93edo: Difference between revisions

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The 93 equal division divides the octave into 93 equal parts of 12.903 cents each. 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.
{{EDO intro|93}}
 
== Theory ==
{{Harmonics in equal|93}}
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{{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¢, 103.226¢, and 296.774¢ respectively), it also allows one to give a clearer harmonic identity to [[31edo]]'s excellent approximation of 13:17:19.
{{harmonics in equal|93}}
== Temperament properties ==
== Temperament properties ==
Since 93edo has a step of 12.903 cents, 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 [http://en.wikipedia.org/wiki/Raden_Machjar_Angga_Koesoemadinata Raden Machjar Angga Koesoemadinata] as a circulating temperament.  
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 [http://en.wikipedia.org/wiki/Raden_Machjar_Angga_Koesoemadinata Raden Machjar Angga Koesoemadinata] as a circulating temperament.  
{| class="wikitable"
{| class="wikitable"
|+Circulating temperaments  in 93edo
|+Circulating temperaments  in 93edo
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|19L 55s
|19L 55s
|}
|}
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->

Revision as of 05:48, 2 July 2022

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.

Temperament properties

Since 93edo has a step of 12.903 ¢, 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 Pelog and the theories of Sundanese composer-musicologist-teacher Raden Machjar Angga Koesoemadinata as a circulating temperament.

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 8:7
13 2L 11s
14 9L 5s 7:6
15 3L 12s
16 13L 3s 6:5
17 8L 9s
18 3L 15s
19 17L 2s 5:4
20 13L 7s
21 9L 12s
22 5L 17s
23 1L 22s
24 21L 3s 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 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 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
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