123edo: Difference between revisions
Cleanup |
+subsets and supersets; relation to 41edo |
||
| Line 2: | Line 2: | ||
{{EDO intro|123}} | {{EDO intro|123}} | ||
The equal temperament [[tempering out|tempers out]] the [[valentine comma]], 1990656/1953125 and the [[misty comma]], 67108864/66430125 in the [[5-limit]]; [[126/125]], [[1029/1024]] and 537824/531441 in the [[7-limit]] | 123 = 3 × 41, and 123edo shares its [[perfect fifth|fifth]] with [[41edo]]. The equal temperament [[tempering out|tempers out]] the [[valentine comma]], 1990656/1953125 and the [[misty comma]], 67108864/66430125 in the [[5-limit]]; [[126/125]], [[1029/1024]] and 537824/531441 in the [[7-limit]]; [[243/242]], [[896/891]], 2401/2376, and [[3388/3375]] in the [[11-limit]]; [[196/195]], [[351/350]], 832/825, [[1575/1573]], and 2197/2178 in the [[13-limit]]. It provides the [[optimal patent val]] for the [[gravid]] temperament. | ||
Given its in[[consistency]] to the [[7-odd-limit]] and higher odd limits, the mapping {{val| 123 195 286 '''346''' }} (123d) is also possible for the 7-limit. Using the 123d val, it tempers out [[2430/2401]], [[3136/3125]], and [[5120/5103]] in the 7-limit; [[176/175]], 243/242, [[1375/1372]], and 2560/2541 in the 11-limit; [[169/168]], [[364/363]], [[640/637]], [[729/728]], and 832/825 in the 13-limit. Using the 123df val, it tempers out [[144/143]], 351/350, [[352/351]], and [[847/845]] in the 13-limit. | Given its in[[consistency]] to the [[7-odd-limit]] and higher odd limits, the mapping {{val| 123 195 286 '''346''' }} (123d) is also possible for the 7-limit. Using the 123d val, it tempers out [[2430/2401]], [[3136/3125]], and [[5120/5103]] in the 7-limit; [[176/175]], 243/242, [[1375/1372]], and 2560/2541 in the 11-limit; [[169/168]], [[364/363]], [[640/637]], [[729/728]], and 832/825 in the 13-limit. Using the 123df val, it tempers out [[144/143]], 351/350, [[352/351]], and [[847/845]] in the 13-limit. | ||
| Line 12: | Line 12: | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|123}} | {{Harmonics in equal|123}} | ||
=== Subsets and supersets === | |||
Since 123 factors into {{factorization|123}}, 123edo contains [[3edo]] and [[41edo]] as its subsets. | |||
[[Category:Gravid]] | [[Category:Gravid]] | ||
Revision as of 08:19, 29 May 2024
| ← 122edo | 123edo | 124edo → |
123 = 3 × 41, and 123edo shares its fifth with 41edo. The equal temperament tempers out the valentine comma, 1990656/1953125 and the misty comma, 67108864/66430125 in the 5-limit; 126/125, 1029/1024 and 537824/531441 in the 7-limit; 243/242, 896/891, 2401/2376, and 3388/3375 in the 11-limit; 196/195, 351/350, 832/825, 1575/1573, and 2197/2178 in the 13-limit. It provides the optimal patent val for the gravid temperament.
Given its inconsistency to the 7-odd-limit and higher odd limits, the mapping ⟨123 195 286 346] (123d) is also possible for the 7-limit. Using the 123d val, it tempers out 2430/2401, 3136/3125, and 5120/5103 in the 7-limit; 176/175, 243/242, 1375/1372, and 2560/2541 in the 11-limit; 169/168, 364/363, 640/637, 729/728, and 832/825 in the 13-limit. Using the 123df val, it tempers out 144/143, 351/350, 352/351, and 847/845 in the 13-limit.
Using the 123ce val, it tempers out 1331/1323 in the 11-limit, as well as 225/224, 245/243, and 1029/1024; 275/273, 352/351, 847/845, 1573/1568, and 3185/3168 in the 13-limit. Using the 123e val, it tempers out 121/120, 176/175, and 441/440 in the 11-limit; 196/195, 351/350, 352/351, 1287/1280, and 2197/2187 in the 13-limit.
5 steps of the 123be val is related to Triple BP.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.48 | +3.93 | -2.97 | +4.78 | -1.50 | +2.36 | -4.83 | -3.88 | +4.57 | -3.57 |
| Relative (%) | +0.0 | +5.0 | +40.3 | -30.5 | +49.0 | -15.4 | +24.2 | -49.5 | -39.8 | +46.8 | -36.6 | |
| Steps (reduced) |
123 (0) |
195 (72) |
286 (40) |
345 (99) |
426 (57) |
455 (86) |
503 (11) |
522 (30) |
556 (64) |
598 (106) |
609 (117) | |
Subsets and supersets
Since 123 factors into 3 × 41, 123edo contains 3edo and 41edo as its subsets.