193edo: Difference between revisions

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== Theory ==
== Theory ==
193edo provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[Swetismic temperaments #Minos|minos]] and [[Mirkwai family #Indra|vish]] temperaments.  
193edo provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limit, and for the 13-limit [[minos]] and [[Mirkwai family #Indra|vish]] temperaments.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|193|columns=11}}
{{Harmonics in equal|193}}


=== Miscellaneous properties ===
=== Subsets and supersets ===
193edo is the 44th [[prime edo]].
193edo is the 44th [[prime edo]].


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| 2.3
| 2.3
| {{monzo| 306 -193 }}
| {{monzo| 306 -193 }}
| [{{val| 193 306 }}]
| {{mapping| 193 306 }}
| -0.2005
| -0.2005
| 0.2005
| 0.2005
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|-
|-
| 2.3.5
| 2.3.5
| 15625/15552, {{monzo|50 -33 1}}
| 15625/15552, {{monzo| 50 -33 1 }}
| [{{val| 193 306 448 }}]
| {{mapping| 193 306 448 }}
| -0.0158
| -0.0158
| 0.3084
| 0.3084
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| 2.3.5.7
| 2.3.5.7
| 5120/5103, 15625/15552, 16875/16807
| 5120/5103, 15625/15552, 16875/16807
| [{{val| 193 306 448 542 }}]
| {{mapping| 193 306 448 542 }}
| -0.1118
| -0.1118
| 0.3146
| 0.3146
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| 2.3.5.7.11
| 2.3.5.7.11
| 540/539, 1375/1372, 4375/4356, 5120/5103
| 540/539, 1375/1372, 4375/4356, 5120/5103
| [{{val| 193 306 448 542 668 }}]
| {{mapping| 193 306 448 542 668 }}
| -0.2080
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| 0.3408
| 0.3408
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| 2.3.5.7.11.13
| 325/324, 364/363, 540/539, 625/624, 4096/4095
| 325/324, 364/363, 540/539, 625/624, 4096/4095
| [{{val| 193 306 448 542 668 714 }}]
| {{mapping| 193 306 448 542 668 714 }}
| -0.1216
| -0.1216
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| 0.3662
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| 2.3.5.7.11.13.17
| 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095
| 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095
| [{{val| 193 306 448 542 668 714 789 }}]
| {{mapping| 193 306 448 542 668 714 789 }}
| -0.1302
| -0.1302
| 0.3397
| 0.3397
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| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 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
| [{{val| 193 306 448 542 668 714 789 820 }}]
| {{mapping| 193 306 448 542 668 714 789 820 }}
| -0.1414
| -0.1414
| 0.3191
| 0.3191
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperament
! Temperament
|-
|-
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| [[Kwai]]
| [[Kwai]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


== Scales ==
== Scales ==
*Approximation of sqrt (π): '''159\193''' (988.60104 cents), and of φ: '''134\193''' (833.16062 cents), both inside in the [[7L 2s|superdiatonic]] scale: 25 25 25 9 25 25 25 25 9
* Approximation of sqrt (π): '''159\193''' (988.60104 cents), and of φ: '''134\193''' (833.16062 cents), both inside in the [[7L 2s|superdiatonic]] scale: 25 25 25 9 25 25 25 25 9


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]
[[Category:Sqrtphi]]
[[Category:Sqrtphi]]

Revision as of 14:11, 16 April 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 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
  • 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
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