190edo

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← 189edo190edo191edo →
Prime factorization 2 × 5 × 19
Step size 6.31579¢ 
Fifth 111\190 (701.053¢)
Semitones (A1:m2) 17:15 (107.4¢ : 94.74¢)
Consistency limit 15
Distinct consistency limit 15

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

Approximation of prime harmonics in 190edo
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

Table of rank-2 temperaments by generator
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

Scales

Music

Chris Vaisvil