Recall the basic Vector of Relations between set elements that Balzano defines:
so the Vector of relations for the second element of
{0, 2, 4, 5, 7, 9, 11} would be {0, 2, 4, 6, 7, 9, 11}.
Symbolically, then, a set satisfies coherence if for any pair of elements:
where j and k are scalestep-counting indices, and take on values from 0 to m-1.
Sets satisfying this equation have scalesteps which form a monotonic increasing function of semitones. Of the 66 essentially different 5-note scales, only 4 satisfy Coherence, one of which is the pentatonic scale. Of the 66 essentially different 7-note scales only the diatonic scale satisfies Coherence.
Balzano
Simplicity of Scalestep Family