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16808edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-08-23 11:54:58 UTC</tt>.<br>
: The original revision id was <tt>557205531</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The 16808 division divides the octave into 16808 steps of size 0.0714 cents each. It is distinctly consistent and highly accurate through the 35 limit, and can be used as a measure of interval size (the [[jinn]]) for most intervals which occur in practice. It is a very, very strong 31-limit division, and both a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak]] and [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral]] tuning, and may also be a zeta gap edo, which someone with a large enough table of zeta zeros might check. In the 23, 29 and 31 limits it has the lowest logflat badness up until at least 200000; in the 19 limit it is beaten out by [[8539edo]], and in the 17 limit by [[72edo]], [[1506edo]], [[3395edo]] and [[7033edo]].


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 and1990656/1990625; in the 17-limit 194481/194480 and 336141/336140; in the 19-limit 43681/43680,89376/89375 and104976/104975. 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.
16808edo's step size is sometimes called a '''jinn''', a term proposed by [[Gene Ward Smith]]<ref>[https://www.huygens-fokker.org/docs/measures.html Stichting Huygens-Fokker: Logarithmic Interval Measures]</ref>, when used as an [[interval size unit]].


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.</pre></div>
== Theory ==
<h4>Original HTML content:</h4>
16808edo is distinctly [[consistent]] and highly accurate through the [[35-odd-limit]], being [[consistency #Generalization|consistent to distance 2]]. It is a very, very strong [[31-limit]] system, and a [[zeta peak edo|zeta peak]], [[zeta peak integer edo|zeta peak integer]] and [[zeta integral edo]]. In the [[23-limit|23-]], [[29-limit|29-]] and 31-limit it has the lowest [[Tenney–Euclidean temperament measures #TE simple badness|relative error]] up until [[148418edo|148418]]; in the [[17-limit|17-]] and [[19-limit]] up until [[20203edo|20203]]; though in the [[13-limit]] it is beaten out by smaller edos {{EDOs| 5585, 6079, 8269, 8539, 13112 and 14618 }}. As such, its step size can be used as an [[interval size unit]] (the jinn) for most intervals which occur in practice.
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;16808edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 16808 division divides the octave into 16808 steps of size 0.0714 cents each. It is distinctly consistent and highly accurate through the 35 limit, and can be used as a measure of interval size (the &lt;a class="wiki_link" href="/jinn"&gt;jinn&lt;/a&gt;) for most intervals which occur in practice. It is a very, very strong 31-limit division, and both a &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&gt;zeta peak&lt;/a&gt; and &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&gt;zeta integral&lt;/a&gt; tuning, and may also be a zeta gap edo, which someone with a large enough table of zeta zeros might check. In the 23, 29 and 31 limits it has the lowest logflat badness up until at least 200000; in the 19 limit it is beaten out by &lt;a class="wiki_link" href="/8539edo"&gt;8539edo&lt;/a&gt;, and in the 17 limit by &lt;a class="wiki_link" href="/72edo"&gt;72edo&lt;/a&gt;, &lt;a class="wiki_link" href="/1506edo"&gt;1506edo&lt;/a&gt;, &lt;a class="wiki_link" href="/3395edo"&gt;3395edo&lt;/a&gt; and &lt;a class="wiki_link" href="/7033edo"&gt;7033edo&lt;/a&gt;.&lt;br /&gt;
 
&lt;br /&gt;
Its [[3/2|perfect fifth]] ultimately comes from [[2101edo]], so it not only has two [[chain of fifths|circles of fifths]] ([[hemipyth]]), but ''eight'', giving itself another edge over similar systems.  
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 and1990656/1990625; in the 17-limit 194481/194480 and 336141/336140; in the 19-limit 43681/43680,89376/89375 and104976/104975. 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.&lt;br /&gt;
 
&lt;br /&gt;
Among the enormous list of 31-limit [[comma]]s it [[tempering out|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.
16808 has proper divisors 1, 2, 4, 8, 11, 22, 44, 88, 191, 382, 764, 1528, 2101, 4202 and 8404, among which &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; and &lt;a class="wiki_link" href="/764edo"&gt;764edo&lt;/a&gt; are particularly notable. One step of 22edo is 764 jinns, and one step of 764edo is 22 jinns.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
=== Prime harmonics ===
{{Harmonics in equal|16808|columns=11}}
{{Harmonics in equal|16808|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 16808edo (continued)}}
 
=== Subsets and supersets ===
16808 has subset edos 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. [[33616edo]], which doubles it, corrects its harmonics 37, [[43/1|43]], and [[47/1|47]] to near-just qualities.
 
== Intervals ==
Below the intervals of the 35-odd-limit [[tonality diamond]] are tabulated, with the sizes listed in both [[cent]]s 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.
 
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ {{nowrap|35-odd-limit intervals}}
! 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
|}
 
== References ==
<references />
 
[[Category:16808edo| ]] <!-- main article -->
[[Category:Interval size measures]]
Retrieved from "https://en.xen.wiki/w/16808edo"