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'''196608EDO''' is the [[EDO|equal division of the octave]] into 196608 parts of exact 0.006103515625 cents each, which is to say 2<sup>(1/196608)</sup> as a frequency ratio. Its adjacent step is known as the '''MIDI Tuning Standard unit''' (MTSU) or '''Tetradecamu''' (fourteenth MIDI-resolution unit, 14mu, 2<sup>14</sup> = 16384 equal divisions of the [[12edo]] semitone). The 14mu is specified in the MIDI spec (1983) as the smallest increment available for the pitch-bend controller, and as the frequency data format for MTS (1999). The 14mu is the smallest unit of tuning resolution which has ever been put into common musical use, and provides extremely accurate tuning in microtonal electronic music.
{{Software tuning}}
{{Infobox ET}}
{{ED intro}}


==Intervals==
== Theory ==
There is a list for just intervals.
196608edo is [[enfactoring|enfactored]] in the 17-limit, having the same tuning as [[98304edo]], which is quite an efficient system in itself. In that regard, 196608edo provides barely anything new apart from most characteristics of what it doubles.
{| class="wikitable"
 
=== As a tuning standard ===
[[File:Tetradecamu Approximation Quality.png|thumb|Approximation quality of smaller edos in tetradecamu, maximum relative error against edo from 1 to 16808. ]]
 
A step of 196608edo is known as a '''MIDI Tuning Standard unit''' ('''MTSU''') or a '''tetradecamu''' (fourteenth MIDI-resolution unit, 14mu, {{nowrap|2<sup>14</sup> {{=}} 16384}} equal divisions of the [[12edo]] semitone). The 14mu is specified in the [[MIDI]] spec (1983) as the smallest increment available for the pitch-bend controller, and as the frequency data format for MTS (1999). The 14mu is the smallest unit of tuning resolution which has ever been put into common musical use, and provides extremely accurate tuning in microtonal electronic music.
 
The main application of 196608edo is thus not as a compositional device, but as a technical tuning standard. If we adopt direct approximation, some JI intervals are indeed improved, which makes sense since we are only quantizing JI to the grid of this edo.
 
Another usage that is not commonly seen in other edos is to approximate smaller edos. It cannot approximate any larger edos or any edos of the same order of magnitude. From the diagram we can observe the maximum relative errors of smallers edos are mostly linear with respect to the edo number. [[16808edo]], a notable zeta edo that is an order of magnitude below, is approximated with a ~4% maximum relative error.
 
=== Odd harmonics ===
{{Harmonics in equal|196608}}
 
== Selected intervals ==
Below is a list for just 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>(Steps)
! rowspan="2"| Error <br>(Cents)
|-
! Ratio
! Cents
|-
| 0
| 0.000000000000
| [[1/1]]
| 0.000000000000
| ±0.000000
| ±0.000000000000
|-
| 18306
| 111.730957031250
| [[16/15]]
| 111.731285269778
| −0.053779
| −0.000328238528
|-
| 19570
| 119.445800781250
| [[15/14]]
| 119.442808261097
| +0.490295
| +0.002992520153
|-
| 21020
| 128.295898437500
| [[14/13]]
| 128.298244699814
| −0.384412
| −0.002346262314
|-
| 22704
| 138.574218750000
| [[13/12]]
| 138.572660903923
| +0.255238
| +0.001557846077
|-
| 24680
| 150.634765625000
| [[12/11]]
| 150.637058500631
| −0.375665
| −0.002292875631
|-
| 27034
| 165.002441406250
| [[11/10]]
| 165.004228499922
| −0.292797
| −0.001787093672
|-
| 29885
| 182.403564453125
| [[10/9]]
| 182.403712134060
| −0.024196
| −0.000147680935
|-
| 33409
| 203.912353515625
| [[9/8]]
| 203.910001730775
| +0.385316
| +0.002351784850
|-
| 37876
| 231.176757812500
| [[8/7]]
| 231.174093530875
| +0.436516
| +0.002664281625
|-
| 40590
| 247.741699218750
| [[15/13]]
| 247.741052960912
| +0.105883
| +0.000646257838
|-
|-
! | 196608EDO <br>steps
| 43724
! | cents <br>value
| 266.870117187500
! | JI interval <br>ratio
| [[7/6]]
! | JI interval <br>cents
| 266.870905603738
! | error <br>(steps)
| −0.129174
! | error <br>(cents)
| −0.000788416238
|-
|-
| | 0
| 47384
| | 0.000000000000
| 289.208984375000
| | [[1/1]]
| [[13/11]]
| | 0.000000000000
| 289.209719404554
| | &plusmn;0.000000
| −0.120427
| | &plusmn;0.000000000000
| −0.000735029554
|-
|-
| | 24680
| 51715
| | 150.634765625
| 315.643310546875
| | [[12/11]]
| [[6/5]]
| | 150.637058500631
| 315.641287000553
| | &minus;0.375665
| +0.331538
| | &minus;0.002292875631
| +0.002023546322
|-
|-
| | 27034
| 56919
| | 165.00244140625
| 347.406005859375
| | [[11/10]]
| [[11/9]]
| | 165.004228499922
| 347.407940633982
| | &minus;0.292797
| −0.316993
| | &minus;0.001787093672
| −0.001934774607
|-
|-
| | 29885
| 58896
| | 182.403564453125
| 359.472656250000
| | [[10/9]]
| [[16/13]]
| | 182.403712134060
| 359.472338230689
| | &minus;0.024196
| +0.052104
| | &minus;0.000147680935
| +0.000318019311
|-
|-
| | 33409
| 63294
| | 203.912353515625
| 386.315917968750
| | [[9/8]]
| [[5/4]]
| | 203.910001730775
| 386.313713864835
| | +0.385316
| +0.361120
| | +0.002351784850
| +0.002204103915
|-
|-
| | 37876
| 68405
| | 231.1767578125
| 417.510986328125
| | [[8/7]]
| [[14/11]]
| | 231.174093530875
| 417.507964104368
| | +0.436516
| +0.495161
| | +0.002664281625
| +0.003022223757
|-
|-
| | 43724
| 71284
| | 266.8701171875
| 435.083007812500
| | [[7/6]]
| [[9/7]]
| | 266.870905603738
| 435.084095261650
| | &minus;0.129174
| −0.178168
| | &minus;0.000788416238
| −0.001087449150
|-
|-
| | 51715
| 74418
| | 315.643310546875
| 454.211425781250
| | [[6/5]]
| [[13/10]]
| | 315.641287000553
| 454.213947904476
| | +0.331538
| −0.413225
| | +0.002023546322
| −0.002522123226
|-
|-
| | 56919
| 81600
| | 347.406005859375
| 498.046875000000
| | [[11/9]]
| [[4/3]]
| | 347.407940633982
| 498.044999134613
| | &minus;0.316993
| +0.307342
| | &minus;0.001934774607
| +0.001875865387
|-
|-
| | 63294
| 87974
| | 386.31591796875
| 536.950683593750
| | [[5/4]]
| [[15/11]]
| | 386.313713864835
| 536.950772365466
| | +0.361120
| −0.014544
| | +0.002204103915
| −0.000088771716
|-
|-
| | 68405
| 90328
| | 417.510986328125
| 551.318359375000
| | [[14/11]]
| [[11/8]]
| | 417.507964104368
| 551.317942364757
| | +0.495161
| +0.068323
| | +0.003022223757
| +0.000417010243
|-
|-
| | 71284
| 92305
| | 435.0830078125
| 563.385009765625
| | [[9/7]]
| [[18/13]]
| | 435.084095261650
| 563.382339961464
| | &minus;0.178168
| +0.437421
| | &minus;0.001087449150
| +0.002669804161
|-
|-
| | 81600
| 95439
| | 498.046875
| 582.513427734375
| | [[4/3]]
| [[7/5]]
| | 498.044999134613
| 582.512192604290
| | +0.307342
| +0.202364
| | +0.001875865387
| +0.001235130085
|-
|-
| | 90328
| 101169
| | 551.318359375
| 617.486572265625
| | [[11/8]]
| [[10/7]]
| | 551.317942364757
| 617.487807395710
| | +0.068323
| −0.202364
| | +0.000417010243
| −0.001235130085
|-
|-
| | 95439
| 106280
| | 582.513427734375
| 648.681640625000
| | [[7/5]]
| [[16/11]]
| | 582.512192604290
| 648.682057635243
| | +0.202364
| −0.068323
| | +0.001235130085
| −0.000417010243
|-
|-
| | 101169
| 115008
| | 617.486572265625
| 701.953125000000
| | [[10/7]]
| [[3/2]]
| | 617.487807395710
| 701.955000865387
| | &minus;0.202364
| −0.307342
| | &minus;0.001235130085
| −0.001875865387
|-
|-
| | 106280
| 125324
| | 648.681640625
| 764.916992187500
| | [[16/11]]
| [[14/9]]
| | 648.682057635243
| 764.915904738350
| | &minus;0.068323
| +0.178168
| | &minus;0.000417010243
| +0.001087449150
|-
|-
| | 115008
| 128203
| | 701.953125
| 782.489013671875
| | [[3/2]]
| [[11/7]]
| | 701.955000865387
| 782.492035895632
| | &minus;0.307342
| −0.495161
| | &minus;0.001875865387
| −0.003022223757
|-
|-
| | 125324
| 133314
| | 764.9169921875
| 813.684082031250
| | [[14/9]]
| [[8/5]]
| | 764.915904738350
| 813.686286135165
| | +0.178168
| −0.361120
| | +0.001087449150
| −0.002204103915
|-
|-
| | 128203
| 139689
| | 782.489013671875
| 852.593994140625
| | [[11/7]]
| [[18/11]]
| | 782.492035895632
| 852.592059366018
| | &minus;0.495161
| +0.316993
| | &minus;0.003022223757
| +0.001934774607
|-
|-
| | 133314
| 144893
| | 813.68408203125
| 884.356689453125
| | [[8/5]]
| [[5/3]]
| | 813.686286135165
| 884.358712999447
| | &minus;0.361120
| −0.331538
| | &minus;0.002204103915
| −0.002023546322
|-
|-
| | 139689
| 152884
| | 852.593994140625
| 933.129882812500
| | [[18/11]]
| [[12/7]]
| | 852.592059366018
| 933.129094396262
| | +0.316993
| +0.129174
| | +0.001934774607
| +0.000788416238
|-
|-
| | 144893
| 158732
| | 884.356689453125
| 968.823242187500
| | [[5/3]]
| [[7/4]]
| | 884.358712999447
| 968.825906469125
| | &minus;0.331538
| −0.436516
| | &minus;0.002023546322
| −0.002664281625
|-
|-
| | 152884
| 163199
| | 933.1298828125
| 996.087646484375
| | [[12/7]]
| [[16/9]]
| | 933.129094396262
| 996.089998269225
| | +0.129174
| −0.385316
| | +0.000788416238
| −0.002351784850
|-
|-
| | 158732
| 166723
| | 968.8232421875
| 1017.596435546875
| | [[7/4]]
| [[9/5]]
| | 968.825906469125
| 1017.596287865940
| | &minus;0.436516
| +0.024196
| | &minus;0.002664281625
| +0.000147680935
|-
|-
| | 163199
| 169574
| | 996.087646484375
| 1034.997558593750
| | [[16/9]]
| [[20/11]]
| | 996.089998269225
| 1034.995771500078
| | &minus;0.385316
| +0.292797
| | &minus;0.002351784850
| +0.001787093672
|-
|-
| | 166723
| 171928
| | 1017.596435546875
| 1049.365234375000
| | [[9/5]]
| [[11/6]]
| | 1017.596287865940
| 1049.362941499369
| | +0.024196
| +0.375665
| | +0.000147680935
| +0.002292875631
|-
|-
| | 196608
| 196608
| | 1200.000000000000
| 1200.000000000000
| | [[Octave|2/1]]
| [[Octave|2/1]]
| | 1200.000000000000
| 1200.000000000000
| | &plusmn;0.000000
| ±0.000000
| | &plusmn;0.000000000000
| ±0.000000000000
|}
|}


==See also==
== See also ==
*[[Interval size measure]]
* [[Interval size measure]]
* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
** [[24edo]] (1mu tuning)
** [[48edo]] (2mu tuning)
** [[96edo]] (3mu tuning)
** [[192edo]] (4mu tuning)
** [[384edo]] (5mu tuning)
** [[768edo]] (6mu tuning)
** [[1536edo]] (7mu tuning)
** [[3072edo]] (8mu tuning)
** [[6144edo]] (9mu tuning)
** [[12288edo]] (10mu tuning)
** [[24576edo]] (11mu tuning)
** [[49152edo]] (12mu tuning)
** [[98304edo]] (13mu tuning)


[[Category:Edo]]
== External links ==
[[Category:Theory]]
* [http://tonalsoft.com/enc/number/14mu.aspx 14mu / tetradekamu] on [[Tonalsoft Encyclopedia]]
Retrieved from "https://en.xen.wiki/w/196608edo"