190edo: Difference between revisions
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{| 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>(Reduced) | ! Generator<br>(Reduced) | ||
! Cents<br>(Reduced) | ! Cents<br>(Reduced) | ||
Revision as of 21:14, 13 September 2023
| ← 189edo | 190edo | 191edo → |
Theory
190edo is interesting because of the utility of its approximations; it tempers out 1029/1024, 4375/4374, 385/384, 441/440, 3025/3024 and 9801/9800. It provides the optimal patent val for both the 7- and 11-limit versions of unidec, the 72 & 118 temperament, which tempers out 1029/1024, 4375/4374, and in the 11-limit, 385/384 and 441/440. It also provides the optimal patent val for the rank-3 11-limit temperament portent, which tempers out 385/384 and 441/440, and gamelan, the rank-3 7-limit temperament which tempers out 1029/1024, as well as slendric, the 2.3.7 subgroup temperament featured in the #Music section. In the 13-limit, 190et tempers out 847/845, 625/624, 729/728, 1575/1573 and 1001/1000, and provides the optimal patent val for the ekadash temperament and the rank-3 portentous temperament.
Prime harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.90 | -1.05 | -2.51 | -1.80 | -1.84 | -0.53 | -1.95 | +2.41 | -0.67 | +2.90 | -3.01 |
| Relative (%) | -14.3 | -16.6 | -39.7 | -28.6 | -29.2 | -8.4 | -30.9 | +38.2 | -10.6 | +46.0 | -47.7 | |
| Steps (reduced) |
301 (111) |
441 (61) |
533 (153) |
602 (32) |
657 (87) |
703 (133) |
742 (172) |
777 (17) |
807 (47) |
835 (75) |
859 (99) | |
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-301 190⟩ | [⟨190 301]] | +0.285 | 0.285 | 4.51 |
| 2.3.5 | 2109375/2097152, [-7 22 -12⟩ | [⟨190 301 441]] | +0.341 | 0.246 | 3.89 |
| 2.3.5.7 | 1029/1024, 4375/4374, 235298/234375 | [⟨190 301 441 533]] | +0.479 | 0.321 | 5.07 |
| 2.3.5.7.11 | 385/384, 441/440, 4375/4374, 234375/234256 | [⟨190 301 441 533 657]] | +0.490 | 0.288 | 4.55 |
| 2.3.5.7.11.13 | 385/384, 441/440, 625/624, 729/728, 847/845 | [⟨190 301 441 533 657 703]] | +0.432 | 0.293 | 4.63 |
Rank-2 temperaments
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 37\190 | 233.68 | 8/7 | Slendric |
| 1 | 43\190 | 271.58 | 75/64 | Orson / sabric |
| 1 | 49\190 | 309.47 | 448/375 | Triwell |
| 1 | 71\190 | 448.42 | 35/27 | Semidimfourth |
| 1 | 83\190 | 524.21 | 65/48 | Widefourth |
| 2 | 28\190 | 176.84 | 195/176 | Quatracot |
| 2 | 29\190 | 183.16 | 10/9 | Unidec / ekadash |
| 2 | 59\190 (36\190) |
372.63 (227.37) |
26/21 (297/260) |
Essence |
| 2 | 71\190 (24\190) |
448.42 (151.58) |
35/27 (12/11) |
Neusec |
| 5 | 79\190 (3\190) |
498.95 (18.95) |
4/3 (81/80) |
Pental |
| 10 | 50\190 (7\190) |
315.79 (45.79) |
6/5 (40/39) |
Deca |
| 10 | 79\190 (3\190) |
498.95 (18.95) |
4/3 (81/80) |
Decal |
| 19 | 79\190 (1\190) |
498.95 (6.32) |
4/3 (225/224) |
Enneadecal |
| 38 | 79\190 (1\190) |
265.26 (6.32) |
4/3 (225/224) |
Hemienneadecal |
| 38 | 42\190 (2\190) |
265.26 (12.63) |
500/429 (144/143) |
Semihemienneadecal |