# Combinatorics of some musical objects

Excerpts from Appendix C (page 164) from Keith [1991].  All results assume a 12-tone equal-tempered scale.

Number of diatonic scale classes: 3
Number of note names (A-G); number of notes in a common scale; number of white keys per octave on a piano:  7
Number of scales one note different from the Major scale: 9
Number of notes in the most common equal-tempered scale:  12
Number of common musical keys (C + 1-6 flats/sharps):  13
Number of 7-note diatonic scales (=7 * 3):  21
Number of elementary 2-fold polychords: 23
A k-fold polychord is an n-note chord sub-divided into k non-empty subchords, for k=1,  . . ., n.  For example, the 6-note chord <C, D, E, F#, G, A> can be subdivided into the 3-note 2-fold polychords, <C, E, G> and <D, F#, A>.
Number of 7-note chords:  66
Number of distinct interval sets (partitions of 12):  77
Number of 7-note triatonic scales (=7*35):  245
Number of notationally-distinct diatonic scales (=13 *21):  273
Number of distinct chord-types (= N(12) – 1):  351
Number of 7-note musical scales (=7*66):  462
Number of scales (=Number of n-note scales, summed over all n)  (=2^(12-1) = 2^11):   2048
Number of chords without rotational isomorphism (= 2^12 – 1):  4095
Number of notationally-distinct scales (=13 * 462):  6006
Number of non-syncopated 8-bar 1/4-note rhythmic patterns:  458,330
Number of non-syncopated 8-bar 1/8-note rhythmic patterns:  210,066,388,901

Reference:
Michael Keith [1991]:  From Polychords to Polya:  Adventures in Musical Combinatorics.  (Princeton, NJ:  Vinculum Press.)

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