174edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 173edo 174edo 175edo →
Prime factorization 2 × 3 × 29
Step size 6.89655 ¢ 
Fifth 102\174 (703.448 ¢) (→ 17\29)
Semitones (A1:m2) 18:12 (124.1 ¢ : 82.76 ¢)
Consistency limit 5
Distinct consistency limit 5

174 equal divisions of the octave (abbreviated 174edo or 174ed2), also called 174-tone equal temperament (174tet) or 174 equal temperament (174et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 174 equal parts of about 6.9 ¢ each. Each step represents a frequency ratio of 21/174, or the 174th root of 2.

Theory

174edo is closely related to 87edo, but the patent vals differ on the mapping for 17 and some higher primes. It is contorted in the 13-limit, tempering out 196/195, 245/243, 352/351, 364/363, and 625/624. Using the patent val, it tempers out 289/288 in the 17-limit; 361/360, 476/475, and 665/663 in the 19-limit; 391/390, 392/391, 460/459, 529/528, and 760/759 in the 23-limit; 1309/1305, 1450/1449, and 4147/4140 in the 29-limit; 496/495 and 1365/1364 in the 31-limit.

The 174b val flat fifth is a meantone fifth very close to the quarter-comma meantone fifth, being only 0.027 cents flat of it.

Odd harmonics

Approximation of prime harmonics in 174edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +1.49 -0.11 -3.31 +0.41 +0.85 -1.51 -0.96 -0.69 -1.99 -0.21
Relative (%) +0.0 +21.7 -1.5 -48.0 +5.9 +12.3 -21.9 -13.9 -10.0 -28.9 -3.0
Steps
(reduced) 		174
(0) 		276
(102) 		404
(56) 		488
(140) 		602
(80) 		644
(122) 		711
(15) 		739
(43) 		787
(91) 		845
(149) 		862
(166)

Subsets and supersets

Since 174 factors into primes as 2 × 3 × 29, 174edo has subset edos 2, 3, 6, 29, 58, and 87.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 6.9 ^D, ^7E♭♭
2 13.79 ^^D, ^8E♭♭
3 20.69 ^3D, v9E♭
4 27.59 63/62 ^4D, v8E♭
5 34.48 51/50, 52/51 ^5D, v7E♭
6 41.38 42/41, 43/42 ^6D, v6E♭
7 48.28 ^7D, v5E♭
8 55.17 32/31 ^8D, v4E♭
9 62.07 57/55 ^9D, v3E♭
10 68.97 26/25 v8D♯, vvE♭
11 75.86 23/22 v7D♯, vE♭
12 82.76 43/41 v6D♯, E♭
13 89.66 20/19 v5D♯, ^E♭
14 96.55 37/35, 55/52 v4D♯, ^^E♭
15 103.45 v3D♯, ^3E♭
16 110.34 vvD♯, ^4E♭
17 117.24 46/43 vD♯, ^5E♭
18 124.14 43/40 D♯, ^6E♭
19 131.03 41/38, 55/51 ^D♯, ^7E♭
20 137.93 13/12 ^^D♯, ^8E♭
21 144.83 25/23, 62/57 ^3D♯, v9E
22 151.72 12/11 ^4D♯, v8E
23 158.62 23/21, 34/31, 57/52 ^5D♯, v7E
24 165.52 11/10 ^6D♯, v6E
25 172.41 21/19 ^7D♯, v5E
26 179.31 51/46 ^8D♯, v4E
27 186.21 ^9D♯, v3E
28 193.1 19/17 v8D𝄪, vvE
29 200 46/41 v7D𝄪, vE
30 206.9 62/55 E
31 213.79 43/38 ^E, ^7F♭
32 220.69 25/22, 42/37 ^^E, ^8F♭
33 227.59 57/50 ^3E, v9F
34 234.48 55/48, 63/55 ^4E, v8F
35 241.38 23/20 ^5E, v7F
36 248.28 15/13 ^6E, v6F
37 255.17 51/44 ^7E, v5F
38 262.07 50/43 ^8E, v4F
39 268.97 ^9E, v3F
40 275.86 34/29 v8E♯, vvF
41 282.76 v7E♯, vF
42 289.66 13/11 F
43 296.55 19/16, 51/43 ^F, ^7G♭♭
44 303.45 31/26, 56/47 ^^F, ^8G♭♭
45 310.34 55/46 ^3F, v9G♭
46 317.24 ^4F, v8G♭
47 324.14 41/34 ^5F, v7G♭
48 331.03 23/19, 63/52 ^6F, v6G♭
49 337.93 62/51 ^7F, v5G♭
50 344.83 50/41 ^8F, v4G♭
51 351.72 38/31 ^9F, v3G♭
52 358.62 16/13 v8F♯, vvG♭
53 365.52 21/17 v7F♯, vG♭
54 372.41 31/25, 57/46 v6F♯, G♭
55 379.31 v5F♯, ^G♭
56 386.21 5/4 v4F♯, ^^G♭
57 393.1 v3F♯, ^3G♭
58 400 63/50 vvF♯, ^4G♭
59 406.9 43/34 vF♯, ^5G♭
60 413.79 33/26, 47/37 F♯, ^6G♭
61 420.69 37/29, 51/40 ^F♯, ^7G♭
62 427.59 32/25 ^^F♯, ^8G♭
63 434.48 ^3F♯, v9G
64 441.38 40/31 ^4F♯, v8G
65 448.28 57/44 ^5F♯, v7G
66 455.17 13/10 ^6F♯, v6G
67 462.07 ^7F♯, v5G
68 468.97 38/29 ^8F♯, v4G
69 475.86 25/19 ^9F♯, v3G
70 482.76 37/28, 41/31 v8F𝄪, vvG
71 489.66 v7F𝄪, vG
72 496.55 G
73 503.45 ^G, ^7A♭♭
74 510.34 43/32, 47/35, 51/38 ^^G, ^8A♭♭
75 517.24 31/23, 58/43 ^3G, v9A♭
76 524.14 23/17 ^4G, v8A♭
77 531.03 34/25 ^5G, v7A♭
78 537.93 15/11 ^6G, v6A♭
79 544.83 63/46 ^7G, v5A♭
80 551.72 11/8 ^8G, v4A♭
81 558.62 29/21 ^9G, v3A♭
82 565.52 43/31 v8G♯, vvA♭
83 572.41 32/23 v7G♯, vA♭
84 579.31 v6G♯, A♭
85 586.21 v5G♯, ^A♭
86 593.1 31/22 v4G♯, ^^A♭
87 600 41/29, 58/41 v3G♯, ^3A♭
88 606.9 44/31 vvG♯, ^4A♭
89 613.79 57/40 vG♯, ^5A♭
90 620.69 63/44 G♯, ^6A♭
91 627.59 23/16 ^G♯, ^7A♭
92 634.48 62/43 ^^G♯, ^8A♭
93 641.38 42/29, 55/38 ^3G♯, v9A
94 648.28 16/11 ^4G♯, v8A
95 655.17 ^5G♯, v7A
96 662.07 22/15, 63/43 ^6G♯, v6A
97 668.97 25/17 ^7G♯, v5A
98 675.86 34/23 ^8G♯, v4A
99 682.76 43/29, 46/31 ^9G♯, v3A
100 689.66 v8G𝄪, vvA
101 696.55 v7G𝄪, vA
102 703.45 A
103 710.34 ^A, ^7B♭♭
104 717.24 56/37, 62/41 ^^A, ^8B♭♭
105 724.14 38/25 ^3A, v9B♭
106 731.03 29/19 ^4A, v8B♭
107 737.93 ^5A, v7B♭
108 744.83 20/13, 63/41 ^6A, v6B♭
109 751.72 ^7A, v5B♭
110 758.62 31/20 ^8A, v4B♭
111 765.52 ^9A, v3B♭
112 772.41 25/16 v8A♯, vvB♭
113 779.31 58/37 v7A♯, vB♭
114 786.21 52/33, 63/40 v6A♯, B♭
115 793.1 v5A♯, ^B♭
116 800 v4A♯, ^^B♭
117 806.9 51/32 v3A♯, ^3B♭
118 813.79 8/5 vvA♯, ^4B♭
119 820.69 vA♯, ^5B♭
120 827.59 50/31 A♯, ^6B♭
121 834.48 34/21 ^A♯, ^7B♭
122 841.38 13/8 ^^A♯, ^8B♭
123 848.28 31/19 ^3A♯, v9B
124 855.17 41/25 ^4A♯, v8B
125 862.07 51/31 ^5A♯, v7B
126 868.97 38/23 ^6A♯, v6B
127 875.86 63/38 ^7A♯, v5B
128 882.76 ^8A♯, v4B
129 889.66 ^9A♯, v3B
130 896.55 47/28, 52/31 v8A𝄪, vvB
131 903.45 32/19 v7A𝄪, vB
132 910.34 22/13 B
133 917.24 ^B, ^7C♭
134 924.14 29/17 ^^B, ^8C♭
135 931.03 ^3B, v9C
136 937.93 43/25, 55/32 ^4B, v8C
137 944.83 ^5B, v7C
138 951.72 26/15 ^6B, v6C
139 958.62 40/23 ^7B, v5C
140 965.52 ^8B, v4C
141 972.41 ^9B, v3C
142 979.31 37/21, 44/25 v8B♯, vvC
143 986.21 v7B♯, vC
144 993.1 55/31 C
145 1000 41/23, 57/32 ^C, ^7D♭♭
146 1006.9 34/19 ^^C, ^8D♭♭
147 1013.79 ^3C, v9D♭
148 1020.69 ^4C, v8D♭
149 1027.59 38/21 ^5C, v7D♭
150 1034.48 20/11 ^6C, v6D♭
151 1041.38 31/17, 42/23 ^7C, v5D♭
152 1048.28 11/6 ^8C, v4D♭
153 1055.17 46/25, 57/31 ^9C, v3D♭
154 1062.07 24/13 v8C♯, vvD♭
155 1068.97 63/34 v7C♯, vD♭
156 1075.86 v6C♯, D♭
157 1082.76 43/23 v5C♯, ^D♭
158 1089.66 v4C♯, ^^D♭
159 1096.55 v3C♯, ^3D♭
160 1103.45 vvC♯, ^4D♭
161 1110.34 19/10 vC♯, ^5D♭
162 1117.24 C♯, ^6D♭
163 1124.14 44/23 ^C♯, ^7D♭
164 1131.03 25/13 ^^C♯, ^8D♭
165 1137.93 ^3C♯, v9D
166 1144.83 31/16 ^4C♯, v8D
167 1151.72 ^5C♯, v7D
168 1158.62 41/21 ^6C♯, v6D
169 1165.52 51/26 ^7C♯, v5D
170 1172.41 63/32 ^8C♯, v4D
171 1179.31 ^9C♯, v3D
172 1186.21 v8C𝄪, vvD
173 1193.1 v7C𝄪, vD
174 1200 2/1 D
Retrieved from "https://en.xen.wiki/index.php?title=174edo&oldid=222115"