56edo: Difference between revisions
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Note on it being a convergent to the bronze metallic mean |
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'''56edo''' divides the octave into 56 parts of 21.429 cents each. | '''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 [[Sycamore_family #Sycamore|sycamore]], and the 11-limit 56d val is close to the POTE tuning for 11-limit pajara. | 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 #Sycamore|sycamore]], and the 11-limit 56d val is close to the POTE tuning for 11-limit pajara. | ||
Revision as of 18:11, 15 October 2020
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
| # | Cents | Approximate Ratios |
|---|---|---|
| 0 | 0.000 | 1/1 |
| 1 | 21.429 | 49/48, 64/63 |
| 2 | 42.857 | 81/80, 28/27, 50/49 |
| 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