494edo: Difference between revisions
Cleanup; clarify the title row of the rank-2 temp table; -redundant categories |
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{monzo| 783 -494 }} | | {{monzo| 783 -494 }} | ||
| {{mapping| 494 783 }} | | {{mapping| 494 783 }} | ||
| | | −0.0219 | ||
| 0.0219 | | 0.0219 | ||
| 0.90 | | 0.90 | ||
| Line 39: | Line 31: | ||
| {{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 | ||
| 0.0318 | | 0.0318 | ||
| 1.31 | | 1.31 | ||
| Line 46: | Line 38: | ||
| 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 | ||
| 0.0670 | | 0.0670 | ||
| 2.76 | | 2.76 | ||
| Line 53: | Line 45: | ||
| 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 | ||
| 0.0600 | | 0.0600 | ||
| 2.47 | | 2.47 | ||
| Line 60: | Line 52: | ||
| 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 | ||
| 0.0576 | | 0.0576 | ||
| 2.37 | | 2.37 | ||
| Line 67: | Line 59: | ||
| 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 | ||
| 0.0752 | | 0.0752 | ||
| 3.09 | | 3.09 | ||
{{comma basis end}} | |||
* 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. | * 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 === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 113: | Line 99: | ||
|- | |- | ||
| 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]] | ||
{{rank-2 end}} | |||
{{orf}} | |||
== Music == | == Music == | ||
| Line 144: | Line 130: | ||
[[Category:Enneadecal]] | [[Category:Enneadecal]] | ||
[[Category:Kwazy]] | [[Category:Kwazy]] | ||
[[Category:Listen]] | |||
[[Category:Tricot]] | [[Category:Tricot]] | ||
Revision as of 05:21, 16 November 2024
| ← 493edo | 494edo | 495edo → |
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
| 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
Regular temperament properties
Template:Comma basis begin |- | 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 Template:Comma basis end
- 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
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf
Music
- Unknown piece in Abigail (2023)