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{{Infobox ET}}
{{Infobox ET}}
'''102edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 102 steps of size 11.765 [[cent]]s each. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo|34edo]]. In the [[7-limit|7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit|11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit|13-limit]] 91/90 and 169/168; in the [[17-limit|17-limit]] 136/135 and 154/153; and in the [[19-limit|19-limit]] 133/132 and 190/189. It is the [[Optimal_patent_val|optimal patent val]] for 13-limit [[Diaschismic_family#Echidnic|echidnic temperament]], and the rank five temperament tempering out 91/90.
{{ED intro}}


===13-limit Echidnic===
== Theory ==
102edo is [[enfactoring|enfactored]] in the [[5-limit]], where it [[tempering out|tempers out]] the same [[comma]]s ([[2048/2025]], [[15625/15552]], [[20000/19683]]) as [[34edo]]. In the [[7-limit]] it tempers out [[686/675]] and [[1029/1024]]; in the [[11-limit]] [[385/384]], [[441/440]] and [[4000/3993]]; in the [[13-limit]] [[91/90]] and [[169/168]]; in the [[17-limit]] [[136/135]] and [[154/153]]; and in the [[19-limit]] [[133/132]] and [[190/189]]. It is the [[optimal patent val]] for 13-limit [[Diaschismic family #Echidnic|echidnic]] temperament, and the rank-5 temperament tempering out 91/90.


{| class="wikitable"
=== Odd harmonics ===
!Degree
{{Harmonics in equal|102}}
!Cents
 
!Difference from 46edo
== Intervals ==
|-
{{Interval table}}
| |2
| |23.529
|  -2.5575¢
|-
| |4
| |47.059
| -5.115¢
|-
| | 7
| |82.353
|4.092¢
|-
| |9
| |105.882
|1.5345¢
|-
| |11
| |129.412
| -1.023¢
|-
| |13
| |152.941
|8.184¢
|-
| |16
| |188.235
|5.627¢
|-
| | 18
| |211.765
|3.069¢
|-
| |20
| |235.294
|0.511¢
|-
| |22
| |258.824
| -2.046¢
|-
| | 24
| |282.353
| -4.604¢
|-
| |27
| |317.647
|4.604¢
|-
| |29
| |341.176
|2.046¢
|-
| |31
| |364.706
| -0.5115¢
|-
| |33
| |388.235
|  -3.069¢
|-
| |35
| |411.765
| -5.627¢
|-
| |38
| |447.059
| 3.581¢
|-
| |40
| |470.588
|1.023¢
|-
| |42
| |494.117
| -1.5345¢
|-
| |44
| |517.647
| -4.092¢
|-
| |47
| |552.941
|5.115¢
|-
| |49
| |576.471
|2.5575¢
|}Since 102edo has a step of 11.765 cents, it also allows one to use its MOS scales as circulating temperaments. It is the first edo which truly allows one to use an MOS scale with 60 or more small steps as a circulating temperament.
{| class="wikitable"
|+Circulating temperaments in 102edo
!Tones
!Pattern
!L:s
|-
|5
|[[2L 3s]]
|21:20
|-
|6
|[[6edo]]
|equal
|-
|7
|[[4L 3s]]
|15:14
|-
|8
|[[6L 2s]]
|13:12
|-
|9
|[[3L 6s]]
|12:11
|-
|10
|[[2L 8s]]
|11:10
|-
|11
|[[3L 8s]]
|10:9
|-
|12
|[[6L 6s]]
|9:8
|-
|13
|[[11L 2s]]
| rowspan="2" |8:7
|-
|14
|[[4L 10s]]
|-
|15
|[[12L 3s]]
| rowspan="2" |7:6
|-
|16
|[[6L 10s]]
|-
|17
|[[17edo]]
|equal
|-
|18
|10L 8s
| rowspan="3" |6:5
|-
|19
|[[7L 12s]]
|-
|20
|2L 18s
|-
|21
|18L 3s
| rowspan="5" |5:4
|-
|22
|14L 8s
|-
|23
|10L 13s
|-
|24
|6L 18s
|-
|25
|2L 23s
|-
|26
|24L 2s
| rowspan="8" |4:3
|-
|27
|21L 6s
|-
|28
|18L 10s
|-
|29
|15L 14s
|-
|30
|12L 18s
|-
|31
|9L 22s
|-
|32
|6L 26s
|-
|33
|3L 30s
|-
|34
|[[34edo]]
|equal
|-
|35
|32L 3s
| rowspan="16" |3:2
|-
|36
|30L 6s
|-
|37
|28L 9s
|-
|38
|26L 12s
|-
|39
|24L 15s
|-
|40
|22L 18s
|-
|41
|20L 21s
|-
|42
|18L 24s
|-
|43
|16L 27s
|-
|44
|14L 30s
|-
|45
|12L 33s
|-
|46
|10L 36s
|-
|47
|8L 39s
|-
|48
|6L 42s
|-
|49
|4L 45s
|-
|50
|2L 48s
|-
|51
|[[51edo]]
|equal
|-
|52
|50L 2s
| rowspan="30" |2:1
|-
|53
|49L 4s
|-
|54
|48L 6s
|-
|55
|47L 8s
|-
|56
|46L 10s
|-
|57
|45L 12s
|-
|58
|44L 14s
|-
|59
|43L 16s
|-
|60
|42L 18s
|-
|61
|41L 20s
|-
|62
|40L 22s
|-
|63
|39L 24s
|-
|64
|38L 26s
|-
|65
|37L 28s
|-
|66
|36L 30s
|-
|67
|35L 32s
|-
|68
|34L 34s
|-
|69
|33L 36s
|-
|70
|32L 38s
|-
|71
|31L 40s
|-
|72
|30L 42s
|-
|73
|29L 44s
|-
|74
|28L 46s
|-
|75
|27L 48s
|-
|76
|26L 50s
|-
|77
|25L 52s
|-
|78
|24L 54s
|-
|79
|23L 56s
|-
|80
|22L 58s
|-
|81
|21L 60s
|}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Echidnic]]
[[Category:Echidnic]]

Latest revision as of 16:46, 18 February 2025

← 101edo 102edo 103edo →
Prime factorization 2 × 3 × 17
Step size 11.7647 ¢ 
Fifth 60\102 (705.882 ¢) (→ 10\17)
Semitones (A1:m2) 12:6 (141.2 ¢ : 70.59 ¢)
Dual sharp fifth 60\102 (705.882 ¢) (→ 10\17)
Dual flat fifth 59\102 (694.118 ¢)
Dual major 2nd 17\102 (200 ¢) (→ 1\6)
Consistency limit 5
Distinct consistency limit 5

102 equal divisions of the octave (abbreviated 102edo or 102ed2), also called 102-tone equal temperament (102tet) or 102 equal temperament (102et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 102 equal parts of about 11.8 ¢ each. Each step represents a frequency ratio of 21/102, or the 102nd root of 2.

Theory

102edo is enfactored in the 5-limit, where it tempers out the same commas (2048/2025, 15625/15552, 20000/19683) as 34edo. In the 7-limit it tempers out 686/675 and 1029/1024; in the 11-limit 385/384, 441/440 and 4000/3993; in the 13-limit 91/90 and 169/168; in the 17-limit 136/135 and 154/153; and in the 19-limit 133/132 and 190/189. It is the optimal patent val for 13-limit echidnic temperament, and the rank-5 temperament tempering out 91/90.

Odd harmonics

Approximation of odd harmonics in 102edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +3.93 +1.92 -4.12 -3.91 +1.62 -5.23 +5.85 +0.93 -3.40 -0.19 -4.74
Relative (%) +33.4 +16.3 -35.0 -33.2 +13.8 -44.5 +49.7 +7.9 -28.9 -1.6 -40.3
Steps
(reduced) 		162
(60) 		237
(33) 		286
(82) 		323
(17) 		353
(47) 		377
(71) 		399
(93) 		417
(9) 		433
(25) 		448
(40) 		461
(53)

Intervals

Steps Cents Approximate ratios Ups and downs notation
(Dual flat fifth 59\102) 		Ups and downs notation
(Dual sharp fifth 60\102)
0 0 1/1 D D
1 11.8 ^D, E♭♭♭ ^D, v5E♭
2 23.5 ^^D, ^E♭♭♭ ^^D, v4E♭
3 35.3 vvD♯, ^^E♭♭♭ ^3D, v3E♭
4 47.1 38/37, 39/38 vD♯, vvE♭♭ ^4D, vvE♭
5 58.8 30/29 D♯, vE♭♭ ^5D, vE♭
6 70.6 25/24 ^D♯, E♭♭ ^6D, E♭
7 82.4 21/20, 22/21, 43/41 ^^D♯, ^E♭♭ v5D♯, ^E♭
8 94.1 37/35 vvD𝄪, ^^E♭♭ v4D♯, ^^E♭
9 105.9 17/16 vD𝄪, vvE♭ v3D♯, ^3E♭
10 117.6 D𝄪, vE♭ vvD♯, ^4E♭
11 129.4 14/13, 41/38 ^D𝄪, E♭ vD♯, ^5E♭
12 141.2 38/35 ^^D𝄪, ^E♭ D♯, v6E
13 152.9 12/11, 35/32 vvD♯𝄪, ^^E♭ ^D♯, v5E
14 164.7 11/10 vD♯𝄪, vvE ^^D♯, v4E
15 176.5 31/28, 41/37 D♯𝄪, vE ^3D♯, v3E
16 188.2 39/35 E ^4D♯, vvE
17 200 ^E, F♭♭ ^5D♯, vE
18 211.8 26/23, 35/31, 43/38 ^^E, ^F♭♭ E
19 223.5 25/22, 33/29 vvE♯, ^^F♭♭ ^E, v5F
20 235.3 39/34 vE♯, vvF♭ ^^E, v4F
21 247.1 E♯, vF♭ ^3E, v3F
22 258.8 29/25, 43/37 ^E♯, F♭ ^4E, vvF
23 270.6 ^^E♯, ^F♭ ^5E, vF
24 282.4 20/17 vvE𝄪, ^^F♭ F
25 294.1 vE𝄪, vvF ^F, v5G♭
26 305.9 31/26, 37/31 E𝄪, vF ^^F, v4G♭
27 317.6 6/5 F ^3F, v3G♭
28 329.4 23/19, 29/24 ^F, G♭♭♭ ^4F, vvG♭
29 341.2 28/23, 39/32 ^^F, ^G♭♭♭ ^5F, vG♭
30 352.9 38/31 vvF♯, ^^G♭♭♭ ^6F, G♭
31 364.7 21/17 vF♯, vvG♭♭ v5F♯, ^G♭
32 376.5 36/29 F♯, vG♭♭ v4F♯, ^^G♭
33 388.2 5/4 ^F♯, G♭♭ v3F♯, ^3G♭
34 400 ^^F♯, ^G♭♭ vvF♯, ^4G♭
35 411.8 vvF𝄪, ^^G♭♭ vF♯, ^5G♭
36 423.5 vF𝄪, vvG♭ F♯, v6G
37 435.3 F𝄪, vG♭ ^F♯, v5G
38 447.1 22/17 ^F𝄪, G♭ ^^F♯, v4G
39 458.8 ^^F𝄪, ^G♭ ^3F♯, v3G
40 470.6 21/16 vvF♯𝄪, ^^G♭ ^4F♯, vvG
41 482.4 33/25, 37/28, 41/31 vF♯𝄪, vvG ^5F♯, vG
42 494.1 F♯𝄪, vG G
43 505.9 G ^G, v5A♭
44 517.6 31/23 ^G, A♭♭♭ ^^G, v4A♭
45 529.4 19/14 ^^G, ^A♭♭♭ ^3G, v3A♭
46 541.2 26/19 vvG♯, ^^A♭♭♭ ^4G, vvA♭
47 552.9 11/8 vG♯, vvA♭♭ ^5G, vA♭
48 564.7 43/31 G♯, vA♭♭ ^6G, A♭
49 576.5 ^G♯, A♭♭ v5G♯, ^A♭
50 588.2 ^^G♯, ^A♭♭ v4G♯, ^^A♭
51 600 vvG𝄪, ^^A♭♭ v3G♯, ^3A♭
52 611.8 37/26 vG𝄪, vvA♭ vvG♯, ^4A♭
53 623.5 G𝄪, vA♭ vG♯, ^5A♭
54 635.3 ^G𝄪, A♭ G♯, v6A
55 647.1 16/11 ^^G𝄪, ^A♭ ^G♯, v5A
56 658.8 19/13, 41/28 vvG♯𝄪, ^^A♭ ^^G♯, v4A
57 670.6 28/19 vG♯𝄪, vvA ^3G♯, v3A
58 682.4 G♯𝄪, vA ^4G♯, vvA
59 694.1 A ^5G♯, vA
60 705.9 ^A, B♭♭♭ A
61 717.6 ^^A, ^B♭♭♭ ^A, v5B♭
62 729.4 32/21 vvA♯, ^^B♭♭♭ ^^A, v4B♭
63 741.2 43/28 vA♯, vvB♭♭ ^3A, v3B♭
64 752.9 17/11 A♯, vB♭♭ ^4A, vvB♭
65 764.7 ^A♯, B♭♭ ^5A, vB♭
66 776.5 ^^A♯, ^B♭♭ ^6A, B♭
67 788.2 41/26 vvA𝄪, ^^B♭♭ v5A♯, ^B♭
68 800 vA𝄪, vvB♭ v4A♯, ^^B♭
69 811.8 8/5 A𝄪, vB♭ v3A♯, ^3B♭
70 823.5 29/18, 37/23 ^A𝄪, B♭ vvA♯, ^4B♭
71 835.3 34/21 ^^A𝄪, ^B♭ vA♯, ^5B♭
72 847.1 31/19 vvA♯𝄪, ^^B♭ A♯, v6B
73 858.8 23/14 vA♯𝄪, vvB ^A♯, v5B
74 870.6 38/23, 43/26 A♯𝄪, vB ^^A♯, v4B
75 882.4 5/3 B ^3A♯, v3B
76 894.1 ^B, C♭♭ ^4A♯, vvB
77 905.9 ^^B, ^C♭♭ ^5A♯, vB
78 917.6 17/10 vvB♯, ^^C♭♭ B
79 929.4 vB♯, vvC♭ ^B, v5C
80 941.2 B♯, vC♭ ^^B, v4C
81 952.9 ^B♯, C♭ ^3B, v3C
82 964.7 ^^B♯, ^C♭ ^4B, vvC
83 976.5 44/25 vvB𝄪, ^^C♭ ^5B, vC
84 988.2 23/13 vB𝄪, vvC C
85 1000 41/23 B𝄪, vC ^C, v5D♭
86 1011.8 C ^^C, v4D♭
87 1023.5 ^C, D♭♭♭ ^3C, v3D♭
88 1035.3 20/11 ^^C, ^D♭♭♭ ^4C, vvD♭
89 1047.1 11/6 vvC♯, ^^D♭♭♭ ^5C, vD♭
90 1058.8 35/19 vC♯, vvD♭♭ ^6C, D♭
91 1070.6 13/7 C♯, vD♭♭ v5C♯, ^D♭
92 1082.4 43/23 ^C♯, D♭♭ v4C♯, ^^D♭
93 1094.1 32/17 ^^C♯, ^D♭♭ v3C♯, ^3D♭
94 1105.9 vvC𝄪, ^^D♭♭ vvC♯, ^4D♭
95 1117.6 21/11, 40/21 vC𝄪, vvD♭ vC♯, ^5D♭
96 1129.4 C𝄪, vD♭ C♯, v6D
97 1141.2 29/15 ^C𝄪, D♭ ^C♯, v5D
98 1152.9 37/19 ^^C𝄪, ^D♭ ^^C♯, v4D
99 1164.7 vvC♯𝄪, ^^D♭ ^3C♯, v3D
100 1176.5 vC♯𝄪, vvD ^4C♯, vvD
101 1188.2 C♯𝄪, vD ^5C♯, vD
102 1200 2/1 D D
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