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COMMENTS
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This list can be enumerated by calculating the number of strings of length m using elements {3,4}, for the pitch-class set of minor and major thirds, that contain no substring divisible by 12, for the twelve Western notes. Any string of length m that contains a substring divisible by 12 is to be subtracted from the total number of strings of length m, given by 2^m. A substring whose length is divisible by 12 would have an enharmonic equivalent to an octave, necessitating a note repetition.
A functional musical analog would be viewing this list as the number of possible "extended" chords using m+1 notes, keeping the rules of construction using only the two functional types of thirds.
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REFERENCES
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Stefan Kostka, Dorothy Payne, and Byron Almén, Tonal Harmony, 8th edition, McGraw Hill, 2018, Glossary.
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EXAMPLE
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For m={0,1,2}, no strings need be omitted from the enumeration, as a valid string with elements {3,4} cannot be equal to a multiple of 12.
For m=3, the first case where strings must be omitted, the valid strings are {333,334,343,344,433,434,443}. The string {444} was omitted in the enumeration because it contains a substring divisible by 12.
For m=4, the valid strings would be {3334,3343,3344,3433,3434,3443,4333,4334,4343,4344,4433,4434}. The strings {3333,3444,4443,4444} were omitted in the enumeration because they contain a substring divisible by 12.
For m=12, there are no valid strings, as when m > 12, since there has to be a substring divisible by 12. Any combination of at least twelve numbers using elements {3,4} must contain at least one string of length m={3,4,7,10,11} that is divisible by 12. This can also be realized from a musical sense, as, since there are only twelve possible notes, stacking m+1 notes when m >= 12 means at least one must appear more than once.
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