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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|93}}
{{ED intro}}


== Theory ==
== Theory ==
Since {{nowrap|93 {{=}} 3 × 31}}, 93edo is a [[contorted]] [[31edo]] through the [[7-limit]]. In the 11-limit the [[patent val]] [[tempering out|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 [[31st-octave_temperaments#Prajapati|prajapati]] and 13-limit [[31st-octave_temperaments#Kumhar|kumhar]] temperaments and the 11- and 13-limit [[Meantone family#Trimean|trimean]] ({{nowrap|43 & 50}}) temperament, and is the 13th no-3s [[zeta peak edo]]. The 93bd val is close to the 9-odd limit minimax tuning for [[superpyth]] and approximates {{nowrap|{{frac|2|7}}-[[64/63|septimal comma]]}} superpyth very well.
Since 93edo has good approximations of [[13/1|13th]], [[17/1|17th]] and [[19/1|19th]] [[harmonic]]s unlike 31edo (as 838.710{{c}}, 103.226{{c}}, and 296.774{{c}} respectively, [[octave-reduced]]), it also allows one to give a clearer harmonic identity to [[31edo]]'s excellent approximation of 13:17:19.
=== Odd harmonics ===
{{Harmonics in equal|93}}
{{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.
=== No-3 approach ===
If prime 3 is ignored, 93edo represents the no-3 35-odd-limit consistently. 93edo is distinctly consistent within the no-3 19-integer-limit.
 
== Intervals ==
{{Interval table}}
 
== Scales ==
* Superpyth[5]: 21 17 17 21 17 ((21 38 55 76 93)\93)
* Superpyth[12]: 4 13 4 13 4 13 4 4 13 4 13 4 ((4 17 21 34 38 51 55 59 72 76 89 93)\93)
* Superpyth Shailaja: 21 34 4 17 17 ((21 55 59 76 93)\93)
* Superpyth Subminor Hexatonic: 17 4 17 17 21 17 ((17 21 38 55 76 93)\93)


== Temperament properties ==
== Instruments ==
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"
|+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 -->
A [[Lumatone mapping for 93edo]] is available.


== See Also ==
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/eknKeDeRlQs ''microtonal improvisation in 93edo''] (2025)


== See also ==
* [[93edo and stretched hemififths]]
* [[93edo and stretched hemififths]]

Latest revision as of 14:56, 2 July 2025

← 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

93 equal divisions of the octave (abbreviated 93edo or 93ed2), also called 93-tone equal temperament (93tet) or 93 equal temperament (93et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 93 equal parts of about 12.9 ¢ each. Each step represents a frequency ratio of 21/93, or the 93rd root of 2.

Theory

Since 93 = 3 × 31, 93edo is a contorted 31edo 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 trimean (43 & 50) temperament, and is the 13th no-3s zeta peak edo. The 93bd val is close to the 9-odd limit minimax tuning for superpyth and approximates 2⁄7-septimal comma superpyth very well.

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.

Odd harmonics

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)

No-3 approach

If prime 3 is ignored, 93edo represents the no-3 35-odd-limit consistently. 93edo is distinctly consistent within the no-3 19-integer-limit.

Intervals

Steps Cents Approximate ratios Ups and downs notation
(Dual flat fifth 54\93) 		Ups and downs notation
(Dual sharp fifth 55\93)
0 0 1/1 D D
1 12.9 ^D, vvE♭♭ ^D, v3E♭
2 25.8 ^^D, vE♭♭ ^^D, vvE♭
3 38.7 ^3D, E♭♭ ^3D, vE♭
4 51.6 33/32, 34/33, 35/34 vvD♯, ^E♭♭ ^4D, E♭
5 64.5 vD♯, ^^E♭♭ ^5D, ^E♭
6 77.4 23/22 D♯, v3E♭ ^6D, ^^E♭
7 90.3 20/19, 39/37 ^D♯, vvE♭ v6D♯, ^3E♭
8 103.2 17/16, 35/33 ^^D♯, vE♭ v5D♯, ^4E♭
9 116.1 31/29 ^3D♯, E♭ v4D♯, ^5E♭
10 129 14/13, 41/38 vvD𝄪, ^E♭ v3D♯, ^6E♭
11 141.9 25/23, 38/35 vD𝄪, ^^E♭ vvD♯, v6E
12 154.8 35/32 D𝄪, v3E vD♯, v5E
13 167.7 32/29 ^D𝄪, vvE D♯, v4E
14 180.6 ^^D𝄪, vE ^D♯, v3E
15 193.5 19/17, 28/25 E ^^D♯, vvE
16 206.5 ^E, vvF♭ ^3D♯, vE
17 219.4 17/15, 25/22, 42/37 ^^E, vF♭ E
18 232.3 8/7 ^3E, F♭ ^E, v3F
19 245.2 15/13, 38/33 vvE♯, ^F♭ ^^E, vvF
20 258.1 29/25 vE♯, ^^F♭ ^3E, vF
21 271 E♯, v3F F
22 283.9 20/17, 33/28 ^E♯, vvF ^F, v3G♭
23 296.8 19/16 ^^E♯, vF ^^F, vvG♭
24 309.7 F ^3F, vG♭
25 322.6 41/34 ^F, vvG♭♭ ^4F, G♭
26 335.5 17/14, 40/33 ^^F, vG♭♭ ^5F, ^G♭
27 348.4 ^3F, G♭♭ ^6F, ^^G♭
28 361.3 16/13, 37/30 vvF♯, ^G♭♭ v6F♯, ^3G♭
29 374.2 31/25, 41/33 vF♯, ^^G♭♭ v5F♯, ^4G♭
30 387.1 5/4 F♯, v3G♭ v4F♯, ^5G♭
31 400 29/23 ^F♯, vvG♭ v3F♯, ^6G♭
32 412.9 33/26 ^^F♯, vG♭ vvF♯, v6G
33 425.8 32/25 ^3F♯, G♭ vF♯, v5G
34 438.7 40/31 vvF𝄪, ^G♭ F♯, v4G
35 451.6 13/10 vF𝄪, ^^G♭ ^F♯, v3G
36 464.5 17/13 F𝄪, v3G ^^F♯, vvG
37 477.4 25/19, 29/22 ^F𝄪, vvG ^3F♯, vG
38 490.3 ^^F𝄪, vG G
39 503.2 G ^G, v3A♭
40 516.1 31/23, 35/26 ^G, vvA♭♭ ^^G, vvA♭
41 529 19/14 ^^G, vA♭♭ ^3G, vA♭
42 541.9 26/19, 41/30 ^3G, A♭♭ ^4G, A♭
43 554.8 40/29 vvG♯, ^A♭♭ ^5G, ^A♭
44 567.7 43/31 vG♯, ^^A♭♭ ^6G, ^^A♭
45 580.6 7/5 G♯, v3A♭ v6G♯, ^3A♭
46 593.5 31/22 ^G♯, vvA♭ v5G♯, ^4A♭
47 606.5 ^^G♯, vA♭ v4G♯, ^5A♭
48 619.4 10/7 ^3G♯, A♭ v3G♯, ^6A♭
49 632.3 vvG𝄪, ^A♭ vvG♯, v6A
50 645.2 29/20 vG𝄪, ^^A♭ vG♯, v5A
51 658.1 19/13, 41/28 G𝄪, v3A G♯, v4A
52 671 28/19 ^G𝄪, vvA ^G♯, v3A
53 683.9 43/29 ^^G𝄪, vA ^^G♯, vvA
54 696.8 A ^3G♯, vA
55 709.7 ^A, vvB♭♭ A
56 722.6 38/25 ^^A, vB♭♭ ^A, v3B♭
57 735.5 26/17 ^3A, B♭♭ ^^A, vvB♭
58 748.4 20/13, 37/24 vvA♯, ^B♭♭ ^3A, vB♭
59 761.3 31/20 vA♯, ^^B♭♭ ^4A, B♭
60 774.2 25/16 A♯, v3B♭ ^5A, ^B♭
61 787.1 41/26 ^A♯, vvB♭ ^6A, ^^B♭
62 800 ^^A♯, vB♭ v6A♯, ^3B♭
63 812.9 8/5 ^3A♯, B♭ v5A♯, ^4B♭
64 825.8 vvA𝄪, ^B♭ v4A♯, ^5B♭
65 838.7 13/8 vA𝄪, ^^B♭ v3A♯, ^6B♭
66 851.6 A𝄪, v3B vvA♯, v6B
67 864.5 28/17, 33/20 ^A𝄪, vvB vA♯, v5B
68 877.4 ^^A𝄪, vB A♯, v4B
69 890.3 B ^A♯, v3B
70 903.2 32/19 ^B, vvC♭ ^^A♯, vvB
71 916.1 17/10 ^^B, vC♭ ^3A♯, vB
72 929 41/24 ^3B, C♭ B
73 941.9 vvB♯, ^C♭ ^B, v3C
74 954.8 26/15, 33/19 vB♯, ^^C♭ ^^B, vvC
75 967.7 7/4 B♯, v3C ^3B, vC
76 980.6 30/17, 37/21 ^B♯, vvC C
77 993.5 ^^B♯, vC ^C, v3D♭
78 1006.5 25/14, 34/19 C ^^C, vvD♭
79 1019.4 ^C, vvD♭♭ ^3C, vD♭
80 1032.3 29/16 ^^C, vD♭♭ ^4C, D♭
81 1045.2 ^3C, D♭♭ ^5C, ^D♭
82 1058.1 35/19 vvC♯, ^D♭♭ ^6C, ^^D♭
83 1071 13/7 vC♯, ^^D♭♭ v6C♯, ^3D♭
84 1083.9 43/23 C♯, v3D♭ v5C♯, ^4D♭
85 1096.8 32/17 ^C♯, vvD♭ v4C♯, ^5D♭
86 1109.7 19/10 ^^C♯, vD♭ v3C♯, ^6D♭
87 1122.6 ^3C♯, D♭ vvC♯, v6D
88 1135.5 vvC𝄪, ^D♭ vC♯, v5D
89 1148.4 33/17 vC𝄪, ^^D♭ C♯, v4D
90 1161.3 43/22 C𝄪, v3D ^C♯, v3D
91 1174.2 ^C𝄪, vvD ^^C♯, vvD
92 1187.1 ^^C𝄪, vD ^3C♯, vD
93 1200 2/1 D D

Scales

  • Superpyth[5]: 21 17 17 21 17 ((21 38 55 76 93)\93)
  • Superpyth[12]: 4 13 4 13 4 13 4 4 13 4 13 4 ((4 17 21 34 38 51 55 59 72 76 89 93)\93)
  • Superpyth Shailaja: 21 34 4 17 17 ((21 55 59 76 93)\93)
  • Superpyth Subminor Hexatonic: 17 4 17 17 21 17 ((17 21 38 55 76 93)\93)

Instruments

A Lumatone mapping for 93edo is available.

Music

Bryan Deister

See also

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