As suggested by comments from Gene Ward Smith, one can replace my arbitrary 2-digit id numbers with a 3 member vector of prime factors (the number of 2s, the number of 3s, and the number of 5s).

And so, one can label the stacks in the 3-unit-vectors that show number of prime factors, as long as one identifies the starting point. From A440, the central stack is the [0,0,0] stack; the octave higher the [1,0,0] stack; the octave lower is the [-1,0,0] stack. The major thirds in relation to the [0,0,0] stack form a [-2, 0, 1] stack; the minor thirds in relation to the [0,0,0] stack form a [1, 1, -1] stack. In relation to the octave higher [1,0,0] stack the major thirds are [-1,0,1] and the minor thirds are [2, 1, -1]. In relation to the octave lower, the major thirds are [-3, 0, 1] and the minor thirds are [0, 1, -1]. In relation to the central stack, the two-major-third group that I’ve called “thirds of secondary dominants” are [-4, 0, 2] and the two-minor-third group, rarely used, are [2, 2, -2]. The very rarely used three-major-third stack would be [-6, 0, 3] and the almost never used three-minor-third stack would be [3, 3, -3]. Then a path from one note to another begins (?always, for convenience?) as a movement of fifths up or down the beginning stack, followed by the necessary octave shifts to different ‘stack groups’, followed by whatever major third or minor third movement is necessary.

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