188edo
| ← 187edo | 188edo | 189edo → |
188 equal divisions of the octave (abbreviated 188edo or 188ed2), also called 188-tone equal temperament (188tet) or 188 equal temperament (188et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 188 equal parts of about 6.38 ¢ each. Each step represents a frequency ratio of 21/188, or the 188th root of 2.
188edo is closely related to 94edo, but the patent vals differ on the mapping for 5. The equal temperament tempers out 129140163/128000000 (graviton) and 268435456/263671875 (mabila comma) in the 5-limit; 2401/2400, 19683/19600, and 110592/109375 in the 7-limit. It supports harry and newt, and provides the optimal patent val for tuskaloosa, the 77 & 111 temperament.
Using the patent val, it tempers out 176/175, 1331/1323, 3773/3750, and 16896/16807 in the 11-limit; 351/350, 352/351, 676/675, 1573/1568, and 16848/16807 in the 13-limit; supporting 13-limit tuskaloosa. Using the alternative 188e val, 243/242, 441/440, 540/539, 8019/8000, and 9801/9800 in the 11-limit; 351/350, 1716/1715, and 2080/2079 in the 13-limit; supporting 13-limit harry.
The 188cef val is a tuning for the hemigari temperament in the 11- and 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.17 | +3.05 | +1.39 | +0.35 | -2.38 | +2.03 | -3.16 | -2.83 | +2.49 | +1.56 | -2.74 |
| relative (%) | +3 | +48 | +22 | +5 | -37 | +32 | -50 | -44 | +39 | +24 | -43 | |
| Steps (reduced) |
298 (110) |
437 (61) |
528 (152) |
596 (32) |
650 (86) |
696 (132) |
734 (170) |
768 (16) |
799 (47) |
826 (74) |
850 (98) | |
Subsets and supersets
Since 188 factors into 22 × 47, 188edo contains subset edos 2, 4, 47, and 94.