123edo: Difference between revisions

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It [[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]]~~, providing the~~ [[~~optimal patent val~~]] ~~for the~~ [[~~gravid]] temperament; 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]].

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.

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.

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.

~~Using~~ the ~~123df~~ val~~, it tempers out 144/143, 351/350, 352/351, and 847/845 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 ===

=== Subsets and supersets ===

Since 123 factors into {{nowrap| 3 × 41 }}, 123edo contains [[3edo]] and [[41edo]] as its subsets.

[[Category:Gravid]]

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 {{Infobox ET}}
-{{EDO intro|123}}
+{{ED intro}}
-It [[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]], providing the [[optimal patent val]] for the [[gravid]] temperament; 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]].
+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.
+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.
+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.
-Using the 123df val, it tempers out 144/143, 351/350, 352/351, and 847/845 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}}
+=== Subsets and supersets ===
+Since 123 factors into {{nowrap| 3 × 41 }}, 123edo contains [[3edo]] and [[41edo]] as its subsets.
 [[Category:Gravid]]