Talk:Kite's thoughts on antipodes: Difference between revisions
Antipodes for multiple-ring EDOs? |
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== Antipodes for multiple-ring EDOs? == | == 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. | 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.) | |||
[[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 21:27, 20 June 2024 (UTC) | [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 21:27, 20 June 2024 (UTC) | ||
Revision as of 21:57, 20 June 2024
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)