Kite's thoughts on antipodes: Difference between revisions

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== 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.

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 [[Fifthspan|multi-ring]] (e.g. [[24edo]]), the antipodes are undefined. Otherwise, there are only three possibilities:

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* 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.

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 ==

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== 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.

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. [[Mohajira|Lulu/Mohajira]] temperament) has antipodes the upmajor 3rd and the downminor 6th.

* [[41edo]] generated by the downmajor 3rd (e.g. [[Magic|Laquinyo/Magic]] temperament) has antipodes the major 3rd and the minor 6th.

* [[13ed3]] generated by 9/7 = 3\13 (e.g. [[Lambda|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 ==

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 == 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.
+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 [[Fifthspan|multi-ring]] (e.g. [[24edo]]), the antipodes are undefined. Otherwise, there are only three possibilities:
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 * 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.
+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 ==
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 == 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.
+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. [[Mohajira|Lulu/Mohajira]] temperament) has antipodes the upmajor 3rd and the downminor 6th.
 * [[41edo]] generated by the downmajor 3rd (e.g. [[Magic|Laquinyo/Magic]] temperament) has antipodes the major 3rd and the minor 6th.
 * [[13ed3]] generated by 9/7 = 3\13 (e.g. [[Lambda|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 ==