175edo
| ← 174edo | 175edo | 176edo → |
175 equal divisions of the octave (abbreviated 175edo or 175ed2), also called 175-tone equal temperament (175tet) or 175 equal temperament (175et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 175 equal parts of about 6.86 ¢ each. Each step represents a frequency ratio of 21/175, or the 175th root of 2.
Theory
175et tempers out 225/224 and 1029/1024, so that it supports 7-limit miracle, and in fact provides an excellent alternative to 72edo for 7-limit miracle with improved 5 and 7 at the cost of a slightly flatter 3. In the 11-limit, it tempers out 243/242, 385/384, 441/440 and 540/539, and supports 11-limit miracle. In the 13-limit, the 175f val, ⟨175 277 406 491 605 647] tempers out 351/350 just as 72 does, and provides a tuning for benediction temperament very close to the POTE tuning.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -2.53 | -2.31 | -1.97 | +1.80 | -2.75 | +2.90 | +2.02 | -2.10 | -2.66 | +2.36 | +2.58 |
| relative (%) | -37 | -34 | -29 | +26 | -40 | +42 | +29 | -31 | -39 | +34 | +38 | |
| Steps (reduced) |
277 (102) |
406 (56) |
491 (141) |
555 (30) |
605 (80) |
648 (123) |
684 (159) |
715 (15) |
743 (43) |
769 (69) |
792 (92) | |
Subsets and supersets
Since 175 factors into 52 × 7, 175edo has subset edos 5, 7, 25, and 35.