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196608edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|196608}}
{{ED intro}}


== Theory ==
== Theory ==
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[[File:Tetradecamu Approximation Quality.png|thumb|Approximation quality of smaller edos in tetradecamu, maximum relative error against edo from 1 to 16808. ]]
[[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, 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.
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.  
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.  
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== Selected intervals ==
== Selected intervals ==
Below is a list for just intervals.
Below is a list for just intervals.
{| class="wikitable right-1 right-2 center-3 right-4 right-5"
{| class="wikitable right-1 right-2 center-3 right-4 right-5"
|-
|-
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| [[1/1]]
| [[1/1]]
| 0.000000000000
| 0.000000000000
| &plusmn;0.000000
| ±0.000000
| &plusmn;0.000000000000
| ±0.000000000000
|-
|-
| 18306
| 18306
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| [[16/15]]
| [[16/15]]
| 111.731285269778
| 111.731285269778
| &minus;0.053779
| −0.053779
| &minus;0.000328238528
| −0.000328238528
|-
|-
| 19570
| 19570
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| [[14/13]]
| [[14/13]]
| 128.298244699814
| 128.298244699814
| &minus;0.384412
| −0.384412
| &minus;0.002346262314
| −0.002346262314
|-
|-
| 22704
| 22704
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| [[12/11]]
| [[12/11]]
| 150.637058500631
| 150.637058500631
| &minus;0.375665
| −0.375665
| &minus;0.002292875631
| −0.002292875631
|-
|-
| 27034
| 27034
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| [[11/10]]
| [[11/10]]
| 165.004228499922
| 165.004228499922
| &minus;0.292797
| −0.292797
| &minus;0.001787093672
| −0.001787093672
|-
|-
| 29885
| 29885
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| [[10/9]]
| [[10/9]]
| 182.403712134060
| 182.403712134060
| &minus;0.024196
| −0.024196
| &minus;0.000147680935
| −0.000147680935
|-
|-
| 33409
| 33409
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| [[7/6]]
| [[7/6]]
| 266.870905603738
| 266.870905603738
| &minus;0.129174
| −0.129174
| &minus;0.000788416238
| −0.000788416238
|-
|-
| 47384
| 47384
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| [[13/11]]
| [[13/11]]
| 289.209719404554
| 289.209719404554
| &minus;0.120427
| −0.120427
| &minus;0.000735029554
| −0.000735029554
|-
|-
| 51715
| 51715
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| [[11/9]]
| [[11/9]]
| 347.407940633982
| 347.407940633982
| &minus;0.316993
| −0.316993
| &minus;0.001934774607
| −0.001934774607
|-
|-
| 58896
| 58896
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| [[9/7]]
| [[9/7]]
| 435.084095261650
| 435.084095261650
| &minus;0.178168
| −0.178168
| &minus;0.001087449150
| −0.001087449150
|-
|-
| 74418
| 74418
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| [[13/10]]
| [[13/10]]
| 454.213947904476
| 454.213947904476
| &minus;0.413225
| −0.413225
| &minus;0.002522123226
| −0.002522123226
|-
|-
| 81600
| 81600
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| [[15/11]]
| [[15/11]]
| 536.950772365466
| 536.950772365466
| &minus;0.014544
| −0.014544
| &minus;0.000088771716
| −0.000088771716
|-
|-
| 90328
| 90328
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| [[10/7]]
| [[10/7]]
| 617.487807395710
| 617.487807395710
| &minus;0.202364
| −0.202364
| &minus;0.001235130085
| −0.001235130085
|-
|-
| 106280
| 106280
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| [[16/11]]
| [[16/11]]
| 648.682057635243
| 648.682057635243
| &minus;0.068323
| −0.068323
| &minus;0.000417010243
| −0.000417010243
|-
|-
| 115008
| 115008
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| [[3/2]]
| [[3/2]]
| 701.955000865387
| 701.955000865387
| &minus;0.307342
| −0.307342
| &minus;0.001875865387
| −0.001875865387
|-
|-
| 125324
| 125324
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| [[11/7]]
| [[11/7]]
| 782.492035895632
| 782.492035895632
| &minus;0.495161
| −0.495161
| &minus;0.003022223757
| −0.003022223757
|-
|-
| 133314
| 133314
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| [[8/5]]
| [[8/5]]
| 813.686286135165
| 813.686286135165
| &minus;0.361120
| −0.361120
| &minus;0.002204103915
| −0.002204103915
|-
|-
| 139689
| 139689
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| [[5/3]]
| [[5/3]]
| 884.358712999447
| 884.358712999447
| &minus;0.331538
| −0.331538
| &minus;0.002023546322
| −0.002023546322
|-
|-
| 152884
| 152884
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| [[7/4]]
| [[7/4]]
| 968.825906469125
| 968.825906469125
| &minus;0.436516
| −0.436516
| &minus;0.002664281625
| −0.002664281625
|-
|-
| 163199
| 163199
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| [[16/9]]
| [[16/9]]
| 996.089998269225
| 996.089998269225
| &minus;0.385316
| −0.385316
| &minus;0.002351784850
| −0.002351784850
|-
|-
| 166723
| 166723
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| [[Octave|2/1]]
| [[Octave|2/1]]
| 1200.000000000000
| 1200.000000000000
| &plusmn;0.000000
| ±0.000000
| &plusmn;0.000000000000
| ±0.000000000000
|}
|}


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