93edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Cmloegcmluin (talk | contribs)
autopatrolled
5,160 edits
add link for defactoring
Contribution (talk | contribs)
autopatrolled
2,880 edits
No edit summary
 
(19 intermediate revisions by 12 users not shown)
Line 1: Line 1:
The 93 equal division divides the octave into 93 equal parts of 12.903 cents each. 93 = 3 * 31, and 93 is a [[contorted]] (or [[enfactored]]) 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.
{{Infobox ET}}
{{ED intro}}


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.
== 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 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 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.
{| class="wikitable"
 
|+Circulating temperaments  in 93edo
=== Odd harmonics ===
!Tones
{{Harmonics in equal|93}}
!Pattern
 
!L:s
=== 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.
|5
 
|[[3L 2s]]
== Intervals ==
|19:18
{{Interval table}}
|-
 
|6
== Scales ==
|[[3L 3s]]
* Superpyth[5]: 21 17 17 21 17 ((21 38 55 76 93)\93)
|16:15
* 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)
|7
* Superpyth Subminor Hexatonic: 17 4 17 17 21 17 ((17 21 38 55 76 93)\93)
|[[2L 5s]]
 
|14:13
== Instruments ==
|-
 
|8
A [[Lumatone mapping for 93edo]] is available.
|[[5L 3s]]
 
|12:11
== Music ==
|-
; [[Bryan Deister]]
|9
* [https://www.youtube.com/shorts/eknKeDeRlQs ''microtonal improvisation in 93edo''] (2025)
|[[3L 6s]]
 
|11:10
== See also ==
|-
* [[93edo and stretched hemififths]]
|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
|}

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

Retrieved from "https://en.xen.wiki/index.php?title=93edo&oldid=204032"