92edo: Difference between revisions

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The 92 divisions of 92edo measure 13.0435 cents each. 92 is contorted through the 17-limit, with the same tuning and commas as [[46edo|46edo]], and hence attracts little interest. The 53\92 generator is a very flat flattone fifth, flatter even than [[26edo|26edo]].  This is the highest in a series of four consecutive EDOs to temper out [[Quartisma|117440512/117406179]].
{{Infobox ET}}
{{ED intro}}


[[Category:Equal divisions of the octave]]
== Theory ==
[[Category:Quartismic]]
The equal temperament is [[contorted]] through the 17-limit, with the same tuning and [[comma]]s as [[46edo]], and hence attracts little interest. That said, the approximation to the [[19/1|19th harmonic]] is much improved. Like 46, the [[patent fifth]] (54\92) is about 2.4{{c}} sharp. The alternate fifth 53\92 is a very flat fifth, flatter even than that of [[26edo]] and only 0.102{{c}} sharp of [[1/2-comma meantone]]; the 92bcccd val [[support]]s [[flattone]], while the 92bcccdd val supports [[Meantone_family#Flattertone|flattertone]]. 92edo is the highest in a series of four consecutive edos to temper out the [[quartisma]] (117440512/117406179).
 
=== Odd harmonics ===
{{Harmonics in equal|92}}
 
=== Subsets and supersets ===
Since 92 factors into 2<sup>2</sup> × 23, 92edo has subset edos {{EDOs| 2, 4, 23, and 46 }}.
 
== Intervals ==
{{Interval table}}
 
== Instruments ==
 
A [[Lumatone mapping for 92edo]] is available.
 
== Music==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/5XFOALAzLiA ''microtonal improvisation in 92edo''] (2025)
* [https://www.youtube.com/watch?v=qWAinBYHwtE ''92edo waltz''] (2025)

Latest revision as of 15:50, 26 September 2025

← 91edo 92edo 93edo →
Prime factorization 22 × 23
Step size 13.0435 ¢ 
Fifth 54\92 (704.348 ¢) (→ 27\46)
Semitones (A1:m2) 10:6 (130.4 ¢ : 78.26 ¢)
Consistency limit 5
Distinct consistency limit 5

92 equal divisions of the octave (abbreviated 92edo or 92ed2), also called 92-tone equal temperament (92tet) or 92 equal temperament (92et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 92 equal parts of about 13 ¢ each. Each step represents a frequency ratio of 21/92, or the 92nd root of 2.

Theory

The equal temperament is contorted through the 17-limit, with the same tuning and commas as 46edo, and hence attracts little interest. That said, the approximation to the 19th harmonic is much improved. Like 46, the patent fifth (54\92) is about 2.4 ¢ sharp. The alternate fifth 53\92 is a very flat fifth, flatter even than that of 26edo and only 0.102 ¢ sharp of 1/2-comma meantone; the 92bcccd val supports flattone, while the 92bcccdd val supports flattertone. 92edo is the highest in a series of four consecutive edos to temper out the quartisma (117440512/117406179).

Odd harmonics

Approximation of odd harmonics in 92edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +2.39 +4.99 -3.61 +4.79 -3.49 -5.75 -5.66 -0.61 +2.49 -1.22 -2.19
Relative (%) +18.3 +38.3 -27.7 +36.7 -26.8 -44.0 -43.4 -4.7 +19.1 -9.3 -16.8
Steps
(reduced) 		146
(54) 		214
(30) 		258
(74) 		292
(16) 		318
(42) 		340
(64) 		359
(83) 		376
(8) 		391
(23) 		404
(36) 		416
(48)

Subsets and supersets

Since 92 factors into 22 × 23, 92edo has subset edos 2, 4, 23, and 46.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 13 ^D, v5E♭
2 26.1 ^^D, v4E♭
3 39.1 42/41, 43/42 ^3D, v3E♭
4 52.2 33/32, 34/33, 35/34 ^4D, vvE♭
5 65.2 ^5D, vE♭
6 78.3 22/21, 23/22 v4D♯, E♭
7 91.3 19/18, 20/19, 39/37 v3D♯, ^E♭
8 104.3 17/16, 35/33 vvD♯, ^^E♭
9 117.4 31/29 vD♯, ^3E♭
10 130.4 14/13, 41/38 D♯, ^4E♭
11 143.5 38/35 ^D♯, v5E
12 156.5 23/21, 35/32 ^^D♯, v4E
13 169.6 32/29, 43/39 ^3D♯, v3E
14 182.6 10/9 ^4D♯, vvE
15 195.7 37/33 ^5D♯, vE
16 208.7 35/31 E
17 221.7 33/29, 42/37 ^E, v5F
18 234.8 ^^E, v4F
19 247.8 ^3E, v3F
20 260.9 36/31, 43/37 ^4E, vvF
21 273.9 34/29, 41/35 ^5E, vF
22 287 13/11, 33/28 F
23 300 19/16 ^F, v5G♭
24 313 6/5 ^^F, v4G♭
25 326.1 29/24, 35/29, 41/34 ^3F, v3G♭
26 339.1 28/23 ^4F, vvG♭
27 352.2 38/31 ^5F, vG♭
28 365.2 21/17 v4F♯, G♭
29 378.3 41/33 v3F♯, ^G♭
30 391.3 vvF♯, ^^G♭
31 404.3 24/19, 43/34 vF♯, ^3G♭
32 417.4 14/11 F♯, ^4G♭
33 430.4 41/32 ^F♯, v5G
34 443.5 31/24, 40/31 ^^F♯, v4G
35 456.5 43/33 ^3F♯, v3G
36 469.6 21/16, 38/29 ^4F♯, vvG
37 482.6 37/28, 41/31 ^5F♯, vG
38 495.7 4/3 G
39 508.7 ^G, v5A♭
40 521.7 23/17, 27/20 ^^G, v4A♭
41 534.8 ^3G, v3A♭
42 547.8 ^4G, vvA♭
43 560.9 29/21 ^5G, vA♭
44 573.9 32/23, 39/28 v4G♯, A♭
45 587 v3G♯, ^A♭
46 600 41/29 vvG♯, ^^A♭
47 613 37/26 vG♯, ^3A♭
48 626.1 23/16, 33/23 G♯, ^4A♭
49 639.1 42/29 ^G♯, v5A
50 652.2 35/24 ^^G♯, v4A
51 665.2 ^3G♯, v3A
52 678.3 34/23, 40/27 ^4G♯, vvA
53 691.3 ^5G♯, vA
54 704.3 3/2 A
55 717.4 ^A, v5B♭
56 730.4 29/19, 32/21 ^^A, v4B♭
57 743.5 43/28 ^3A, v3B♭
58 756.5 31/20 ^4A, vvB♭
59 769.6 ^5A, vB♭
60 782.6 11/7 v4A♯, B♭
61 795.7 19/12 v3A♯, ^B♭
62 808.7 vvA♯, ^^B♭
63 821.7 37/23 vA♯, ^3B♭
64 834.8 34/21 A♯, ^4B♭
65 847.8 31/19 ^A♯, v5B
66 860.9 23/14 ^^A♯, v4B
67 873.9 ^3A♯, v3B
68 887 5/3 ^4A♯, vvB
69 900 32/19, 37/22 ^5A♯, vB
70 913 22/13, 39/23 B
71 926.1 29/17, 41/24 ^B, v5C
72 939.1 31/18 ^^B, v4C
73 952.2 ^3B, v3C
74 965.2 ^4B, vvC
75 978.3 37/21 ^5B, vC
76 991.3 39/22 C
77 1004.3 ^C, v5D♭
78 1017.4 9/5 ^^C, v4D♭
79 1030.4 29/16 ^3C, v3D♭
80 1043.5 42/23 ^4C, vvD♭
81 1056.5 35/19 ^5C, vD♭
82 1069.6 13/7 v4C♯, D♭
83 1082.6 43/23 v3C♯, ^D♭
84 1095.7 32/17 vvC♯, ^^D♭
85 1108.7 19/10, 36/19 vC♯, ^3D♭
86 1121.7 21/11 C♯, ^4D♭
87 1134.8 ^C♯, v5D
88 1147.8 33/17 ^^C♯, v4D
89 1160.9 41/21, 43/22 ^3C♯, v3D
90 1173.9 ^4C♯, vvD
91 1187 ^5C♯, vD
92 1200 2/1 D

Instruments

A Lumatone mapping for 92edo is available.

Music

Bryan Deister
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