764edo: Difference between revisions

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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 }}
| [{{val| 764 1211 }}]
| {{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 }}
| [{{val| 764 1211 1774 }}]
| {{mapping| 764 1211 1774 }}
| -0.0399
| -0.0399
| 0.0363
| 0.0363
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| 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 }}
| [{{val| 764 1211 1774 2145 }}]
| {{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
| [{{val| 764 1211 1774 2145 2643 }}]
| {{mapping| 764 1211 1774 2145 2643 }}
| -0.0436
| -0.0436
| 0.0435
| 0.0435
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| 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
| [{{val| 764 1211 1774 2145 2643 2827 }}]
| {{mapping| 764 1211 1774 2145 2643 2827 }}
| -0.0267
| -0.0267
| 0.0548
| 0.0548
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| 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
| [{{val| 764 1211 1774 2145 2643 2827 3123 }}]
| {{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<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio
! Temperaments
! Temperaments
Line 101: Line 101:
| [[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]]

Revision as of 12:03, 20 October 2023

← 763edo 764edo 765edo →
Prime factorization 22 × 191
Step size 1.57068 ¢ 
Fifth 447\764 (702.094 ¢)
Semitones (A1:m2) 73:57 (114.7 ¢ : 89.53 ¢)
Consistency limit 17
Distinct consistency limit 17

Template:EDO intro

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.

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%.

Prime harmonics

Approximation of prime harmonics in 764edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) +0.000 +0.139 +0.074 +0.284 -0.009 -0.214 +0.280 -0.654 -0.002 -0.781 -0.009 -0.035
Relative (%) +0.0 +8.9 +4.7 +18.1 -0.6 -13.6 +17.8 -41.7 -0.1 -49.7 -0.6 -2.2
Steps
(reduced) 		764
(0) 		1211
(447) 		1774
(246) 		2145
(617) 		2643
(351) 		2827
(535) 		3123
(67) 		3245
(189) 		3456
(400) 		3711
(655) 		3785
(729) 		3980
(160)

Subsets and supersets

Since 764 factors into 22 × 191, 764edo has subset edos 2, 4, 191, and 382. In addition, one step of 764edo is exactly 22 jinns.

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 has lower absolute errors than any previous equal temperaments in the 13- and 17-limit. In the 13-limit it beats 684 and is only bettered by 935. In the 17-limit it beats 742 and is only bettered by 814.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio 		Temperaments
1 123\764 193.19 262144/234375 Lunatic (7-limit)
1 277\764 435.08 9/7 Supermajor
2 133\764 208.90 44/39 Abigail
2 277\764
(105\764) 		435.08
(164.92) 		9/7
(11/10) 		Semisupermajor

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

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