
COMMENTS

This list can be enumerated by calculating the number of strings of length m using elements {3,4}, for the pitchclass 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.


REFERENCES

Stefan Kostka, Dorothy Payne, and Byron Almén, Tonal Harmony, 8th edition, McGraw Hill, 2018, Glossary.


EXAMPLE

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.
