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Stellated CPS: How many tones? 2000.06.12
Okay, I'm ready to publish. Define the following variables:
M: combination count
N: number of factors
Tr: number of tones in the raw CPS, N!/M!(N-M)!
Co: cardinality of the basic otonal chord, 1+(N-M)
Cu: cardinality of the basic utonal chord, 1+M
So: number of tones needed to saturate the basic otonal chord, N-Co
Su: number of tones needed to saturate the basic utonal chord, N-Cu
No: number of basic otonal chords in the CPS, N!/So!(N-So)!
Nu: number of basic utonal chords in the CPS, N!/Su!(N-Su)!
Ts: number of tones in the stellated CPS, Tr + No(So) + Nu(Su)
Backing out and simplifying:
N! N! N!
Ts = ------- + -------------- + --------------
M!(N-M)! (M-2)!(1+N-M)! (N-M-2)!(1+M)!
N! (M^3 + (N-M)^3 - N) N!
= ------- + ----------------------
M!(N-M)! (M+1)!(N-M+1)!
[If the factors weren't mutually prime, remove doubled pitches here.]