Kite's thoughts on antipodes: Difference between revisions
→Generalizations: the generalized antipodes of multi-ring edos |
m TallKite moved page Antipodes to Kite's thoughts on antipodes: Editor VectorGraphics has repeatedly edited pages I've written about my own research and inserted misinformation. On discord he is openly hostile to me. This move is necessary to avoid a toxic work environment for me. |
||
| (3 intermediate revisions by 3 users not shown) | |||
| Line 8: | Line 8: | ||
* 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. | * 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. Only one of these exists; or all three exist, in which case the edo is multi-ring. | ||
== Applications == | == Applications == | ||
| Line 18: | Line 18: | ||
[[File:Lumatone 31edo 2.jpg|left|thumb]] | [[File:Lumatone 31edo 2.jpg|left|thumb]] | ||
[[File:Lumatone antipodes for single-ring edos 12-53.jpg|none|thumb]] | [[File:Lumatone antipodes for single-ring edos 12-53.jpg|none|thumb]] | ||
== Generalizations == | == Generalizations == | ||
| Line 24: | Line 26: | ||
* [[31edo]] generated by the mid 3rd (e.g. [[Mohajira|Lulu/Mohajira]] temperament) has antipodes the upmajor 3rd and the downminor 6th. | * [[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. | * [[41edo]] generated by the downmajor 3rd (e.g. [[Magic|Laquinyo/Magic]] temperament) has antipodes the major 3rd and the minor 6th. | ||
* [[ | * [[13edt]] 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. | 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. | ||