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. 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 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.

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

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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. 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 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.
+{{Harmonics in equal|136}}
 [[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->