494edo: Difference between revisions

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The '''494 equal divisions of the octave''' ('''494edo'''), or the '''494(-tone) equal temperament''' ('''494tet''', '''494et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 494 [[equal]] parts of about 2.43 [[cent]]s each.

== Theory ==

494 is a very strong [[13-limit|13]]- and [[17-limit]] equal temperament. 494edo is a [[~~The Riemann~~ zeta ~~function and tuning #Zeta EDO lists~~|zeta peak and zeta peak integer edo]] and ~~uniquely~~ [[consistent]] through the [[17-odd-limit]]. It [[tempering out|tempers out]] the [[enneadeca]], {{monzo| -14 -19 19 }}, the [[~~tricot~~ comma]], {{monzo| 39 -29 3 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }} in the [[5-limit]]. In the [[7-limit]], it tempers out [[4375/4374]] and [[703125/702464]]; in the [[11-limit]] [[3025/3024]] and [[9801/9800]]; in the [[13-limit]] [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and [[6656/6655]]; and in the 17-limit, [[1156/1155]], 1275/1274, 2431/2430, and 2500/2499.

494 is a very strong [[13-limit|13]]- and [[17-limit]] equal temperament. 494edo is a [[zeta edo|zeta peak and zeta peak integer edo]] and [[consistency|distinctly consistent]] through the [[17-odd-limit]]. It [[tempering out|tempers out]] the [[enneadeca]], {{monzo| -14 -19 19 }}, the [[alphatricot comma]], {{monzo| 39 -29 3 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }} in the [[5-limit]]. In the [[7-limit]], it tempers out [[4375/4374]] and [[703125/702464]]; in the [[11-limit]] [[3025/3024]] and [[9801/9800]]; in the [[13-limit]] [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and [[6656/6655]]; and in the 17-limit, [[1156/1155]], 1275/1274, 2431/2430, and 2500/2499.

Since the step size is close to [[729/728]], the squbema, the accepted name for 494edo's step is ''squb''.

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=== ~~Divisors~~ and ~~multipliers~~ ===

=== Subsets and supersets ===

Since 494 ~~= 2 × 13 × 19~~, 494edo has subset edos {{EDOs| 2, 13, 19, 26, 38, and 247 }}.

Since 494 factors into {{factorization|494}}, 494edo has subset edos {{EDOs| 2, 13, 19, 26, 38, and 247 }}.

[[988edo]], which slices the edostep in two, provides a good correction of the 19th harmonic. [[2964edo]], which slices the edostep in six, provides an extremely precise correction of the 7th harmonic.

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== Intervals ==

== 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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|-

| 2.3

| {{~~monzo~~| 783 -494 }}

| {{Monzo| 783 -494 }}

| [{{~~val~~| 494 783 }}]

| {{Mapping| 494 783 }}

| -0.0219

| −0.0219

| 0.0219

| 0.90

|-

| 2.3.5

| {{~~monzo~~| -14 -19 19 }}, {{monzo| 39 -23 3 }}

| {{Monzo| -14 -19 19 }}, {{monzo| 39 -23 3 }}

| [{{~~val~~| 494 783 1147 }}]

| {{Mapping| 494 783 1147 }}

| -0.0032

| −0.0032

| 0.0318

| 1.31

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| 2.3.5.7

| 4375/4374, 703125/702464, {{monzo| 21 3 1 -10 }}

| [{{~~val~~| 494 783 1147 1387 }}]

| {{Mapping| 494 783 1147 1387 }}

| -0.0385

| −0.0385

| 0.0670

| 2.76

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| 2.3.5.7.11

| 3025/3024, 4375/4374, 131072/130977, 234375/234256

| [{{~~val~~| 494 783 1147 1387 1709 }}]

| {{Mapping| 494 783 1147 1387 1709 }}

| -0.0365

| −0.0365

| 0.0600

| 2.47

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| 2.3.5.7.11.13

| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 31250/31213

| [{{~~val~~| 494 783 1147 1387 1709 1828 }}]

| {{Mapping| 494 783 1147 1387 1709 1828 }}

| -0.0286

| −0.0286

| 0.0576

| 2.37

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| 2.3.5.7.11.13.17

| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2431/2430, 4096/4095

| [{{~~val~~| 494 783 1147 1387 1709 1828 2019 }}]

| {{Mapping| 494 783 1147 1387 1709 1828 2019 }}

| -0.0069

| −0.0069

| 0.0752

| 3.09

|}

* 494et has a lower [[Tenney-Euclidean temperament measures #TE simple badness|relative ~~error~~]] than any previous equal temperaments in the 13- and 17-limit. It is the first past [[270edo|270]] with a lower 13-limit relative error, and the first past [[72edo|72]] with a lower 17-limit relative error. It is narrowly beaten by [[684edo|684]] in terms of 13-limit absolute error and by [[581edo|581]] in terms of 17-limit absolute error. Not until [[1506edo|1506]] do we reach an equal temperament with a lower relative error in either subgroup.

* 494et has lower [[Tenney-Euclidean temperament measures #TE simple badness|relative errors]] than any previous equal temperaments in the 13- and 17-limit. It is the first past [[270edo|270]] with a lower 13-limit relative error, and the first past [[72edo|72]] with a lower 17-limit relative error. It is narrowly beaten by [[684edo|684]] in terms of 13-limit absolute error and by [[581edo|581]] in terms of 17-limit absolute error. Not until [[1506edo|1506]] do we reach an equal temperament with a lower relative error in either subgroup.

=== 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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| 565.99

| 104/75

| [[~~Tricot]] / [[trillium~~]]

| [[Alphatrillium]]

|-

| 2

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| 162.75

| 1125/1024

| [[~~Kwazy~~]]

| [[Crazy]]

|-

| 2

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| 13

| 205\494 (15\494)

| 497.98 (36.43)

| 497.98 (36.43)

| 4/3 (?)

| [[Aluminium]]

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| [[Semihemienneadecal]]

|}

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

== Music ==

* [https://www.youtube.com/watch?v=JGdBFEz7Fq8 Unknown piece in Abigail (~~Op.2, No. 4~~)~~] by [[Eliora]]~~

; [[Eliora]]

* [https://www.youtube.com/watch?v=JGdBFEz7Fq8 ''Unknown piece in Abigail''] (2023)

~~[[Category:17-limit]]~~

[[Category:Enneadecal]]

[[Category:Kwazy]]

~~[[Category:Tricot]]~~

[[Category:Listen]]

[[Category:Alphatricot]]

@@ Line 1: / Line 1: @@
-{{novelty}}{{stub}}{{Infobox ET}}
+{{Infobox ET}}
-The '''494 equal divisions of the octave''' ('''494edo'''), or the '''494(-tone) equal temperament''' ('''494tet''', '''494et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 494 [[equal]] parts of about 2.43 [[cent]]s each.
+{{ED intro}}
 == Theory ==
-is a very strong [[13-limit|13]]- and [[17-limit]] equal temperament. 494edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak and zeta peak integer edo]] and uniquely [[consistent]] through the [[17-odd-limit]]. It [[tempering out|tempers out]] the [[enneadeca]], {{monzo| -14 -19 19 }}, the [[tricot comma]], {{monzo| 39 -29 3 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }} in the [[5-limit]]. In the [[7-limit]], it tempers out [[4375/4374]] and [[703125/702464]]; in the [[11-limit]] [[3025/3024]] and [[9801/9800]]; in the [[13-limit]] [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and [[6656/6655]]; and in the 17-limit, [[1156/1155]], 1275/1274, 2431/2430, and 2500/2499.
+is a very strong [[13-limit|13]]- and [[17-limit]] equal temperament. 494edo is a [[zeta edo|zeta peak and zeta peak integer edo]] and [[consistency|distinctly consistent]] through the [[17-odd-limit]]. It [[tempering out|tempers out]] the [[enneadeca]], {{monzo| -14 -19 19 }}, the [[alphatricot comma]], {{monzo| 39 -29 3 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }} in the [[5-limit]]. In the [[7-limit]], it tempers out [[4375/4374]] and [[703125/702464]]; in the [[11-limit]] [[3025/3024]] and [[9801/9800]]; in the [[13-limit]] [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and [[6656/6655]]; and in the 17-limit, [[1156/1155]], 1275/1274, 2431/2430, and 2500/2499.
 Since the step size is close to [[729/728]], the squbema, the accepted name for 494edo's step is ''squb''.
@@ Line 10: / Line 10: @@
 {{Harmonics in equal|494|prec=3}}
-=== Divisors and multipliers ===
+=== Subsets and supersets ===
-Since 494 = 2 × 13 × 19, 494edo has subset edos {{EDOs| 2, 13, 19, 26, 38, and 247 }}.
+Since 494 factors into {{factorization|494}}, 494edo has subset edos {{EDOs| 2, 13, 19, 26, 38, and 247 }}.
 [[988edo]], which slices the edostep in two, provides a good correction of the 19th harmonic. [[2964edo]], which slices the edostep in six, provides an extremely precise correction of the 7th harmonic.
@@ Line 17: / Line 17: @@
 == Intervals ==
 {{Main| Table of 494edo intervals }}
+{{Q-odd-limit intervals|494}}
 == 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 30: / Line 32: @@
 |-
 | 2.3
-| {{monzo| 783 -494 }}
+| {{Monzo| 783 -494 }}
-| [{{val| 494 783 }}]
+| {{Mapping| 494 783 }}
-| -0.0219
+| −0.0219
 | 0.0219
 | 0.90
 |-
 | 2.3.5
-| {{monzo| -14 -19 19 }}, {{monzo| 39 -23 3 }}
+| {{Monzo| -14 -19 19 }}, {{monzo| 39 -23 3 }}
-| [{{val| 494 783 1147 }}]
+| {{Mapping| 494 783 1147 }}
-| -0.0032
+| −0.0032
 | 0.0318
 | 1.31
@@ Line 45: / Line 47: @@
 | 2.3.5.7
 | 4375/4374, 703125/702464, {{monzo| 21 3 1 -10 }}
-| [{{val| 494 783 1147 1387 }}]
+| {{Mapping| 494 783 1147 1387 }}
-| -0.0385
+| −0.0385
 | 0.0670
 | 2.76
@@ Line 52: / Line 54: @@
 | 2.3.5.7.11
 | 3025/3024, 4375/4374, 131072/130977, 234375/234256
-| [{{val| 494 783 1147 1387 1709 }}]
+| {{Mapping| 494 783 1147 1387 1709 }}
-| -0.0365
+| −0.0365
 | 0.0600
 | 2.47
@@ Line 59: / Line 61: @@
 | 2.3.5.7.11.13
 | 1716/1715, 2080/2079, 3025/3024, 4096/4095, 31250/31213
-| [{{val| 494 783 1147 1387 1709 1828 }}]
+| {{Mapping| 494 783 1147 1387 1709 1828 }}
-| -0.0286
+| −0.0286
 | 0.0576
 | 2.37
@@ Line 66: / Line 68: @@
 | 2.3.5.7.11.13.17
 | 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2431/2430, 4096/4095
-| [{{val| 494 783 1147 1387 1709 1828 2019 }}]
+| {{Mapping| 494 783 1147 1387 1709 1828 2019 }}
-| -0.0069
+| −0.0069
 | 0.0752
 | 3.09
 |}
-* 494et has a lower [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any previous equal temperaments in the 13- and 17-limit. It is the first past [[270edo|270]] with a lower 13-limit relative error, and the first past [[72edo|72]] with a lower 17-limit relative error. It is narrowly beaten by [[684edo|684]] in terms of 13-limit absolute error and by [[581edo|581]] in terms of 17-limit absolute error. Not until [[1506edo|1506]] do we reach an equal temperament with a lower relative error in either subgroup.
+* 494et has lower [[Tenney-Euclidean temperament measures #TE simple badness|relative errors]] than any previous equal temperaments in the 13- and 17-limit. It is the first past [[270edo|270]] with a lower 13-limit relative error, and the first past [[72edo|72]] with a lower 17-limit relative error. It is narrowly beaten by [[684edo|684]] in terms of 13-limit absolute error and by [[581edo|581]] in terms of 17-limit absolute error. Not until [[1506edo|1506]] do we reach an equal temperament with a lower relative error in either subgroup.
 === 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 98: / Line 101: @@
 | 565.99
 | 104/75
-| [[Tricot]] / [[trillium]]
+| [[Alphatrillium]]
 |-
 | 2
@@ Line 104: / Line 107: @@
 | 162.75
 | 1125/1024
-| [[Kwazy]]
+| [[Crazy]]
 |-
 | 2
@@ Line 114: / Line 117: @@
 | 13
 | 205\494<br>(15\494)
-| 497.98<br>(36.43)
+| 497.98<br/>(36.43)
 | 4/3<br>(?)
 | [[Aluminium]]
@@ Line 136: / Line 139: @@
 | [[Semihemienneadecal]]
 |}
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 == Music ==
-* [https://www.youtube.com/watch?v=JGdBFEz7Fq8 Unknown piece in Abigail (Op.2, No. 4)] by [[Eliora]]
+; [[Eliora]]
+* [https://www.youtube.com/watch?v=JGdBFEz7Fq8 ''Unknown piece in Abigail''] (2023)
-[[Category:17-limit]]
 [[Category:Enneadecal]]
 [[Category:Kwazy]]
-[[Category:Tricot]]
 [[Category:Listen]]
+[[Category:Alphatricot]]