1178edo
| ← 1177edo | 1178edo | 1179edo → |
1178 equal divisions of the octave (abbreviated 1178edo or 1178ed2), also called 1178-tone equal temperament (1178tet) or 1178 equal temperament (1178et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1178 equal parts of about 1.02 ¢ each. Each step represents a frequency ratio of 21/1178, or the 1178th root of 2.
Theory
1178edo is a very strong 19-limit system, and is a zeta peak, integral and gap edo. It is also distinctly consistent through to the 21-odd-limit, and is the first edo past 742 with a lower 19-limit relative error. A basis for its 19-limit commas consists of 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, and 4914/4913. It supports and provides a great tuning for semihemienneadecal.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.087 | -0.236 | -0.065 | -0.214 | -0.120 | -0.032 | -0.060 | +0.249 | +0.304 | -0.044 |
| Relative (%) | +0.0 | -8.6 | -23.1 | -6.4 | -21.0 | -11.8 | -3.1 | -5.9 | +24.4 | +29.8 | -4.3 | |
| Steps (reduced) |
1178 (0) |
1867 (689) |
2735 (379) |
3307 (951) |
4075 (541) |
4359 (825) |
4815 (103) |
5004 (292) |
5329 (617) |
5723 (1011) |
5836 (1124) | |
Subsets and supersets
Since 1178 factors into 2 × 19 × 31, 1178edo is notable for containing both 19 and 31. Its subset edos are 2, 19, 31, 38, 62, and 589.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-1867 1178⟩ | [⟨1178 1867]] | +0.0276 | 0.0276 | 2.71 |
| 2.3.5 | [-14 -19-19⟩, [-99 61 1⟩ | [⟨1178 1867 2735]] | +0.0522 | 0.0415 | 4.07 |
| 2.3.5.7 | 4375/4374, 703125/702464, [-52 -5 -2 23⟩ | [⟨1178 1867 2735 3307]] | +0.0450 | 0.0380 | 3.73 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 234375/234256, [-27 3 -4 10 1⟩ | [⟨1178 1867 2735 3307 4075]] | +0.0484 | 0.0347 | 3.41 |
| 2.3.5.7.11.13 | 3025/3024, 4225/4224, 4375/4374, 78125/78078, 1664000/1663893 | [⟨1178 1867 2735 3307 4075 4359]] | +0.0457 | 0.0322 | 3.16 |
| 2.3.5.7.11.13.17 | 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872 | [⟨1178 1867 2735 3307 4075 4359 4815]] | +0.0403 | 0.0327 | 3.21 |
| 2.3.5.7.11.13.17.19 | 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913 | [⟨1178 1867 2735 3307 4075 4359 4815 5004]] | +0.0370 | 0.0318 | 3.12 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 337\1178 | 343.29 | 8000/6561 | Raider |
| 19 | 489\1178 (7\1178) |
498.13 (7.13) |
4/3 (225/224) |
Enneadecal |
| 31 | 581\1178 (11\1178) |
591.851 (11.205) |
936/665 (?) |
217 & 1178 |
| 38 | 260\1178 (12\1178) |
264.86 (12.22) |
500/429 (144/143) |
Semihemienneadecal |
| 38 | 489\1178 (7\1178) |
498.13 (7.13) |
4/3 (225/224) |
Hemienneadecal |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Music
- Listening (2023) – 217 & 1178 and enneadecal in 1178edo tuning