136edo: Difference between revisions

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~~'''~~136edo~~''' is the [[EDO|equal division of the octave]] into 136 parts of 8.8235 cents each. It~~ is closely related to [[68edo]], but the patent ~~vals~~ differ on the mapping for 13. It is [[~~contorted~~]] in the 11-limit, tempering out 121/120, 176/175, 245/243, and 1375/1372. ~~Using the patent val, it~~ tempers out 169/168 and 847/845 in the 13-limit; 136/135, 154/153, 256/255, 561/560, and 1089/1088 in the 17-limit; 190/189, 343/342, 361/360, 363/361, and 400/399 in the 19-limit. Using the 136b val, it tempers out 81/80, 99/98, 126/125, and 136410197/134217728 in the 11-limit; 847/845, 2704/2695, 3042/3025, 5445/5408, and 15379/15360 in the 13-limit, making it close to optimal as an 11-limit meantone tuning [http://www.tonalsoft.com/enc/m/meantone-error.aspx| by some metrics]. Using the 136bcd val, it tempers out 540/539, 1375/1372, 2079/2048, and 3125/3072 in the 11-limit; 105/104, 847/845, 1188/1183, 1287/1280, and 6561/6500 in the 13-limit. Using the 136e val, it tempers out 245/243, 2048/2025, 2401/2400, and 2560/2541 in the 11-limit; 169/168, 352/351, 832/825, 1001/1000, and 1716/1715 in the 13-limit. Using the 136ef val, it tempers out 196/195, 325/324, 364/363, 512/507, and 625/624 in the 13-limit.

136edo is closely related to [[68edo]], but the [[patent val]]s differ on the [[mapping]] for 13. Using this val, it is [[enfactoring|enfactored]] in the 11-limit, [[tempering out]] [[121/120]], [[176/175]], [[245/243]], and [[1375/1372]]. It tempers out [[169/168]] and [[847/845]] in the 13-limit; [[136/135]], [[154/153]], [[256/255]], [[561/560]], and [[1089/1088]] in the 17-limit; [[190/189]], [[343/342]], [[361/360]], 363/361, and [[400/399]] in the 19-limit.

Using the 136e val, it tempers out 2560/2541 in the 11-limit; [[169/168]], [[352/351]], [[832/825]], [[1001/1000]], and [[1716/1715]] in the 13-limit. Using the 136ef val, it tempers out [[196/195]], [[325/324]], [[364/363]], [[512/507]], and [[625/624]] in the 13-limit.

Using the 136b val, it tempers out [[81/80]], [[99/98]], [[126/125]], and 136410197/134217728 in the 11-limit; [[847/845]], 2704/2695, 3042/3025, 5445/5408, and 15379/15360 in the 13-limit, making it close to optimal as an 11-limit [[meantone]] tuning [http://www.tonalsoft.com/enc/m/meantone-error.aspx by some metrics].

Using the 136bcd val, it tempers out [[540/539]], [[1375/1372]], 2079/2048, and [[3125/3072]] in the 11-limit; [[105/104]], 847/845, [[1188/1183]], [[1287/1280]], and 6561/6500 in the 13-limit.

=== Odd harmonics ===

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

=== Subsets and supersets ===

Since 136 factors into {{factorization|136}}, 136edo has subset edos {{EDOs| 2, 4, 8, 17, 34, and 68 }}.

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-'''136edo''' is the [[EDO|equal division of the octave]] into 136 parts of 8.8235 cents each. It is closely related to [[68edo]], but the patent vals differ on the mapping for 13. It is [[contorted]] in the 11-limit, tempering out 121/120, 176/175, 245/243, and 1375/1372. Using the patent val, it tempers out 169/168 and 847/845 in the 13-limit; 136/135, 154/153, 256/255, 561/560, and 1089/1088 in the 17-limit; 190/189, 343/342, 361/360, 363/361, and 400/399 in the 19-limit. Using the 136b val, it tempers out 81/80, 99/98, 126/125, and 136410197/134217728 in the 11-limit; 847/845, 2704/2695, 3042/3025, 5445/5408, and 15379/15360 in the 13-limit, making it close to optimal as an 11-limit meantone tuning [http://www.tonalsoft.com/enc/m/meantone-error.aspx| by some metrics]. Using the 136bcd val, it tempers out 540/539, 1375/1372, 2079/2048, and 3125/3072 in the 11-limit; 105/104, 847/845, 1188/1183, 1287/1280, and 6561/6500 in the 13-limit. Using the 136e val, it tempers out 245/243, 2048/2025, 2401/2400, and 2560/2541 in the 11-limit; 169/168, 352/351, 832/825, 1001/1000, and 1716/1715 in the 13-limit. Using the 136ef val, it tempers out 196/195, 325/324, 364/363, 512/507, and 625/624 in the 13-limit.
+{{Infobox ET}}
+{{ED intro}}
+edo is closely related to [[68edo]], but the [[patent val]]s differ on the [[mapping]] for 13. Using this val, it is [[enfactoring|enfactored]] in the 11-limit, [[tempering out]] [[121/120]], [[176/175]], [[245/243]], and [[1375/1372]]. It tempers out [[169/168]] and [[847/845]] in the 13-limit; [[136/135]], [[154/153]], [[256/255]], [[561/560]], and [[1089/1088]] in the 17-limit; [[190/189]], [[343/342]], [[361/360]], 363/361, and [[400/399]] in the 19-limit.
+Using the 136e val, it tempers out 2560/2541 in the 11-limit; [[169/168]], [[352/351]], [[832/825]], [[1001/1000]], and [[1716/1715]] in the 13-limit. Using the 136ef val, it tempers out [[196/195]], [[325/324]], [[364/363]], [[512/507]], and [[625/624]] in the 13-limit.
+Using the 136b val, it tempers out [[81/80]], [[99/98]], [[126/125]], and 136410197/134217728 in the 11-limit; [[847/845]], 2704/2695, 3042/3025, 5445/5408, and 15379/15360 in the 13-limit, making it close to optimal as an 11-limit [[meantone]] tuning [http://www.tonalsoft.com/enc/m/meantone-error.aspx by some metrics].
+Using the 136bcd val, it tempers out [[540/539]], [[1375/1372]], 2079/2048, and [[3125/3072]] in the 11-limit; [[105/104]], 847/845, [[1188/1183]], [[1287/1280]], and 6561/6500 in the 13-limit.
+=== Odd harmonics ===
 {{Harmonics in equal|136}}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+=== Subsets and supersets ===
+Since 136 factors into {{factorization|136}}, 136edo has subset edos {{EDOs| 2, 4, 8, 17, 34, and 68 }}.