1080edo
| ← 1079edo | 1080edo | 1081edo → |
1080 equal divisions of the octave (abbreviated 1080edo or 1080ed2), also called 1080-tone equal temperament (1080tet) or 1080 equal temperament (1080et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1080 equal parts of about 1.11 ¢ each. Each step represents a frequency ratio of 21/1080, or the 1080th root of 2.
Theory
1080edo is enfactored in the 13-limit, with the same tuning as 270edo. Aside from the patent val, there are a number of mappings to be cosidered. The 1080e val, [⟨1080 1712 2508 3032 3737]], tempers out 114345/114244, and the 1080ef val, [⟨1080 1712 2508 3032 3737 3997]] it tempers out 2080/2079.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.267 | +0.353 | +0.063 | +0.534 | -0.207 | -0.528 | -0.491 | -0.511 | +0.265 | +0.330 | -0.497 |
| relative (%) | +24 | +32 | +6 | +48 | -19 | -47 | -44 | -46 | +24 | +30 | -45 | |
| Steps (reduced) |
1712 (632) |
2508 (348) |
3032 (872) |
3424 (184) |
3736 (496) |
3996 (756) |
4219 (979) |
4414 (94) |
4588 (268) |
4744 (424) |
4885 (565) | |
Subsets and supersets
1080 is a largely composite edo, meaning it is notable for its divisors. Its 32 divisors are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 27, 30, 36, 40, 45, 54, 60, 72, 90, 108, 120, 135, 180, 216, 270, 360, 540, and 1080. 1080's abundancy index is 2.33…, or exactly 7/3.
Notable subsets of 1080edo are 270edo and 72edo, as they both belong to the zeta peak edos, zeta integral edos and zeta gap edos. However, the patent val of 1080edo does not consist of their approximation alone, as the 17th harmonic comes from 540edo. In addition, 12edo is the dominant tuning system in the world, and 360edo is a highly composite edo.
As every 4th step of 4320edo, it is a good tuning for the 2.5/3.7 subgroup, and has strong representation for 19/12, 19/10, 17/13, 23/13, and 23/17.
Selected intervals
| Step | Eliora's Naming System | Approximate Ratio | Comments |
|---|---|---|---|
| 0 | Prime | ||
| 3 | Degree | Derives from 360edo. | |
| 4 | Ducentiseptuagesima | Derives from 270edo | |
| 7 | Septimal kelisma | ||
| 15 | Moria | Derives form 72edo. | |
| 90 | Dodecaphonic semitone | ||
| 94 | Septendecimal semitone | 17/16 | |
| 240 | Septimal submajor second | 7/6 | Derives form 9edo. |
| 360 | Landscape major third | 63/50 | |
| 495 | 24-phonic superfourth | Derives from 24edo. | |
| 496 | Undecimal superfourth | 11/8 | |
| 630 | Dodecaphonic fifth | ||
| 632 | 135-phonic Fifth | 3/2 | |
| 756 | Tridecimal neutral sixth, 13th harmonic | 13/8 | Derives from 10edo. |
| 1080 | Octave |