173ed49

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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
    • 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.