757edo: 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 stretch (¢) | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
| Line 25: | Line 26: | ||
| {{monzo| 1200 -757 }} | | {{monzo| 1200 -757 }} | ||
| {{mapping| 757 1200 }} | | {{mapping| 757 1200 }} | ||
| | | −0.0917 | ||
| 0.0917 | | 0.0917 | ||
| 5.78 | | 5.78 | ||
| Line 32: | Line 33: | ||
| {{monzo| 24 -21 4 }}, {{monzo| 78 5 -37 }} | | {{monzo| 24 -21 4 }}, {{monzo| 78 5 -37 }} | ||
| {{mapping| 757 1200 1758 }} | | {{mapping| 757 1200 1758 }} | ||
| | | −0.1295 | ||
| 0.0920 | | 0.0920 | ||
| 5.80 | | 5.80 | ||
| Line 39: | Line 40: | ||
| 4375/4374, 67108864/66976875, 282475249/281250000 | | 4375/4374, 67108864/66976875, 282475249/281250000 | ||
| {{mapping| 757 1200 1758 2125 }} | | {{mapping| 757 1200 1758 2125 }} | ||
| | | −0.0735 | ||
| 0.1256 | | 0.1256 | ||
| 7.92 | | 7.92 | ||
| Line 46: | Line 47: | ||
| 4375/4374, 5632/5625, 160083/160000, 5764801/5749920 | | 4375/4374, 5632/5625, 160083/160000, 5764801/5749920 | ||
| {{mapping| 757 1200 1758 2125 2619 }} | | {{mapping| 757 1200 1758 2125 2619 }} | ||
| | | −0.0780 | ||
| 0.1127 | | 0.1127 | ||
| 7.11 | | 7.11 | ||
| Line 53: | Line 54: | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br>ratio* | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 66: | Line 68: | ||
| [[Vulture]] | | [[Vulture]] | ||
|} | |} | ||
<nowiki>* | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
Revision as of 12:59, 21 February 2025
| ← 756edo | 757edo | 758edo → |
757 equal divisions of the octave (abbreviated 757edo or 757ed2), also called 757-tone equal temperament (757tet) or 757 equal temperament (757et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 757 equal parts of about 1.59 ¢ each. Each step represents a frequency ratio of 21/757, or the 757th root of 2.
Theory
757edo is consistent to the 7-odd-limit. As an equal temperament, it tempers out 4375/4374, 67108864/66976875 and 282475249/281250000. Using the patent val, it tempers out 5632/5625, 160083/160000 and 5764801/5749920 in the 11-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.291 | +0.476 | -0.266 | +0.333 | -0.369 | -0.332 | +0.505 | -0.533 | -0.779 | -0.518 |
| Relative (%) | +0.0 | +18.3 | +30.0 | -16.8 | +21.0 | -23.3 | -20.9 | +31.9 | -33.6 | -49.2 | -32.7 | |
| Steps (reduced) |
757 (0) |
1200 (443) |
1758 (244) |
2125 (611) |
2619 (348) |
2801 (530) |
3094 (66) |
3216 (188) |
3424 (396) |
3677 (649) |
3750 (722) | |
Subsets and supersets
757edo is the 134th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1200 -757⟩ | [⟨757 1200]] | −0.0917 | 0.0917 | 5.78 |
| 2.3.5 | [24 -21 4⟩, [78 5 -37⟩ | [⟨757 1200 1758]] | −0.1295 | 0.0920 | 5.80 |
| 2.3.5.7 | 4375/4374, 67108864/66976875, 282475249/281250000 | [⟨757 1200 1758 2125]] | −0.0735 | 0.1256 | 7.92 |
| 2.3.5.7.11 | 4375/4374, 5632/5625, 160083/160000, 5764801/5749920 | [⟨757 1200 1758 2125 2619]] | −0.0780 | 0.1127 | 7.11 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 300\757 | 475.561 | 320/243 | Vulture |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct