494edo: Difference between revisions

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== Intervals ==
== Intervals ==
{{Main| Table of 494edo intervals }}
{{Main| Table of 494edo intervals }}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 783 -494 }}
| [{{val| 494 783 }}]
| -0.0219
| 0.0219
| 0.90
|-
| 2.3.5
| {{monzo| -14 -19 19 }}, {{monzo| 39 -23 3 }}
| [{{val| 494 783 1147 }}]
| -0.0032
| 0.0318
| 1.31
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| 21 3 1 -10 }}
| [{{val| 494 783 1147 1387 }}]
| -0.0385
| 0.0670
| 2.76
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 131072/130977, 234375/234256
| [{{val| 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
| [{{val| 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
| [{{val| 494 783 1147 1387 1709 1828 2019 }}]
| -0.0069
| 0.0752
| 3.09
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
| 1
| 27\494
| 65.59
| 27/26
| [[Luminal]]
|-
| 1
| 233\422
| 565.99
| 104/75
| [[Tricot]] / [[trillium]]
|-
| 2
| 67\494
| 162.75
| 1125/1024
| [[Kwazy]]
|-
| 2
| 86\494
| 208.91
| 44/39
| [[Abigail]]
|-
| 19
| 205\494<br>(3\494)
| 497.98<br>(7.29)
| 4/3<br>(225/224)
| [[Enneadecal]]
|-
| 38
| 205\494<br>(3\494)
| 497.98<br>(7.29)
| 4/3<br>(225/224)
| [[Hemienneadecal]]
|}


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

Revision as of 16:05, 28 January 2022

← 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

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 cents each. It is a very strong 13- and 17-limit equal temperament. The step size is close to 729/728, the squbema, and a step is a squb.

Theory

494edo is a zeta peak and zeta peak integer edo and uniquely 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. Not until 1506 do we reach a division with a lower 13- or 17-limit relative error, and it is the first past 72 with a lower 17-limit relative error. 494 is divisible by 2, 13, 19, 26, 38 and 247.

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)

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

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 27\494 65.59 27/26 Luminal
1 233\422 565.99 104/75 Tricot / trillium
2 67\494 162.75 1125/1024 Kwazy
2 86\494 208.91 44/39 Abigail
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
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