123edo: Difference between revisions

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Since {{nowrap| 123 {{=}} 3 × 41 }}, 123edo shares its [[perfect fifth|fifth]] with [[41edo]]. As an ~~equl~~ 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.

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.

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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 123ce val can be used as a generator for the [[triple BP]] scale.

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 ===

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 {{ED intro}}
-Since {{nowrap| 123 {{=}} 3 × 41 }}, 123edo shares its [[perfect fifth|fifth]] with [[41edo]]. As an equl 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.
+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.
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 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.
-steps of the 123ce val can be used as a generator for the [[triple BP]] scale.
+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 ===