180edo
| ← 179edo | 180edo | 181edo → |
180 equal divisions of the octave (abbreviated 180edo or 180ed2), also called 180-tone equal temperament (180tet) or 180 equal temperament (180et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 180 equal parts of about 6.67 ¢ each. Each step represents a frequency ratio of 21/180, or the 180th root of 2.
The equal temperament tempers out 531441/524288 (pythagorean comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit, as well as 31381059609/30517578125 (mowgli comma) and 274877906944/274658203125 (hemithirds comma); 1029/1024, 3136/3125, and 118098/117649 in the 7-limit.
Using the patent val, it tempers out 540/539, 2835/2816, 4000/3993, and 6912/6875 in the 11-limit; 351/350, 364/363, 1001/1000, and 1701/1690 in the 13-limit. Using the 180e val, it tempers out 385/384, 441/440, 3388/3375, and 216513/214375 in the 11-limit; 351/350, 1188/1183, 1287/1280, 1573/1568, and 3146/3125 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -1.96 | +0.35 | -2.16 | +2.76 | +2.02 | -0.53 | -1.60 | +1.71 | +2.49 | +2.55 | -1.61 |
| relative (%) | -29 | +5 | -32 | +41 | +30 | -8 | -24 | +26 | +37 | +38 | -24 | |
| Steps (reduced) |
285 (105) |
418 (58) |
505 (145) |
571 (31) |
623 (83) |
666 (126) |
703 (163) |
736 (16) |
765 (45) |
791 (71) |
814 (94) | |
Subsets and supersets
180edo is the 11th highly composite edo; its nontrivial subsets are: 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, and 90.