123edo: Difference between revisions

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~~'''123edo''' is the [[~~EDO|~~equal division of the octave]] into~~ 123 ~~parts of 9.7561 [[cent]]s each.~~

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 [[~~Gravity family|~~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]].

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]].

Given its ~~inconsistency~~ to the 7-limit and higher ~~limit~~, 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.

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 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.

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Using the 123df val, it tempers out 144/143, 351/350, 352/351, and 847/845 in the 13-limit.

=== Prime harmonics ===

[[Category:~~Equal divisions of the octave|###~~]] ~~~~

[[Category:Gravid]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-'''123edo''' is the [[EDO|equal division of the octave]] into 123 parts of 9.7561 [[cent]]s each.
+{{EDO intro|123}}
-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 [[Gravity family|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]].
+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]].
-Given its inconsistency to the 7-limit and higher limit, the mapping &lt;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 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.
@@ Line 10: / Line 10: @@
 Using the 123df val, it tempers out 144/143, 351/350, 352/351, and 847/845 in the 13-limit.
+=== Prime harmonics ===
 {{Harmonics in equal|123}}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+[[Category:Gravid]]