764edo: Difference between revisions

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~~{{novelty}}{{stub}}~~{{Infobox ET}}

== 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, [[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 [[tempering out|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%.

=== Prime harmonics ===

{{Harmonics in equal|764|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 764edo (continued)}}

=== 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 [[jinn]]s.

Since 764 factors into primes as {{nowrap| 22 × 191 }}, 764edo has subset edos 2, 4, 191, and 382. In addition, one step of 764edo is exactly 22 [[jinn]]s ([[16808edo|22\16808]]).

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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Line 27:

|-

| 2.3

| {{~~monzo~~| 1211 -764 }}

| {{Monzo| 1211 -764 }}

| [{{~~val~~| 764 1211 }}]

| {{Mapping| 764 1211 }}

| -0.0439

| −0.0439

| 0.0439

| 2.80

|-

| 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

| 2.31

Line 38:

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| 2.3.5.7

| 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

| 2.62

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| 2.3.5.7.11

| 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

| 2.77

Line 52:

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| 2.3.5.7.11.13

| 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

| 3.49

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| 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 }}]

| {{Mapping| 764 1211 1774 2145 2643 2827 3123 }}

| -0.0327

| −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

| {{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]].

* 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 ===

{| 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 per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated ~~Ratio~~

! Associated ratio*

! Temperaments

|-

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| 435.08

| 9/7

| [[Supermajor]]

| [[Supermajor (temperament)|Supermajor]]

|-

| 2

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| [[Semisupermajor]]

|}

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms|minimal form]] in parentheses if distinct

[[Category:Abigail]]

@@ Line 1: / Line 1: @@
-{{novelty}}{{stub}}{{Infobox ET}}
+{{Infobox ET}}
-{{EDO intro|764}}
+{{ED intro}}
 == Theory ==
-edo 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.
+edo 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 [[tempering out|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%.
 === Prime harmonics ===
 {{Harmonics in equal|764|columns=11}}
+{{Harmonics in equal|764|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 764edo (continued)}}
 === Subsets and supersets ===
-Since 764 factors into 2<sup>2</sup> × 191, 764edo has subset edos 2, 4, 191, and 382. In addition, one step of 764edo is exactly 22 [[jinn]]s.
+Since 764 factors into primes as {{nowrap| 2<sup>2</sup> × 191 }}, 764edo has subset edos 2, 4, 191, and 382. In addition, one step of 764edo is exactly 22 [[jinn]]s ([[16808edo|22\16808]]).
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! 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]] (¢)
@@ Line 23: / Line 27: @@
 |-
 | 2.3
-| {{monzo| 1211 -764 }}
+| {{Monzo| 1211 -764 }}
-| [{{val| 764 1211 }}]
+| {{Mapping| 764 1211 }}
-| -0.0439
+| −0.0439
 | 0.0439
 | 2.80
 |-
 | 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
 | 2.31
@@ Line 38: / Line 42: @@
 | 2.3.5.7
 | 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
 | 2.62
@@ Line 45: / Line 49: @@
 | 2.3.5.7.11
 | 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
 | 2.77
@@ Line 52: / Line 56: @@
 | 2.3.5.7.11.13
 | 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
 | 3.49
@@ Line 59: / Line 63: @@
 | 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 }}]
+| {{Mapping| 764 1211 1774 2145 2643 2827 3123 }}
-| -0.0327
+| −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
+| {{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]].
+* 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 ===
 {| 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
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 85: / Line 98: @@
 | 435.08
 | 9/7
-| [[Supermajor]]
+| [[Supermajor (temperament)|Supermajor]]
 |-
 | 2
@@ Line 99: / Line 112: @@
 | [[Semisupermajor]]
 |}
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms|minimal form]] in parentheses if distinct
 [[Category:Abigail]]