Harmonic entropy of just intervals

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This is a procedurally generated table of several hundred just intonation intervals between 1/1 and 8/1: roughly evenly spaced with about one interval every 4-5 cents.

The table shows how wide each interval is in cents, as well as showing its harmonic entropy.

For a more general-purpose list of intervals, see Gallery of just intervals.

Methodology

This table was made with Scale Workshop. It was made by starting with 810ed8 (270edo but treating 8/1 as the period), then using "convert interval values>fraction" with a tolerance of 4.4 cents to convert all the intervals into the nearest simple just interval (according to Scale Workshop's default settings). Then Scale Workshop's built-in harmonic entropy table was used to get the cents and the harmonic entropy (again, everything left on its default settings).

Table of intervals

1/1 to 3/2

Harmonic entropy of just intervals
Ratio Size (cents) Entropy
(%, Rényi order 1.0) 		Entropy
(%, Rényi order 7.0)		 Comments
1/1 0.000 0.000
390/389 4.445 3.760
195/194 8.901 14.289
130/129 13.369 29.536
97/96 17.940 47.113
78/77 22.339 63.285
65/64 26.841 77.192
56/55 31.194 87.233
49/48 35.697 94.083
43/42 40.737 98.235
39/38 44.970 99.690
35/34 50.184 99.958
32/31 54.964 99.447
30/29 58.692 98.855
28/27 62.961 98.131
26/25 67.900 97.339
24/23 73.681 96.526
23/22 76.956 96.119
22/21 80.537 95.722
21/20 84.467 95.323
20/19 88.801 94.928
19/18 93.603 94.534
18/17 98.955 94.138
17/16 104.955 93.748
16/15 111.731 93.375
15/14 119.443 93.016
14/13 128.298 92.675
27/25 133.238 92.514
13/12 138.573 92.347
12/11 150.637 92.045
23/21 157.493 91.881
34/31 159.920 91.829
11/10 165.004 91.727
21/19 173.268 91.551
41/37 177.718 91.462
10/9 182.404 91.367
19/17 192.558 91.175
9/8 203.910 91.000
35/31 210.104 90.909
17/15 216.687 90.809
25/22 221.310 90.746
49/43 226.134 90.706
8/7 231.174 90.701
23/20 241.961 90.726
15/13 247.741 90.623
22/19 253.805 90.413
29/25 256.950 90.286
57/49 261.816 90.135
7/6 266.871 90.099
27/23 277.591 90.529
20/17 281.358 90.732
13/11 289.210 90.867
19/16 297.513 90.317
25/21 301.847 89.806
37/31 306.309 89.268
73/61 310.905 88.838
6/5 315.641 88.670
35/29 325.562 89.348
23/19 330.761 90.058
17/14 336.130 90.813
28/23 340.552 91.327
11/9 347.408 91.786
16/13 359.472 91.181
21/17 365.826 90.053
31/25 372.408 88.498
46/37 376.930 87.477
5/4 386.314 86.433
89/71 391.184 86.736
44/35 396.178 87.574
29/23 401.303 88.739
24/19 404.442 89.458
19/15 409.244 90.412
14/11 417.508 91.306
23/18 424.364 91.415
9/7 435.084 91.284
49/38 440.139 91.361
22/17 446.363 91.641
13/10 454.214 92.079
17/13 464.428 91.930
21/16 470.781 90.841
25/19 475.114 89.528
29/22 478.259 88.318
41/31 484.027 85.801
61/46 488.611 83.904
121/91 493.282 82.505
4/3 498.045 81.976
67/50 506.680 83.591
43/32 511.518 85.531
31/23 516.761 87.848
23/17 523.319 90.284
19/14 528.687 91.567
15/11 536.951 92.317
26/19 543.015 92.254
37/27 545.479 92.154
11/8 551.318 91.819
18/13 563.382 90.784
25/18 568.717 90.231
32/23 571.726 89.927
67/48 577.352 89.498
7/5 582.512 89.352
38/27 591.648 89.727
24/17 597.000 90.174
17/12 603.000 90.642
27/19 608.352 90.950
10/7 617.488 91.230
43/30 623.249 91.380
23/16 628.274 91.555
13/9 636.618 92.012
16/11 648.682 92.948
19/13 656.985 93.440
22/15 663.049 93.268
25/17 667.672 92.484
31/21 674.255 89.925
37/25 678.717 87.087
49/33 684.379 82.440
64/43 688.482 78.794
100/67 693.320 74.956
3/2 701.955 71.779

3/2 to 2/1

Harmonic entropy of just intervals
Ratio Size (cents) Entropy
(%, Rényi order 1.0) 		Entropy
(%, Rényi order 7.0)		 Comments
3/2 701.955 71.779
182/121 706.718 72.810
92/61 711.390 75.553
62/41 715.973 79.322
47/31 720.471 83.329
38/25 724.886 86.938
32/21 729.219 89.774
26/17 735.572 92.399
23/15 740.006 93.257
20/13 745.786 93.525
17/11 753.638 93.117
31/20 758.722 92.691
14/9 764.916 92.202
25/16 772.627 91.717
47/30 777.238 91.493
11/7 782.492 91.272
30/19 790.756 90.881
19/12 795.558 90.588
27/17 800.910 90.222
35/22 803.822 90.033
67/42 808.526 89.806
8/5 813.686 89.732
37/23 823.070 90.195
21/13 830.253 90.896
34/21 834.175 91.296
13/8 840.528 91.817
18/11 852.592 91.703
23/14 859.448 90.557
28/17 863.871 89.342
43/26 870.990 86.943
63/38 875.223 85.597
5/3 884.359 84.167
127/76 888.909 84.530
62/37 893.692 85.623
42/25 898.154 87.045
32/19 902.487 88.529
22/13 910.790 90.839
17/10 918.642 91.911
29/17 924.622 92.114
12/7 933.129 91.936
43/25 938.890 91.611
19/11 946.195 90.913
26/15 952.259 90.085
47/27 959.642 89.027
7/4 968.826 88.381
93/53 973.486 88.539
44/25 978.691 89.061
30/17 983.313 89.680
23/13 987.747 90.241
16/9 996.090 90.836
25/14 1003.802 90.734
34/19 1007.442 90.562
9/5 1017.596 90.182
74/41 1022.282 90.185
38/21 1026.732 90.292
20/11 1034.996 90.598
31/17 1040.080 90.750
53/29 1043.927 90.833
11/6 1049.363 90.918
24/13 1061.427 91.130
37/20 1065.030 91.211
13/7 1071.702 91.394
28/15 1080.557 91.657
15/8 1088.269 91.912
32/17 1095.045 92.173
17/9 1101.045 92.438
36/19 1106.397 92.704
19/10 1111.199 92.968
21/11 1119.463 93.512
23/12 1126.319 94.067
25/13 1132.100 94.649
27/14 1137.039 95.215
29/15 1141.309 95.749
31/16 1145.036 96.208
33/17 1148.318 96.553
37/19 1153.831 96.788
41/21 1158.282 96.316
47/24 1163.552 94.342
55/28 1168.806 90.065
63/32 1172.736 84.977
77/39 1177.661 76.268
95/48 1181.872 67.226
129/65 1186.631 56.465
193/97 1191.053 47.543
389/195 1195.555 41.234
2/1 1200.000 39.006

2/1 to 3/1

Harmonic entropy of just intervals
Ratio Size (cents) Entropy
(%, Rényi order 1.0) 		Entropy
(%, Rényi order 7.0)		 Comments
2/1 1200.000 39.006
389/194 1204.456 41.246
195/97 1208.901 47.467
129/64 1213.473 56.700
97/48 1217.940 66.810
77/38 1222.631 76.861
65/32 1226.841 84.341
55/27 1231.767 90.672
49/24 1235.697 93.888
43/21 1240.737 96.081
39/19 1244.970 96.733
35/17 1250.184 96.662
31/15 1256.767 95.974
29/14 1260.751 95.470
27/13 1265.337 94.913
25/12 1270.672 94.335
23/11 1276.956 93.767
21/10 1284.467 93.226
40/19 1288.801 92.971
19/9 1293.603 92.713
17/8 1304.955 92.212
32/15 1311.731 91.962
15/7 1319.443 91.713
43/20 1325.204 91.549
28/13 1328.298 91.468
54/25 1333.238 91.338
13/6 1338.573 91.201
24/11 1350.637 90.891
35/16 1355.140 90.798
68/31 1359.920 90.737
11/5 1365.004 90.741
31/14 1376.210 90.799
20/9 1382.404 90.645
29/13 1389.050 90.242
47/21 1394.726 89.813
9/4 1403.910 89.442
79/35 1409.397 89.622
43/19 1414.005 89.985
25/11 1421.310 90.755
41/18 1425.152 91.119
16/7 1431.174 91.447
23/10 1441.961 90.803
30/13 1447.741 89.775
37/16 1451.344 88.987
58/25 1456.950 87.767
121/52 1462.108 86.960
7/3 1466.871 86.699
61/26 1476.357 87.725
40/17 1481.358 88.788
33/14 1484.447 89.469
26/11 1489.210 90.374
19/8 1497.513 91.263
31/13 1504.508 91.358
67/28 1510.481 91.232
12/5 1515.641 91.162
29/12 1527.622 91.519
17/7 1536.130 92.120
39/16 1542.483 92.492
22/9 1547.408 92.527
27/11 1554.547 91.809
32/13 1559.472 90.533
37/15 1563.075 89.159
47/19 1567.994 86.769
62/25 1572.408 84.387
97/39 1577.413 81.915
5/2 1586.314 79.831
178/71 1591.184 80.481
93/37 1595.647 82.086
63/25 1600.109 84.302
43/17 1606.562 87.719
38/15 1609.244 88.973
28/11 1617.508 91.634
23/9 1624.364 92.434
18/7 1635.084 92.172
49/19 1640.139 91.840
31/12 1643.081 91.658
57/22 1648.150 91.431
13/5 1654.214 91.300
34/13 1664.428 91.237
21/8 1670.781 90.960
29/11 1678.259 90.106
37/14 1682.518 89.391
61/23 1688.611 88.342
117/44 1693.120 87.764
8/3 1698.045 87.525
67/25 1706.680 88.237
43/16 1711.518 89.069
27/10 1719.551 90.435
19/7 1728.687 91.142
49/18 1733.742 91.064
30/11 1736.951 90.897
41/15 1740.794 90.654
85/31 1746.234 90.327
11/4 1751.318 90.168
47/17 1760.551 90.312
25/9 1768.717 90.677
39/14 1773.657 90.867
67/24 1777.352 90.985
14/5 1782.512 91.109
31/11 1793.718 91.367
17/6 1803.000 91.650
37/13 1810.816 91.957
20/7 1817.488 92.272
43/15 1823.249 92.602
23/8 1828.274 92.938
26/9 1836.618 93.633
29/10 1843.264 94.306
32/11 1848.682 94.862
35/12 1853.185 95.221
38/13 1856.985 95.305
41/14 1860.237 95.088
47/16 1865.507 93.821
53/18 1869.595 91.704
65/22 1875.523 86.386
77/26 1879.616 81.103
98/33 1884.379 73.722
131/44 1888.790 66.544
200/67 1893.320 60.026
3/1 1901.955 54.245

3/1 to 4/1

Harmonic entropy of just intervals
Ratio Size (cents) Entropy
(%, Rényi order 1.0) 		Entropy
(%, Rényi order 7.0)		 Comments
3/1 1901.955 54.245
367/122 1906.679 56.071
187/62 1911.238 60.881
127/42 1915.641 67.404
94/31 1920.471 75.286
76/25 1924.886 81.991
64/21 1929.219 87.364
55/18 1933.722 91.364
46/15 1940.006 94.439
43/14 1942.692 95.041
37/12 1949.389 95.375
34/11 1953.638 95.100
31/10 1958.722 94.593
28/9 1964.916 93.929
25/8 1972.627 93.206
47/15 1977.238 92.840
22/7 1982.492 92.487
60/19 1990.756 92.024
19/6 1995.558 91.813
54/17 2000.910 91.606
35/11 2003.822 91.516
67/21 2008.526 91.378
16/5 2013.686 91.231
29/9 2025.667 90.874
42/13 2030.253 90.752
81/25 2035.193 90.679
13/4 2040.528 90.693
49/15 2049.383 90.886
36/11 2052.592 90.938
23/7 2059.448 90.832
33/10 2066.959 90.254
43/13 2070.990 89.797
63/19 2075.223 89.312
10/3 2084.359 88.773
127/38 2088.909 88.907
57/17 2094.513 89.429
37/11 2100.026 90.130
27/8 2105.865 90.864
44/13 2110.790 91.364
17/5 2118.642 91.902
58/17 2124.622 92.131
24/7 2133.129 92.030
31/9 2141.126 91.024
38/11 2146.195 89.734
45/13 2149.696 88.572
59/17 2154.216 86.904
94/27 2159.642 85.001
7/2 2168.826 83.459
186/53 2173.486 83.881
95/27 2177.962 84.993
60/17 2183.313 86.860
46/13 2187.747 88.508
39/11 2191.165 89.654
32/9 2196.090 90.939
25/7 2203.802 91.985
43/12 2209.563 92.143
18/5 2217.596 91.868
65/18 2222.931 91.484
29/8 2229.577 90.831
40/11 2234.996 90.215
51/14 2238.085 89.874
117/32 2244.438 89.346
11/3 2249.363 89.211
59/16 2259.172 89.729
37/10 2265.030 90.282
26/7 2271.702 90.802
41/11 2277.745 91.024
15/4 2288.269 91.072
79/21 2293.756 91.101
34/9 2301.045 91.261
72/19 2306.397 91.442
19/5 2311.199 91.637
42/11 2319.463 92.015
23/6 2326.319 92.389
27/7 2337.039 93.165
31/8 2345.036 93.896
35/9 2351.230 94.391
39/10 2356.169 94.508
47/12 2363.552 93.521
55/14 2368.806 91.325
63/16 2372.736 88.598
75/19 2377.069 84.455
95/24 2381.872 78.764
127/32 2386.422 72.968
191/48 2390.960 67.739
387/97 2395.532 64.080
4/1 2400.000 62.798

4/1 to 6/1

Harmonic entropy of just intervals
Ratio Size (cents) Entropy
(%, Rényi order 1.0) 		Entropy
(%, Rényi order 7.0)		 Comments
4/1 2400.000 62.798
389/97 2404.456 64.076
193/48 2408.993 67.714
129/32 2413.473 72.852
97/24 2417.940 78.558
77/19 2422.631 84.164
65/16 2426.841 88.282
53/13 2432.977 92.286
49/12 2435.697 93.337
41/10 2442.749 94.509
37/9 2447.434 94.478
33/8 2453.273 94.049
29/7 2460.751 93.336
25/6 2470.672 92.507
46/11 2476.956 92.096
67/16 2479.307 91.958
21/5 2484.467 91.696
59/14 2490.346 91.443
38/9 2493.603 91.333
55/13 2497.104 91.229
17/4 2504.955 91.063
64/15 2511.731 90.911
30/7 2519.443 90.589
43/10 2525.204 90.237
56/13 2528.298 90.045
108/25 2533.238 89.798
13/3 2538.573 89.729
61/14 2548.059 90.228
35/8 2555.140 90.942
57/13 2558.940 91.316
22/5 2565.004 91.732
31/7 2576.210 91.266
40/9 2582.404 90.118
49/11 2586.334 89.120
67/15 2591.038 87.824
103/23 2595.526 86.696
9/2 2603.910 85.700
167/37 2609.101 86.069
86/19 2614.005 87.026
59/13 2618.644 88.237
41/9 2625.152 89.931
32/7 2631.174 91.047
23/5 2641.961 91.557
37/8 2651.344 90.918
51/11 2655.593 90.538
121/26 2662.108 90.078
14/3 2666.871 89.943
61/13 2676.357 90.178
47/10 2679.193 90.338
33/7 2684.447 90.654
52/11 2689.210 90.913
19/4 2697.513 91.250
81/17 2702.865 91.441
43/9 2707.608 91.620
67/14 2710.481 91.756
24/5 2715.641 92.033
29/6 2727.622 92.932
34/7 2736.130 93.640
39/8 2742.483 93.885
44/9 2747.408 93.589
54/11 2754.547 91.677
64/13 2759.472 88.918
74/15 2763.075 86.111
94/19 2767.994 81.362
129/26 2772.945 76.173
199/40 2777.636 71.787
5/1 2786.314 68.009
361/72 2791.116 69.236
186/37 2795.647 72.363
126/25 2800.109 76.655
91/18 2805.444 82.237
76/15 2809.244 85.883
61/12 2814.930 90.123
51/10 2820.597 92.661
46/9 2824.364 93.525
41/8 2829.062 93.937
36/7 2835.084 93.804
31/6 2843.081 93.178
57/11 2848.150 92.750
26/5 2854.214 92.307
47/9 2861.597 91.870
21/4 2870.781 91.425
79/15 2876.268 91.152
37/7 2882.518 90.820
53/10 2887.191 90.579
117/22 2893.120 90.378
16/3 2898.045 90.360
59/11 2907.854 90.739
43/8 2911.518 90.933
27/5 2919.551 91.115
65/12 2924.886 90.826
38/7 2928.687 90.366
49/9 2933.742 89.479
60/11 2936.951 88.814
93/17 2942.035 87.822
181/33 2946.542 87.184
11/2 2951.318 86.954
94/17 2960.551 87.918
61/11 2965.567 88.943
50/9 2968.717 89.611
39/7 2973.657 90.516
67/12 2977.352 91.004
28/5 2982.512 91.342
45/8 2990.224 91.271
62/11 2993.718 91.121
17/3 3003.000 90.796
91/16 3009.354 90.823
40/7 3017.488 91.140
63/11 3021.418 91.351
23/4 3028.274 91.759
52/9 3036.618 92.295
29/5 3043.264 92.782
35/6 3053.185 93.524
41/7 3060.237 93.677
47/8 3065.507 93.159
53/9 3069.595 92.138
65/11 3075.523 89.379
77/13 3079.616 86.553
95/16 3083.827 83.035
131/22 3088.790 78.596
197/33 3093.189 75.071
6/1 3101.955 71.731

6/1 to 8/1

Harmonic entropy of just intervals
Ratio Size (cents) Entropy
(%, Rényi order 1.0) 		Entropy
(%, Rényi order 7.0)		 Comments
6/1 3101.955 71.731
367/61 3106.679 72.714
187/31 3111.238 75.340
127/21 3115.641 78.905
91/15 3121.085 83.711
73/12 3125.835 87.463
61/10 3130.571 90.313
55/9 3133.722 91.647
49/8 3137.652 92.720
43/7 3142.692 93.315
37/6 3149.389 93.281
31/5 3158.722 92.634
56/9 3164.916 92.175
25/4 3172.627 91.679
69/11 3178.911 91.336
44/7 3182.492 91.164
63/10 3186.422 91.034
120/19 3190.756 90.949
19/3 3195.558 90.954
127/20 3200.108 91.016
51/8 3206.910 91.113
32/5 3213.686 90.949
45/7 3221.398 90.200
58/9 3225.667 89.582
84/13 3230.253 88.880
162/25 3235.193 88.287
13/2 3240.528 88.036
98/15 3249.383 88.679
59/9 3255.262 89.542
46/7 3259.448 90.160
33/5 3266.959 90.915
53/8 3273.505 91.119
20/3 3284.359 91.038
127/19 3288.909 91.050
47/7 3296.681 91.284
27/4 3305.865 91.845
61/9 3312.975 92.403
34/5 3318.642 92.846
41/6 3327.107 93.164
48/7 3333.129 92.684
55/8 3337.632 91.625
62/9 3341.126 90.272
76/11 3346.195 87.464
97/14 3351.070 83.990
125/18 3355.031 80.945
195/28 3359.971 77.500
7/1 3368.826 74.586
379/54 3373.400 75.394
190/27 3377.962 77.639
127/18 3382.512 80.826
92/13 3387.747 84.821
78/11 3391.165 87.227
64/9 3396.090 90.022
50/7 3403.802 92.504
43/6 3409.563 93.087
36/5 3417.596 92.889
65/9 3422.931 92.505
29/4 3429.577 91.982
51/7 3438.085 91.477
117/16 3444.438 91.266
22/3 3449.363 91.188
59/8 3459.172 91.036
37/5 3465.030 90.729
52/7 3471.702 90.110
67/9 3475.397 89.685
97/13 3479.368 89.251
15/2 3488.269 88.748
158/21 3493.756 88.946
83/11 3498.729 89.408
53/7 3504.679 90.087
38/5 3511.199 90.725
84/11 3519.463 91.165
23/3 3526.319 91.334
77/10 3533.830 91.570
54/7 3537.039 91.735
31/4 3545.036 92.279
70/9 3551.230 92.737
39/5 3556.169 92.976
47/6 3563.552 92.726
55/7 3568.806 91.736
63/8 3572.736 90.390
71/9 3575.787 88.962
95/12 3581.872 85.279
127/16 3586.422 82.182
191/24 3590.960 79.377
383/48 3595.486 77.393
8/1 3600.000 76.672

See also


Todo: complete table

Fill in the values for the Rényi order 7.0 column of each table.

Retrieved from "https://en.xen.wiki/index.php?title=Harmonic_entropy_of_just_intervals&oldid=177711"