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174edo is closely related to [[87edo]], but the [[patent val]]s differ on the mapping for [[17/1|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. | 174edo is closely related to [[87edo]], but the [[patent val]]s differ on the mapping for [[17/1|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 | 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 === | === Odd harmonics === | ||
Latest revision as of 18:32, 21 January 2026
| ← 173edo | 174edo | 175edo → |
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
| 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 |