764edo: Difference between revisions
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Cleanup; clarify the title row of the rank-2 temp table |
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== Theory == | == 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. | ||
In higher limits, it is a strong no-19 and no-29 37-limit tuning, and an exceptional 2.11.23.31.37 subgroup system, with errors less than 2%. | In higher limits, it is a strong no-19 and no-29 37-limit tuning, and an exceptional 2.11.23.31.37 subgroup system, with errors less than 2%. | ||
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| 2.3 | | 2.3 | ||
| {{monzo| 1211 -764 }} | | {{monzo| 1211 -764 }} | ||
| | | {{mapping| 764 1211 }} | ||
| -0.0439 | | -0.0439 | ||
| 0.0439 | | 0.0439 | ||
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| 2.3.5 | | 2.3.5 | ||
| {{monzo| 38 -2 -15 }}, {{monzo| 25 -48 22 }} | | {{monzo| 38 -2 -15 }}, {{monzo| 25 -48 22 }} | ||
| | | {{mapping| 764 1211 1774 }} | ||
| -0.0399 | | -0.0399 | ||
| 0.0363 | | 0.0363 | ||
| Line 40: | Line 40: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 4375/4374, 52734375/52706752, {{monzo| 31 -6 -2 -6 }} | | 4375/4374, 52734375/52706752, {{monzo| 31 -6 -2 -6 }} | ||
| | | {{mapping| 764 1211 1774 2145 }} | ||
| -0.0552 | | -0.0552 | ||
| 0.0412 | | 0.0412 | ||
| Line 47: | Line 47: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 3025/3024, 4375/4374, 131072/130977, 35156250/35153041 | | 3025/3024, 4375/4374, 131072/130977, 35156250/35153041 | ||
| | | {{mapping| 764 1211 1774 2145 2643 }} | ||
| -0.0436 | | -0.0436 | ||
| 0.0435 | | 0.0435 | ||
| Line 54: | Line 54: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 10549994/10546875 | | 1716/1715, 2080/2079, 3025/3024, 4096/4095, 10549994/10546875 | ||
| | | {{mapping| 764 1211 1774 2145 2643 2827 }} | ||
| -0.0267 | | -0.0267 | ||
| 0.0548 | | 0.0548 | ||
| Line 61: | Line 61: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4096/4095, 4914/4913 | | 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4096/4095, 4914/4913 | ||
| | | {{mapping| 764 1211 1774 2145 2643 2827 3123 }} | ||
| -0.0327 | | -0.0327 | ||
| 0.0528 | | 0.0528 | ||
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|+Table of rank-2 temperaments by generator | |+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 | ||
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| [[Semisupermajor]] | | [[Semisupermajor]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
[[Category:Abigail]] | [[Category:Abigail]] | ||