954edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
954edo is a very strong 17-limit system, | 954edo is a very strong [[17-limit]] system, [[consistency|distinctly consistent]] in the 17-limit, and is a [[zeta edo|zeta peak, integral and gap edo]]. The tuning of the primes to 17 are all flat, and the equal temperament [[tempering out|tempers out]] the [[ennealimma]], {{monzo| 1 -27 18 }}, in the 5-limit and [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s the [[ennealimmal]] temperament. In the 11-limit it tempers out [[3025/3024]], [[9801/9800]], 43923/43904, and 151263/151250 so that it supports hemiennealimmal. In the 13-limit it tempers out [[4225/4224]] and [[10648/10647]] and in the 17-limit 2431/2430 and [[2601/2600]]. It supports and gives the [[optimal patent val]] for the [[semihemiennealimmal]] temperament. | ||
Beyond the 17-limit, the 954hj [[val]] is the most accurate, with a lower [[relative error]] than any previous equal temperaments in the [[29-limit|29-]] and [[31-limit]]. In the 954hj val, [[19/16]], [[29/16]], and their [[octave complement]]s are the only inconsistent intervals in the [[35-odd-limit]], which are in fact the very primes with [[wart]]s. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|954| | {{Harmonics in equal|954}} | ||
{{Harmonics in equal|954|start=12|collapsed=1|title=Approximation of prime harmonics in 954edo (continued)}} | |||
=== | === Subsets and supersets === | ||
Since 954 | Since 954 factors into primes as {{nowrap| 2 × 3<sup>2</sup> × 53 }}, 954edo has subset edos {{EDOs| 2, 3, 6, 9, 18, 53, 106, 159, 318, 477 }}. | ||
[[Category:Ennealimmal]] | [[Category:Ennealimmal]] | ||
[[Category:Hemiennealimmal]] | [[Category:Hemiennealimmal]] | ||
[[Category:Semihemiennealimmal]] | [[Category:Semihemiennealimmal]] | ||
Latest revision as of 00:17, 17 January 2026
| ← 953edo | 954edo | 955edo → |
954 equal divisions of the octave (abbreviated 954edo or 954ed2), also called 954-tone equal temperament (954tet) or 954 equal temperament (954et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 954 equal parts of about 1.26 ¢ each. Each step represents a frequency ratio of 21/954, or the 954th root of 2.
954edo is a very strong 17-limit system, distinctly consistent in the 17-limit, and is a zeta peak, integral and gap edo. The tuning of the primes to 17 are all flat, and the equal temperament tempers out the ennealimma, [1 -27 18⟩, in the 5-limit and 2401/2400 and 4375/4374 in the 7-limit, so that it supports the ennealimmal temperament. In the 11-limit it tempers out 3025/3024, 9801/9800, 43923/43904, and 151263/151250 so that it supports hemiennealimmal. In the 13-limit it tempers out 4225/4224 and 10648/10647 and in the 17-limit 2431/2430 and 2601/2600. It supports and gives the optimal patent val for the semihemiennealimmal temperament.
Beyond the 17-limit, the 954hj val is the most accurate, with a lower relative error than any previous equal temperaments in the 29- and 31-limit. In the 954hj val, 19/16, 29/16, and their octave complements are the only inconsistent intervals in the 35-odd-limit, which are in fact the very primes with warts.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.068 | -0.150 | -0.272 | -0.375 | -0.276 | -0.553 | +0.600 | -0.601 | +0.611 | -0.381 |
| Relative (%) | +0.0 | -5.4 | -11.9 | -21.7 | -29.8 | -21.9 | -44.0 | +47.7 | -47.8 | +48.6 | -30.3 | |
| Steps (reduced) |
954 (0) |
1512 (558) |
2215 (307) |
2678 (770) |
3300 (438) |
3530 (668) |
3899 (83) |
4053 (237) |
4315 (499) |
4635 (819) |
4726 (910) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.228 | -0.132 | +0.432 | -0.098 | -0.549 | -0.052 | +0.096 | -0.062 | +0.178 | -0.117 | +0.243 |
| Relative (%) | +18.1 | -10.5 | +34.3 | -7.8 | -43.6 | -4.1 | +7.7 | -4.9 | +14.1 | -9.3 | +19.3 | |
| Steps (reduced) |
4970 (200) |
5111 (341) |
5177 (407) |
5299 (529) |
5464 (694) |
5612 (842) |
5658 (888) |
5787 (63) |
5867 (143) |
5905 (181) |
6014 (290) | |
Subsets and supersets
Since 954 factors into primes as 2 × 32 × 53, 954edo has subset edos 2, 3, 6, 9, 18, 53, 106, 159, 318, 477.