Xenharmonic Wiki

16808edo: Difference between revisions

← Older edit
VisualWikitext
Fredg999 category edits (talk | contribs)
autopatrolled, Bots, emailconfirmed
4,405 edits
m Removing from Category:Theory using Cat-a-lot
Overthink (talk | contribs)
autopatrolled
2,726 edits
Undo revision 230160 by Domin (talk) please give a better reason
Tag: Undo
 
(26 intermediate revisions by 10 users not shown)
Line 1: Line 1:
The '''16808 equal division''' divides the octave into 16808 steps of size 0.071395 [[cent]]s each. It is distinctly consistent and highly accurate through the 35 limit, and can be used as a [[interval size measure|measure of interval size]] (the [[jinn]]) for most intervals which occur in practice. It is a very, very strong 31-limit division, and a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak]], [[zeta peak integer edo|zeta peak integer]], [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral]] and zeta gap tuning. In the [[23-limit|23]], [[29-limit|29]] and [[31-limit|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]].
{{Infobox ET}}
{{ED intro}}


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 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.
== Theory ==
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.  


{{Primes in edo|16808|prec=5|columns=11}}
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.
[[Category:Equal divisions of the octave]]
 
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.
 
=== 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"