764edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|764}} | {{EDO intro|764}} | ||
== Theory == | |||
764edo is a very strong 17-limit system distinctly [[consistent]] to the 17-odd-limit, and is the fourteenth [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]]. In the 5-limit it tempers out the hemithirds comma, {{monzo| 38 -2 -15 }}; in the 7-limit [[4375/4374]]; in the 11-limit [[3025/3024]] and [[9801/9800]]; in the 13-limit [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]]; and in the 17-limit 2431/2430, 2500/2499, 4914/4913 and [[5832/5831]]. It provides the [[optimal patent val]] for the [[abigail]] temperament in the 11-limit. | 764edo is a very strong 17-limit system distinctly [[consistent]] to the 17-odd-limit, and is the fourteenth [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]]. In the 5-limit it tempers out the hemithirds comma, {{monzo| 38 -2 -15 }}; in the 7-limit [[4375/4374]]; in the 11-limit [[3025/3024]] and [[9801/9800]]; in the 13-limit [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]]; and in the 17-limit 2431/2430, 2500/2499, 4914/4913 and [[5832/5831]]. It provides the [[optimal patent val]] for the [[abigail]] temperament in the 11-limit. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|764|columns=11}} | {{Harmonics in equal|764|columns=11}} | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list|Comma List]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve Stretch (¢) | |||
! colspan="2" | Tuning Error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| 1211 -764 }} | |||
| [{{val| 764 1211 }}] | |||
| -0.0439 | |||
| 0.0439 | |||
| 2.80 | |||
|- | |||
| 2.3.5 | |||
| {{monzo| 38 -2 -15 }}, {{monzo| 25 -48 22 }} | |||
| [{{val| 764 1211 1774 }}] | |||
| -0.0399 | |||
| 0.0363 | |||
| 2.31 | |||
|- | |||
| 2.3.5.7 | |||
| 4375/4374, 52734375/52706752, {{monzo| 31 -6 -2 -6 }} | |||
| [{{val| 764 1211 1774 2145 }}] | |||
| -0.0552 | |||
| 0.0412 | |||
| 2.62 | |||
|- | |||
| 2.3.5.7.11 | |||
| 3025/3024, 4375/4374, 131072/130977, 35156250/35153041 | |||
| [{{val| 764 1211 1774 2145 2643 }}] | |||
| -0.0436 | |||
| 0.0435 | |||
| 2.77 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 10549994/10546875 | |||
| [{{val| 764 1211 1774 2145 2643 2827 }}] | |||
| -0.0267 | |||
| 0.0548 | |||
| 3.49 | |||
|- | |||
| 2.3.5.7.11.13.17 | |||
| 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4096/4095, 4914/4913 | |||
| [{{val| 764 1211 1774 2145 2643 2827 3123 }}] | |||
| -0.0327 | |||
| 0.0528 | |||
| 3.36 | |||
|} | |||
764et is the first equal temperament past [[684edo|684]] with a lower 13-limit absolute error, and is only bettered by [[935edo|935]]. It is also the first equal temperament past [[742edo|742]] with a lower 17-limit absolute error, and is only bettered by [[814edo|814]]. | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
[[Category:Zeta]] | [[Category:Zeta]] | ||
[[Category:Abigail]] | [[Category:Abigail]] | ||
Revision as of 09:55, 6 October 2022
| ← 763edo | 764edo | 765edo → |
Theory
764edo is a very strong 17-limit system distinctly consistent to the 17-odd-limit, and is the fourteenth zeta integral edo. In the 5-limit it tempers out the hemithirds comma, [38 -2 -15⟩; in the 7-limit 4375/4374; in the 11-limit 3025/3024 and 9801/9800; in the 13-limit 1716/1715, 2080/2079, 4096/4095, 4225/4224, 6656/6655 and 10648/10647; and in the 17-limit 2431/2430, 2500/2499, 4914/4913 and 5832/5831. It provides the optimal patent val for the abigail temperament in the 11-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.139 | +0.074 | +0.284 | -0.009 | -0.214 | +0.280 | -0.654 | -0.002 | -0.781 | -0.009 |
| Relative (%) | +0.0 | +8.9 | +4.7 | +18.1 | -0.6 | -13.6 | +17.8 | -41.7 | -0.1 | -49.7 | -0.6 | |
| Steps (reduced) |
764 (0) |
1211 (447) |
1774 (246) |
2145 (617) |
2643 (351) |
2827 (535) |
3123 (67) |
3245 (189) |
3456 (400) |
3711 (655) |
3785 (729) | |
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1211 -764⟩ | [⟨764 1211]] | -0.0439 | 0.0439 | 2.80 |
| 2.3.5 | [38 -2 -15⟩, [25 -48 22⟩ | [⟨764 1211 1774]] | -0.0399 | 0.0363 | 2.31 |
| 2.3.5.7 | 4375/4374, 52734375/52706752, [31 -6 -2 -6⟩ | [⟨764 1211 1774 2145]] | -0.0552 | 0.0412 | 2.62 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 131072/130977, 35156250/35153041 | [⟨764 1211 1774 2145 2643]] | -0.0436 | 0.0435 | 2.77 |
| 2.3.5.7.11.13 | 1716/1715, 2080/2079, 3025/3024, 4096/4095, 10549994/10546875 | [⟨764 1211 1774 2145 2643 2827]] | -0.0267 | 0.0548 | 3.49 |
| 2.3.5.7.11.13.17 | 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4096/4095, 4914/4913 | [⟨764 1211 1774 2145 2643 2827 3123]] | -0.0327 | 0.0528 | 3.36 |
764et is the first equal temperament past 684 with a lower 13-limit absolute error, and is only bettered by 935. It is also the first equal temperament past 742 with a lower 17-limit absolute error, and is only bettered by 814.