193edo: Difference between revisions

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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.  
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 6.22 [[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 provides the [[optimal patent val]] for the [[sqrtphi]] temperament in the 13-, 17- and 19-limits, and for the 13-limit [[Swetismic temperaments #Minos|minos]] and [[Mirkwai family #Indra|vish]] temperaments. It is the 44th [[prime edo]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|193}}
{{Harmonics in equal|193}}


== Regular temperament properties ==
== Regular temperament properties ==
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| 3.225
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| 2.3.5
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| -0.0158
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| 4.960
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| 2.3.5.7
| 2.3.5.7
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| -0.1118
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|-
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| 2.3.5.7.11
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
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| 5.89
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
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| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
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| 5.13
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== Sqrtphi scale in 193edo ==
== Scales ==
Approximation of the intervals:
* 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
 
Square root of Pi: '''159\193''' (988.60104 cents), and
 
Phi: '''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:Equal divisions of the octave]]
[[Category:Prime EDO]]
[[Category:Prime EDO]]

Revision as of 15:27, 6 March 2022

← 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

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 equal division of the octave into 193 parts of about 6.22 cents each.

Theory

193edo provides the optimal patent val for the sqrtphi temperament in the 13-, 17- and 19-limits, and for the 13-limit minos and vish temperaments. It is the 44th prime edo.

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)

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

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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