190edo
| ← 189edo | 190edo | 191edo → |
190 equal divisions of the octave (abbreviated 190edo or 190ed2), also called 190-tone equal temperament (190tet) or 190 equal temperament (190et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 190 equal parts of about 6.32 ¢ each. Each step represents a frequency ratio of 21/190, or the 190th root of 2.
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 | -14 | -17 | -40 | -29 | -8 | +38 | -11 | -48 | -2 | -30 | |
| 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
| 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 |
| 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 |
- 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
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 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 |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct