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 [[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.

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.

Line 31:

| 2.3

| {{monzo| 783 -494 }}

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

| {{mapping| 494 783 }}

| -0.0219

| 0.0219

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| 2.3.5

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

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

| {{mapping| 494 783 1147 }}

| -0.0032

| 0.0318

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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

| 0.0670

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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

| 0.0600

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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

| 0.0576

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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

| 0.0752

| 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 ===

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|+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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| [[Semihemienneadecal]]

|}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is 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]]

@@ Line 1: / Line 1: @@
 {{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.
+{{EDO intro|494}}
 == 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 [[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.
 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 31: / Line 31: @@
 | 2.3
 | {{monzo| 783 -494 }}
-| [{{val| 494 783 }}]
+| {{mapping| 494 783 }}
 | -0.0219
 | 0.0219
@@ Line 38: / Line 38: @@
 | 2.3.5
 | {{monzo| -14 -19 19 }}, {{monzo| 39 -23 3 }}
-| [{{val| 494 783 1147 }}]
+| {{mapping| 494 783 1147 }}
 | -0.0032
 | 0.0318
@@ Line 45: / Line 45: @@
 | 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
 | 0.0670
@@ Line 52: / Line 52: @@
 | 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
 | 0.0600
@@ Line 59: / Line 59: @@
 | 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
 | 0.0576
@@ Line 66: / Line 66: @@
 | 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
 | 0.0752
 | 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 ===
@@ Line 77: / Line 77: @@
 |+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 136: / Line 136: @@
 | [[Semihemienneadecal]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is 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]]