123edo: Difference between revisions
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{{ED intro}} | |||
Since {{nowrap| 123 {{=}} 3 × 41 }}, 123edo shares its [[perfect fifth|fifth]] with [[41edo]]. As an equal temperament, it [[tempering out|tempers out]] 1990656/1953125 ([[valentine comma]]), 67108864/66430125 ([[misty comma]]), and {{monzo| -13 17 -6 }} ([[graviton]]) 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. | |||
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. | |||
Whereas 5 steps of 41edo can be used as a generator for the [[Bohlen–Pierce]] scale, 5 steps of 123edo with the 123ce val can be used as a generator for the [[triple BP]] scale. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|123}} | {{Harmonics in equal|123}} | ||
[[Category: | |||
=== Subsets and supersets === | |||
Since 123 factors into {{nowrap| 3 × 41 }}, 123edo contains [[3edo]] and [[41edo]] as its subsets. | |||
[[Category:Gravid]] | |||
Latest revision as of 18:41, 13 March 2025
| ← 122edo | 123edo | 124edo → |
123 equal divisions of the octave (abbreviated 123edo or 123ed2), also called 123-tone equal temperament (123tet) or 123 equal temperament (123et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 123 equal parts of about 9.76 ¢ each. Each step represents a frequency ratio of 21/123, or the 123rd root of 2.
Since 123 = 3 × 41, 123edo shares its fifth with 41edo. As an equal temperament, it tempers out 1990656/1953125 (valentine comma), 67108864/66430125 (misty comma), and [-13 17 -6⟩ (graviton) 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.
Whereas 5 steps of 41edo can be used as a generator for the Bohlen–Pierce scale, 5 steps of 123edo with the 123ce val can be used as a generator for the triple BP scale.
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.