193edo

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← 192edo193edo194edo →
Prime factorization 193 (prime)
Step size 6.21762¢ 
Fifth 113\193 (702.591¢)
Semitones (A1:m2) 19:14 (118.1¢ : 87.05¢)
Consistency limit 11
Distinct consistency limit 11

193 equal divisions of the octave (abbreviated 193edo or 193ed2), also called 193-tone equal temperament (193tet) or 193 equal temperament (193et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 193 equal parts of about 6.22 ¢ each. Each step represents a frequency ratio of 21/193, or the 193rd root of 2.

Theory

193edo is consistent to the 11-odd-limit, and almost consistent to the 23-odd-limit, the only failure being 13/11 and its octave complement. This makes it a strong 23-limit system.

As an equal temperament, 193et tempers out the kleisma in the 5-limit; 5120/5103 and 16875/16807 in the 7-limit; 540/539, 1375/1372, 3025/3024, 4375/4356 in the 11-limit; 325/324, 364/363, 625/624, 676/675, 1575/1573, 1716/1715, 4096/4095 in the 13-limit; 375/374, 442/441, 595/594, 715/714, 936/935, 1156/1155, 1225/1224, 2058/2057, 2431/2430 in the 17-limit; 400/399, 969/968, 1216/1215, 1445/1444, 1521/1520, 1540/1539, 1729/1728 in the 19-limit; and 460/459, 507/506, 529/528 in the 23-limit.

It provides the optimal patent val for the sqrtphi temperament in the 13-, 17- and 19-limit, and for the 13-limit minos and vish temperaments.

Prime harmonics

Approximation of prime harmonics in 193edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.64 -0.82 +1.12 +2.05 -1.15 +0.74 +0.93 -0.30 +2.55 -0.99
Relative (%) +0.0 +10.2 -13.2 +18.1 +33.0 -18.5 +12.0 +15.0 -4.7 +41.0 -16.0
Steps
(reduced)
193
(0)
306
(113)
448
(62)
542
(156)
668
(89)
714
(135)
789
(17)
820
(48)
873
(101)
938
(166)
956
(184)

Subsets and supersets

193edo is the 44th prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal 8ve
Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [306 -193 [193 306]] -0.2005 0.2005 3.23
2.3.5 15625/15552, [50 -33 1 [193 306 448]] -0.0158 0.3084 4.96
2.3.5.7 5120/5103, 15625/15552, 16875/16807 [193 306 448 542]] -0.1118 0.3146 5.06
2.3.5.7.11 540/539, 1375/1372, 4375/4356, 5120/5103 [193 306 448 542 668]] -0.2080 0.3408 5.48
2.3.5.7.11.13 325/324, 364/363, 540/539, 625/624, 4096/4095 [193 306 448 542 668 714]] -0.1216 0.3662 5.89
2.3.5.7.11.13.17 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095 [193 306 448 542 668 714 789]] -0.1302 0.3397 5.46
2.3.5.7.11.13.17.19 325/324, 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215 [193 306 448 542 668 714 789 820]] -0.1414 0.3191 5.13
2.3.5.7.11.13.17.19.23 325/324, 364/363, 375/374, 400/399, 442/441, 460/459, 507/506, 529/528 [193 306 448 542 668 714 789 820 873]] -0.1184 0.3078 4.95
  • 193et has a lower relative error in the 23-limit than any previous equal temperaments, past 190g and followed by 217.
  • 193et is also notable in the 19-limit, where it has a lower absolute error than any previous equal temperaments, past 190g and followed by 212gh.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperament
1 16\193 99.48 18/17 Quintakwai / quintakwoid
1 18\193 111.92 16/15 Vavoom
1 39\193 242.49 147/128 Septiquarter
1 51\193 317.10 6/5 Countercata (7-limit)
1 56\193 348.19 11/9 Eris
1 61\193 379.28 56/45 Marthirds
1 67\193 416.58 14/11 Sqrtphi
1 79\193 491.19 3645/2744 Fifthplus
1 80\193 497.41 4/3 Kwai

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Scales

  • Approximation of sqrt (π): 159\193 (988.60104 cents), and of φ: 134\193 (833.16062 cents), both inside in the superdiatonic scale: 25 25 25 9 25 25 25 25 9