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]] | ||