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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|108}}
{{ED intro}}


==Theory==
== Theory ==
108edo tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.
108et [[tempering out|tempers out]] the [[Pythagorean comma]] in the 3-limit and 1990656/1953125 ([[valentine comma]]) in the 5-limit. In the 7-limit it tempers out [[126/125]] and [[1029/1024]], [[support]]ing [[valentine]], and making for a good tuning for it and for [[starling]] temperament, the planar temperament tempering out 126/125. In the 11-limit the [[patent val]] tempers out [[540/539]] and 1344/1331, and the 108e val tempers out [[121/120]], [[176/175]], [[385/384]], and [[441/440]], supporting 11-limit valentine for which it is again a good tuning.
 
108edo is the smallest 12''n''-edo which offers a decent alternative fifth to 12edo's fifth, that is [[27edo]]'s superpyth fifth.


108edo is the smallest 12n-edo which offers a decent alternative fifth to 12edo's fifth, that is [[27edo]]'s superpyth fifth.
=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|108}}
{{Harmonics in equal|108}}


=== Miscellany ===
=== Subsets and supersets ===
Aside from tuning, 108 is the smallest number with a prime factorization of the form <math>p^p \cdot q^q</math>. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.
Since 108 factors into {{factorization|108}}, 108edo has subset edos {{EDOs| 2, 3, 4, 6, 9, 12, 18, 27, 36, and 54 }}.  


Since 108edo has a step of 11.111 cents, it also allows one to use its MOS scales as circulating temperaments.
== Intervals ==
{| class="wikitable"
{{Interval table}}
|+Circulating temperaments in 108edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|22:21
|-
|6
|[[6edo]]
|equal
|-
|7
|[[3L 4s]]
|16:15
|-
|8
|[[4L 4s]]
|14:13
|-
|9
|[[9edo]]
|equal
|-
|10
|[[8L 2s]]
|11:10
|-
|11
|[[9L 2s]]
|10:9
|-
|12
|[[12edo]]
|equal
|-
|13
|[[3L 10s]]
|9:8
|-
|14
|[[10L 4s]]
| rowspan="2" |8:7
|-
|15
|[[3L 12s]]
|-
|16
|12L 4s
| rowspan="2" |7:6
|-
|17
|[[6L 11s]]
|-
|18
|[[18edo]]
|equal
|-
|19
|[[13L 6s]]
| rowspan="3" |6:5
|-
|20
|8L 12s
|-
|21
|3L 18s
|-
|22
|20L 2s
| rowspan="5" |5:4
|-
|23
|16L 7s
|-
|24
|12L 12s
|-
|25
|8L 17s
|-
|26
|4L 22s
|-
|27
|[[27edo]]
|equal
|-
|28
|24L 4s
| rowspan="8" |4:3
|-
|29
|21L 8s
|-
|30
|18L 12s
|-
|31
|15L 16s
|-
|32
|12L 20s
|-
|33
|9L 24s
|-
|34
|6L 28s
|-
|35
|3L 32s
|-
|36
|[[36edo]]
|equal
|-
|37
|34L 3s
| rowspan="17" |3:2
|-
|38
|32L 6s
|-
|39
|30L 9s
|-
|40
|28L 12s
|-
|41
|26L 15s
|-
|42
|24L 18s
|-
|43
|22L 21s
|-
|44
|20L 24s
|-
|45
|18L 27s
|-
|46
|16L 30s
|-
|47
|14L 33s
|-
|48
|12L 36s
|-
|49
|10L 39s
|-
|50
|8L 42s
|-
|51
|6L 45s
|-
|52
|4L 48s
|-
|53
|2L 51s
|-
|54
|[[54edo]]
|equal
|-
|55
|53L 2s
| rowspan="32" |2:1
|-
|56
|52L 4s
|-
|57
|51L 6s
|-
|58
|50L 8s
|-
|59
|49L 10s
|-
|60
|48L 12s
|-
|61
|47L 14s
|-
|62
|46L 16s
|-
|63
|45L 18s
|-
|64
|44L 20s
|-
|65
|43L 22s
|-
|66
|42L 24s
|-
|67
|41L 26s
|-
|68
|40L 28s
|-
|69
|39L 30s
|-
|70
|38L 32s
|-
|71
|37L 34s
|-
|72
|36L 36s
|-
|73
|35L 38s
|-
|74
|34L 40s
|-
|75
|33L 42s
|-
|76
|32L 44s
|-
|77
|31L 46s
|-
|78
|30L 48s
|-
|79
|29L 50s
|-
|80
|28L 52s
|-
|81
|27L 54s
|-
|82
|26L 56s
|-
|83
|25L 58s
|-
|84
|24L 60s
|-
|85
|23L 62s
|-
|86
|22L 64s
|}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
== Music ==
; [[Billy Stiltner]]
* [https://billystiltner.bandcamp.com/track/108-tone-equal 108 tone equal]
[[Category:Valentine]]
[[Category:Valentine]]

Latest revision as of 11:27, 17 August 2025

← 107edo 108edo 109edo →
Prime factorization 22 × 33
Step size 11.1111 ¢ 
Fifth 63\108 (700 ¢) (→ 7\12)
Semitones (A1:m2) 9:9 (100 ¢ : 100 ¢)
Consistency limit 7
Distinct consistency limit 7

108 equal divisions of the octave (abbreviated 108edo or 108ed2), also called 108-tone equal temperament (108tet) or 108 equal temperament (108et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 108 equal parts of about 11.1 ¢ each. Each step represents a frequency ratio of 21/108, or the 108th root of 2.

Theory

108et tempers out the Pythagorean comma in the 3-limit and 1990656/1953125 (valentine comma) in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting valentine, and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and 1344/1331, and the 108e val tempers out 121/120, 176/175, 385/384, and 441/440, supporting 11-limit valentine for which it is again a good tuning.

108edo is the smallest 12n-edo which offers a decent alternative fifth to 12edo's fifth, that is 27edo's superpyth fifth.

Prime harmonics

Approximation of prime harmonics in 108edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -1.96 +2.58 -2.16 +4.24 +3.92 -4.96 +2.49 +5.06 +3.76 -0.59
Relative (%) +0.0 -17.6 +23.2 -19.4 +38.1 +35.3 -44.6 +22.4 +45.5 +33.8 -5.3
Steps
(reduced) 		108
(0) 		171
(63) 		251
(35) 		303
(87) 		374
(50) 		400
(76) 		441
(9) 		459
(27) 		489
(57) 		525
(93) 		535
(103)

Subsets and supersets

Since 108 factors into 22 × 33, 108edo has subset edos 2, 3, 4, 6, 9, 12, 18, 27, 36, and 54.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 11.1 ^D, ^E♭♭
2 22.2 ^^D, ^^E♭♭
3 33.3 ^3D, ^3E♭♭
4 44.4 38/37, 39/38, 40/39, 41/40 ^4D, ^4E♭♭
5 55.6 31/30, 32/31 v4D♯, v4E♭
6 66.7 26/25 v3D♯, v3E♭
7 77.8 23/22, 45/43 vvD♯, vvE♭
8 88.9 20/19, 39/37 vD♯, vE♭
9 100 18/17, 35/33 D♯, E♭
10 111.1 16/15 ^D♯, ^E♭
11 122.2 44/41 ^^D♯, ^^E♭
12 133.3 40/37, 41/38 ^3D♯, ^3E♭
13 144.4 25/23, 38/35 ^4D♯, ^4E♭
14 155.6 35/32, 47/43 v4D𝄪, v4E
15 166.7 11/10 v3D𝄪, v3E
16 177.8 31/28, 41/37 vvD𝄪, vvE
17 188.9 29/26, 39/35 vD𝄪, vE
18 200 37/33, 46/41 E
19 211.1 26/23, 35/31 ^E, ^F♭
20 222.2 25/22, 33/29 ^^E, ^^F♭
21 233.3 8/7 ^3E, ^3F♭
22 244.4 38/33 ^4E, ^4F♭
23 255.6 22/19, 29/25 v4E♯, v4F
24 266.7 7/6 v3E♯, v3F
25 277.8 47/40 vvE♯, vvF
26 288.9 13/11 vE♯, vF
27 300 44/37 F
28 311.1 ^F, ^G♭♭
29 322.2 47/39 ^^F, ^^G♭♭
30 333.3 40/33 ^3F, ^3G♭♭
31 344.4 39/32 ^4F, ^4G♭♭
32 355.6 43/35 v4F♯, v4G♭
33 366.7 21/17, 47/38 v3F♯, v3G♭
34 377.8 41/33, 46/37 vvF♯, vvG♭
35 388.9 vF♯, vG♭
36 400 29/23, 34/27 F♯, G♭
37 411.1 19/15, 33/26 ^F♯, ^G♭
38 422.2 37/29 ^^F♯, ^^G♭
39 433.3 9/7 ^3F♯, ^3G♭
40 444.4 31/24 ^4F♯, ^4G♭
41 455.6 13/10 v4F𝄪, v4G
42 466.7 38/29 v3F𝄪, v3G
43 477.8 29/22 vvF𝄪, vvG
44 488.9 vF𝄪, vG
45 500 4/3 G
46 511.1 39/29, 43/32, 47/35 ^G, ^A♭♭
47 522.2 ^^G, ^^A♭♭
48 533.3 ^3G, ^3A♭♭
49 544.4 26/19 ^4G, ^4A♭♭
50 555.6 40/29 v4G♯, v4A♭
51 566.7 43/31 v3G♯, v3A♭
52 577.8 vvG♯, vvA♭
53 588.9 45/32 vG♯, vA♭
54 600 41/29 G♯, A♭
55 611.1 37/26, 47/33 ^G♯, ^A♭
56 622.2 43/30 ^^G♯, ^^A♭
57 633.3 ^3G♯, ^3A♭
58 644.4 29/20, 45/31 ^4G♯, ^4A♭
59 655.6 19/13 v4G𝄪, v4A
60 666.7 47/32 v3G𝄪, v3A
61 677.8 37/25 vvG𝄪, vvA
62 688.9 vG𝄪, vA
63 700 3/2 A
64 711.1 ^A, ^B♭♭
65 722.2 44/29, 47/31 ^^A, ^^B♭♭
66 733.3 29/19 ^3A, ^3B♭♭
67 744.4 20/13, 43/28 ^4A, ^4B♭♭
68 755.6 v4A♯, v4B♭
69 766.7 14/9 v3A♯, v3B♭
70 777.8 47/30 vvA♯, vvB♭
71 788.9 30/19, 41/26 vA♯, vB♭
72 800 27/17, 46/29 A♯, B♭
73 811.1 ^A♯, ^B♭
74 822.2 37/23, 45/28 ^^A♯, ^^B♭
75 833.3 34/21 ^3A♯, ^3B♭
76 844.4 ^4A♯, ^4B♭
77 855.6 41/25 v4A𝄪, v4B
78 866.7 33/20 v3A𝄪, v3B
79 877.8 vvA𝄪, vvB
80 888.9 vA𝄪, vB
81 900 37/22 B
82 911.1 22/13 ^B, ^C♭
83 922.2 ^^B, ^^C♭
84 933.3 12/7 ^3B, ^3C♭
85 944.4 19/11 ^4B, ^4C♭
86 955.6 33/19 v4B♯, v4C
87 966.7 7/4 v3B♯, v3C
88 977.8 44/25 vvB♯, vvC
89 988.9 23/13 vB♯, vC
90 1000 41/23 C
91 1011.1 43/24 ^C, ^D♭♭
92 1022.2 ^^C, ^^D♭♭
93 1033.3 20/11 ^3C, ^3D♭♭
94 1044.4 ^4C, ^4D♭♭
95 1055.6 35/19, 46/25 v4C♯, v4D♭
96 1066.7 37/20 v3C♯, v3D♭
97 1077.8 41/22 vvC♯, vvD♭
98 1088.9 15/8 vC♯, vD♭
99 1100 17/9 C♯, D♭
100 1111.1 19/10 ^C♯, ^D♭
101 1122.2 44/23 ^^C♯, ^^D♭
102 1133.3 25/13 ^3C♯, ^3D♭
103 1144.4 31/16 ^4C♯, ^4D♭
104 1155.6 37/19, 39/20 v4C𝄪, v4D
105 1166.7 v3C𝄪, v3D
106 1177.8 vvC𝄪, vvD
107 1188.9 vC𝄪, vD
108 1200 2/1 D

Music

Billy Stiltner
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