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Since 93edo has good approximations of 13th, 17th and 19th harmonics unlike 31edo (as 838.710{{cent}}, 103.226{{cent}}, and 296.774{{cent}} respectively, [[octave-reduced]]), it also allows one to give a clearer harmonic identity to [[31edo]]'s excellent approximation of 13:17:19.
Since 93edo has good approximations of 13th, 17th and 19th harmonics unlike 31edo (as 838.710{{cent}}, 103.226{{cent}}, and 296.774{{cent}} respectively, [[octave-reduced]]), it also allows one to give a clearer harmonic identity to [[31edo]]'s excellent approximation of 13:17:19.
== Temperament properties ==
Since 93edo has a step of 12.903{{cent}}, it also allows one to use its MOS scales as circulating temperaments, which it is the first edo to do. It is also the first edo to allow one to use a syntonic or Mavila MOS scale or a 17 tone MOS scale similar to a median between [http://www.neuroscience-of-music.se/pelog_main.htm Pelog] and the theories of Sundanese composer-musicologist-teacher [https://en.wikipedia.org/wiki/Raden_Machjar_Angga_Koesoemadinata Raden Machjar Angga Koesoemadinata] as a circulating temperament.
{| class="wikitable"
|+Circulating temperaments  in 93edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|19:18
|-
|6
|[[3L 3s]]
|16:15
|-
|7
|[[2L 5s]]
|14:13
|-
|8
|[[5L 3s]]
|12:11
|-
|9
|[[3L 6s]]
|11:10
|-
|10
|[[3L 7s]]
|10:9
|-
|11
|[[5L 6s]]
|9:8
|-
|12
|[[9L 3s]]
| rowspan="2" |8:7
|-
|13
|[[2L 11s]]
|-
|14
|[[9L 5s]]
| rowspan="2" |7:6
|-
|15
|[[3L 12s]]
|-
|16
|13L 3s
| rowspan="3" |6:5
|-
|17
|[[8L 9s]]
|-
|18
|3L 15s
|-
|19
|[[17L 2s]]
| rowspan="5" |5:4
|-
|20
|[[13L 7s]]
|-
|21
|9L 12s
|-
|22
|[[5L 17s]]
|-
|23
|1L 22s
|-
|24
|21L 3s
| rowspan="7" |4:3
|-
|25
|18L 7s
|-
|26
|15L 11s
|-
|27
|12L 15s
|-
|28
|9L 19s
|-
|29
|6L 23s
|-
|30
|3L 27s
|-
|31
|[[31edo]]
|equal
|-
|32
|29L 3s
| rowspan="15" |3:2
|-
|33
|27L 6s
|-
|34
|25L 9s
|-
|35
|23L 12s
|-
|36
|21L 15s
|-
|37
|19L 18s
|-
|38
|17L 21s
|-
|39
|15L 24s
|-
|40
|13L 27s
|-
|41
|12L 29s
|-
|42
|9L 33s
|-
|43
|7L 36s
|-
|44
|5L 39s
|-
|45
|3L 42s
|-
|46
|1L 45s
|-
|47
|46L 1s
| rowspan="28" |2:1
|-
|48
|45L 3s
|-
|49
|44L 5s
|-
|50
|43L 7s
|-
|51
|42L 9s
|-
|52
|41L 11s
|-
|53
|40L 13s
|-
|54
|39L 15s
|-
|55
|38L 17s
|-
|56
|37L 19s
|-
|57
|36L 21s
|-
|58
|35L 23s
|-
|59
|34L 25s
|-
|60
|33L 27s
|-
|61
|32L 29s
|-
|62
|31L 31s
|-
|63
|30L 33s
|-
|64
|29L 35s
|-
|65
|28L 37s
|-
|66
|27L 39s
|-
|67
|26L 41s
|-
|68
|25L 43s
|-
|69
|24L 45s
|-
|70
|23L 47s
|-
|71
|22L 49s
|-
|72
|21L 51s
|-
|73
|20L 53s
|-
|74
|19L 55s
|}


[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->

Revision as of 13:41, 30 May 2023

← 92edo 93edo 94edo →
Prime factorization 3 × 31
Step size 12.9032 ¢ 
Fifth 54\93 (696.774 ¢) (→ 18\31)
Semitones (A1:m2) 6:9 (77.42 ¢ : 116.1 ¢)
Dual sharp fifth 55\93 (709.677 ¢)
Dual flat fifth 54\93 (696.774 ¢) (→ 18\31)
Dual major 2nd 16\93 (206.452 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

Approximation of odd harmonics in 93edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -5.18 +0.78 -1.08 +2.54 +3.52 -1.82 -4.40 -1.73 -0.74 -6.26 +3.98
Relative (%) -40.2 +6.1 -8.4 +19.7 +27.3 -14.1 -34.1 -13.4 -5.7 -48.6 +30.9
Steps
(reduced) 		147
(54) 		216
(30) 		261
(75) 		295
(16) 		322
(43) 		344
(65) 		363
(84) 		380
(8) 		395
(23) 		408
(36) 		421
(49)

93 = 3 * 31, and 93 is a contorted 31 through the 7 limit. In the 11-limit the patent val tempers out 4000/3993 and in the 13-limit 144/143, 1188/1183 and 364/363. It provides the optimal patent val for the 11-limit prajapati and 13-limit kumhar temperaments, and the 11 and 13 limit 43&50 temperament. It is the 13th no-3s zeta peak edo.

Since 93edo has good approximations of 13th, 17th and 19th harmonics unlike 31edo (as 838.710 ¢, 103.226 ¢, and 296.774 ¢ respectively, octave-reduced), it also allows one to give a clearer harmonic identity to 31edo's excellent approximation of 13:17:19.

Scales

Meantone Chromatic

  • 116.129
  • 193.548
  • 309.677
  • 387.097
  • 503.226
  • 580.645
  • 696.774
  • 812.903
  • 890.323
  • 1006.452
  • 1083.871
  • 1200.000


Superpyth Chromatic

  • 51.613
  • 219.355
  • 270.968
  • 438.710
  • 490.323
  • 658.065
  • 709.677
  • 761.290
  • 929.032
  • 980.645
  • 1148.387
  • 1200.000


Superpyth Shailaja

  • 270.968
  • 709.677
  • 761.290
  • 980.645
  • 1200.000


Superpyth Subminor Hexatonic

  • 219.355
  • 270.968
  • 490.323
  • 709.677
  • 980.645
  • 1200.000


Superpyth Subminor Pentatonic

  • 270.968
  • 490.323
  • 709.677
  • 980.645
  • 1200.000

See Also

Retrieved from "https://en.xen.wiki/index.php?title=93edo&oldid=115163"