181edo: Difference between revisions
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Sectioning, +links and -incorrect part (9/5 is inconsistent) |
+RTT table and rank-2 temperaments |
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=== Prime harmonics === | === Prime harmonics === | ||
{{ | {{Harmonics in equal|181}} | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | Subgroup | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| 287 -181 }} | |||
| [{{val| 181 287 }}] | |||
| -0.255 | |||
| 0.255 | |||
| 3.84 | |||
|- | |||
| 2.3.5 | |||
| 2109375/2097152, {{monzo| 14 -22 9 }} | |||
| [{{val| 181 287 420 }}] | |||
| +0.086 | |||
| 0.525 | |||
| 7.92 | |||
|- | |||
| 2.3.5.7 | |||
| 2401/2400, 5120/5103, 390625/387072 | |||
| [{{val| 181 287 420 508 }}] | |||
| +0.142 | |||
| 0.465 | |||
| 7.01 | |||
|- | |||
| 2.3.5.7.11 | |||
| 385/384, 1375/1372, 2200/2187, 4000/3993 | |||
| [{{val| 181 287 420 508 626 }}] | |||
| +0.174 | |||
| 0.421 | |||
| 6.35 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 325/324, 352/351, 385/384, 1375/1372, 1575/1573 | |||
| [{{val| 181 287 420 508 626 670 }}] | |||
| +0.079 | |||
| 0.439 | |||
| 6.62 | |||
|- | |||
| 2.3.5.7.11.13.17 | |||
| 325/324, 352/351, 375/374, 385/384, 595/594, 1275/1274 | |||
| [{{val| 181 287 420 508 626 670 740 }}] | |||
| +0.028 | |||
| 0.425 | |||
| 6.40 | |||
|- | |||
| 2.3.5.7.11.13.17.19 | |||
| 325/324, 352/351, 375/374, 385/384, 400/399, 595/594, 1275/1274 | |||
| [{{val| 181 287 420 508 626 670 740 769 }}] | |||
| +0.000 | |||
| 0.404 | |||
| 6.09 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per octave | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 35\181 | |||
| 232.04 | |||
| 8/7 | |||
| [[Quadrawell]] | |||
|- | |||
| 1 | |||
| 39\181 | |||
| 258.56 | |||
| {{monzo| -32 13 5 }} | |||
| [[Lafa]] | |||
|- | |||
| 1 | |||
| 41\181 | |||
| 271.82 | |||
| 75/64 | |||
| [[Orson]] | |||
|- | |||
| 1 | |||
| 53\181 | |||
| 351.38 | |||
| 49/40 | |||
| [[Hemififths]] (7-limit) | |||
|- | |||
| 1 | |||
| 88\181 | |||
| 583.43 | |||
| 7/5 | |||
| [[Cotritone]] (181f) | |||
|} | |||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
[[Category:Prime EDO]] | [[Category:Prime EDO]] | ||
Revision as of 15:10, 6 March 2022
181edo is the equal division of the octave into 181 parts of 6.6298 cents each.
Theory
181et tempers out 2109375/2097152 (semicomma) and 32000000000/31381059609 in the 5-limit; 2401/2400, 5120/5103, and 390625/387072 in the 7-limit (supporting the hemififths and the cotritone). Using the patent val, it tempers out 385/384, 1375/1372, 2200/2197, and 4000/3993 in the 11-limit; 325/324, 352/351, 847/845, and 1575/1573 in the 13-limit.
181edo is the 42nd prime EDO.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.81 | -1.78 | -0.87 | -1.04 | +1.46 | +1.12 | +0.83 | +1.56 | -1.95 | +1.93 |
| Relative (%) | +0.0 | +12.2 | -26.9 | -13.1 | -15.7 | +22.0 | +16.9 | +12.5 | +23.5 | -29.5 | +29.0 | |
| Steps (reduced) |
181 (0) |
287 (106) |
420 (58) |
508 (146) |
626 (83) |
670 (127) |
740 (16) |
769 (45) |
819 (95) |
879 (155) |
897 (173) | |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [287 -181⟩ | [⟨181 287]] | -0.255 | 0.255 | 3.84 |
| 2.3.5 | 2109375/2097152, [14 -22 9⟩ | [⟨181 287 420]] | +0.086 | 0.525 | 7.92 |
| 2.3.5.7 | 2401/2400, 5120/5103, 390625/387072 | [⟨181 287 420 508]] | +0.142 | 0.465 | 7.01 |
| 2.3.5.7.11 | 385/384, 1375/1372, 2200/2187, 4000/3993 | [⟨181 287 420 508 626]] | +0.174 | 0.421 | 6.35 |
| 2.3.5.7.11.13 | 325/324, 352/351, 385/384, 1375/1372, 1575/1573 | [⟨181 287 420 508 626 670]] | +0.079 | 0.439 | 6.62 |
| 2.3.5.7.11.13.17 | 325/324, 352/351, 375/374, 385/384, 595/594, 1275/1274 | [⟨181 287 420 508 626 670 740]] | +0.028 | 0.425 | 6.40 |
| 2.3.5.7.11.13.17.19 | 325/324, 352/351, 375/374, 385/384, 400/399, 595/594, 1275/1274 | [⟨181 287 420 508 626 670 740 769]] | +0.000 | 0.404 | 6.09 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 35\181 | 232.04 | 8/7 | Quadrawell |
| 1 | 39\181 | 258.56 | [-32 13 5⟩ | Lafa |
| 1 | 41\181 | 271.82 | 75/64 | Orson |
| 1 | 53\181 | 351.38 | 49/40 | Hemififths (7-limit) |
| 1 | 88\181 | 583.43 | 7/5 | Cotritone (181f) |