307edo: 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" |[[Comma list | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" |Optimal<br>8ve | ! rowspan="2" | [[Mapping]] | ||
! colspan="2" |Tuning | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
|- | ! colspan="2" | Tuning error | ||
|- | |||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
| Line 34: | Line 35: | ||
| 32805/32768, {{monzo| 2 47 -65 }} | | 32805/32768, {{monzo| 2 47 -65 }} | ||
| {{mapping|307 973 713 }} | | {{mapping|307 973 713 }} | ||
| | | −0.0257 | ||
| 0.2004 | | 0.2004 | ||
| 5.13 | | 5.13 | ||
| Line 41: | Line 42: | ||
| 32805/32768, 118098/117649, 589824/588245 | | 32805/32768, 118098/117649, 589824/588245 | ||
| {{mapping| 307 973 713 862 }} | | {{mapping| 307 973 713 862 }} | ||
| | | −0.0687 | ||
| 0.1889 | | 0.1889 | ||
| 4.87 | | 4.87 | ||
| Line 48: | Line 49: | ||
| 5632/5625, 8019/8000, 32805/32768, 46656/46585 | | 5632/5625, 8019/8000, 32805/32768, 46656/46585 | ||
| {{mapping| 307 973 713 862 1062 }} | | {{mapping| 307 973 713 862 1062 }} | ||
| | | −0.0447 | ||
| 0.1756 | | 0.1756 | ||
| 4.49 | | 4.49 | ||
| Line 55: | Line 56: | ||
| 729/728, 1001/1000, 4096/4095, 6656/6655, 10648/10647 | | 729/728, 1001/1000, 4096/4095, 6656/6655, 10648/10647 | ||
| {{mapping| 307 973 713 862 1062 1136 }} | | {{mapping| 307 973 713 862 1062 1136 }} | ||
| | | −0.0311 | ||
| 0.1632 | | 0.1632 | ||
| 4.18 | | 4.18 | ||
| Line 62: | Line 63: | ||
| 729/728, 936/935, 1001/1000, 1377/1375, 2025/2023, 7744/7735 | | 729/728, 936/935, 1001/1000, 1377/1375, 2025/2023, 7744/7735 | ||
| {{mapping| 307 973 713 862 1062 1136 1255 }} | | {{mapping| 307 973 713 862 1062 1136 1255 }} | ||
| | | −0.0470 | ||
| 0.1560 | | 0.1560 | ||
| 3.99 | | 3.99 | ||
Latest revision as of 12:42, 21 February 2025
| ← 306edo | 307edo | 308edo → |
307 equal divisions of the octave (abbreviated 307edo or 307ed2), also called 307-tone equal temperament (307tet) or 307 equal temperament (307et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 307 equal parts of about 3.91 ¢ each. Each step represents a frequency ratio of 21/307, or the 307th root of 2.
Theory
307edo is consistent to the 7-odd-limit, but harmonic 3 is about halfway between its steps. It can be considered in either the 2.9.5.7 subgroup or the 2.9.15.21 subgroup, but the former is more flexible as it lends itself to an extension to the 2.9.5.7.11.13.17.19.23.
Using the full 7-limit patent val nonetheless, the equal temperament tempers out 2401/2400 in the 7-limit, and in the 11-limit extension, 3388/3375, 6250/6237, 15488/15435, 16384/16335, and 43923/43904.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.63 | +0.66 | +0.56 | -0.65 | -0.18 | -0.14 | -1.62 | +0.58 | -0.44 | -1.73 | +1.04 |
| Relative (%) | +41.7 | +16.8 | +14.2 | -16.7 | -4.6 | -3.5 | -41.5 | +14.9 | -11.4 | -44.1 | +26.6 | |
| Steps (reduced) |
487 (180) |
713 (99) |
862 (248) |
973 (52) |
1062 (141) |
1136 (215) |
1199 (278) |
1255 (27) |
1304 (76) |
1348 (120) |
1389 (161) | |
Subsets and supersets
307edo is the 63rd prime edo. 614edo, 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 | [-973 307⟩ | [⟨307 973]] | +0.1029 | 0.1030 | 2.64 |
| 2.9.5 | 32805/32768, [2 47 -65⟩ | [⟨307 973 713]] | −0.0257 | 0.2004 | 5.13 |
| 2.9.5.7 | 32805/32768, 118098/117649, 589824/588245 | [⟨307 973 713 862]] | −0.0687 | 0.1889 | 4.87 |
| 2.9.5.7.11 | 5632/5625, 8019/8000, 32805/32768, 46656/46585 | [⟨307 973 713 862 1062]] | −0.0447 | 0.1756 | 4.49 |
| 2.9.5.7.11.13 | 729/728, 1001/1000, 4096/4095, 6656/6655, 10648/10647 | [⟨307 973 713 862 1062 1136]] | −0.0311 | 0.1632 | 4.18 |
| 2.9.5.7.11.13.17 | 729/728, 936/935, 1001/1000, 1377/1375, 2025/2023, 7744/7735 | [⟨307 973 713 862 1062 1136 1255]] | −0.0470 | 0.1560 | 3.99 |