166edo: Difference between revisions
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== Theory == | == Theory == | ||
166edo is consistent through the [[13-odd-limit]], | 166edo is [[consistent]] through the [[13-odd-limit]]. It has a flat tendency, with [[harmonic]]s 3 to 13 all tuned flat. Its principal interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], it [[tempering out|tempers out]] [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and [[1573/1568]], leading to marvell, and tempering out both gives [[gizzard]], the {{nowrap|72 & 94}} temperament, for which 166 is an excellent tuning through the [[19-limit]]. | ||
166edo (as 83edo) contains a very good approximation of the [[7/4|harmonic 7th]], of which it is only flat by 0.15121 cent. | 166edo (as 83edo) contains a very good approximation of the [[7/4|harmonic 7th]], of which it is only flat by 0.15121 cent. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{ | {{Harmonics in equal|166|intervals=prime}} | ||
=== Octave stretch === | |||
166edo's approximated harmonics 3, 5, 7, 11, and 13 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[263edt]] or [[429ed6]]. | |||
=== Subsets and supersets === | |||
Since 166 factors into primes as 2 × 83, 166edo contains [[2edo]] and [[83edo]] as subsets. | |||
== 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]] | ||
! 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 | ||
|- | |- | ||
Line 24: | Line 30: | ||
| 2.3 | | 2.3 | ||
| {{monzo| -263 166 }} | | {{monzo| -263 166 }} | ||
| | | {{mapping| 166 263 }} | ||
| +0.237 | | +0.237 | ||
| 0.237 | | 0.237 | ||
Line 31: | Line 37: | ||
| 2.3.5 | | 2.3.5 | ||
| 1600000/1594323, {{monzo| -31 2 12 }} | | 1600000/1594323, {{monzo| -31 2 12 }} | ||
| | | {{mapping| 166 263 385 }} | ||
| +0.615 | | +0.615 | ||
| 0.568 | | 0.568 | ||
Line 38: | Line 44: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 225/224, 118098/117649, 1250000/1240029 | | 225/224, 118098/117649, 1250000/1240029 | ||
| | | {{mapping| 166 263 385 466 }} | ||
| +0.474 | | +0.474 | ||
| 0.549 | | 0.549 | ||
Line 45: | Line 51: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 225/224, 385/384, 4000/3993, 322102/321489 | | 225/224, 385/384, 4000/3993, 322102/321489 | ||
| | | {{mapping| 166 263 385 466 574 }} | ||
| +0.490 | | +0.490 | ||
| 0.492 | | 0.492 | ||
Line 52: | Line 58: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 225/224, 325/324, 385/384, 1573/1568, 2200/2197 | | 225/224, 325/324, 385/384, 1573/1568, 2200/2197 | ||
| | | {{mapping| 166 263 385 466 574 614 }} | ||
| +0.498 | | +0.498 | ||
| 0.449 | | 0.449 | ||
Line 60: | Line 66: | ||
=== 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 | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br>ratio | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
Line 77: | Line 84: | ||
| 339.76 | | 339.76 | ||
| 243/200 | | 243/200 | ||
| [[ | | [[Houborizic]] | ||
|- | |- | ||
| 1 | | 1 | ||
Line 83: | Line 90: | ||
| 585.54 | | 585.54 | ||
| 7/5 | | 7/5 | ||
| [[Merman]] | | [[Merman]] (7-limit) | ||
|- | |- | ||
| 2 | | 2 | ||
Line 91: | Line 98: | ||
| [[Wizard]] / gizzard | | [[Wizard]] / gizzard | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
== Scales == | == Scales == | ||
* [[ | * [[Prisun]] | ||
* [[Hecatehex]] | |||
* [[Marvel1]] | |||
* [[Marvel2]] | |||
* [[Marvel3]] | |||
* [[Marvel11max7a]] | |||
* [[Marvel11max7b]] | |||
* [[Marveldene]] | |||
* [[Fifteentofourteen]] | |||
[[Category:Wizard]] | [[Category:Wizard]] | ||
[[Category:Houborizic]] | [[Category:Houborizic]] | ||
[[Category:Marvel]] | [[Category:Marvel]] |