954edo: Difference between revisions
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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. | 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 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 | Beyond the 17-limit, the 954hj [[val]] is the most accurate, with a lower [[relative error]] than any previous equal temperaments in the [[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 === | ||
Revision as of 00:31, 26 December 2025
| ← 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 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) | |
| Harmonic | 83 | 89 | 97 | 101 | 103 | 107 | 109 | 113 | 127 | 131 | 137 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.267 | +0.189 | -0.399 | +0.083 | +0.099 | -0.452 | +0.191 | -0.567 | -0.258 | +0.144 | +0.600 |
| Relative (%) | +21.2 | +15.0 | -31.7 | +6.6 | +7.8 | -36.0 | +15.2 | -45.1 | -20.5 | +11.4 | +47.7 | |
| Steps (reduced) |
6082 (358) |
6178 (454) |
6296 (572) |
6352 (628) |
6379 (655) |
6431 (707) |
6457 (733) |
6506 (782) |
6667 (943) |
6710 (32) |
6772 (94) | |
| Harmonic | 139 | 149 | 151 | 157 | 163 | 167 | 173 | 179 | 181 | 191 | 193 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.591 | -0.109 | -0.551 | -0.086 | +0.394 | -0.068 | +0.459 | +0.553 | +0.185 | +0.167 | -0.257 |
| Relative (%) | -47.0 | -8.7 | -43.8 | -6.8 | +31.3 | -5.4 | +36.5 | +44.0 | +14.7 | +13.3 | -20.4 | |
| Steps (reduced) |
6791 (113) |
6887 (209) |
6905 (227) |
6959 (281) |
7011 (333) |
7044 (366) |
7093 (415) |
7140 (462) |
7155 (477) |
7229 (551) |
7243 (565) | |
| Harmonic | 197 | 199 | 211 | 223 | 227 | 229 | 233 | 239 | 241 | 251 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.550 | -0.428 | +0.090 | -0.074 | +0.595 | +0.503 | -0.545 | -0.537 | +0.130 | +0.185 |
| Relative (%) | -43.7 | -34.0 | +7.1 | -5.9 | +47.3 | +40.0 | -43.4 | -42.7 | +10.4 | +14.7 | |
| Steps (reduced) |
7271 (593) |
7285 (607) |
7366 (688) |
7442 (764) |
7467 (789) |
7479 (801) |
7502 (824) |
7537 (859) |
7549 (871) |
7605 (927) | |
Subsets and supersets
Since 954 = 2 × 32 × 53, 954edo has subset edos 2, 3, 6, 9, 18, 53, 106, 159, 318, 477.