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'''98304EDO''' is the [[EDO|equal division of the octave]] into 98304 parts of exact 0.01220703125 cents each, which is to say 2<sup>(1/98304)</sup> as a frequency ratio. Its adjacent step is known as '''Tridecamu''' (thirteenth MIDI-resolution unit, 13mu, 2<sup>13</sup> = 8192 equal divisions of the [[12edo]] semitone). The internal data structure of the 13mu requires two bytes, with the first bits of each byte reserved as a flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or &minus;) showing the direction of the pitch-bend up or down; all bits are used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.
{{Infobox ET}}
{{ED intro}}


==Intervals==
== Theory ==
98304EDO is [[consistent]] to the [[19-odd-limit|19-limit]], tempering out |45 2 -28 6&gt;, |-54 15 -16 24&gt;, and |-29 135 -18 -51&gt; in the 7-limit; |-4 15 -14 7 -2&gt;, |-3 28 2 -9 -6&gt;, |-50 0 -2 17 2&gt;, and |30 10 8 -2 -17&gt; in the 11-limit; 123201/123200, 32427005625/32426652544, 278924131584/278916015625, 37744795080531/37744172597248, and 156905298045000/156904157228819 in the 13-limit; 2000033/2000000, 154002541/154001250, 303464448/303460625, 338676338/338671875, 791249550/791243563, and 176846618624/176846076825 in the 17-limit; 89376/89375, 104976/104975, 709632/709631, 5836831/5836800, 494190983/494190000, 1206902781/1206878450, and 21867094832/21867015625 in the 19-limit.
98304edo is [[consistent]] to the [[19-odd-limit]]. The equal temperament [[tempering out|tempers out]] {{monzo| 45 2 -28 6 }}, {{monzo| -54 15 -16 24 }}, and {{monzo| -29 135 -18 -51 }} in the 7-limit; {{monzo| -4 15 -14 7 -2 }}, {{monzo| -3 28 2 -9 -6 }}, {{monzo| -50 0 -2 17 2 }}, and {{monzo| 30 10 8 -2 -17}} in the 11-limit; 123201/123200, 32427005625/32426652544, 278924131584/278916015625, 37744795080531/37744172597248, and 156905298045000/156904157228819 in the 13-limit; 2000033/2000000, 154002541/154001250, 303464448/303460625, 338676338/338671875, 791249550/791243563, and 176846618624/176846076825 in the 17-limit; 89376/89375, 104976/104975, 709632/709631, 5836831/5836800, 494190983/494190000, 1206902781/1206878450, and 21867094832/21867015625 in the 19-limit.


{| class="wikitable"
=== As a tuning standard ===
A step of 98304edo is known as a '''tridecamu''' (thirteenth MIDI-resolution unit, 13mu, 2<sup>13</sup> = 8192 equal divisions of the [[12edo]] semitone). The internal data structure of the 13mu requires two bytes, with the first bits of each byte reserved as flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or −) showing the direction of the pitch-bend up or down; all bits are used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.
 
=== Prime harmonics ===
{{Harmonics in equal|98304}}
 
== Selected intervals ==
{| class="wikitable right-1 right-2 center-3 right-4 right-5"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="2" | JI interval
! rowspan="2" | Error<br>(cents)
|-
! Ratio
! Cents
|-
| 0
| 0.0000
| [[1/1]]
| 0.0000
| ±0.00000
|-
| 3995
|  48.7671
| [[36/35]]
| 48.7704
| −0.00329
|-
| 4111
| 50.1831
| 35/34
| 50.1842
| −0.00111
|-
| 4234
| 51.6846
| 34/33
| 51.6825
| +0.00210
|-
| 4364
| 53.2715
| [[33/32]]
| 53.2729
| −0.00146
|-
| 5158
| 62.9639
| [[28/27]]
| 62.9609
| +0.00296
|-
| 5352
| 65.3320
| [[27/26]]
| 65.3373
| −0.00531
|-
| 5562
| 67.8955
| [[26/25]]
| 67.9002
| −0.00473
|-
| 6598
| 80.5420
| [[22/21]]
| 80.5370
| +0.00496
|-
| 7275
| 88.8062
| [[20/19]]
| 88.8007
| +0.00545
|-
| 7668
| 93.6035
| [[19/18]]
| 93.6030
| +0.00050
|-
| 8106
| 98.9502
| [[18/17]]
| 98.9546
| −0.00440
|-
| 8598
| 104.9561
| [[17/16]]
| 104.9554
| +0.00065
|-
| 9153
| 111.7310
| [[16/15]]
| 111.7313
| −0.00033
|-
| 9785
| 119.4458
| [[15/14]]
| 119.4428
| +0.00299
|-
| 10510
| 128.2959
| [[14/13]]
| 128.2982
| −0.00235
|-
| 11352
| 138.5742
| [[13/12]]
| 138.5727
| +0.00156
|-
| 12340
| 150.6348
| [[12/11]]
| 150.6371
| −0.00229
|-
| 13517
| 165.0024
| [[11/10]]
| 165.0042
| −0.00179
|-
| 14943
| 182.4097
| [[10/9]]
| 182.4037
| +0.00596
|-
| 15774
| 192.5537
| [[19/17]]
| 192.5576
| −0.00390
|-
|-
! rowspan="2"| 98304EDO <br>steps
| 16704
! rowspan="2"| cents <br>value
| 203.9063
! colspan="2"| JI interval
| [[9/8]]
! rowspan="2"| error <br>(cents)
| 203.9100
| −0.00375
|-
|-
! | ratio
| 17751
! | cents
| 216.6870
| [[17/15]]
| 216.6867
| +0.00032
|-
|-
| | 0
| 18938
| | 0.0000
| 231.1768
| | [[1/1]]
| [[8/7]]
| | 0.0000
| 231.1741
| | &plusmn;0.0000
| +0.00266
|-
|-
| | 3995
| 20295
| | 48.7671
| 247.7417
| | [[36/35]]
| [[15/13]]
| | 48.7704
| 247.7411
| | &minus;0.00329
| +0.00065
|-
|-
| | 4111
| 20792
| | 50.1831
| 253.8086
| | 35/34
| [[22/19]]
| | 50.1842
| 253.8049
| | &minus;0.00111
| +0.00367
|-
|-
| | 4234
| 21862
| | 51.6846
| 266.8701
| | 34/33
| [[7/6]]
| | 51.6825
| 266.8709
| | +0.00210
| −0.00079
|-
|-
| | 4364
| 23049
| | 53.2715
| 281.3599
| | [[33/32]]
| [[20/17]]
| | 53.2729
| 281.3583
| | &minus;0.00146
| +0.00156
|-
|-
| | 5158
| 23692
| | 62.9639
| 289.2090
| | [[28/27]]
| [[13/11]]
| | 62.9609
| 289.2097
| | +0.00296
| −0.00074
|-
|-
| | 5352
| 24372
| | 65.3320
| 297.5098
| | [[27/26]]
| [[19/16]]
| | 65.3373
| 297.5130
| | &minus;0.00531
| −0.00325
|-
|-
| | 5562
| 25857
| | 67.8955
| 315.6372
| | [[26/25]]
| [[6/5]]
| | 67.9002
| 315.6413
| | &minus;0.00473
| −0.00408
|-
|-
| | 6598
| 27536
| | 80.5420
| 336.1328
| | [[22/21]]
| [[17/14]]
| | 80.5370
| 336.1295
| | +0.00496
| +0.00331
|-
|-
| | 7275
| 28460
| | 88.8062
| 347.4121
| | [[20/19]]
| [[11/9]]
| | 88.8007
| 347.4079
| | +0.00545
| +0.00417
|-
|-
| | 7668
| 29448
| | 93.6035
| 359.4727
| | [[19/18]]
| [[16/13]]
| | 93.6030
| 359.4723
| | +0.00050
| +0.00032
|-
|-
| | 8106
| 31647
| | 98.9502
| 386.3159
| | [[18/17]]
| [[5/4]]
| | 98.9546
| 386.3137
| | &minus;0.00440
| +0.00220
|-
|-
| | 8598
| 33132
| | 104.9561
| 404.4434
| | [[17/16]]
| [[24/19]]
| | 104.9554
| 404.4420
| | +0.00065
| +0.00137
|-
|-
| | 9153
| 33525
| | 111.7310
| 409.2407
| | [[16/15]]
| [[19/15]]
| | 111.7313
| 409.2443
| | &minus;0.00033
| −0.00358
|-
|-
| | 9785
| 34202
| | 119.4458
| 417.5049
| | [[15/14]]
| [[14/11]]
| | 119.4428
| 417.5080
| | +0.00299
| −0.00308
|-
|-
| | 10510
| 35642
| | 128.2959
| 435.0830
| | [[14/13]]
| [[9/7]]
| | 128.2982
| 435.0841
| | &minus;0.00235
| −0.00109
|-
|-
| | 11352
| 36566
| | 138.5742
| 446.3623
| | [[13/12]]
| [[22/17]]
| | 138.5727
| 446.3625
| | +0.00156
| −0.00023
|-
|-
| | 12340
| 37209
| | 150.6348
| 454.2114
| | [[12/11]]
| [[13/10]]
| | 150.6371
| 454.2139
| | &minus;0.00229
| −0.00252
|-
|-
| | 13517
| 38046
| | 165.0024
| 464.4287
| | [[11/10]]
| [[17/13]]
| | 165.0042
| 464.4277
| | &minus;0.00179
| +0.00096
|-
|-
| | 14943
| 38566
| | 182.4097
| 470.7764
| | [[10/9]]
| [[21/16]]
| | 182.4037
| 470.7809
| | +0.00596
| −0.00454
|-
|-
| | 15774
| 40800
| | 192.5537
| 498.0469
| | [[19/17]]
| [[4/3]]
| | 192.5576
| 498.0450
| | &minus;0.00390
| +0.00188
|-
|-
| | 16704
| 43310
| | 203.9063
| 528.6865
| | [[9/8]]
| [[19/14]]
| | 203.9100
| 528.6871
| | &minus;0.00375
| −0.00059
|-
|-
| | 17751
| 43987
| | 216.6870
| 536.9507
| | [[17/15]]
| [[15/11]]
| | 216.6867
| 536.9508
| | +0.00032
| −0.00009
|-
|-
| | 18938
| 44484
| | 231.1768
| 543.0176
| | [[8/7]]
| [[26/19]]
| | 231.1741
| 543.0146
| | +0.00266
| +0.00293
|-
|-
| | 20295
| 45164
| | 247.7417
| 551.3184
| | [[15/13]]
| [[11/8]]
| | 247.7411
| 551.3179
| | +0.00065
| +0.00042
|-
|-
| | 20792
| 46152
| | 253.8086
| 563.3789
| | [[22/19]]
| [[18/13]]
| | 253.8049
| 563.3823
| | +0.00367
| −0.00343
|-
|-
| | 21862
| 47719
| | 266.8701
| 582.5073
| | [[7/6]]
| [[7/5]]
| | 266.8709
| 582.5122
| | &minus;0.00079
| −0.00487
|-
|-
| | 23049
| 48906
| | 281.3599
| 596.9971
| | [[20/17]]
| [[24/17]]
| | 281.3583
| 596.9996
| | +0.00156
| −0.00252
|-
|-
| | 23692
| 49398
| | 289.2090
| 603.0029
| | [[13/11]]
| [[17/12]]
| | 289.2097
| 603.0004
| | &minus;0.00074
| +0.00252
|-
|-
| | 24372
| 50585
| | 297.5098
| 617.4927
| | [[19/16]]
| [[10/7]]
| | 297.5130
| 617.4878
| | &minus;0.00325
| +0.00487
|-
|-
| | 25857
| 57504
| | 315.6372
| 701.9531
| | [[6/5]]
| [[3/2]]
| | 315.6413
| 701.9550
| | &minus;0.00408
| −0.00188
|-
|-
| | 27536
| 62662
| | 336.1328
| 764.9170
| | [[17/14]]
| [[14/9]]
| | 336.1295
| 764.9159
| | +0.00331
| +0.00109
|-
|-
| | 28460
| 66657
| | 347.4121
| 813.6841
| | [[11/9]]
| [[8/5]]
| | 347.4079
| 813.6863
| | +0.00417
| −0.00220
|-
|-
| | 29448
| 72447
| | 359.4727
| 884.3628
| | [[16/13]]
| [[5/3]]
| | 359.4723
| 884.3587
| | +0.00032
| +0.00408
|-
|-
| | 31647
| 76442
| | 386.3159
| 933.1299
| | [[5/4]]
| [[12/7]]
| | 386.3137
| 933.1291
| | +0.00220
| +0.00079
|-
|-
| | 40800
| 79366
| | 498.0469
| 968.8232
| | [[4/3]]
| [[7/4]]
| | 498.0450
| 968.8259
| | +0.00188
| −0.00266
|-
|-
| | 98304
| 98304
| | 1200.0000
| 1200.0000
| | [[Octave|2/1]]
| [[Octave|2/1]]
| | 1200.0000
| 1200.0000
| | &plusmn;0.0000
| ±0.00000
|}
|}


==See also==
== See also ==
*[[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
**[[24edo]] (1mu tuning)
** [[24edo]] (1mu tuning)
**[[48edo]] (2mu tuning)
** [[48edo]] (2mu tuning)
**[[96edo]] (3mu tuning)
** [[96edo]] (3mu tuning)
**[[192edo]] (4mu tuning)
** [[192edo]] (4mu tuning)
**[[384edo]] (5mu tuning)
** [[384edo]] (5mu tuning)
**[[768edo]] (6mu tuning)
** [[768edo]] (6mu tuning)
**[[1536edo]] (7mu tuning)
** [[1536edo]] (7mu tuning)
**[[3072edo]] (8mu tuning)
** [[3072edo]] (8mu tuning)
**[[6144edo]] (9mu tuning)
** [[6144edo]] (9mu tuning)
**[[12288edo]] (10mu tuning)
** [[12288edo]] (10mu tuning)
**[[24576edo]] (11mu tuning)
** [[24576edo]] (11mu tuning)
**[[49152edo]] (12mu tuning)
** [[49152edo]] (12mu tuning)
**[[196608edo]] (14mu tuning)
** [[196608edo]] (14mu tuning)


[[Category:Edo]]
== External links ==
[[Category:Theory]]
* [http://tonalsoft.com/enc/number/13mu.aspx 13mu / tridekamu] on [[Tonalsoft Encyclopedia]]
Retrieved from "https://en.xen.wiki/w/98304edo"