494edo: Difference between revisions

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{{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.
{{EDO intro|494}}


== Theory ==
== 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''.
Since the step size is close to [[729/728]], the squbema, the accepted name for 494edo's step is ''squb''.
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{{Harmonics in equal|494|prec=3}}
{{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.
[[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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| 2.3
| 2.3
| {{monzo| 783 -494 }}
| {{monzo| 783 -494 }}
| [{{val| 494 783 }}]
| {{mapping| 494 783 }}
| -0.0219
| -0.0219
| 0.0219
| 0.0219
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| 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
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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
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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
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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
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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 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 ===
=== Rank-2 temperaments ===
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Semihemienneadecal]]
| [[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 ==
== 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:Enneadecal]]
[[Category:Kwazy]]
[[Category:Kwazy]]
[[Category:Tricot]]
[[Category:Tricot]]
[[Category:Listen]]
[[Category:Listen]]

Revision as of 14:03, 2 November 2023

← 493edo 494edo 495edo →
Prime factorization 2 × 13 × 19
Step size 2.42915 ¢ 
Fifth 289\494 (702.024 ¢)
Semitones (A1:m2) 47:37 (114.2 ¢ : 89.88 ¢)
Consistency limit 17
Distinct consistency limit 17

Template:EDO intro

Theory

494 is a very strong 13- and 17-limit equal temperament. 494edo is a zeta peak and zeta peak integer edo and distinctly consistent through the 17-odd-limit. It tempers out the enneadeca, [-14 -19 19⟩, the tricot comma, [39 -29 3⟩, and the kwazy comma, [-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

Approximation of prime harmonics in 494edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.069 -0.079 +0.405 +0.099 -0.042 -0.502 -1.157 +0.875 +0.382 -0.906
Relative (%) +0.0 +2.9 -3.2 +16.7 +4.1 -1.7 -20.7 -47.6 +36.0 +15.7 -37.3
Steps
(reduced) 		494
(0) 		783
(289) 		1147
(159) 		1387
(399) 		1709
(227) 		1828
(346) 		2019
(43) 		2098
(122) 		2235
(259) 		2400
(424) 		2447
(471)

Subsets and supersets

Since 494 factors into 2 × 13 × 19, 494edo has subset 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

Main article: Table of 494edo intervals

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [783 -494⟩ [⟨494 783]] -0.0219 0.0219 0.90
2.3.5 [-14 -19 19⟩, [39 -23 3⟩ [⟨494 783 1147]] -0.0032 0.0318 1.31
2.3.5.7 4375/4374, 703125/702464, [21 3 1 -10⟩ [⟨494 783 1147 1387]] -0.0385 0.0670 2.76
2.3.5.7.11 3025/3024, 4375/4374, 131072/130977, 234375/234256 [⟨494 783 1147 1387 1709]] -0.0365 0.0600 2.47
2.3.5.7.11.13 1716/1715, 2080/2079, 3025/3024, 4096/4095, 31250/31213 [⟨494 783 1147 1387 1709 1828]] -0.0286 0.0576 2.37
2.3.5.7.11.13.17 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2431/2430, 4096/4095 [⟨494 783 1147 1387 1709 1828 2019]] -0.0069 0.0752 3.09
  • 494et has lower relative errors than any previous equal temperaments in the 13- and 17-limit. It is the first past 270 with a lower 13-limit relative error, and the first past 72 with a lower 17-limit relative error. It is narrowly beaten by 684 in terms of 13-limit absolute error and by 581 in terms of 17-limit absolute error. Not until 1506 do we reach an equal temperament with a lower relative error in either subgroup.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 27\494 65.59 27/26 Luminal
1 119\494 289.07 13/11 Moulin
1 233\494 565.99 104/75 Tricot / trillium
2 67\494 162.75 1125/1024 Kwazy
2 86\494 208.91 44/39 Abigail
13 205\494
(15\494) 		497.98
(36.43) 		4/3
(?) 		Aluminium
19 205\494
(3\494) 		497.98
(7.29) 		4/3
(225/224) 		Enneadecal
38 205\494
(3\494) 		497.98
(7.29) 		4/3
(225/224) 		Hemienneadecal
38 109\494
(5\494) 		264.78
(12.15) 		500/429
(144/143) 		Semihemienneadecal

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Music

Eliora
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