Antipodes
Definition
The antipode of a note in an edo is the note that is farthest away in the circle of fifths. In 12edo, the antipode of F is B. If the edo is odd, there are two antipodes. The interval(s) from a note to its antipode(s) are also called the antipodes. By symmetry, an antipode's octave-complement is also an antipode.
If the edo is multi-ring (e.g. 24edo), the antipodes are undefined. Otherwise, there are only three possibilities:
- If the edo is even, the antipode is the half-octave. For example, 12edo's antipode is the tritone 6\12.
- If the edo is odd and the 4th is even, the antipodes are the half-4th and its octave complement a 5th higher. For example, 19edo's 4th is 8\19, and its antipodes are 4\19 (A2/d3) and 15\19 (A6/d7).
- If the edo is odd and the 5th is even, the antipodes are the mid-3rd (half-5th) and its octave complement a 4th higher. For example, 17edo's 5th is 10\17, and its antipodes are 5\17 and 12\17.
In summary, the antipode is always the half-octave, the half-4th or the half-5th, whichever one exists in the edo. The proof that one and only only one of these will exist in any single-ring edo is left as an exercise for the reader.
Applications
The antipodes key is the least closely related key, and thus presumably the most shocking modulation. In 12edo, the antipodes of C major is F# major. But in 19edo, it's D# major or Bbb major.
The antipodes of D is where extended pythagorean notation might logically switch from sharps to flats. In 31edo, it's Gbb and Ax. Despite appearances, Gbb and Ax are a perfect 4th apart.
On a Bosanquet-layout keyboard like the lumatone, the antipodes is the interval that spans the greatest vertical distance, making it perhaps one of the more difficult intervals to play.
Generalizations
Any edo or edonoi can be thought of as generated by any interval in it, as long as the equave and the generator have coprime edostepspans. That generator implies an equivalent generator that is the generator's equave-complement. The farthest point or points in a circle of such generators is the antipode with respect to that generator. The antipode is always the half-equave, the half-generator or the half-equivalent-generator, whichever one exists. The antipode's equave-complement is also an antipode.
- 31edo generated by the mid 3rd (e.g. Lulu/Mohajira temperament) has antipodes the upmajor 3rd and the downminor 6th.
- 41edo generated by the downmajor 3rd (e.g. Laquinyo/Magic temperament) has antipodes the major 3rd and the minor 6th.
- 13edt generated by 9/7 = 3\13 (e.g. Zozoyo/Lambda temperament) has antipodes 5\13 and its tritave-complement 8\13.
While not all edos have conventional antipodes, all edos have generalized antipodes. For every even edo, the generalized antipodes is always the half-octave. For example, 24edo has generators 1\24, 5\24, 7\24 and 11\24 (plus their octave complements). In all 4 circles of generators, the farthest point is 12\24.
Every odd edo >= 5 has multiple generalized antipodes. For example, 15edo has generators 1\15, 2\15, 4\15 and 7\15. These generators imply antipodes of 7\15, 1\15, 2\15 and 4\15 respectively. Thus every generator is also an antipodes, and vice versa.
See also
The concept of antipodes was developed by Kite Giedraitis in 2022.