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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2 × 13 × 19
{{ED intro}}
| Step size = 2.42915¢
| Fifth = 289\494 (702.02¢)
| Semitones = 47:37 (114.17¢ : 89.88¢)
| Consistency = 17
}}
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. It is a very strong [[13-limit|13]]- and [[17-limit]] equal temperament. The step size is close to [[729/728]], the squbema, and a step is a '''squb'''.


== Theory ==
== Theory ==
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. Not until [[1506edo|1506]] do we reach a division with a lower 13- or 17-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], and it is the first past [[72edo|72]] with a lower 17-limit relative error. 494 is divisible by 2, 13, 19, 26, 38 and 247.
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''.


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|494|prec=3}}
{{Harmonics in equal|494|prec=3}}
=== Subsets and supersets ===
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.


== Intervals ==
== Intervals ==
{{Main| Table of 494edo intervals }}
{{Main| Table of 494edo intervals }}
{{Q-odd-limit intervals|494}}


== 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]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
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|-
|-
| 2.3
| 2.3
| {{monzo| 783 -494 }}
| {{Monzo| 783 -494 }}
| [{{val| 494 783 }}]
| {{Mapping| 494 783 }}
| -0.0219
| −0.0219
| 0.0219
| 0.0219
| 0.90
| 0.90
|-
|-
| 2.3.5
| 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
| 0.0318
| 1.31
| 1.31
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| 2.3.5.7
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| 21 3 1 -10 }}
| 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
| 0.0670
| 2.76
| 2.76
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| 2.3.5.7.11
| 2.3.5.7.11
| 3025/3024, 4375/4374, 131072/130977, 234375/234256
| 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
| 0.0600
| 2.47
| 2.47
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 31250/31213
| 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
| 0.0576
| 2.37
| 2.37
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2431/2430, 4096/4095
| 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
| 0.0752
| 3.09
| 3.09
|}
|}
* 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 ===
=== 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 octave
|-
! Generator<br>(reduced)
! Periods<br>per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 1
| 1
| 233\422
| 119\494
| 289.07
| 13/11
| [[Moulin]]
|-
| 1
| 233\494
| 565.99
| 565.99
| 104/75
| 104/75
| [[Tricot]] / [[trillium]]
| [[Alphatrillium]]
|-
|-
| 2
| 2
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| 162.75
| 162.75
| 1125/1024
| 1125/1024
| [[Kwazy]]
| [[Crazy]]
|-
|-
| 2
| 2
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| 44/39
| 44/39
| [[Abigail]]
| [[Abigail]]
|-
| 13
| 205\494<br>(15\494)
| 497.98<br/>(36.43)
| 4/3<br>(?)
| [[Aluminium]]
|-
|-
| 19
| 19
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| 4/3<br>(225/224)
| 4/3<br>(225/224)
| [[Hemienneadecal]]
| [[Hemienneadecal]]
|-
| 38
| 109\494<br>(5\494)
| 264.78<br>(12.15)
| 500/429<br>(144/143)
| [[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 ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=JGdBFEz7Fq8 ''Unknown piece in Abigail''] (2023)


[[Category:494edo]]
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:17-limit]]
[[Category:Enneadecal]]
[[Category:Enneadecal]]
[[Category:Kwazy]]
[[Category:Kwazy]]
[[Category:Tricot]]
[[Category:Listen]]
[[Category:Zeta]]
[[Category:Alphatricot]]
Retrieved from "https://en.xen.wiki/w/494edo"