494edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|494}}
{{ED intro}}


== Theory ==
== Theory ==
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.  
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''.
Since the step size is close to [[729/728]], the squbema, the accepted name for 494edo's step is ''squb''.
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| 783 -494 }}
| {{Monzo| 783 -494 }}
| {{mapping| 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 }}
| {{mapping| 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 }}
| {{mapping| 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
| {{mapping| 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
| {{mapping| 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
| {{mapping| 494 783 1147 1387 1709 1828 2019 }}
| {{Mapping| 494 783 1147 1387 1709 1828 2019 }}
| −0.0069
| −0.0069
| 0.0752
| 0.0752
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 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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|-
|-
| 13
| 13
| 205\494<br />(15\494)
| 205\494<br>(15\494)
| 497.98<br />(36.43)
| 497.98<br/>(36.43)
| 4/3<br />(?)
| 4/3<br>(?)
| [[Aluminium]]
| [[Aluminium]]
|-
|-
| 19
| 19
| 205\494<br />(3\494)
| 205\494<br>(3\494)
| 497.98<br />(7.29)
| 497.98<br>(7.29)
| 4/3<br />(225/224)
| 4/3<br>(225/224)
| [[Enneadecal]]
| [[Enneadecal]]
|-
|-
| 38
| 38
| 205\494<br />(3\494)
| 205\494<br>(3\494)
| 497.98<br />(7.29)
| 497.98<br>(7.29)
| 4/3<br />(225/224)
| 4/3<br>(225/224)
| [[Hemienneadecal]]
| [[Hemienneadecal]]
|-
|-
| 38
| 38
| 109\494<br />(5\494)
| 109\494<br>(5\494)
| 264.78<br />(12.15)
| 264.78<br>(12.15)
| 500/429<br />(144/143)
| 500/429<br>(144/143)
| [[Semihemienneadecal]]
| [[Semihemienneadecal]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<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 ==
== Music ==
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[[Category:Kwazy]]
[[Category:Kwazy]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Tricot]]
[[Category:Alphatricot]]

Latest revision as of 13:38, 13 March 2026

← 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

494 equal divisions of the octave (abbreviated 494edo or 494ed2), also called 494-tone equal temperament (494tet) or 494 equal temperament (494et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 494 equal parts of about 2.43 ¢ each. Each step represents a frequency ratio of 21/494, or the 494th root of 2.

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 alphatricot 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

The following table shows how 15-odd-limit intervals are represented in 494edo. Prime harmonics are in bold.

As 494edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 494edo
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
15/8, 16/15 0.010 0.4
11/6, 12/11 0.030 1.2
15/13, 26/15 0.032 1.3
13/10, 20/13 0.037 1.5
11/9, 18/11 0.040 1.6
13/8, 16/13 0.042 1.7
3/2, 4/3 0.069 2.9
5/4, 8/5 0.079 3.2
11/8, 16/11 0.099 4.1
15/11, 22/15 0.109 4.5
13/12, 24/13 0.111 4.6
9/8, 16/9 0.139 5.7
13/11, 22/13 0.141 5.8
5/3, 6/5 0.148 6.1
11/10, 20/11 0.178 7.3
13/9, 18/13 0.180 7.4
9/5, 10/9 0.217 9.0
9/7, 14/9 0.266 11.0
11/7, 14/11 0.306 12.6
7/6, 12/7 0.336 13.8
7/4, 8/7 0.405 16.7
15/14, 28/15 0.414 17.1
13/7, 14/13 0.447 18.4
7/5, 10/7 0.484 19.9

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 Alphatrillium
2 67\494 162.75 1125/1024 Crazy
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 distinct

Music

Eliora
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