494edo
| ← 493edo | 494edo | 495edo → |
494 equal divisions of the octave (abbreviated 494edo or 494ed2), also called 494-tone equal temperament (494tet) or 494 equal temperament (494et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 494 equal parts of about 2.43 ¢ each. Each step represents a frequency ratio of 21/494, or the 494th root of 2.
Theory
494 is a very strong 13- and 17-limit equal temperament. 494edo is a zeta peak and zeta peak integer edo and distinctly consistent through the 17-odd-limit. It tempers out the enneadeca, [-14 -19 19⟩, the tricot comma, [39 -29 3⟩, and the kwazy comma, [-53 10 16⟩ in the 5-limit. In the 7-limit, it tempers out 4375/4374 and 703125/702464; in the 11-limit 3025/3024 and 9801/9800; in the 13-limit 1716/1715, 2080/2079, 4096/4095, 4225/4224 and 6656/6655; and in the 17-limit, 1156/1155, 1275/1274, 2431/2430, and 2500/2499.
Since the step size is close to 729/728, the squbema, the accepted name for 494edo's step is squb.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.069 | -0.079 | +0.405 | +0.099 | -0.042 | -0.502 | -1.157 | +0.875 | +0.382 | -0.906 |
| Relative (%) | +0.0 | +2.9 | -3.2 | +16.7 | +4.1 | -1.7 | -20.7 | -47.6 | +36.0 | +15.7 | -37.3 | |
| Steps (reduced) |
494 (0) |
783 (289) |
1147 (159) |
1387 (399) |
1709 (227) |
1828 (346) |
2019 (43) |
2098 (122) |
2235 (259) |
2400 (424) |
2447 (471) | |
Subsets and supersets
Since 494 factors into 2 × 13 × 19, 494edo has subset edos 2, 13, 19, 26, 38, and 247.
988edo, which slices the edostep in two, provides a good correction of the 19th harmonic. 2964edo, which slices the edostep in six, provides an extremely precise correction of the 7th harmonic.
Intervals
The following table shows how 15-odd-limit intervals are represented in 494edo. Prime harmonics are in bold.
As 494edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 15/8, 16/15 | 0.010 | 0.4 |
| 11/6, 12/11 | 0.030 | 1.2 |
| 15/13, 26/15 | 0.032 | 1.3 |
| 13/10, 20/13 | 0.037 | 1.5 |
| 11/9, 18/11 | 0.040 | 1.6 |
| 13/8, 16/13 | 0.042 | 1.7 |
| 3/2, 4/3 | 0.069 | 2.9 |
| 5/4, 8/5 | 0.079 | 3.2 |
| 11/8, 16/11 | 0.099 | 4.1 |
| 15/11, 22/15 | 0.109 | 4.5 |
| 13/12, 24/13 | 0.111 | 4.6 |
| 9/8, 16/9 | 0.139 | 5.7 |
| 13/11, 22/13 | 0.141 | 5.8 |
| 5/3, 6/5 | 0.148 | 6.1 |
| 11/10, 20/11 | 0.178 | 7.3 |
| 13/9, 18/13 | 0.180 | 7.4 |
| 9/5, 10/9 | 0.217 | 9.0 |
| 9/7, 14/9 | 0.266 | 11.0 |
| 11/7, 14/11 | 0.306 | 12.6 |
| 7/6, 12/7 | 0.336 | 13.8 |
| 7/4, 8/7 | 0.405 | 16.7 |
| 15/14, 28/15 | 0.414 | 17.1 |
| 13/7, 14/13 | 0.447 | 18.4 |
| 7/5, 10/7 | 0.484 | 19.9 |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [783 -494⟩ | [⟨494 783]] | −0.0219 | 0.0219 | 0.90 |
| 2.3.5 | [-14 -19 19⟩, [39 -23 3⟩ | [⟨494 783 1147]] | −0.0032 | 0.0318 | 1.31 |
| 2.3.5.7 | 4375/4374, 703125/702464, [21 3 1 -10⟩ | [⟨494 783 1147 1387]] | −0.0385 | 0.0670 | 2.76 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 131072/130977, 234375/234256 | [⟨494 783 1147 1387 1709]] | −0.0365 | 0.0600 | 2.47 |
| 2.3.5.7.11.13 | 1716/1715, 2080/2079, 3025/3024, 4096/4095, 31250/31213 | [⟨494 783 1147 1387 1709 1828]] | −0.0286 | 0.0576 | 2.37 |
| 2.3.5.7.11.13.17 | 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2431/2430, 4096/4095 | [⟨494 783 1147 1387 1709 1828 2019]] | −0.0069 | 0.0752 | 3.09 |
- 494et has lower relative errors than any previous equal temperaments in the 13- and 17-limit. It is the first past 270 with a lower 13-limit relative error, and the first past 72 with a lower 17-limit relative error. It is narrowly beaten by 684 in terms of 13-limit absolute error and by 581 in terms of 17-limit absolute error. Not until 1506 do we reach an equal temperament with a lower relative error in either subgroup.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 27\494 | 65.59 | 27/26 | Luminal |
| 1 | 119\494 | 289.07 | 13/11 | Moulin |
| 1 | 233\494 | 565.99 | 104/75 | Tricot / trillium |
| 2 | 67\494 | 162.75 | 1125/1024 | Kwazy |
| 2 | 86\494 | 208.91 | 44/39 | Abigail |
| 13 | 205\494 (15\494) |
497.98 (36.43) |
4/3 (?) |
Aluminium |
| 19 | 205\494 (3\494) |
497.98 (7.29) |
4/3 (225/224) |
Enneadecal |
| 38 | 205\494 (3\494) |
497.98 (7.29) |
4/3 (225/224) |
Hemienneadecal |
| 38 | 109\494 (5\494) |
264.78 (12.15) |
500/429 (144/143) |
Semihemienneadecal |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Music
- Unknown piece in Abigail (2023)