309edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|309}} == Theory == 309edo is only consistent to the 3-limit, with five possible mappings in the 7-limit: * {{val|309 490 717 867}} (patent val) * {{..." |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|309}} | {{EDO intro|309}} | ||
309edo is | 309edo is in[[consistent]] to the [[5-odd-limit]], with four possible mappings in the [[7-limit]]: | ||
* {{val|309 490 717 867}} (patent val | * {{val| 309 490 717 867 }} ([[patent val]]) | ||
* {{val| 309 490 '''718''' 867 }} (309c) | |||
* {{val|309 490 '''718''' 867}} (309c) | * {{val| 309 490 717 '''868''' }} (309d) | ||
* {{val|309 490 717 '''868'''}} (309d) | * {{val| 309 490 '''718''' '''868''' }} (309cd) | ||
* {{val|309 490 '''718''' '''868'''}} (309cd) | |||
Using the patent val, it tempers out [[5120/5103]], 117649/116640, 390625/387072, 537824/531441, [[823543/819200]] and 1953125/1928934 in the 7-limit, supporting the [[hemifamity]] temperament. | Using the patent val, it tempers out [[5120/5103]], 117649/116640, 390625/387072, 537824/531441, [[823543/819200]] and 1953125/1928934 in the 7-limit, supporting the [[hemifamity]] temperament. | ||
Using the 309c val, it tempers out [[4375/4374]], 153664/151875, 395136/390625, 537824/531441, [[2100875/2097152]] and 5250987/5242880 in the 7-limit, supporting [[mitonic]] and [[ragismic]]. | Using the 309c val, it tempers out [[4375/4374]], 153664/151875, 395136/390625, 537824/531441, [[2100875/2097152]] and 5250987/5242880 in the 7-limit, supporting [[mitonic]] and [[ragismic]]. | ||
Using the 309d val, it tempers out [[2109375/2097152]] in the 5-limit; [[4000/3969]], 420175/419904, 829440/823543, 1071875/1062882 and 9765625/9633792 in the 7-limit. It supports the [[octagari]] temperament. | Using the 309d val, it tempers out [[2109375/2097152]] in the 5-limit; [[4000/3969]], [[420175/419904]], 829440/823543, 1071875/1062882 and 9765625/9633792 in the 7-limit. It supports the [[octagari]] temperament. | ||
Using the 309cd val, it tempers out [[67108864/66430125]] in the 5-limit; [[3136/3125]], [[5120/5103]], [[250047/250000]], 458752/455625, 49009212/48828125 and 725594112/720600125 in the 7-limit. It supports [[misty]], [[hemifamity]] and [[landscape]]. | Using the 309cd val, it tempers out [[67108864/66430125]] in the 5-limit; [[3136/3125]], [[5120/5103]], [[250047/250000]], 458752/455625, 49009212/48828125 and 725594112/720600125 in the 7-limit. It supports [[misty]], [[hemifamity]] and [[landscape]]. | ||
Additionally, the 309b val {{val| 309 '''489''' 717 867 }} (309b) is [[enfactoring|enfactored]] [[103edo]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
Revision as of 10:49, 21 February 2024
| ← 308edo | 309edo | 310edo → |
309edo is inconsistent to the 5-odd-limit, with four possible mappings in the 7-limit:
- ⟨309 490 717 867] (patent val)
- ⟨309 490 718 867] (309c)
- ⟨309 490 717 868] (309d)
- ⟨309 490 718 868] (309cd)
Using the patent val, it tempers out 5120/5103, 117649/116640, 390625/387072, 537824/531441, 823543/819200 and 1953125/1928934 in the 7-limit, supporting the hemifamity temperament.
Using the 309c val, it tempers out 4375/4374, 153664/151875, 395136/390625, 537824/531441, 2100875/2097152 and 5250987/5242880 in the 7-limit, supporting mitonic and ragismic.
Using the 309d val, it tempers out 2109375/2097152 in the 5-limit; 4000/3969, 420175/419904, 829440/823543, 1071875/1062882 and 9765625/9633792 in the 7-limit. It supports the octagari temperament.
Using the 309cd val, it tempers out 67108864/66430125 in the 5-limit; 3136/3125, 5120/5103, 250047/250000, 458752/455625, 49009212/48828125 and 725594112/720600125 in the 7-limit. It supports misty, hemifamity and landscape.
Additionally, the 309b val ⟨309 489 717 867] (309b) is enfactored 103edo.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.96 | -1.85 | -1.84 | +0.14 | -1.69 | -0.10 | +1.52 | +0.85 | -0.45 | +0.60 |
| Relative (%) | +0.0 | +24.7 | -47.6 | -47.3 | +3.6 | -43.6 | -2.6 | +39.0 | +21.9 | -11.6 | +15.3 | |
| Steps (reduced) |
309 (0) |
490 (181) |
717 (99) |
867 (249) |
1069 (142) |
1143 (216) |
1263 (27) |
1313 (77) |
1398 (162) |
1501 (265) |
1531 (295) | |
Subsets and supersets
309 factors into 3 × 103, with 3edo and 103edo as its subset edos. 927edo, which triples it, gives a good correction to the harmonic 5.