246edo: Difference between revisions
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{{Infobox ET}} | |||
{{ED intro}} | |||
== Theory == | |||
246 = 6 × 41, and 246edo shares its [[perfect fifth|fifth]] with 41edo. It is only [[consistent]] to the [[5-odd-limit]], but the [[patent val]] offers excellent approximations (within half a cent) of [[prime harmonic]]s [[11/1|11]], [[19/1|19]], and [[29/1|29]], and quite good approximations (within one cent) of [[5/1|5]] and [[23/1|23]]. The same 11 and 19 are straight-up inherited by the monstrous [[2460edo]]. | |||
As an equal temperament, 246et [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the 5-limit; [[5120/5103]] and 118098/117649 in the 7-limit; and [[540/539]], [[9801/9800]] in the 11-limit; [[325/324]], [[625/624]] in the 13-limit. It provides the [[optimal patent val]] for [[cata]], the 2.3.5.13 [[subgroup]] temperament tempering out 325/324 and 625/624. The 246d val [[support]]s [[tritikleismic]]. The 246ee val supports [[countercata]]. The 246f val supports [[supers]]. | |||
[[ | === Prime harmonics === | ||
[[ | {{Harmonics in equal|246}} | ||
[[ | |||
[[ | === Subsets and supersets === | ||
Since 246 factors into {{factorization|246}}, 246edo has subset edos {{EDOs| 2, 3, 6, 41, 82, and 123 }}. | |||
A step of 246edo is exactly 10 [[mina]]s. | |||
== Scales == | |||
[[File:cata_246edo.jpg|thumb|alt=cata_246edo.jpg|Cata in 246edo]] | |||
* [[Cata7]] | |||
* [[Cata11]] | |||
* [[Cata15]] | |||
* [[Cata19]] | |||
[[Category:Kleismic]] | |||
Latest revision as of 04:14, 5 June 2025
| ← 245edo | 246edo | 247edo → |
246 equal divisions of the octave (abbreviated 246edo or 246ed2), also called 246-tone equal temperament (246tet) or 246 equal temperament (246et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 246 equal parts of about 4.88 ¢ each. Each step represents a frequency ratio of 21/246, or the 246th root of 2.
Theory
246 = 6 × 41, and 246edo shares its fifth with 41edo. It is only consistent to the 5-odd-limit, but the patent val offers excellent approximations (within half a cent) of prime harmonics 11, 19, and 29, and quite good approximations (within one cent) of 5 and 23. The same 11 and 19 are straight-up inherited by the monstrous 2460edo.
As an equal temperament, 246et tempers out 15625/15552 (kleisma) in the 5-limit; 5120/5103 and 118098/117649 in the 7-limit; and 540/539, 9801/9800 in the 11-limit; 325/324, 625/624 in the 13-limit. It provides the optimal patent val for cata, the 2.3.5.13 subgroup temperament tempering out 325/324 and 625/624. The 246d val supports tritikleismic. The 246ee val supports countercata. The 246f val supports supers.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.48 | -0.95 | +1.91 | -0.10 | -1.50 | +2.36 | +0.05 | +0.99 | -0.31 | +1.31 |
| Relative (%) | +0.0 | +9.9 | -19.4 | +39.1 | -2.0 | -30.8 | +48.4 | +1.0 | +20.4 | -6.3 | +26.8 | |
| Steps (reduced) |
246 (0) |
390 (144) |
571 (79) |
691 (199) |
851 (113) |
910 (172) |
1006 (22) |
1045 (61) |
1113 (129) |
1195 (211) |
1219 (235) | |
Subsets and supersets
Since 246 factors into 2 × 3 × 41, 246edo has subset edos 2, 3, 6, 41, 82, and 123.
A step of 246edo is exactly 10 minas.
Scales
