494edo: Difference between revisions
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== 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 | 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. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|494|columns=11|prec=3}} | {{Harmonics in equal|494|columns=11|prec=3}} | ||
=== Miscellaneous properties === | |||
Since 494 = 2 × 13 × 19, 494edo has subset edos {{EDOs| 2, 13, 19, 26, 38, and 247 }}. | |||
== Intervals == | == Intervals == | ||
| Line 13: | Line 16: | ||
== 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|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 64: | Line 67: | ||
| 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. Not until [[1506edo|1506]] do we reach a division with a lower 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 | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per | ! Periods<br>per 8ve | ||
! Generator<br>( | ! Generator<br>(Reduced) | ||
! Cents<br>( | ! Cents<br>(Reduced) | ||
! Associated<br> | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
Revision as of 13:55, 31 October 2022
| ← 493edo | 494edo | 495edo → |
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.
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) | |
Miscellaneous properties
Since 494 = 2 × 13 × 19, 494edo has subset edos 2, 13, 19, 26, 38, and 247.
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 a lower relative error 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. Not until 1506 do we reach a division with a lower error in either subgroup.
Rank-2 temperaments
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 27\494 | 65.59 | 27/26 | Luminal |
| 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 |
| 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 |