16808edo
| ← 16807edo | 16808edo | 16809edo → |
16808 equal divisions of the octave (abbreviated 16808edo or 16808ed2), also called 16808-tone equal temperament (16808tet) or 16808 equal temperament (16808et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 16808 equal parts of about 0.0714 ¢ each. Each step represents a frequency ratio of 21/16808, or the 16808th root of 2.
16808edo's step size is sometimes called a jinn, a term proposed by Gene Ward Smith[1], when used as an interval size unit.
Theory
16808edo is distinctly consistent and highly accurate through the 35-odd-limit, and its step size can be used as an interval size unit (the jinn) for most intervals which occur in practice. It is a very, very strong 31-limit division, and a zeta peak, zeta peak integer and zeta integral edo. In the 23-, 29- and 31-limit it has the lowest relative error up until 148418; in the 17- and 19-limit up until 20203; though in the 13-limit it is beaten out by smaller edos 5585, 6079, 8269, 8539, 13112 and 14618.
Among the enormous list of 31-limit commas it tempers out, the simplest are 43681/43680, 49011/49010, 52326/52325 and 53361/53360. In the 13-limit it tempers out 123201/123200 and 1990656/1990625; in the 17-limit 194481/194480 and 336141/336140; in the 19-limit 43681/43680, 89376/89375 and 104976/104975. Since 43681/43680 is both the simplest comma it tempers out and the limit is as low (in this context) as 19, it may be regarded as rather characteristic of 16808.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00000 | -0.00355 | +0.00233 | -0.00154 | -0.00904 | +0.00066 | -0.00539 | -0.01183 | -0.00210 | +0.00396 | -0.00939 |
| Relative (%) | +0.0 | -5.0 | +3.3 | -2.2 | -12.7 | +0.9 | -7.5 | -16.6 | -2.9 | +5.5 | -13.2 | |
| Steps (reduced) |
16808 (0) |
26640 (9832) |
39027 (5411) |
47186 (13570) |
58146 (7722) |
62197 (11773) |
68702 (1470) |
71399 (4167) |
76032 (8800) |
81653 (14421) |
83270 (16038) | |
Subsets and supersets
16808 has proper divisors 1, 2, 4, 8, 11, 22, 44, 88, 191, 382, 764, 1528, 2101, 4202 and 8404, among which 22edo and 764edo are particularly notable. One step of 22edo is 764 jinns, and one step of 764edo is 22 jinns.
Intervals
Below the intervals of the 35-odd-limit tonality diamond are tabulated, with the sizes listed in both cents and jinns. The worst error occurs for 33/25 and 50/33, which is less than 1/4 of a jinn off. The measure in jinns can be rounded to the next integer to find the corresponding degree of 16808edo.
| Interval | Size in cents | Size in jinns |
|---|---|---|
| 36/35 | 48.7700 | 683.110 |
| 35/34 | 50.1840 | 702.914 |
| 34/33 | 51.6820 | 723.899 |
| 33/32 | 53.273 | 746.176 |
| 32/31 | 54.964 | 769.868 |
| 31/30 | 56.767 | 795.114 |
| 30/29 | 58.692 | 822.073 |
| 29/28 | 60.751 | 850.923 |
| 28/27 | 62.961 | 881.872 |
| 27/26 | 65.337 | 915.158 |
| 26/25 | 67.900 | 951.056 |
| 25/24 | 70.672 | 989.885 |
| 24/23 | 73.681 | 1032.020 |
| 23/22 | 76.956 | 1077.903 |
| 22/21 | 80.537 | 1128.055 |
| 21/20 | 84.467 | 1183.104 |
| 20/19 | 88.801 | 1243.802 |
| 19/18 | 93.603 | 1311.066 |
| 18/17 | 98.955 | 1386.024 |
| 35/33 | 101.867 | 1426.813 |
| 17/16 | 104.955 | 1470.075 |
| 33/31 | 108.237 | 1516.045 |
| 16/15 | 111.731 | 1564.983 |
| 31/29 | 115.458 | 1617.187 |
| 15/14 | 119.443 | 1672.996 |
| 29/27 | 123.712 | 1732.795 |
| 14/13 | 128.298 | 1797.031 |
| 27/25 | 133.238 | 1866.214 |
| 13/12 | 138.573 | 1940.941 |
| 38/35 | 142.373 | 1994.177 |
| 25/23 | 144.353 | 2021.905 |
| 12/11 | 150.637 | 2109.923 |
| 35/32 | 155.140 | 2172.989 |
| 23/21 | 157.493 | 2205.958 |
| 34/31 | 159.920 | 2239.944 |
| 11/10 | 165.004 | 2311.159 |
| 32/29 | 170.423 | 2387.055 |
| 21/19 | 173.268 | 2426.906 |
| 31/28 | 176.210 | 2468.110 |
| 10/9 | 182.404 | 2554.868 |
| 29/26 | 189.050 | 2647.954 |
| 19/17 | 192.558 | 2697.090 |
| 28/25 | 196.198 | 2748.087 |
| 9/8 | 203.910 | 2856.099 |
| 35/31 | 210.104 | 2942.857 |
| 26/23 | 212.253 | 2972.961 |
| 17/15 | 216.687 | 3035.058 |
| 25/22 | 221.309 | 3099.808 |
| 33/29 | 223.696 | 3133.232 |
| 8/7 | 231.174 | 3237.978 |
| 31/27 | 239.171 | 3349.982 |
| 23/20 | 241.961 | 3389.062 |
| 38/33 | 244.240 | 3420.989 |
| 15/13 | 247.741 | 3470.026 |
| 22/19 | 253.805 | 3554.961 |
| 29/25 | 256.950 | 3599.010 |
| 36/31 | 258.874 | 3625.968 |
| 7/6 | 266.871 | 3737.972 |
| 34/29 | 275.378 | 3857.131 |
| 27/23 | 277.591 | 3888.120 |
| 20/17 | 281.358 | 3940.892 |
| 33/28 | 284.447 | 3984.155 |
| 13/11 | 289.210 | 4050.864 |
| 32/27 | 294.135 | 4119.851 |
| 19/16 | 297.513 | 4167.166 |
| 25/21 | 301.847 | 4227.864 |
| 31/26 | 304.508 | 4265.141 |
| 6/5 | 315.641 | 4421.082 |
| 35/29 | 325.562 | 4560.044 |
| 29/24 | 327.622 | 4588.895 |
| 23/19 | 330.761 | 4632.864 |
| 40/33 | 333.041 | 4664.791 |
| 17/14 | 336.130 | 4708.054 |
| 28/23 | 340.552 | 4769.992 |
| 11/9 | 347.408 | 4866.027 |
| 38/31 | 352.477 | 4937.034 |
| 27/22 | 354.547 | 4966.022 |
| 16/13 | 359.472 | 5035.009 |
| 21/17 | 365.825 | 5123.996 |
| 26/21 | 369.747 | 5178.920 |
| 31/25 | 372.408 | 5216.197 |
| 36/29 | 374.333 | 5243.155 |
| 5/4 | 386.314 | 5410.967 |
| 44/35 | 396.178 | 5549.138 |
| 34/27 | 399.090 | 5589.926 |
| 29/23 | 401.303 | 5620.915 |
| 24/19 | 404.442 | 5664.884 |
| 19/15 | 409.244 | 5732.149 |
| 33/26 | 412.745 | 5781.186 |
| 14/11 | 417.508 | 5847.895 |
| 23/18 | 424.364 | 5943.930 |
| 32/25 | 427.373 | 5986.065 |
| 9/7 | 435.084 | 6094.078 |
| 40/31 | 441.278 | 6180.836 |
| 31/24 | 443.081 | 6206.082 |
| 22/17 | 446.363 | 6252.051 |
| 35/27 | 449.275 | 6292.840 |
| 13/10 | 454.214 | 6362.023 |
| 30/23 | 459.994 | 6442.988 |
| 17/13 | 464.428 | 6505.085 |
| 38/29 | 467.936 | 6554.221 |
| 21/16 | 470.781 | 6594.071 |
| 46/35 | 473.135 | 6627.040 |
| 25/19 | 475.114 | 6654.769 |
| 29/22 | 478.259 | 6698.818 |
| 33/25 | 480.646 | 6732.242 |
| 4/3 | 498.045 | 6975.950 |
| 35/26 | 514.612 | 7207.998 |
| 31/23 | 516.761 | 7238.102 |
| 27/20 | 519.551 | 7277.182 |
| 23/17 | 523.319 | 7329.954 |
| 42/31 | 525.745 | 7363.940 |
| 19/14 | 528.687 | 7405.144 |
| 34/25 | 532.328 | 7456.141 |
| 15/11 | 536.951 | 7520.890 |
| 26/19 | 543.015 | 7605.825 |
| 48/35 | 546.815 | 7659.061 |
| 11/8 | 551.318 | 7722.127 |
| 40/29 | 556.737 | 7798.023 |
| 29/21 | 558.796 | 7826.873 |
| 18/13 | 563.382 | 7891.109 |
| 25/18 | 568.717 | 7965.835 |
| 32/23 | 571.726 | 8007.971 |
| 46/33 | 575.001 | 8053.853 |
| 7/5 | 582.512 | 8159.054 |
| 38/27 | 591.648 | 8287.017 |
| 31/22 | 593.718 | 8316.005 |
| 24/17 | 597.000 | 8361.974 |
| 17/12 | 603.000 | 8446.026 |
| 44/31 | 606.282 | 8491.995 |
| 27/19 | 608.352 | 8520.983 |
| 10/7 | 617.488 | 8648.946 |
| 33/23 | 624.999 | 8754.147 |
| 23/16 | 628.274 | 8800.029 |
| 36/25 | 631.283 | 8842.165 |
| 13/9 | 636.618 | 8916.891 |
| 42/29 | 641.204 | 8981.127 |
| 29/20 | 643.263 | 9009.977 |
| 16/11 | 648.682 | 9085.873 |
| 35/24 | 653.185 | 9148.939 |
| 19/13 | 656.985 | 9202.175 |
| 22/15 | 663.049 | 9287.110 |
| 25/17 | 667.672 | 9351.859 |
| 28/19 | 671.313 | 9402.856 |
| 31/21 | 674.255 | 9444.060 |
| 34/23 | 676.681 | 9478.046 |
| 40/27 | 680.449 | 9530.818 |
| 46/31 | 683.239 | 9569.898 |
| 52/35 | 685.388 | 9600.002 |
| 3/2 | 701.955 | 9832.050 |
| 50/33 | 719.354 | 10075.758 |
| 44/29 | 721.741 | 10109.182 |
| 38/25 | 724.886 | 10153.231 |
| 35/23 | 726.865 | 10180.960 |
| 32/21 | 729.219 | 10213.929 |
| 29/19 | 732.064 | 10253.779 |
| 26/17 | 735.572 | 10302.915 |
| 23/15 | 740.006 | 10365.012 |
| 20/13 | 745.786 | 10445.977 |
| 54/35 | 750.725 | 10515.160 |
| 17/11 | 753.637 | 10555.949 |
| 48/31 | 756.919 | 10601.918 |
| 31/20 | 758.722 | 10627.164 |
| 14/9 | 764.916 | 10713.922 |
| 25/16 | 772.627 | 10821.935 |
| 36/23 | 775.636 | 10864.070 |
| 11/7 | 782.492 | 10960.105 |
| 52/33 | 787.255 | 11026.814 |
| 30/19 | 790.756 | 11075.851 |
| 19/12 | 795.558 | 11143.116 |
| 46/29 | 798.697 | 11187.085 |
| 27/17 | 800.910 | 11218.074 |
| 35/22 | 803.822 | 11258.862 |
| 8/5 | 813.686 | 11397.033 |
| 29/18 | 825.667 | 11564.845 |
| 50/31 | 827.592 | 11591.803 |
| 21/13 | 830.253 | 11629.080 |
| 34/21 | 834.175 | 11684.004 |
| 13/8 | 840.528 | 11772.991 |
| 44/27 | 845.453 | 11841.978 |
| 31/19 | 847.523 | 11870.966 |
| 18/11 | 852.592 | 11941.973 |
| 23/14 | 859.448 | 12038.008 |
| 28/17 | 863.870 | 12099.946 |
| 33/20 | 866.959 | 12143.209 |
| 38/23 | 869.239 | 12175.136 |
| 48/29 | 872.378 | 12219.105 |
| 58/35 | 874.438 | 12247.956 |
| 5/3 | 884.359 | 12386.918 |
| 52/31 | 895.492 | 12542.859 |
| 42/25 | 898.153 | 12580.136 |
| 32/19 | 902.487 | 12640.834 |
| 27/16 | 905.865 | 12688.149 |
| 22/13 | 910.790 | 12757.136 |
| 56/33 | 915.553 | 12823.845 |
| 17/10 | 918.642 | 12867.108 |
| 46/27 | 922.409 | 12919.880 |
| 29/17 | 924.622 | 12950.869 |
| 12/7 | 933.129 | 13070.028 |
| 31/18 | 941.126 | 13182.032 |
| 50/29 | 943.050 | 13208.990 |
| 19/11 | 946.195 | 13253.039 |
| 26/15 | 952.259 | 13337.974 |
| 33/19 | 955.760 | 13387.011 |
| 40/23 | 958.039 | 13418.938 |
| 54/31 | 960.829 | 13458.018 |
| 7/4 | 968.826 | 13570.022 |
| 58/33 | 976.304 | 13674.768 |
| 44/25 | 978.691 | 13708.192 |
| 30/17 | 983.313 | 13772.942 |
| 23/13 | 987.747 | 13835.039 |
| 62/35 | 989.896 | 13865.143 |
| 16/9 | 996.090 | 13951.901 |
| 25/14 | 1003.802 | 14059.913 |
| 34/19 | 1007.442 | 14110.910 |
| 52/29 | 1010.950 | 14160.046 |
| 9/5 | 1017.596 | 14253.132 |
| 56/31 | 1023.790 | 14339.890 |
| 38/21 | 1026.732 | 14381.094 |
| 29/16 | 1029.577 | 14420.945 |
| 20/11 | 1034.996 | 14496.841 |
| 31/17 | 1040.080 | 14568.056 |
| 42/23 | 1042.507 | 14602.042 |
| 64/35 | 1044.860 | 14635.011 |
| 11/6 | 1049.363 | 14698.077 |
| 46/25 | 1055.647 | 14786.095 |
| 35/19 | 1057.627 | 14813.823 |
| 24/13 | 1061.427 | 14867.059 |
| 50/27 | 1066.762 | 14941.786 |
| 13/7 | 1071.702 | 15010.969 |
| 54/29 | 1076.288 | 15075.205 |
| 28/15 | 1080.557 | 15135.004 |
| 58/31 | 1084.542 | 15190.813 |
| 15/8 | 1088.269 | 15243.017 |
| 62/33 | 1091.763 | 15291.955 |
| 32/17 | 1095.045 | 15337.925 |
| 66/35 | 1098.133 | 15381.187 |
| 17/9 | 1101.045 | 15421.976 |
| 36/19 | 1106.397 | 15496.934 |
| 19/10 | 1111.199 | 15564.198 |
| 40/21 | 1115.533 | 15624.896 |
| 21/11 | 1119.463 | 15679.945 |
| 44/23 | 1123.044 | 15730.097 |
| 23/12 | 1126.319 | 15775.980 |
| 48/25 | 1129.328 | 15818.115 |
| 25/13 | 1132.100 | 15856.944 |
| 52/27 | 1134.663 | 15892.842 |
| 27/14 | 1137.039 | 15926.128 |
| 56/29 | 1139.249 | 15957.077 |
| 29/15 | 1141.308 | 15985.927 |
| 60/31 | 1143.233 | 16012.886 |
| 31/16 | 1145.036 | 16038.132 |
| 64/33 | 1146.727 | 16061.824 |
| 33/17 | 1148.318 | 16084.101 |
| 68/35 | 1149.816 | 16105.086 |
| 35/18 | 1151.230 | 16124.890 |
| 2 | 1200.000 | 16808.000 |