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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 193 (prime)
{{ED intro}}
| Step size = 6.21762¢
| Fifth = 113\193 (702.59¢)
| Semitones = 19:14 (118.13¢ : 87.05¢)
| Consistency = 11
}}
The '''193 equal divisions of the octave''' ('''193edo'''), or the '''193(-tone) equal temperament''' ('''193tet''', '''193et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 193 parts of about 6.21762 [[cent]]s each.


== Theory ==
== Theory ==
193edo provides the [[optimal patent val]] for [[Kleismic_family #Sqrtphi|sqrtphi temperament]] in the 13-, 17- and 19- limits, and for 13-limit [[Swetismic_temperaments #Minos|minos]] and [[Mirkwai_family #Indra|vish]] temperaments. It is the 44th [[prime_numbers|prime]] EDO.
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, it [[tempering out|tempers out]] the [[15625/15552|kleisma]] in the [[5-limit]]; [[5120/5103]] and [[16875/16807]] in the [[7-limit]]; [[540/539]], [[1375/1372]], [[3025/3024]], and 4375/4356 in the [[11-limit]]; [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1575/1573]], [[1716/1715]], and [[4096/4095]] in the [[13-limit]]; [[375/374]], [[442/441]], [[595/594]], [[715/714]], [[936/935]], [[1156/1155]], [[1225/1224]], [[2058/2057]], and [[2431/2430]] in the [[17-limit]]; [[400/399]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]], [[1540/1539]], and [[1729/1728]] in the [[19-limit]]; and [[460/459]], [[507/506]], and [[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 [[Mirkwai family #Indra|vish]] temperaments.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|193}}
{{Harmonics in equal|193}}
 
=== Subsets and supersets ===
193edo is the 44th [[prime edo]].


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 27: Line 29:
| 2.3
| 2.3
| {{monzo| 306 -193 }}
| {{monzo| 306 -193 }}
| [{{val| 193 306 }}]
| {{mapping| 193 306 }}
| -0.2005
| −0.2005
| 0.2005
| 0.2005
| 3.225
| 3.23
|-
|-
| 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
| 4.960
| 4.96
|-
|-
| 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
| 5.059
| 5.06
|-
|-
| 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
| −0.2080
| 0.3408
| 0.3408
| 5.481
| 5.48
|-
|-
| 2.3.5.7.11.13
| 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
| 0.3662
| 0.3662
| 5.890
| 5.89
|-
|-
| 2.3.5.7.11.13.17
| 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
| 5.464
| 5.46
|-
|-
| 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
| 5.133
| 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
| {{mapping| 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 [[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]].


== Sqrtphi scale in 193edo ==
=== Rank-2 temperaments ===
Approximation of the intervals:
{| class="wikitable center-all left-5"
 
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
Square root of Pi: '''159\193''' (988.60104 cents), and
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
| 1
| 16\193
| 99.48
| 18/17
| [[Quindromeda family#Quintakwai|Quintakwai]] / [[Quindromeda family#Quintakwoid|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]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


Phi: '''134\193''' (833.16062 cents), both inside in the [[7L_2s|superdiatonic]] scale: 25 25 25 9 25 25 25 25 9
== 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


[[Category:Sqrtphi]]
[[Category:Sqrtphi]]
[[Category:Equal divisions of the octave]]
[[Category:Prime EDO]]

Latest revision as of 14:11, 20 February 2025

← 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

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