764edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|764}}
{{ED intro}}


== Theory ==
== Theory ==
Line 15: Line 15:
== 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|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]] (¢)
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| {{monzo| 1211 -764 }}
| {{monzo| 1211 -764 }}
| {{mapping| 764 1211 }}
| {{mapping| 764 1211 }}
| -0.0439
| −0.0439
| 0.0439
| 0.0439
| 2.80
| 2.80
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| {{monzo| 38 -2 -15 }}, {{monzo| 25 -48 22 }}
| {{monzo| 38 -2 -15 }}, {{monzo| 25 -48 22 }}
| {{mapping| 764 1211 1774 }}
| {{mapping| 764 1211 1774 }}
| -0.0399
| −0.0399
| 0.0363
| 0.0363
| 2.31
| 2.31
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| 4375/4374, 52734375/52706752, {{monzo| 31 -6 -2 -6 }}
| 4375/4374, 52734375/52706752, {{monzo| 31 -6 -2 -6 }}
| {{mapping| 764 1211 1774 2145 }}
| {{mapping| 764 1211 1774 2145 }}
| -0.0552
| −0.0552
| 0.0412
| 0.0412
| 2.62
| 2.62
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| 3025/3024, 4375/4374, 131072/130977, 35156250/35153041
| 3025/3024, 4375/4374, 131072/130977, 35156250/35153041
| {{mapping| 764 1211 1774 2145 2643 }}
| {{mapping| 764 1211 1774 2145 2643 }}
| -0.0436
| −0.0436
| 0.0435
| 0.0435
| 2.77
| 2.77
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| 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 }}
| {{mapping| 764 1211 1774 2145 2643 2827 }}
| -0.0267
| −0.0267
| 0.0548
| 0.0548
| 3.49
| 3.49
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| 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 }}
| {{mapping| 764 1211 1774 2145 2643 2827 3123 }}
| -0.0327
| −0.0327
| 0.0528
| 0.0528
| 3.36
| 3.36
|-
| 2.3.5.7.11.13.17.23
| 1716/1715, 2080/2079, 2024/2023, 2431/2430, 2500/2499, 3520/3519, 4096/4095
| {{mapping| 764 1211 1774 2145 2643 2827 3123 3456 }}
| −0.0286
| 0.0506
| 3.22
|}
|}
* 764et has lower absolute errors than any previous equal temperaments in the 13- and 17-limit. In the 13-limit it beats [[684edo|684]] and is only bettered by [[935edo|935]]. In the 17-limit it beats [[742edo|742]] and is only bettered by [[814edo|814]].  
* 764et has lower absolute errors than any previous equal temperaments in the 13- and 17-limit. In the 13-limit it beats [[684edo|684]] and is only bettered by [[935edo|935]]. In the 17-limit it beats [[742edo|742]] and is only bettered by [[814edo|814]].  
* It is best at the no-19 23-limit, where it has a lower relative error than any previous equal temperaments, past [[494edo|494]] and before [[1578edo|1578]].


=== 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
|-
|-
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|-
|-
| 2
| 2
| 277\764<br>(105\764)
| 277\764<br />(105\764)
| 435.08<br>(164.92)
| 435.08<br />(164.92)
| 9/7<br>(11/10)
| 9/7<br />(11/10)
| [[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
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


[[Category:Abigail]]
[[Category:Abigail]]

Latest revision as of 16:14, 20 February 2025

← 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

764 equal divisions of the octave (abbreviated 764edo or 764ed2), also called 764-tone equal temperament (764tet) or 764 equal temperament (764et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 764 equal parts of about 1.57 ¢ each. Each step represents a frequency ratio of 21/764, or the 764th root of 2.

Theory

764edo is a very strong 17-limit system, consistent to the 17-odd-limit or the no-19 no-29 41-odd-limit. It 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 41 43 47
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 -0.267 +0.524 +0.462
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 -17.0 +33.4 +29.4
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) 		4093
(273) 		4146
(326) 		4244
(424)

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 (22\16808).

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
2.3.5.7.11.13.17.23 1716/1715, 2080/2079, 2024/2023, 2431/2430, 2500/2499, 3520/3519, 4096/4095 [⟨764 1211 1774 2145 2643 2827 3123 3456]] −0.0286 0.0506 3.22
  • 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.
  • It is best at the no-19 23-limit, where it has a lower relative error than any previous equal temperaments, past 494 and before 1578.

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 distinct

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