basic Definitions

1.1 Pitches

• A "pitch" is any note that we hear.
• The standard piano can play 88 pitches:  A0 to C8, where C4 = middle-C.
• For example, the notes above middle-C are as follows:

               C4 (B3), C4(D4), D4, D4(E4), E4(F4), F4(E4),
               F4(G4), G4, G4(A4), A4, A4(B4), B4(C5)

1.2 Pitch Classes

• Pitch classes are used to discuss pitches independent of octave displacement and enharmonic spelling.
• Any two pitches which sound the same on an equal tempered scale (for example, C and D) or are only different due to octave displacement are said to belong to the same "pitch class".
• For example, the following pitches all belong to Pitch Class C:
                  B4, D4 (enharmonic equivalents), C0, C1, C2, C3, C4, C5, C6 (octave displacements)
• There are only 12 pitch classes in a system where each octave has 12 chromatic notes.
• Pitch classes can also be numbered:  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.
  These numbers are sometimes called "Pitch Class Representatives".
• For this tutorial, 0 = Pitch Class C   (i.e. "fixed Do").
• All other pitch classes will by numbered by counting the half steps from pitch-class C.
• Therefore, C = 0,  C = 1,  D = 2,  D = 3,  E = 4,  F = 5,  F = 6,  G = 7,  G = 8,  A = 9,  A = 10,  B = 11
• Sometimes the letter 'T' (for Ten) or 'A' is used instead of the number 10, and 'E' (for Eleven) or 'B' instead of 11.

1.3 Pitch Class Sets

• A "Pitch Class Set" is a list of pitch class numbers:   [0, 4, 7, 10]   (note the square brackets)
• These are also called  "PC Sets".
• The PC Set for a C minor triad:  [0, 3, 7]
• The PC Set for a G major triad:  [7, 11, 2]
• In Pitch Class sets, octave doublings and displacements are ignored:
• [0, 3, 7, 12]  =>  [0, 3, 7]    (see the section below on modulo math)
• [14, 7, 11]  =>  [2, 7, 11]
• For example, all of the following can be described with Pitch Class Set  [0, 1,4]. The only difference in these chords are octave displacements or enharmonic spellings in the pitches.

• Oftentimes PC Set notation is shown without the commas:  [037]
  (here is where A=T=10 and B=E=11 comes in handy, for example:
                  [0,4,7,10] = [047T] = [047A]   (note:  this is a C dominant 7^th chord)

Copyright © 2004 by Paul Nelson, all rights reserved.