166edo: Difference between revisions
Jump to navigation
Jump to search
m Infobox ET now computes most parameters automatically |
m Cleanup |
||
| Line 3: | Line 3: | ||
== Theory == | == Theory == | ||
166edo is consistent through the [[13-odd-limit]], yet its principle interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], | 166edo is [[consistent]] through the [[13-odd-limit]], yet its principle interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], the equal temperament 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 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}} | ||
=== Subsets and supersets === | |||
Since 166 = 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|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 62: | Line 63: | ||
{| 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 | ||
|- | |- | ||
| Line 104: | Line 105: | ||
* [[Fifteentofourteen]] | * [[Fifteentofourteen]] | ||
[[Category:Wizard]] | [[Category:Wizard]] | ||
[[Category:Gizzard]] | [[Category:Gizzard]] | ||
[[Category:Houborizic]] | [[Category:Houborizic]] | ||
[[Category:Marvel]] | [[Category:Marvel]] | ||
Revision as of 08:39, 17 July 2023
| ← 165edo | 166edo | 167edo → |
The 166 equal divisions of the octave (166edo), or the 166(-tone) equal temperament (166tet, 166et) when viewed from a regular temperament perspective, divides the octave into 166 equal steps of about 7.23 cents each, a size close to 243/242, the rastma.
Theory
166edo is consistent through the 13-odd-limit, yet its principle interest lies in the usefulness of its approximations. In addition to the 5-limit amity comma, the equal temperament 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 72 & 94 temperament, for which 166 is an excellent tuning through the 19-limit.
166edo (as 83edo) contains a very good approximation of the harmonic 7th, of which it is only flat by 0.15121 cent.
Prime harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.75 | -3.18 | -0.15 | -1.50 | -1.92 | -1.97 | +3.30 | +3.48 | -1.13 | -0.90 | +0.64 |
| Relative (%) | -10.4 | -44.0 | -2.1 | -20.8 | -26.6 | -27.3 | +45.6 | +48.1 | -15.6 | -12.5 | +8.9 | |
| Steps (reduced) |
263 (97) |
385 (53) |
466 (134) |
526 (28) |
574 (76) |
614 (116) |
649 (151) |
679 (15) |
705 (41) |
729 (65) |
751 (87) | |
Subsets and supersets
Since 166 = 2 × 83, 166edo contains 2edo and 83edo as subsets.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-263 166⟩ | [⟨166 263]] | +0.237 | 0.237 | 3.27 |
| 2.3.5 | 1600000/1594323, [-31 2 12⟩ | [⟨166 263 385]] | +0.615 | 0.568 | 7.86 |
| 2.3.5.7 | 225/224, 118098/117649, 1250000/1240029 | [⟨166 263 385 466]] | +0.474 | 0.549 | 7.59 |
| 2.3.5.7.11 | 225/224, 385/384, 4000/3993, 322102/321489 | [⟨166 263 385 466 574]] | +0.490 | 0.492 | 6.80 |
| 2.3.5.7.11.13 | 225/224, 325/324, 385/384, 1573/1568, 2200/2197 | [⟨166 263 385 466 574 614]] | +0.498 | 0.449 | 6.21 |
Rank-2 temperaments
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 33\166 | 238.55 | 147/128 | Tokko |
| 1 | 47\166 | 339.76 | 243/200 | Amity / houborizic |
| 1 | 81\166 | 585.54 | 7/5 | Merman |
| 2 | 30\166 | 216.87 | 17/15 | Wizard / gizzard |