56edo: Difference between revisions
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KingHyperio (talk | contribs) →Intervals: bruh the step is like almost exactly a syntonic comma and you didn’t even put it in the right place tf Tags: Mobile edit Mobile web edit |
Dude if you want that kind of approximation plz make another column |
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'''56edo''' divides the octave into 56 parts of 21.429 cents each. It shares it's near perfect major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th that is a convergent towards the [[ | '''56edo''' divides the octave into 56 parts of 21.429 cents each. It shares it's near perfect major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th that is a convergent towards the [[Metallic harmonic series|bronze metallic mean]], following [[17edo]] and preceding [[185edo]]. | ||
56edo can be used to tune [[hemithirds]], [[superkleismic]], [[sycamore]] and [[keen]] temperaments, and using {{val|56 89 130 158}} (56d) as the equal temperament val, for [[pajara]]. It provides the optimal patent val for 7-, 11- and 13-limit [[ | 56edo can be used to tune [[hemithirds]], [[superkleismic]], [[sycamore]] and [[keen]] temperaments, and using {{val| 56 89 130 158 }} (56d) as the equal temperament val, for [[pajara]]. It provides the [[optimal patent val]] for 7-, 11- and 13-limit [[Sycamore family #Septimal sycamore|sycamore]], and the 11-limit 56d val is close to the [[POTE tuning]] for 11-limit pajara. | ||
== Intervals == | == Intervals == | ||
The following table assumes the [[patent val]] {{val| 56 89 130 157 194 207 }}. Other approaches are possible. | |||
{| class="wikitable center-all right-2 left-3" | {| class="wikitable center-all right-2 left-3" | ||
! # | ! # | ||
| Line 15: | Line 17: | ||
| 1 | | 1 | ||
| 21.429 | | 21.429 | ||
| [[49/48]], [[64/63 | | [[49/48]], [[64/63]] | ||
|- | |- | ||
| 2 | | 2 | ||
| 42.857 | | 42.857 | ||
| [[28/27]], [[50/49]] | | [[28/27]], [[50/49]], [[81/80]] | ||
|- | |- | ||
| 3 | | 3 | ||
| Line 131: | Line 133: | ||
== Commas == | == Commas == | ||
* 5-limit commas: 2048/2025, |-5 -10 9 | * 5-limit commas: 2048/2025, {{monzo| -5 -10 9 }}; | ||
* 7-limit commas: 686/675, 875/864, 1029/1024 | * 7-limit commas: 686/675, 875/864, 1029/1024 | ||
* 11-limit commas: 100/99, 245/242, 385/384, 686/675 | * 11-limit commas: 100/99, 245/242, 385/384, 686/675 | ||
== Scales == | == Scales == | ||
* [[ | * [[Supra7]] | ||
* [[ | * [[Supra12]] | ||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
Revision as of 06:08, 10 May 2021
56edo divides the octave into 56 parts of 21.429 cents each. It shares it's near perfect major third with 28edo, which it doubles, while also adding a superpythagorean 5th that is a convergent towards the bronze metallic mean, following 17edo and preceding 185edo.
56edo can be used to tune hemithirds, superkleismic, sycamore and keen temperaments, and using ⟨56 89 130 158] (56d) as the equal temperament val, for pajara. It provides the optimal patent val for 7-, 11- and 13-limit sycamore, and the 11-limit 56d val is close to the POTE tuning for 11-limit pajara.
Intervals
The following table assumes the patent val ⟨56 89 130 157 194 207]. Other approaches are possible.
| # | Cents | Approximate Ratios |
|---|---|---|
| 0 | 0.000 | 1/1 |
| 1 | 21.429 | 49/48, 64/63 |
| 2 | 42.857 | 28/27, 50/49, 81/80 |
| 3 | 64.286 | 25/24, 36/35, 33/32 |
| 4 | 85.714 | 21/20, 22/21 |
| 5 | 107.143 | 16/15 |
| 6 | 128.571 | 15/14, 13/12, 14/13 |
| 7 | 150.000 | 12/11 |
| 8 | 171.429 | 10/9, 11/10 |
| 9 | 192.857 | 28/25 |
| 10 | 214.286 | 9/8 |
| 11 | 235.714 | 8/7 |
| 12 | 257.143 | 7/6, 15/13 |
| 13 | 278.571 | 75/64, 13/11 |
| 14 | 300.000 | 25/21 |
| 15 | 321.429 | 6/5 |
| 16 | 342.857 | 11/9, 39/32 |
| 17 | 364.286 | 27/22, 16/13, 26/21 |
| 18 | 385.714 | 5/4 |
| 19 | 407.143 | 14/11 |
| 20 | 428.571 | 32/25, 33/26 |
| 21 | 450.000 | 9/7, 13/10 |
| 22 | 471.429 | 21/16 |
| 23 | 492.857 | 4/3 |
| 24 | 514.286 | |
| 25 | 535.714 | 27/20, 15/11 |
| 26 | 557.143 | 11/8 |
| 27 | 578.571 | 7/5 |
| 28 | 600.000 | 45/32, 64/45 |
| … | … | … |
Commas
- 5-limit commas: 2048/2025, [-5 -10 9⟩;
- 7-limit commas: 686/675, 875/864, 1029/1024
- 11-limit commas: 100/99, 245/242, 385/384, 686/675