1080edo: Difference between revisions
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1080 is a largely composite EDO, meaning it's notable for its divisors. Its [[number of the divisors|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. 1080's abundancy index is 2.33..., or exactly 7/3. | 1080 is a largely composite EDO, meaning it's notable for its divisors. Its [[number of the divisors|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. 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 Riemann Zeta Function and Tuning#Zeta EDO lists|the ''zeta peak edos'', ''zeta integral edos'' and ''zeta gap edos'' sequences]], however | Notable subsets of 1080edo are [[270edo]] and [[72edo]], as they both belong to [[The Riemann Zeta Function and Tuning#Zeta EDO lists|the ''zeta peak edos'', ''zeta integral edos'' and ''zeta gap edos'' sequences]], 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. | ||
=== Regular temperament theory === | === Regular temperament theory === | ||
In the 13-limit, 1080edo is contorted order-4, with the same tuning as [[270edo]]. In the 1080e val, which puts the 11th harmonic on 3737, it tempers out 114345/114244, and in the 1080ef val it tempers out [[2080/2079]]. | In the 13-limit, 1080edo is contorted order-4, with the same tuning as [[270edo]]. In the 1080e val, which puts the 11th harmonic on 3737, it tempers out 114345/114244, and in the 1080ef val it tempers out [[2080/2079]]. | ||
=== Harmonics === | |||
{{Harmonics in equal|1080}} | |||
== Table of intervals == | == Table of intervals == | ||
Revision as of 19:03, 29 November 2022
| ← 1079edo | 1080edo | 1081edo → |
Theory
1080 is a largely composite EDO, meaning it's 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. 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 sequences, 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.
Regular temperament theory
In the 13-limit, 1080edo is contorted order-4, with the same tuning as 270edo. In the 1080e val, which puts the 11th harmonic on 3737, it tempers out 114345/114244, and in the 1080ef val it tempers out 2080/2079.
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 (%) | +24.0 | +31.8 | +5.7 | +48.1 | -18.6 | -47.5 | -44.2 | -46.0 | +23.8 | +29.7 | -44.7 | |
| Steps (reduced) |
1712 (632) |
2508 (348) |
3032 (872) |
3424 (184) |
3736 (496) |
3996 (756) |
4219 (979) |
4414 (94) |
4588 (268) |
4744 (424) |
4885 (565) | |
Table of intervals
| Step | Name | Associated 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 |