166edo: Difference between revisions
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m Odd -> prime errors since there's not much improvement of odd harmonics by direct approximation |
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== 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]], 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 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 [[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 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. | ||
| Line 26: | Line 26: | ||
| 2.3 | | 2.3 | ||
| {{monzo| -263 166 }} | | {{monzo| -263 166 }} | ||
| | | {{mapping| 166 263 }} | ||
| +0.237 | | +0.237 | ||
| 0.237 | | 0.237 | ||
| Line 33: | Line 33: | ||
| 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 40: | Line 40: | ||
| 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 47: | Line 47: | ||
| 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 54: | Line 54: | ||
| 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 64: | Line 64: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 79: | Line 79: | ||
| 339.76 | | 339.76 | ||
| 243/200 | | 243/200 | ||
| [[ | | [[Houborizic]] | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 85: | Line 85: | ||
| 585.54 | | 585.54 | ||
| 7/5 | | 7/5 | ||
| [[Merman]] | | [[Merman]] (7-limit) | ||
|- | |- | ||
| 2 | | 2 | ||
| Line 93: | Line 93: | ||
| [[Wizard]] / gizzard | | [[Wizard]] / gizzard | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
== Scales == | == Scales == | ||
| Line 106: | Line 107: | ||
[[Category:Wizard]] | [[Category:Wizard]] | ||
[[Category:Houborizic]] | [[Category:Houborizic]] | ||
[[Category:Marvel]] | [[Category:Marvel]] | ||
Revision as of 08:50, 5 May 2024
| ← 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 | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.75 | -3.18 | -0.15 | -1.92 | -1.97 | +3.48 | -1.13 | +0.64 | -3.07 | -2.87 |
| Relative (%) | +0.0 | -10.4 | -44.0 | -2.1 | -26.6 | -27.3 | +48.1 | -15.6 | +8.9 | -42.5 | -39.7 | |
| Steps (reduced) |
166 (0) |
263 (97) |
385 (53) |
466 (134) |
574 (76) |
614 (116) |
679 (15) |
705 (41) |
751 (87) |
806 (142) |
822 (158) | |
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* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 33\166 | 238.55 | 147/128 | Tokko |
| 1 | 47\166 | 339.76 | 243/200 | Houborizic |
| 1 | 81\166 | 585.54 | 7/5 | Merman (7-limit) |
| 2 | 30\166 | 216.87 | 17/15 | Wizard / gizzard |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct