104edo: Difference between revisions
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== Theory == | == Theory == | ||
104edo | 104edo is a strong no-fives system, with good approximations up to the no-5 19-limit. In the [[2.3.7.11.13 subgroup|2.3.7.11.13-subgroup]], it tempers out [[352/351]], [[364/363]], [[896/891]], [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364.<!-- Add commas in 2.3.7.11.13.17.19 as well --> It is an excellent tuning for the 2.3.7.11.13-subgroup [[rank]]-3 [[parapyth]] temperament tempering out 352/351, 364/363, and 896/891, which maps [[14/11]] to the diatonic major third and [[13/11]] to the diatonic minor third, in fact providing the [[optimal patent val]]. Additionally, it supports the extension to prime 17 known as [[etypyth]], which maps 17/14 to the augmented second, though [[121edo]] is a more optimal tuning of it. It also provides the optimal patent val for the 2.3.7.11.13-subgroup {{nowrap| 17 & 87 }} temperament tempering out 352/351, 364/363 and 2197/2187, which splits 3/1 into three ~13/9's, and can be considered a rank-2 reduction of parapyth. | ||
104edo | Notably, 104edo inherits [[26edo]]'s accurate representation of the [[2.7.11 subgroup|2.7.11-subgroup]], and thus supports [[orgone]] temperament in that subgroup. | ||
If prime 5 is desired, 104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament. Additionally, it is viable to treat 104edo as dual-5, or as a 2.3.25.7.11.13.17.19 subgroup temperament. | |||
104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], 896/891, [[385/384]] and [[540/539]]. It also provides an excellent tuning for the rank-3 temperament pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, and the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val. | |||
104edo with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], 352/351 and 364/363 in the 13-limit, and [[136/135]] and [[256/255]] in the 17-limit. | |||
=== Prime harmonics === | === Prime harmonics === | ||
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| [[Bosonic]] | | [[Bosonic]] | ||
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<nowiki />* [[Normal | <nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
== Intervals == | == Intervals == | ||
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! colspan="3" | Approximate ratios | ! colspan="3" | Approximate ratios | ||
|- | |- | ||
! Of 2.3.7.11.13.17.19 | ! Of 2.3.25.7.11.13.17.19<br>subgroup | ||
! Additional ratios of 5<br>tending sharp (104c val) | ! Additional ratios of 5<br>tending sharp (104c val) | ||
! Additional ratios of 5<br>tending flat (patent val) | ! Additional ratios of 5<br>tending flat (patent val) | ||
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| 5 | | 5 | ||
| 57.7 | | 57.7 | ||
| | | [[28/27]], [[33/32]] | ||
| ''[[26/25]]'' | | ''[[26/25]]'' | ||
| ''[[25/24]]'', ''[[36/35]]'' | | ''[[25/24]]'', ''[[36/35]]'' | ||
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| | | | ||
| [[6/5]], ''[[40/33]]'' | | ''[[6/5]]'', ''[[40/33]]'' | ||
|- | |- | ||
| 29 | | 29 | ||