1080edo: Difference between revisions
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== Theory == | == Theory == | ||
1080edo is [[enfactoring|enfactored]] in the 13-limit, with the same tuning as [[270edo]]. In the 1080e val, which puts the 11th harmonic on 3737 steps, it [[Tempering out|tempers out]] 114345/114244, and in the 1080ef val it tempers out [[2080/2079]]. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|1080}} | |||
=== Subsets and supersets === | |||
1080 is a largely composite edo, meaning it is notable for its divisors. Its 32 [[number of the divisors|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 [[The Riemann zeta function and tuning #Zeta EDO lists|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 == | ||
{| class="wikitable" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|+ | |+ | ||
!Step | ! Step | ||
! | ! Eliora's Naming System | ||
! | ! Approximate Ratio | ||
!Comments | ! Comments | ||
|- | |- | ||
|0 | |0 | ||
Revision as of 15:19, 19 October 2023
| ← 1079edo | 1080edo | 1081edo → |
Theory
1080edo is enfactored in the 13-limit, with the same tuning as 270edo. In the 1080e val, which puts the 11th harmonic on 3737 steps, it tempers out 114345/114244, and in the 1080ef val 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 (%) | +24.0 | +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) | |
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 |