770edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
| Line 25: | Line 26: | ||
| {{monzo| 2441 -770 }} | | {{monzo| 2441 -770 }} | ||
| {{mapping| 770 2441 }} | | {{mapping| 770 2441 }} | ||
| | | −0.0388 | ||
| 0.0388 | | 0.0388 | ||
| 2.49 | | 2.49 | ||
| Line 32: | Line 33: | ||
| {{monzo| 61 2 -29 }}, {{monzo| 40 -28 21 }} | | {{monzo| 61 2 -29 }}, {{monzo| 40 -28 21 }} | ||
| {{mapping| 770 2441 1788 }} | | {{mapping| 770 2441 1788 }} | ||
| | | −0.0517 | ||
| 0.0365 | | 0.0365 | ||
| 2.34 | | 2.34 | ||
| Line 39: | Line 40: | ||
| 43046721/43025920, 134217728/133984375, 1220703125/1219784832 | | 43046721/43025920, 134217728/133984375, 1220703125/1219784832 | ||
| {{mapping| 770 2441 1788 2162 }} | | {{mapping| 770 2441 1788 2162 }} | ||
| | | −0.0855 | ||
| 0.0666 | | 0.0666 | ||
| 4.27 | | 4.27 | ||
| Line 46: | Line 47: | ||
| 9801/9800, 820125/819896, 1296000/1294139, 1362944/1361367 | | 9801/9800, 820125/819896, 1296000/1294139, 1362944/1361367 | ||
| {{mapping| 770 2441 1788 2162 2664 }} | | {{mapping| 770 2441 1788 2162 2664 }} | ||
| | | −0.0898 | ||
| 0.0602 | | 0.0602 | ||
| 3.86 | | 3.86 | ||
Latest revision as of 12:32, 21 February 2025
| ← 769edo | 770edo | 771edo → |
770 equal divisions of the octave (abbreviated 770edo or 770ed2), also called 770-tone equal temperament (770tet) or 770 equal temperament (770et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 770 equal parts of about 1.56 ¢ each. Each step represents a frequency ratio of 21/770, or the 770th root of 2.
Theory
770edo is inconsistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. It has a reasonable approximation of the 2.9.5.7.11 subgroup, where it notably tempers out 9801/9800.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.656 | +0.180 | +0.525 | +0.246 | +0.370 | -0.528 | -0.477 | -0.540 | +0.149 | -0.132 | -0.222 |
| Relative (%) | -42.1 | +11.5 | +33.7 | +15.8 | +23.8 | -33.9 | -30.6 | -34.6 | +9.6 | -8.4 | -14.3 | |
| Steps (reduced) |
1220 (450) |
1788 (248) |
2162 (622) |
2441 (131) |
2664 (354) |
2849 (539) |
3008 (698) |
3147 (67) |
3271 (191) |
3382 (302) |
3483 (403) | |
Subsets and supersets
Since 770 factors into 2 × 5 × 7 × 11, 770edo has subset edos 2, 5, 7, 10, 11, 14, 22, 35, 55, 70, 77, 110, 154, and 385. 1540edo, which doubles it, gives a good correction to the harmonic 3.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [2441 -770⟩ | [⟨770 2441]] | −0.0388 | 0.0388 | 2.49 |
| 2.9.5 | [61 2 -29⟩, [40 -28 21⟩ | [⟨770 2441 1788]] | −0.0517 | 0.0365 | 2.34 |
| 2.9.5.7 | 43046721/43025920, 134217728/133984375, 1220703125/1219784832 | [⟨770 2441 1788 2162]] | −0.0855 | 0.0666 | 4.27 |
| 2.9.5.7.11 | 9801/9800, 820125/819896, 1296000/1294139, 1362944/1361367 | [⟨770 2441 1788 2162 2664]] | −0.0898 | 0.0602 | 3.86 |