190edo
| ← 189edo | 190edo | 191edo → |
Theory
190edo is distinctly consistent in the 15-odd-limit with a flat tendency, as harmonics 3 through 13 are all tuned flat.
The equal temperament 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 625/624, 729/728, 847/845, 1001/1000 and 1575/1573, and provides the optimal patent val for the ekadash temperament and the rank-3 portentous temperament.
The 190g val shows us a smooth path to the even higher limits. This extension tempers out 289/288, 561/560, 595/594 in the 17-limit; 343/342, 476/475, 495/494 in the 19-limit; and 391/390, 529/528 in the 23-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.90 | -1.05 | -2.51 | -1.84 | -0.53 | +2.41 | -0.67 | -3.01 | -0.10 | -1.88 |
| Relative (%) | +0.0 | -14.3 | -16.6 | -39.7 | -29.2 | -8.4 | +38.2 | -10.6 | -47.7 | -1.6 | -29.7 | |
| Steps (reduced) |
190 (0) |
301 (111) |
441 (61) |
533 (153) |
657 (87) |
703 (133) |
777 (17) |
807 (47) |
859 (99) |
923 (163) |
941 (181) | |
Subsets and supersets
Since 190 factors into 2 × 5 × 19, 190edo has subset edos 2, 5, 10, 19, 38, and 95.
Regular temperament properties
Template:Comma basis begin |- | 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 |- | 2.3.5.7.11.13.17 | 289/288, 385/384, 441/440, 561/560, 625/624, 847/845 | [⟨190 301 441 533 657 703 776]] (190g) | +0.507 | 0.327 | 5.18 |- | 2.3.5.7.11.13.17.19 | 289/288, 343/342, 385/384, 441/440, 476/475, 495/494, 847/845 | [⟨190 301 441 533 657 703 776 807]] (190g) | +0.463 | 0.327 | 5.17 |- | 2.3.5.7.11.13.17.19.23 | 289/288, 343/342, 385/384, 391/390, 441/440, 476/475, 495/494, 529/528 | [⟨190 301 441 533 657 703 776 807 859]] (190g) | +0.486 | 0.315 | 4.98 Template:Comma basis end
- 190et (190g val) has a lower relative error in the 23-limit than any previous equal temperaments, being the first to beat 94. However, 193, only slightly larger, beats it.
- It is also prominent in the 13- and 19-limit, with lower absolute errors than any previous equal temperaments. It beats 183 in either subgroup and is bettered by 198 in the 13-limit, and by 193 in the 19-limit.
Rank-2 temperaments
Template:Rank-2 begin
|-
| 1
| 37\190
| 233.68
| 8/7
| Slendric
|-
| 1
| 43\190
| 271.58
| 75/64
| 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
Template:Rank-2 end
Template:Orf