117edo: Difference between revisions

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Using the 117bd val, it tempers out 15625/15552 ([[15625/15552|kleisma]]) and {{monzo| 34 -17 -3 }} in the 5-limit; [[245/243]], [[3136/3125]], and 51200/50421 in the 7-limit; 176/175, 1232/1215, [[1375/1372]], and 2560/2541 in the 11-limit; [[169/168]], [[364/363]], 640/637, [[832/825]], and 3200/3159 in the 13-limit.  
Using the 117bd val, it tempers out 15625/15552 ([[15625/15552|kleisma]]) and {{monzo| 34 -17 -3 }} in the 5-limit; [[245/243]], [[3136/3125]], and 51200/50421 in the 7-limit; 176/175, 1232/1215, [[1375/1372]], and 2560/2541 in the 11-limit; [[169/168]], [[364/363]], 640/637, [[832/825]], and 3200/3159 in the 13-limit.  


=== Odd harmonics ===
== Odd harmonics ==
{{Harmonics in equal|117}}
{{Harmonics in equal|117}}


=== Subsets and supersets ===
== Subsets and supersets ==
Since 117 factors into {{factorization|117}}, 117edo has subset edos {{EDOs| 3, 9, 13, and 39 }}. [[234edo]], which doubles it, provides a correction for the approximation to harmonic 3.
Since 117 factors into {{factorization|117}}, 117edo has subset edos {{EDOs| 3, 9, 13, and 39 }}. [[234edo]], which doubles it, provides a correction for the approximation to harmonic 3.
== Intervals ==
{{Interval table}}

Revision as of 12:45, 2 October 2024

← 116edo 117edo 118edo →
Prime factorization 32 × 13
Step size 10.2564 ¢ 
Fifth 68\117 (697.436 ¢)
Semitones (A1:m2) 8:11 (82.05 ¢ : 112.8 ¢)
Dual sharp fifth 69\117 (707.692 ¢) (→ 23\39)
Dual flat fifth 68\117 (697.436 ¢)
Dual major 2nd 20\117 (205.128 ¢)
Consistency limit 3
Distinct consistency limit 3

Template:EDO intro

117edo is inconsistent to the 5-odd-limit and higher odd limits, with four mappings possible for the 11-limit: ⟨117 185 272 328 405] (patent val), ⟨117 186 272 329 405] (117bd), ⟨117 185 271 328 404] (117ce), and ⟨117 185 272 329 405] (117d).

Using the patent val, it tempers out 81/80 (syntonic comma) and [69 -1 -29⟩ in the 5-limit; 6144/6125, 31104/30625, and 403368/390625 in the 7-limit, supporting the 7-limit mohajira temperament; 540/539, 1344/1331, 1617/1600, and 3168/3125 in the 11-limit, supporting the rank-3 terpsichore temperament; 144/143, 196/195, 364/363, 729/715, and 3146/3125 in the 13-limit.

Using the 117d val, it tempers out 126/125, 225/224, and [29 3 0 -12⟩ in the 7-limit; 99/98, 176/175, 441/440, and 12582912/12400927 in the 11-limit; 144/143, 640/637, 648/637, 1001/1000, and 43940/43923 in the 13-limit, supporting the 13-limit grosstone temperament.

Using the 117ce val, it tempers out 3125/3072 (magic comma) and [-31 24 -3⟩ in the 5-limit; 2401/2400, 3645/3584, and 4375/4374 in the 7-limit; 243/242, 441/440, and 1815/1792 in the 11-limit; 105/104, 275/273, 1287/1280, and 2025/2002 in the 13-limit.

Using the 117bd val, it tempers out 15625/15552 (kleisma) and [34 -17 -3⟩ in the 5-limit; 245/243, 3136/3125, and 51200/50421 in the 7-limit; 176/175, 1232/1215, 1375/1372, and 2560/2541 in the 11-limit; 169/168, 364/363, 640/637, 832/825, and 3200/3159 in the 13-limit.

Odd harmonics

Approximation of odd harmonics in 117edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -4.52 +3.43 -4.72 +1.22 +2.53 +0.50 -1.09 -2.39 -0.08 +1.01 -2.63
Relative (%) -44.1 +33.4 -46.1 +11.9 +24.7 +4.9 -10.6 -23.3 -0.8 +9.9 -25.7
Steps
(reduced) 		185
(68) 		272
(38) 		328
(94) 		371
(20) 		405
(54) 		433
(82) 		457
(106) 		478
(10) 		497
(29) 		514
(46) 		529
(61)

Subsets and supersets

Since 117 factors into 32 × 13, 117edo has subset edos 3, 9, 13, and 39. 234edo, which doubles it, provides a correction for the approximation to harmonic 3.

Intervals

Steps Cents Approximate ratios Ups and downs notation
(Dual flat fifth 68\117) 		Ups and downs notation
(Dual sharp fifth 69\117)
0 0 1/1 D D
1 10.3 ^D, vvE♭♭ ^D, v5E♭
2 20.5 ^^D, vE♭♭ ^^D, v4E♭
3 30.8 ^3D, E♭♭ ^3D, v3E♭
4 41 41/40, 44/43 ^4D, ^E♭♭ ^4D, vvE♭
5 51.3 33/32, 34/33, 35/34 v3D♯, ^^E♭♭ ^5D, vE♭
6 61.5 29/28 vvD♯, ^3E♭♭ ^6D, E♭
7 71.8 24/23 vD♯, v4E♭ ^7D, ^E♭
8 82.1 43/41 D♯, v3E♭ v7D♯, ^^E♭
9 92.3 ^D♯, vvE♭ v6D♯, ^3E♭
10 102.6 35/33 ^^D♯, vE♭ v5D♯, ^4E♭
11 112.8 16/15, 47/44 ^3D♯, E♭ v4D♯, ^5E♭
12 123.1 44/41 ^4D♯, ^E♭ v3D♯, ^6E♭
13 133.3 40/37, 41/38 v3D𝄪, ^^E♭ vvD♯, ^7E♭
14 143.6 38/35 vvD𝄪, ^3E♭ vD♯, v7E
15 153.8 35/32, 47/43 vD𝄪, v4E D♯, v6E
16 164.1 11/10 D𝄪, v3E ^D♯, v5E
17 174.4 ^D𝄪, vvE ^^D♯, v4E
18 184.6 ^^D𝄪, vE ^3D♯, v3E
19 194.9 E ^4D♯, vvE
20 205.1 ^E, vvF♭ ^5D♯, vE
21 215.4 17/15, 43/38 ^^E, vF♭ E
22 225.6 33/29 ^3E, F♭ ^E, v5F
23 235.9 39/34, 47/41 ^4E, ^F♭ ^^E, v4F
24 246.2 15/13, 38/33 v3E♯, ^^F♭ ^3E, v3F
25 256.4 vvE♯, ^3F♭ ^4E, vvF
26 266.7 7/6 vE♯, v4F ^5E, vF
27 276.9 34/29 E♯, v3F F
28 287.2 13/11, 46/39 ^E♯, vvF ^F, v5G♭
29 297.4 19/16 ^^E♯, vF ^^F, v4G♭
30 307.7 37/31 F ^3F, v3G♭
31 317.9 ^F, vvG♭♭ ^4F, vvG♭
32 328.2 29/24 ^^F, vG♭♭ ^5F, vG♭
33 338.5 ^3F, G♭♭ ^6F, G♭
34 348.7 ^4F, ^G♭♭ ^7F, ^G♭
35 359 16/13 v3F♯, ^^G♭♭ v7F♯, ^^G♭
36 369.2 47/38 vvF♯, ^3G♭♭ v6F♯, ^3G♭
37 379.5 vF♯, v4G♭ v5F♯, ^4G♭
38 389.7 F♯, v3G♭ v4F♯, ^5G♭
39 400 29/23 ^F♯, vvG♭ v3F♯, ^6G♭
40 410.3 19/15 ^^F♯, vG♭ vvF♯, ^7G♭
41 420.5 ^3F♯, G♭ vF♯, v7G
42 430.8 41/32 ^4F♯, ^G♭ F♯, v6G
43 441 40/31 v3F𝄪, ^^G♭ ^F♯, v5G
44 451.3 vvF𝄪, ^3G♭ ^^F♯, v4G
45 461.5 30/23 vF𝄪, v4G ^3F♯, v3G
46 471.8 46/35 F𝄪, v3G ^4F♯, vvG
47 482.1 41/31 ^F𝄪, vvG ^5F♯, vG
48 492.3 ^^F𝄪, vG G
49 502.6 G ^G, v5A♭
50 512.8 35/26, 39/29, 43/32 ^G, vvA♭♭ ^^G, v4A♭
51 523.1 23/17 ^^G, vA♭♭ ^3G, v3A♭
52 533.3 49/36 ^3G, A♭♭ ^4G, vvA♭
53 543.6 26/19 ^4G, ^A♭♭ ^5G, vA♭
54 553.8 v3G♯, ^^A♭♭ ^6G, A♭
55 564.1 vvG♯, ^3A♭♭ ^7G, ^A♭
56 574.4 39/28, 46/33 vG♯, v4A♭ v7G♯, ^^A♭
57 584.6 G♯, v3A♭ v6G♯, ^3A♭
58 594.9 31/22 ^G♯, vvA♭ v5G♯, ^4A♭
59 605.1 44/31 ^^G♯, vA♭ v4G♯, ^5A♭
60 615.4 ^3G♯, A♭ v3G♯, ^6A♭
61 625.6 33/23 ^4G♯, ^A♭ vvG♯, ^7A♭
62 635.9 v3G𝄪, ^^A♭ vG♯, v7A
63 646.2 vvG𝄪, ^3A♭ G♯, v6A
64 656.4 19/13 vG𝄪, v4A ^G♯, v5A
65 666.7 47/32 G𝄪, v3A ^^G♯, v4A
66 676.9 34/23, 37/25 ^G𝄪, vvA ^3G♯, v3A
67 687.2 ^^G𝄪, vA ^4G♯, vvA
68 697.4 A ^5G♯, vA
69 707.7 ^A, vvB♭♭ A
70 717.9 ^^A, vB♭♭ ^A, v5B♭
71 728.2 35/23 ^3A, B♭♭ ^^A, v4B♭
72 738.5 23/15 ^4A, ^B♭♭ ^3A, v3B♭
73 748.7 v3A♯, ^^B♭♭ ^4A, vvB♭
74 759 31/20, 45/29 vvA♯, ^3B♭♭ ^5A, vB♭
75 769.2 vA♯, v4B♭ ^6A, B♭
76 779.5 A♯, v3B♭ ^7A, ^B♭
77 789.7 30/19, 41/26 ^A♯, vvB♭ v7A♯, ^^B♭
78 800 46/29 ^^A♯, vB♭ v6A♯, ^3B♭
79 810.3 ^3A♯, B♭ v5A♯, ^4B♭
80 820.5 45/28 ^4A♯, ^B♭ v4A♯, ^5B♭
81 830.8 v3A𝄪, ^^B♭ v3A♯, ^6B♭
82 841 13/8 vvA𝄪, ^3B♭ vvA♯, ^7B♭
83 851.3 vA𝄪, v4B vA♯, v7B
84 861.5 A𝄪, v3B A♯, v6B
85 871.8 43/26, 48/29 ^A𝄪, vvB ^A♯, v5B
86 882.1 ^^A𝄪, vB ^^A♯, v4B
87 892.3 B ^3A♯, v3B
88 902.6 32/19 ^B, vvC♭ ^4A♯, vvB
89 912.8 22/13, 39/23 ^^B, vC♭ ^5A♯, vB
90 923.1 29/17 ^3B, C♭ B
91 933.3 12/7 ^4B, ^C♭ ^B, v5C
92 943.6 v3B♯, ^^C♭ ^^B, v4C
93 953.8 26/15, 33/19 vvB♯, ^3C♭ ^3B, v3C
94 964.1 vB♯, v4C ^4B, vvC
95 974.4 B♯, v3C ^5B, vC
96 984.6 30/17 ^B♯, vvC C
97 994.9 ^^B♯, vC ^C, v5D♭
98 1005.1 C ^^C, v4D♭
99 1015.4 ^C, vvD♭♭ ^3C, v3D♭
100 1025.6 47/26 ^^C, vD♭♭ ^4C, vvD♭
101 1035.9 20/11 ^3C, D♭♭ ^5C, vD♭
102 1046.2 ^4C, ^D♭♭ ^6C, D♭
103 1056.4 35/19 v3C♯, ^^D♭♭ ^7C, ^D♭
104 1066.7 37/20 vvC♯, ^3D♭♭ v7C♯, ^^D♭
105 1076.9 41/22 vC♯, v4D♭ v6C♯, ^3D♭
106 1087.2 15/8 C♯, v3D♭ v5C♯, ^4D♭
107 1097.4 ^C♯, vvD♭ v4C♯, ^5D♭
108 1107.7 ^^C♯, vD♭ v3C♯, ^6D♭
109 1117.9 ^3C♯, D♭ vvC♯, ^7D♭
110 1128.2 23/12 ^4C♯, ^D♭ vC♯, v7D
111 1138.5 v3C𝄪, ^^D♭ C♯, v6D
112 1148.7 33/17 vvC𝄪, ^3D♭ ^C♯, v5D
113 1159 43/22 vC𝄪, v4D ^^C♯, v4D
114 1169.2 C𝄪, v3D ^3C♯, v3D
115 1179.5 ^C𝄪, vvD ^4C♯, vvD
116 1189.7 ^^C𝄪, vD ^5C♯, vD
117 1200 2/1 D D
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