136edo: Difference between revisions
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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. | 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. | ||
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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 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 | 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 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. | ||
Latest revision as of 19:28, 20 February 2025
| ← 135edo | 136edo | 137edo → |
136 equal divisions of the octave (abbreviated 136edo or 136ed2), also called 136-tone equal temperament (136tet) or 136 equal temperament (136et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 136 equal parts of about 8.82 ¢ each. Each step represents a frequency ratio of 21/136, or the 136th root of 2.
136edo is closely related to 68edo, but the patent vals differ on the mapping for 13. Using this val, it is 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 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
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.93 | +1.92 | +1.76 | -0.97 | -4.26 | -2.29 | -2.97 | +0.93 | +2.49 | -3.13 | -1.80 |
| Relative (%) | +44.5 | +21.8 | +20.0 | -11.0 | -48.3 | -26.0 | -33.7 | +10.5 | +28.2 | -35.5 | -20.4 | |
| Steps (reduced) |
216 (80) |
316 (44) |
382 (110) |
431 (23) |
470 (62) |
503 (95) |
531 (123) |
556 (12) |
578 (34) |
597 (53) |
615 (71) | |
Subsets and supersets
Since 136 factors into 23 × 17, 136edo has subset edos 2, 4, 8, 17, 34, and 68.