Kite's thoughts on antipodes

Definition

The antipode of a note in an edo is the note that is furthest 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-fourth and its octave complement, the half-12th. The antipodes are a perfect 5th apart. 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, the mid-6th (half-11th). The antipodes are a perfect 4th apart. 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.

Applications

On a bosanquet-layout keyboard, in general the antipodes is the most awkward interval to play, in the sense that it spans the greatest vertical distance.

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 and 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.

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 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.
13ed3 generated by 5/3 = 6\13 (e.g. Zozoyo/Lambda temperament) has antipodes 3\13 and its tritave-complement 10\13.