98edo: Difference between revisions

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Move Music section below Instruments to conform to most other EDO pages; add Bryan Deister's ''98edo prelude'' (2025); add Lumatone mapping for 98edo page
 
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'''98 EDO''', the 98 equal temperament divides the octave into equal parts of 12.245 cents each. The [[patent val]] has a flat 3, a sharp 5 and a slightly flat 7, and tempers out 81/80 in the 5-limit, making it a system of [[meantone family]] with a 4-cent-flat fifth. In the 7-limit it tempers out 1029/1024, 1728/1715, supporting [[mothra]] temperament, in the 11-limit 176/175 and 540/539, supporting [[mosura]], and in the 13-limit 144/143 and 196/195. It provides the optimal patent val for 13-limit mosura temperament.
{{Infobox ET}}
{{ED intro}}


{{Primes in edo|98}}
== Theory ==
The [[patent val]] of 98edo has a flat [[3/1|3]], a sharp [[5/1|5]] and a slightly flat [[7/1|7]], and [[tempering out|tempers out]] [[81/80]] in the 5-limit, making it a system of [[meantone family]] with a 4-cent-flat fifth. In the 7-limit it tempers out [[1029/1024]], [[1728/1715]], [[support]]ing [[mothra]] temperament, in the 11-limit [[176/175]] and [[540/539]], supporting [[mosura]], and in the 13-limit [[144/143]] and [[196/195]]. It provides the optimal patent val for 13-limit mosura temperament.


Since 98 EDO has a step of 12.245 cents, it also allows one to use its MOS scales as circulating temperaments. As ''2*7*[[7edo]]'', It is the first ''km<sup>n</sup>'' EDO which does this and the first EDO which allows one to use a Magic MOS scale as a circulating temperament.
=== Odd harmonics ===
{{Harmonics in equal|98}}


{| class="wikitable center-all"
=== Subsets and supersets ===
|+ Circulating temperaments in 98 EDO
Since 98 factors into {{factorization|98}}, 98edo has subset edos {{EDOs| 2, 7, 14, and 49 }}.
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[3L 2s]]
| 20:19
|-
| 6
| [[2L 4s]]
| 17:16
|-
| 7
| ''[[7 EDO]]''
| ''equal''
|-
| 8
| [[2L 6s]]
| 13:12
|-
| 9
| [[8L 1s]]
| 11:10
|-
| 10
| [[8L 2s]]
| 10:9
|-
| 11
| [[10L 1s]]
| rowspan="2" | 9:8
|-
| 12
| [[2L 10s]]
|-
| 13
| [[7L 6s]]
| 8:7
|-
| 14
| ''[[14 EDO]]''
| ''equal''
|-
| 15
| [[8L 7s]]
| rowspan="2" | 7:6
|-
| 16
| 2L 14s
|-
| 17
| 13L 4s
| rowspan="3" | 6:5
|-
| 18
| 8L 10s
|-
| 19
| [[3L 16s]]
|-
| 20
| 18L 2s
| rowspan="5" | 5:4
|-
| 21
| 14L 7s
|-
| 22
| 10L 12s
|-
| 23
| 6L 17s
|-
| 24
| 2L 22s
|-
| 25
| 23L 2s
| rowspan="8" | 4:3
|-
| 26
| 20L 6s
|-
| 27
| 17L 10s
|-
| 28
| 14L 14s
|-
| 29
| 11L 18s
|-
| 30
| 8L 22s
|-
| 31
| 5L 26s
|-
| 32
| 2L 30s
|-
| 33
| 32L 1s
| rowspan="16" | 3:2
|-
| 34
| 30L 4s
|-
| 35
| 28L 7s
|-
| 36
| 26L 10s
|-
| 37
| 24L 13s
|-
| 38
| 22L 16s
|-
| 39
| 20L 19s
|-
| 40
| 18L 22s
|-
| 41
| 16L 25s
|-
| 42
| 14L 28s
|-
| 43
| 12L 31s
|-
| 44
| 10L 34s
|-
| 45
| 8L 37s
|-
| 46
| 6L 40s
|-
| 47
| 4L 43s
|-
| 48
| 2L 46s
|-
| 49
| ''[[49 EDO]]''
| ''equal''
|-
| 50
| 48L 2s
| rowspan="29" | 2:1
|-
| 51
| 47L 4s
|-
| 52
| 46L 6s
|-
| 53
| 45L 8s
|-
| 54
| 44L 10s
|-
| 55
| 43L 12s
|-
| 56
| 42L 14s
|-
| 57
| 41L 16s
|-
| 58
| 40L 18s
|-
| 59
| 39L 20s
|-
| 60
| 38L 22s
|-
| 61
| 37L 24s
|-
| 62
| 36L 26s
|-
| 63
| 35L 28s
|-
| 64
| 34L 30s
|-
| 65
| 33L 32s
|-
| 66
| 32L 34s
|-
| 67
| 31L 36s
|-
| 68
| 30L 38s
|-
| 69
| 29L 40s
|-
| 70
| 28L 42s
|-
| 71
| 27L 44s
|-
| 72
| 26L 46s
|-
| 73
| 25L 48s
|-
| 74
| 24L 50s
|-
| 75
| 23L 52s
|-
| 76
| 22L 54s
|-
| 77
| 21L 56s
|-
| 78
| 20L 58s
|}


[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
== Intervals ==
[[Category:Meantone]]
{{Interval table}}
 
== Instruments ==
=== Keyboards ===
[[Lumatone mapping for 98edo|Lumatone mappings for 98edo]] are available.
 
=== Skip fretting ===
'''Skip fretting system 98 9 11''' is a [[skip fretting]] system for [[98edo]]. All examples on this page are for 7-string [[guitar]].
 
; Odd harmonics
 
1/1: string 2 open
 
2/1: string 6 fret 6
 
3/2: string 3 fret 16
 
5/4: string 4 fret 12
 
7/4: string 1 fret 10
 
9/8: string 1 fret 14
 
11/8: string 2 fret 5 and string 6 fret 11
 
13/8: string 5 fret 4
 
15/8: string 6 fret 5
 
17/16: string 2 fret 1
 
19/16: string 4 fret 22
 
21/16: string 3 fret 3
 
23/16: string 5 fret 2
 
25/16: string 2 fret 7
 
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=pR4li7aIUZE ''microtonal improvisation in 98edo''] (2023)
* [https://www.youtube.com/shorts/4NR3KFBd720 ''98edo waltz''] (2025)
* [https://www.youtube.com/shorts/EUR8t8ynORg ''98edo prelude''] (2025)
 
[[Category:Mothra]]
[[Category:Listen]]

Latest revision as of 09:02, 6 August 2025

← 97edo 98edo 99edo →
Prime factorization 2 × 72
Step size 12.2449 ¢ 
Fifth 57\98 (697.959 ¢)
Semitones (A1:m2) 7:9 (85.71 ¢ : 110.2 ¢)
Consistency limit 3
Distinct consistency limit 3

98 equal divisions of the octave (abbreviated 98edo or 98ed2), also called 98-tone equal temperament (98tet) or 98 equal temperament (98et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 98 equal parts of about 12.2 ¢ each. Each step represents a frequency ratio of 21/98, or the 98th root of 2.

Theory

The patent val of 98edo has a flat 3, a sharp 5 and a slightly flat 7, and tempers out 81/80 in the 5-limit, making it a system of meantone family with a 4-cent-flat fifth. In the 7-limit it tempers out 1029/1024, 1728/1715, supporting mothra temperament, in the 11-limit 176/175 and 540/539, supporting mosura, and in the 13-limit 144/143 and 196/195. It provides the optimal patent val for 13-limit mosura temperament.

Odd harmonics

Approximation of odd harmonics in 98edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -4.00 +5.52 -1.48 +4.25 -0.30 +4.37 +1.53 +5.25 -3.64 -5.47 -3.78
Relative (%) -32.6 +45.1 -12.1 +34.7 -2.4 +35.7 +12.5 +42.9 -29.7 -44.7 -30.9
Steps
(reduced) 		155
(57) 		228
(32) 		275
(79) 		311
(17) 		339
(45) 		363
(69) 		383
(89) 		401
(9) 		416
(24) 		430
(38) 		443
(51)

Subsets and supersets

Since 98 factors into 2 × 72, 98edo has subset edos 2, 7, 14, and 49.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 12.2 ^D, vE♭♭
2 24.5 ^^D, E♭♭
3 36.7 ^3D, ^E♭♭
4 49 35/34 v3D♯, ^^E♭♭
5 61.2 29/28, 30/29 vvD♯, ^3E♭♭
6 73.5 24/23 vD♯, v3E♭
7 85.7 41/39 D♯, vvE♭
8 98 37/35 ^D♯, vE♭
9 110.2 16/15 ^^D♯, E♭
10 122.4 44/41 ^3D♯, ^E♭
11 134.7 40/37 v3D𝄪, ^^E♭
12 146.9 37/34 vvD𝄪, ^3E♭
13 159.2 23/21, 34/31 vD𝄪, v3E
14 171.4 21/19, 32/29, 43/39 D𝄪, vvE
15 183.7 ^D𝄪, vE
16 195.9 E
17 208.2 35/31, 44/39 ^E, vF♭
18 220.4 ^^E, F♭
19 232.7 8/7 ^3E, ^F♭
20 244.9 38/33 v3E♯, ^^F♭
21 257.1 vvE♯, ^3F♭
22 269.4 7/6 vE♯, v3F
23 281.6 20/17 E♯, vvF
24 293.9 ^E♯, vF
25 306.1 31/26, 37/31 F
26 318.4 ^F, vG♭♭
27 330.6 23/19 ^^F, G♭♭
28 342.9 28/23, 39/32 ^3F, ^G♭♭
29 355.1 43/35 v3F♯, ^^G♭♭
30 367.3 vvF♯, ^3G♭♭
31 379.6 vF♯, v3G♭
32 391.8 F♯, vvG♭
33 404.1 24/19, 43/34 ^F♯, vG♭
34 416.3 14/11 ^^F♯, G♭
35 428.6 41/32 ^3F♯, ^G♭
36 440.8 40/31 v3F𝄪, ^^G♭
37 453.1 13/10 vvF𝄪, ^3G♭
38 465.3 17/13 vF𝄪, v3G
39 477.6 29/22 F𝄪, vvG
40 489.8 ^F𝄪, vG
41 502 G
42 514.3 35/26, 39/29 ^G, vA♭♭
43 526.5 19/14 ^^G, A♭♭
44 538.8 15/11, 41/30 ^3G, ^A♭♭
45 551 11/8 v3G♯, ^^A♭♭
46 563.3 vvG♯, ^3A♭♭
47 575.5 39/28 vG♯, v3A♭
48 587.8 G♯, vvA♭
49 600 41/29 ^G♯, vA♭
50 612.2 37/26 ^^G♯, A♭
51 624.5 33/23, 43/30 ^3G♯, ^A♭
52 636.7 v3G𝄪, ^^A♭
53 649 16/11 vvG𝄪, ^3A♭
54 661.2 22/15, 41/28 vG𝄪, v3A
55 673.5 28/19 G𝄪, vvA
56 685.7 ^G𝄪, vA
57 698 A
58 710.2 ^A, vB♭♭
59 722.4 44/29 ^^A, B♭♭
60 734.7 26/17 ^3A, ^B♭♭
61 746.9 20/13 v3A♯, ^^B♭♭
62 759.2 31/20 vvA♯, ^3B♭♭
63 771.4 vA♯, v3B♭
64 783.7 11/7 A♯, vvB♭
65 795.9 19/12 ^A♯, vB♭
66 808.2 ^^A♯, B♭
67 820.4 ^3A♯, ^B♭
68 832.7 v3A𝄪, ^^B♭
69 844.9 vvA𝄪, ^3B♭
70 857.1 23/14 vA𝄪, v3B
71 869.4 38/23, 43/26 A𝄪, vvB
72 881.6 ^A𝄪, vB
73 893.9 B
74 906.1 ^B, vC♭
75 918.4 17/10 ^^B, C♭
76 930.6 12/7 ^3B, ^C♭
77 942.9 v3B♯, ^^C♭
78 955.1 33/19 vvB♯, ^3C♭
79 967.3 7/4 vB♯, v3C
80 979.6 B♯, vvC
81 991.8 39/22 ^B♯, vC
82 1004.1 C
83 1016.3 ^C, vD♭♭
84 1028.6 29/16, 38/21 ^^C, D♭♭
85 1040.8 31/17, 42/23 ^3C, ^D♭♭
86 1053.1 v3C♯, ^^D♭♭
87 1065.3 37/20 vvC♯, ^3D♭♭
88 1077.6 41/22 vC♯, v3D♭
89 1089.8 15/8 C♯, vvD♭
90 1102 ^C♯, vD♭
91 1114.3 ^^C♯, D♭
92 1126.5 23/12 ^3C♯, ^D♭
93 1138.8 29/15 v3C𝄪, ^^D♭
94 1151 vvC𝄪, ^3D♭
95 1163.3 vC𝄪, v3D
96 1175.5 C𝄪, vvD
97 1187.8 ^C𝄪, vD
98 1200 2/1 D

Instruments

Keyboards

Lumatone mappings for 98edo are available.

Skip fretting

Skip fretting system 98 9 11 is a skip fretting system for 98edo. All examples on this page are for 7-string guitar.

Odd harmonics

1/1: string 2 open

2/1: string 6 fret 6

3/2: string 3 fret 16

5/4: string 4 fret 12

7/4: string 1 fret 10

9/8: string 1 fret 14

11/8: string 2 fret 5 and string 6 fret 11

13/8: string 5 fret 4

15/8: string 6 fret 5

17/16: string 2 fret 1

19/16: string 4 fret 22

21/16: string 3 fret 3

23/16: string 5 fret 2

25/16: string 2 fret 7

Music

Bryan Deister
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