166edo: Difference between revisions
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{{ | {{ED intro}} | ||
== 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. | ||
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=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|166|intervals=prime}} | {{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 === | === Subsets and supersets === | ||
Since 166 factors into | Since 166 factors into primes as 2 × 83, 166edo contains [[2edo]] and [[83edo]] as subsets. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
Latest revision as of 14:16, 24 March 2025
| ← 165edo | 166edo | 167edo → |
166 equal divisions of the octave (abbreviated 166edo or 166ed2), also called 166-tone equal temperament (166tet) or 166 equal temperament (166et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 166 equal parts of about 7.23 ¢ each. Each step represents a frequency ratio of 21/166, or the 166th root of 2.
Theory
166edo is consistent through the 13-odd-limit. It has a flat tendency, with harmonics 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 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) | |
Octave stretch
166edo's approximated harmonics 3, 5, 7, 11, and 13 can all be improved by slightly 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
| 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 distinct