Talk:Antipodes

Antipodes for multiple-ring EDOs?

It would be possible to define antipodes for multi-ring EDOs if fractional distances are allowed. For instance, 34EDO has 2 rings of fifths, so with only integer fifthwards and fourthwards distances, the fifthwards/fourthwards antipode is undefined. But if we divide the perfect fifth (20\34) in 4 (5\34), we get the odd number of 1\34 increments we need to move between rings of fifths. Since 34EDO has (by definition) 34 steps, it has to have 34 quarter-fifths, and (by definition) proceeding through 34 of these (17 fifths) returns to the starting point. Half that (17) number of quarter-fifths is then the greatest fifthward or fourthward distance, and so the antipodes become ±17/4 fifths: ±85\34, which octave-reduces to ±17\34, which is also the half-octave generalized antipode. In (34EDO-modified) conventional 5L 2s notation, this would make the antipode of C major be G↑♭ major = F↓♯ major.

(Also see Talk:Fifthspan for another example of potentially useful fractional distances.)

Lucius Chiaraviglio (talk) 21:27, 20 June 2024 (UTC)

See my comments at Talk:Fifthspan --TallKite (talk) 22:00, 29 June 2024 (UTC)