193edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo| 306 -193 }}
| {{monzo| 306 -193 }}
| {{mapping| 193 306 }}
| {{mapping| 193 306 }}
| -0.2005
| &minus;0.2005
| 0.2005
| 0.2005
| 3.23
| 3.23
Line 36: Line 28:
| 15625/15552, {{monzo| 50 -33 1 }}
| 15625/15552, {{monzo| 50 -33 1 }}
| {{mapping| 193 306 448 }}
| {{mapping| 193 306 448 }}
| -0.0158
| &minus;0.0158
| 0.3084
| 0.3084
| 4.96
| 4.96
Line 43: Line 35:
| 5120/5103, 15625/15552, 16875/16807
| 5120/5103, 15625/15552, 16875/16807
| {{mapping| 193 306 448 542 }}
| {{mapping| 193 306 448 542 }}
| -0.1118
| &minus;0.1118
| 0.3146
| 0.3146
| 5.06
| 5.06
Line 50: Line 42:
| 540/539, 1375/1372, 4375/4356, 5120/5103
| 540/539, 1375/1372, 4375/4356, 5120/5103
| {{mapping| 193 306 448 542 668 }}
| {{mapping| 193 306 448 542 668 }}
| -0.2080
| &minus;0.2080
| 0.3408
| 0.3408
| 5.48
| 5.48
Line 57: Line 49:
| 325/324, 364/363, 540/539, 625/624, 4096/4095
| 325/324, 364/363, 540/539, 625/624, 4096/4095
| {{mapping| 193 306 448 542 668 714 }}
| {{mapping| 193 306 448 542 668 714 }}
| -0.1216
| &minus;0.1216
| 0.3662
| 0.3662
| 5.89
| 5.89
Line 64: Line 56:
| 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095
| 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095
| {{mapping| 193 306 448 542 668 714 789 }}
| {{mapping| 193 306 448 542 668 714 789 }}
| -0.1302
| &minus;0.1302
| 0.3397
| 0.3397
| 5.46
| 5.46
Line 71: Line 63:
| 325/324, 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215
| 325/324, 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215
| {{mapping| 193 306 448 542 668 714 789 820 }}
| {{mapping| 193 306 448 542 668 714 789 820 }}
| -0.1414
| &minus;0.1414
| 0.3191
| 0.3191
| 5.13
| 5.13
Line 78: Line 70:
| 325/324, 364/363, 375/374, 400/399, 442/441, 460/459, 507/506, 529/528
| 325/324, 364/363, 375/374, 400/399, 442/441, 460/459, 507/506, 529/528
| {{mapping| 193 306 448 542 668 714 789 820 873 }}
| {{mapping| 193 306 448 542 668 714 789 820 873 }}
| -0.1184
| &minus;0.1184
| 0.3078
| 0.3078
| 4.95
| 4.95
|}
{{comma basis end}}
* 193et has a lower relative error in the 23-limit than any previous equal temperaments, past [[190edo|190g]] and followed by [[217edo|217]].  
* 193et has a lower relative error in the 23-limit than any previous equal temperaments, past [[190edo|190g]] and followed by [[217edo|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 [[212edo|212gh]].  
* 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 [[212edo|212gh]].  


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperament
|-
|-
| 1
| 1
Line 147: Line 133:
| 4/3
| 4/3
| [[Kwai]]
| [[Kwai]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


== Scales ==
== Scales ==

Revision as of 05:17, 16 November 2024

← 192edo 193edo 194edo →
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

Template:EDO intro

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

Template:Comma basis begin |- | 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 Template:Comma basis end

  • 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

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

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
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