1178edo
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 = 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
Template:Comma basis begin |- | 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 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf
Music
- Listening (2023) – 217 & 1178 and enneadecal in 1178edo tuning