173ed49
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Prime factorization
173 (prime)
Step size
38.946¢
Octave
31\173ed49 (1207.32¢)
Twelfth
49\173ed49 (1908.35¢)
Consistency limit
6
Distinct consistency limit
6
| ← 172ed49 | 173ed49 | 174ed49 → |
Division of the just 49/1 interval into 173 equal parts yields a remarkable non-octave-equivalent equally tempered scale with step width very close to 31edo, yet it is much more suited to melodies and harmonies spanning more than one octave. The octaves in this scale are stretched by 7.3 cents, and the step width is about 0.23 cents wider than in 31edo (which has 174 equal steps in its tempered version of 49:1).
This scale and 18edf perform almost identically except over very large distances.
Out of all the harmonics between 1 and 49 and other low-harmonic entropy intervals within this range, this tuning matches the overwhelming majority with tolerable accuracy. The only harmonics that aren't matched well are 27, 33, and 37. This scale also has very good perfect fifths (within a cent of just intonation), although its fourths are not as good.
The 49:1 interval, 6737.6518 cents, could be called a "wide fortieth" since it consists of five octaves plus a 49:32 "wide fifth". This is more than half the average hearing range of a human, and thus actual instruments would find it seldom necessary to cover much more than this range. Extending too far beyond this range offers diminishing returns anyway, since harmonics above the 51st become very poorly matched compared to those below. Thus, this temperament is best used over a finite range (173 or perhaps 175 steps at the maximum).
The intervals within this scale are:
| Interval | Width in steps | Width in cents | Approximations (error) |
| Diesis | 1 | 38.946 | 49:48
45:44 |
| Chromatic semitone | 2 | 77.891 | 25:24
22:21 |
| Diatonic semitone, secor | 3 | 116.838 | 16:15
15:14 |
| Neutral second | 4 | 155.784 | 12:11 |
| Whole tone | 5 | 194.730 | 10:9
9:8 |
| Septimal whole tone | 6 | 233.676 | 8:7 |
| Septimal minor third | 7 | 272.622 | 7:6 |
| Minor third | 8 | 311.568 | 6:5 |
| Neutral third | 9 | 350.514 | 11:9 |
| Major third | 10 | 389.460 | 5:4 |
| Septimal major third | 11 | 428.406 | 9:7 |
| Subfourth | 12 | 467.351 | 21:16 |
| Perfect fourth | 13 | 506.298 | 4:3 |
| Superfourth | 14 | 545.243 | 11:8
15:11 |
| Lesser septimal tritone | 15 | 584.189 | 7:5 |
| Greater tritone | 16 | 623.135 | 10:7
13:9 |
| Subfifth | 17 | 662.081 | 22:15 |
| Perfect fifth | 18 | 701.027 | 3:2 |
| Superfifth | 19 | 739.973 | 20:13 |
| Undecimal minor sixth | 20 | 778.919 | 11:7 |
| Minor sixth, golden ratio | 21 | 817.865 | 8:5
φ:1 |
| Undecimal neutral sixth | 22 | 856.811 | 18:11 |
| Major sixth | 23 | 895.757 | 5:3 |
| Septimal major sixth | 24 | 934.703 | 12:7 |
| Septimal minor seventh | 25 | 973.649 | 7:4 |
| Minor seventh | 26 | 1012.595 | 9:5 |
| Neutral seventh | 27 | 1051.541 | 11:6 |
| Major seventh | 28 | 1090.487 | 15:8 |
| Supermajor seventh | 29 | 1129.433 | |
| Diminished octave | 30 | 1168.379 | |
| (Stretched) octave | 31 | 1207.325 | 2:1 |
| Augmented octave | 32 | 1246.271 | |
| Enneadecimal minor ninth | 33 | 1285.217 | 19:10 |
| Minor ninth | 34 | 1324.163 | 15:7 |
| Neutral ninth | 35 | 1363.109 | 11:5 |
| Major ninth | 36 | 1402.055 | 9:4 |
| Supermajor ninth | 37 | 1441.001 | |
| Septimal minor tenth | 38 | 1479.947 | 7:3 |
| Minor tenth | 39 | 1518.893 | 12:5 |
| Neutral tenth | 40 | 1557.839 | 22:9 |
| Major tenth | 41 | 1596.785 | 5:2 |
| Septimal major tenth | 42 | 1635.730 | 14:7 |
| Sub-eleventh | 43 | 1674.676 | 21:8 |
| Perfect eleventh | 44 | 1713.622 | 8:3 |
| Tridecimal super-eleventh | 45 | 1752.568 | 13:7 |
| Lesser eka-tritone* | 46 | 1791.514 | |
| Greater eka-tritone* | 47 | 1830.460 | 23:8 |
| Diminished twelfth | 48 | 1869.406 | |
| Tritave; perfect twelfth | 49 | 1908.352 | 3:1 |
| Augmented twelfth | 50 | 1947.298 | |
| Subminor thirteenth, pi | 51 | 1986.244 | 22:7
π:1 |
| Minor thirteenth | 52 | 2025.190 | 16:5
13:4 |
| Neutral-major thirteenth | 53 | 2064.136 | 10:3 (flat) |
| Major thirteenth | 54 | 2103.082 | 10:3 (sharp) |
| Supermajor thirteenth | 55 | 2142.028 | |
| Septimal minor fourteenth | 56 | 2180.974 | 7:2 |
| Minor fourteenth | 57 | 2219.920 | 18:5 |
| Neutral fourteenth | 58 | 2258.866 | 11:3 |
| Major fourteenth | 59 | 2297.819 | 15:4 |
| Supermajor fourteenth | 60 | 2336.758 | |
| Diminished double octave | 61 | 2375.704 | |
| Double octave (fifteenth) | 62 | 2414.650 | 4:1 |
| Augmented double octave | 63 | 2453.596 | 33:8 |
| Minor sixteenth | 64 | 2492.542 | 21:5
17:4 |
| Neutral sixteenth | 65 | 2531.488 | 13:3 |
| Neutral-major sixteenth | 66 | 2570.434 | 22:5 |
| Major sixteenth | 67 | 2609.380 | 9:2 |
| Supermajor sixteenth | 68 | 2648.326 | 14:3 (flat) |
| Minor seventeenth | 69 | 2687.272 | 14:3 (sharp)
19:4 |
| Neutral seventeenth | 70 | 2726.217 | |
| Major seventeenth, 5th harmonic (narrow) | 71 | 2765.153 | 5:1 |
| Major seventeenth, 5th harmonic (wide) | 72 | 2804.109 | 5:1 |
| Supermajor seventeenth | 73 | 2843.055 | 13:5 |
| Eighteenth (narrow) | 74 | 2882.001 | 21:4
16:3 (flat) |
| Eighteenth (wide) | 75 | 2920.947 | 16:3 (sharp)
27:5 |
| Augmented eighteenth | 76 | 2959.893 | 11:2 |
| Septendecimal dvi-tritone | 77 | 2998.839 | 17:3 |
| Greater dvi-tritone | 78 | 3037.785 | |
| Diminished nineteenth | 79 | 3076.731 | |
| Nineteenth; 6th harmonic | 80 | 3115.677 | 6:1 |
| Augmented nineteenth | 81 | 3154.623 | |
| Minor twentieth | 82 | 3183.569 | |
| Minor-neutral twentieth | 83 | 3232.515 | 13:2 |
| Major-neutral twentieth | 84 | 3271.461 | 20:3 |
| Major twentieth | 85 | 3310.407 | 27:4 |
| 7th harmonic (narrow) | 86 | 3349.353 | 7:1 |
| 7th harmonic (wide), subminor twenty-first | 87 | 3388.299 | 7:1 |
| Minor twenty-first | 88 | 3427.245 | 29:4 |
| Neutral twenty-first | 89 | 3466.191 | 22:3
15:2 (flat) |
| Major twenty-first | 90 | 3505.137 | 15:2 (sharp) |
| Supermajor twenty-first | 91 | 3544.083 | |
| Triple octave; twenty-second; 8th harmonic (narrow) | 92 | 3583.029 | 8:1 |
| Triple octave; twenty-second; 8th harmonic (wide) | 93 | 3621.975 | 8:1 |
| Subminor twenty-third | 94 | 3660.921 | 25:3 |
| Minor twenty-third | 95 | 3699.867 | 17:2 |
| Minor-neutral twenty-third | 96 | 3738.813 | 26:3 |
| Major-neutral twenty-third | 97 | 3777.759 | |
| Major twenty-third; 9th harmonic | 98 | 3816.704 | 9:1 |
| Subminor twenty-fourth | 99 | 3855.650 | |
| Minor twenty-fourth | 100 | 3894.597 | 19:2 |
| Minor-neutral twenty-fourth | 101 | 3933.543 | |
| Major-neutral twenty-fourth; 10th harmonic | 102 | 3972.489 | 10:1 |
| Major twenty-fourth | 103 | 4011.434 | |
| Supermajor twenty-fourth; diminished twenty-fifth | 104 | 4050.380 | 21:2 (flat) |
| Twenty-fifth | 105 | 4089.326 | 21:2 (sharp)
32:3 |
| Augmented twenty-fifth | 106 | 4128.272 | |
| 11th harmonic | 107 | 4167.218 | 11:1 |
| Tri-tritone | 108 | 4206.164 | 34:3 |
| Diminished twenty-sixth | 109 | 4245.110 | 23:2 |
| Twenty-sixth; 12th harmonic (flat) | 110 | 4284.056 | 12:1 |
| Twenty-sixth; 12th harmonic (sharp) | 111 | 4323.002 | 12:1 |
| Subminor twenty-seventh | 112 | 4361.948 | 25:2 |
| Minor twenty-seventh | 113 | 4400.894 | |
| 13th harmonic | 114 | 4439.840 | 13:1 |
| Major-neutral twenty-seventh | 115 | 4478.786 | |
| Major twenty-seventh | 116 | 4517:732 | 27:2 |
| 14th harmonic | 117 | 4556.678 | 14:1 |
| Minor twenty-eighth | 118 | 4595.624 | |
| Minor-neutral twenty-eighth | 119 | 4634.57 | 29:2 |
| 15th harmonic | 120 | 4673.516 | 15:1 |
| Major twenty-eighth | 121 | 4712.462 | |
| Diminished quadruple octave | 122 | 4751.408 | |
| Twenty-ninth; quadruple octave; 16th harmonic | 123 | 4790.354 | 16:1 |
| Augmented quadruple octave | 124 | 4829.300 | |
| Subminor thirtieth | 125 | 4868.246 | |
| Minor thirtieth; 17th harmonic | 126 | 4907.191 | 17:1 |
| Neutral thirtieth | 127 | 4946.137 | |
| Major thirtieth, 18th harmonic (narrow) | 128 | 4985.083 | 18:1 |
| Major thirtieth, 18th harmonic (wide) | 129 | 5024.029 | 18:1 |
| Subminor thirty-first | 130 | 5062.975 | 56:3 |
| 19th harmonic | 131 | 5101.921 | 19:1 |
| Neutral thirty-first | 132 | 5140.867 | 39:2 |
| 20th harmonic | 133 | 5179.813 | 20:1 |
| Major thirty-first | 134 | 5218.759 | 41:2 |
| 21st harmonic | 135 | 5257.705 | 21:1 |
| 136 | 5296.651 | 43:2 | |
| 22nd harmonic | 137 | 5335.597 | 22:1 |
| 138 | 5374.543 | 45:2 | |
| 23rd harmonic | 139 | 5413.489 | 23:1 |
| 140 | 5452.435 | 47:2 | |
| 24th harmonic | 141 | 5491.381 | 24:1 |
| 142 | 5530.327 | 49:2 | |
| 25th harmonic | 143 | 5569.273 | 25:1 |
| 144 | 5608.22 | 51:2 | |
| 26th harmonic | 145 | 5647.16 | 26:1 |
| flat 27th harmonic | 146 | 5686.11 | 27:1 (flat) |
| sharp 27th harmonic | 147 | 5725.06 | 27:1 (sharp) |
| 28th harmonic | 148 | 5764.00 | 28:1 |
| 149 | 5802.95 | ||
| 29th harmonic | 150 | 5841.89 | 29:1 |
| 30th harmonic | 151 | 5880.84 | 30:1 |
| 152 | 5919.79 | 61:2 | |
| 31st harmonic | 153 | 5958.73 | 31:1 |
| Quintuple octave, 32nd harmonic | 154 | 5997.68 | 32:1 |
| Flat 33rd harmonic | 155 | 6036.62 | 33:1 (flat) |
| Sharp 33rd harmonic | 156 | 6075.57 | 33:1 (sharp) |
| 34th harmonic | 157 | 6114.52 | 34:1 |
| 35th harmonic | 158 | 6153.46 | 35:1 |
| 36th harmonic | 159 | 6192.41 | 36:1 |
| Flat 37th harmonic | 160 | 6231.35 | 37:1 (flat) |
| Sharp 37th harmonic | 161 | 6270.30 | 37:1 (flat) |
| 38th harmonic | 162 | 6309.25 | 38:1 |
| 39th harmonic | 163 | 6348.19 | 39:1 |
| 40th harmonic | 164 | 6387.14 | 40:1 |
| 41st harmonic | 165 | 6426.08 | 41:1 |
| 42nd harmonic | 166 | 6465.08 | 42:1 |
| 43rd harmonic | 167 | 6503.98 | 43:1 |
| 44th harmonic | 168 | 6542.92 | 44:1 |
| 45th harmonic | 169 | 6581.87 | 45:1 |
| 46th harmonic | 170 | 6620.81 | 46:1 |
| 47th harmonic | 171 | 6659.76 | 47:1 |
| 48th harmonic | 172 | 6698.71 | 48:1 |
| 49th harmonic | 173 | 6737.65 | 49:1 ===(just)=== |
| 50th harmonic | 174 | 6776.60 | 50:1 |
| 51st harmonic | 175 | 6815.55 | 51:1 |
- An eka-tritone (named by analogy with the periodic table) is an octave plus a tritone. A dvi-tritone is two octaves plus a tritone.
- The 5:1, 7:1, 8:1, 12:1, and 18:1 intervals are split, yet all have a relatively high tolerance for mistuning, so in each case, both approximations are reasonable. When designing instruments to play in this tuning, it might be a good idea to dampen the 5th, 7th, 8th, and 12th harmonics while detuning the others slightly toward their corresponding scale degrees.