935edo
| ← 934edo | 935edo | 936edo → |
935 equal divisions of the octave (abbreviated 935edo or 935ed2), also called 935-tone equal temperament (935tet) or 935 equal temperament (935et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 935 equal parts of about 1.28 ¢ each. Each step represents a frequency ratio of 21/935, or the 935th root of 2.
Theory
935edo is a very strong 23-limit system, and is consistent through to the 27-odd-limit. It does reasonably well in the higher limits, though the sharply tuned 11 and 23 and the flatly tuned 29 and 31 create inconsistencies together, those being 29/22, 29/23, 31/22, 31/23 and their octave complements; it is otherwise consistent to the 39-odd-limit. It is a zeta peak edo.
As an equal temperament, it tempers out the [39 -29 3⟩ (alphatricot comma), [-52 -17 34⟩ (septendecima), and [91 -12 -31⟩ (astro comma) in the 5-limit; 4375/4374 and 52734375/52706752 in the 7-limit; 117649/117612, 151263/151250, 161280/161051 in the 11-limit; 2080/2079, 4096/4095, 4225/4224 in the 13-limit; 2058/2057, 2500/2499, 4914/4913 in the 17-limit; 2432/2431, 3136/3135, 3250/3249, 4200/4199 in the 19-limit; and 2025/2024, 2300/2299, 2646/2645 among others in the 23-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.077 | -0.004 | +0.158 | +0.554 | +0.114 | +0.285 | +0.241 | +0.603 | -0.272 | -0.223 |
| Relative (%) | +0.0 | +6.0 | -0.3 | +12.3 | +43.1 | +8.9 | +22.2 | +18.8 | +47.0 | -21.2 | -17.4 | |
| Steps (reduced) |
935 (0) |
1482 (547) |
2171 (301) |
2625 (755) |
3235 (430) |
3460 (655) |
3822 (82) |
3972 (232) |
4230 (490) |
4542 (802) |
4632 (892) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.207 | -0.399 | +0.568 | +0.590 | +0.506 | -0.348 | -0.307 | +0.265 | -0.017 | -0.624 | -0.045 |
| Relative (%) | +16.1 | -31.1 | +44.2 | +45.9 | +39.4 | -27.1 | -23.9 | +20.7 | -1.4 | -48.6 | -3.5 | |
| Steps (reduced) |
4871 (196) |
5009 (334) |
5074 (399) |
5194 (519) |
5356 (681) |
5500 (825) |
5545 (870) |
5672 (62) |
5750 (140) |
5787 (177) |
5894 (284) | |
Subsets and supersets
Since 935 factors into primes as 5 × 11 × 17, 935edo has subset edos 5, 11, 17, 55, 85, and 187.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1482 -935⟩ | [⟨935 1482]] | −0.0243 | 0.0243 | 1.89 |
| 2.3.5 | [39 -29 3⟩, [-52 -17 34⟩ | [⟨935 1482 2171]] | −0.0157 | 0.0233 | 1.82 |
| 2.3.5.7 | 4375/4374, 52734375/52706752, [36 -5 0 -10⟩ | [⟨935 1482 2171 2625]] | −0.0259 | 0.0268 | 2.08 |
| 2.3.5.7.11 | 4375/4374, 117649/117612, 131072/130977, 161280/161051 | [⟨935 1482 2171 2625 3235]] | −0.0527 | 0.0588 | 4.58 |
| 2.3.5.7.11.13 | 2080/2079, 4096/4095, 4375/4374, 78125/78078, 117649/117612 | [⟨935 1482 2171 2625 3235 3460]] | −0.0490 | 0.0543 | 4.23 |
| 2.3.5.7.11.13.17 | 2058/2057, 2080/2079, 2500/2499, 4096/4095, 4375/4374, 4914/4913 | [⟨935 1482 2171 2625 3235 3460 3822]] | −0.0520 | 0.0508 | 3.96 |
| 2.3.5.7.11.13.17.19 | 2058/2057, 2080/2079, 2432/2431, 2500/2499, 3136/3135, 4375/4374, 4914/4913 | [⟨935 1482 2171 2625 3235 3460 3822 3972]] | −0.0526 | 0.0475 | 3.70 |
| 2.3.5.7.11.13.17.19.23 | 2025/2024, 2058/2057, 2080/2079, 2300/2299, 2432/2431, 2500/2499, 2646/2645, 4375/4374 | [⟨935 1482 2171 2625 3235 3460 3822 3972 4230]] | −0.0616 | 0.0515 | 4.01 |