I was just trying to show that changing the indexing from zero- to one-based makes no difference in this case. The same-octave interval will always be the multi-octave interval mod 7, regardless of whether you call the tonic 0 or 1.
Granted, but I was replying to a post stating that "it" is not better in practice, where "it" is using 0 base, mod 7 arithmetic.
Anyway, yes, you've shown correctly that mod 7 arithmetic is useful, proving half the great-great-somethingth-parent's point. Now, the remaining interesting cases are the inversion case and the multiplication case, where 0 based is simpler (for me) than 1 based.
Also, the idea of using the word "octave" for "mod 7" is already pretty broken, and is what caused me my initial hesitation.
