166edo: Difference between revisions
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[[Category:Wizard]] | [[Category:Wizard]] | ||
[[Category:Gizzard]] | [[Category:Gizzard]] | ||
[[Category:Houborizic]] | [[Category:Houborizic]] | ||
[[Category:Marvel]] | [[Category:Marvel]] | ||
Revision as of 21:15, 2 July 2022
| ← 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, 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.
Its prime factorization is 166 = 2 × 83.
166edo (as 83edo) contains a very good approximation of the harmonic 7th, of which it is only flat by 0.15121 cent.
Prime harmonics
Script error: No such module "primes_in_edo".
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 octave |
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 |