193edo: Difference between revisions
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= | {{Infobox ET | ||
| Prime factorization = 193 (prime) | |||
| 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 == | |||
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. | |||
=== Prime harmonics === | |||
{{Primes in edo|193}} | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | Subgroup | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal 8ve <br>stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| 306 -193 }} | |||
| [{{val| 193 306 }}] | |||
| -0.2005 | |||
| 0.2005 | |||
| 3.225 | |||
|- | |||
| 2.3.5 | |||
| 15625/15552, {{monzo|50 -33 1}} | |||
| [{{val| 193 306 448 }}] | |||
| -0.0158 | |||
| 0.3084 | |||
| 4.960 | |||
|- | |||
| 2.3.5.7 | |||
| 5120/5103, 15625/15552, 16875/16807 | |||
| [{{val| 193 306 448 542 }}] | |||
| -0.1118 | |||
| 0.3146 | |||
| 5.059 | |||
|- | |||
| 2.3.5.7.11 | |||
| 540/539, 1375/1372, 4375/4356, 5120/5103 | |||
| [{{val| 193 306 448 542 668 }}] | |||
| -0.2080 | |||
| 0.3408 | |||
| 5.481 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 325/324, 364/363, 540/539, 625/624, 4096/4095 | |||
| [{{val| 193 306 448 542 668 714 }}] | |||
| -0.1216 | |||
| 0.3662 | |||
| 5.890 | |||
|- | |||
| 2.3.5.7.11.13.17 | |||
| 325/324, 364/363, 375/374, 442/441, 595/594, 4096/4095 | |||
| [{{val| 193 306 448 542 668 714 789 }}] | |||
| -0.1302 | |||
| 0.3397 | |||
| 5.464 | |||
|- | |||
| 2.3.5.7.11.13.17.19 | |||
| 325/324, 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215 | |||
| [{{val| 193 306 448 542 668 714 789 820 }}] | |||
| -0.1414 | |||
| 0.3191 | |||
| 5.133 | |||
|} | |||
== Sqrtphi scale in 193edo == | |||
Approximation of the intervals: | Approximation of the intervals: | ||
| Line 9: | Line 82: | ||
Phi: '''134\193''' (833.16062 cents), both inside in the [[7L_2s|superdiatonic]] scale: 25 25 25 9 25 25 25 25 9 | 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: | |||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
[[Category:Prime EDO]] | [[Category:Prime EDO]] | ||
Revision as of 12:28, 20 January 2022
| ← 192edo | 193edo | 194edo → |
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 6.21762 cents each.
Theory
193edo provides the optimal patent val for sqrtphi temperament in the 13-, 17- and 19- limits, and for 13-limit minos and vish temperaments. It is the 44th prime EDO.
Prime harmonics
Script error: No such module "primes_in_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 | 3.225 |
| 2.3.5 | 15625/15552, [50 -33 1⟩ | [⟨193 306 448]] | -0.0158 | 0.3084 | 4.960 |
| 2.3.5.7 | 5120/5103, 15625/15552, 16875/16807 | [⟨193 306 448 542]] | -0.1118 | 0.3146 | 5.059 |
| 2.3.5.7.11 | 540/539, 1375/1372, 4375/4356, 5120/5103 | [⟨193 306 448 542 668]] | -0.2080 | 0.3408 | 5.481 |
| 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 | 5.890 |
| 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 | 5.464 |
| 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 | 5.133 |
Sqrtphi scale in 193edo
Approximation of the intervals:
Square root of Pi: 159\193 (988.60104 cents), and
Phi: 134\193 (833.16062 cents), both inside in the superdiatonic scale: 25 25 25 9 25 25 25 25 9