751edo
| ← 750edo | 751edo | 752edo → |
751 equal divisions of the octave (abbreviated 751edo or 751ed2), also called 751-tone equal temperament (751tet) or 751 equal temperament (751et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 751 equal parts of about 1.6 ¢ each. Each step represents a frequency ratio of 21/751, or the 751st root of 2.
Theory
751edo is inconsistent in the 5-odd-limit, with rather large errors in the harmonics 3, 7 and 17. It has two mappings possible for the 7-limit:
- ⟨751 1190 1744 2108] (patent val)
- ⟨751 1190 1743 2108] (751c val)
Using the patent val, it tempers out 2460375/2458624 (breeze comma), 26873856/26796875 and [-14 7 -6 6⟩ in the 7-limit.
Using the 751c val, it tempers out 420175/419904, 2109375/2097152 and 1640558367/1638400000 in the 7-limit, supporting quinwell.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.490 | +0.371 | -0.517 | +0.617 | -0.053 | -0.048 | -0.120 | +0.504 | -0.309 | +0.591 | -0.312 |
| Relative (%) | -30.7 | +23.2 | -32.4 | +38.6 | -3.3 | -3.0 | -7.5 | +31.5 | -19.4 | +37.0 | -19.5 | |
| Steps (reduced) |
1190 (439) |
1744 (242) |
2108 (606) |
2381 (128) |
2598 (345) |
2779 (526) |
2934 (681) |
3070 (66) |
3190 (186) |
3299 (295) |
3397 (393) | |
Subsets and supersets
751edo is the 133rd prime edo. 1502edo, which doubles it, gives a good correction to the harmonic 5.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-1190 751⟩ | [⟨751 1190]] | +0.1547 | 0.1547 | 9.68 |
|
Todo: explain its xenharmonic value |