196608edo: Difference between revisions

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{{Software tuning}}
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|196608}} 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.
{{ED intro}}


== Theory ==
== Theory ==
Line 6: Line 7:


=== As a tuning standard ===
=== As a tuning standard ===
The main application of 196608edo is not as a compositional device, but as a technical tuning standard. If we adopt direct approximation, some JI intervals are indeed improved. Another usage that is not commonly seen in other edos is to approximate smaller edos, since they are quantized into the grid of this edo. It cannot approximate any larger edos or any edos of the same order of magnitude.  
[[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 ===
=== Odd harmonics ===
Line 13: Line 20:
== Selected intervals ==
== Selected intervals ==
Below is a list for just intervals.
Below is a list for just intervals.
{| class="wikitable"
 
{| class="wikitable right-1 right-2 center-3 right-4 right-5"
|-
|-
! rowspan="2"| #
! rowspan="2"| #
Line 24: Line 32:
! Cents
! Cents
|-
|-
| | 0
| 0
| | 0.000000000000
| 0.000000000000
| | [[1/1]]
| [[1/1]]
| | 0.000000000000
| 0.000000000000
| | &plusmn;0.000000
| ±0.000000
| | &plusmn;0.000000000000
| ±0.000000000000
|-
|-
| | 18306
| 18306
| | 111.73095703125
| 111.730957031250
| | [[16/15]]
| [[16/15]]
| | 111.731285269778
| 111.731285269778
| | &minus;0.053779
| −0.053779
| | &minus;0.000328238528
| −0.000328238528
|-
|-
| | 19570
| 19570
| | 119.44580078125
| 119.445800781250
| | [[15/14]]
| [[15/14]]
| | 119.442808261097
| 119.442808261097
| | +0.490295
| +0.490295
| | +0.002992520153
| +0.002992520153
|-
|-
| | 21020
| 21020
| | 128.2958984375
| 128.295898437500
| | [[14/13]]
| [[14/13]]
| | 128.298244699814
| 128.298244699814
| | &minus;0.384412
| −0.384412
| | &minus;0.002346262314
| −0.002346262314
|-
|-
| | 22704
| 22704
| | 138.57421875
| 138.574218750000
| | [[13/12]]
| [[13/12]]
| | 138.572660903923
| 138.572660903923
| | +0.255238
| +0.255238
| | +0.001557846077
| +0.001557846077
|-
|-
| | 24680
| 24680
| | 150.634765625
| 150.634765625000
| | [[12/11]]
| [[12/11]]
| | 150.637058500631
| 150.637058500631
| | &minus;0.375665
| −0.375665
| | &minus;0.002292875631
| −0.002292875631
|-
|-
| | 27034
| 27034
| | 165.00244140625
| 165.002441406250
| | [[11/10]]
| [[11/10]]
| | 165.004228499922
| 165.004228499922
| | &minus;0.292797
| −0.292797
| | &minus;0.001787093672
| −0.001787093672
|-
|-
| | 29885
| 29885
| | 182.403564453125
| 182.403564453125
| | [[10/9]]
| [[10/9]]
| | 182.403712134060
| 182.403712134060
| | &minus;0.024196
| −0.024196
| | &minus;0.000147680935
| −0.000147680935
|-
|-
| | 33409
| 33409
| | 203.912353515625
| 203.912353515625
| | [[9/8]]
| [[9/8]]
| | 203.910001730775
| 203.910001730775
| | +0.385316
| +0.385316
| | +0.002351784850
| +0.002351784850
|-
|-
| | 37876
| 37876
| | 231.1767578125
| 231.176757812500
| | [[8/7]]
| [[8/7]]
| | 231.174093530875
| 231.174093530875
| | +0.436516
| +0.436516
| | +0.002664281625
| +0.002664281625
|-
|-
| | 40590
| 40590
| | 247.74169921875
| 247.741699218750
| | [[15/13]]
| [[15/13]]
| | 247.741052960912
| 247.741052960912
| | +0.105883
| +0.105883
| | +0.000646257838
| +0.000646257838
|-
|-
| | 43724
| 43724
| | 266.8701171875
| 266.870117187500
| | [[7/6]]
| [[7/6]]
| | 266.870905603738
| 266.870905603738
| | &minus;0.129174
| −0.129174
| | &minus;0.000788416238
| −0.000788416238
|-
|-
| | 47384
| 47384
| | 289.208984375
| 289.208984375000
| | [[13/11]]
| [[13/11]]
| | 289.209719404554
| 289.209719404554
| | &minus;0.120427
| −0.120427
| | &minus;0.000735029554
| −0.000735029554
|-
|-
| | 51715
| 51715
| | 315.643310546875
| 315.643310546875
| | [[6/5]]
| [[6/5]]
| | 315.641287000553
| 315.641287000553
| | +0.331538
| +0.331538
| | +0.002023546322
| +0.002023546322
|-
|-
| | 56919
| 56919
| | 347.406005859375
| 347.406005859375
| | [[11/9]]
| [[11/9]]
| | 347.407940633982
| 347.407940633982
| | &minus;0.316993
| −0.316993
| | &minus;0.001934774607
| −0.001934774607
|-
|-
| | 58896
| 58896
| | 359.47265625
| 359.472656250000
| | [[16/13]]
| [[16/13]]
| | 359.472338230689
| 359.472338230689
| | +0.052104
| +0.052104
| | +0.000318019311
| +0.000318019311
|-
|-
| | 63294
| 63294
| | 386.31591796875
| 386.315917968750
| | [[5/4]]
| [[5/4]]
| | 386.313713864835
| 386.313713864835
| | +0.361120
| +0.361120
| | +0.002204103915
| +0.002204103915
|-
|-
| | 68405
| 68405
| | 417.510986328125
| 417.510986328125
| | [[14/11]]
| [[14/11]]
| | 417.507964104368
| 417.507964104368
| | +0.495161
| +0.495161
| | +0.003022223757
| +0.003022223757
|-
|-
| | 71284
| 71284
| | 435.0830078125
| 435.083007812500
| | [[9/7]]
| [[9/7]]
| | 435.084095261650
| 435.084095261650
| | &minus;0.178168
| −0.178168
| | &minus;0.001087449150
| −0.001087449150
|-
|-
| | 74418
| 74418
| | 454.21142578125
| 454.211425781250
| | [[13/10]]
| [[13/10]]
| | 454.213947904476
| 454.213947904476
| | &minus;0.413225
| −0.413225
| | &minus;0.002522123226
| −0.002522123226
|-
|-
| | 81600
| 81600
| | 498.046875
| 498.046875000000
| | [[4/3]]
| [[4/3]]
| | 498.044999134613
| 498.044999134613
| | +0.307342
| +0.307342
| | +0.001875865387
| +0.001875865387
|-
|-
| | 87974
| 87974
| | 536.95068359375
| 536.950683593750
| | [[15/11]]
| [[15/11]]
| | 536.950772365466
| 536.950772365466
| | &minus;0.014544
| −0.014544
| | &minus;0.000088771716
| −0.000088771716
|-
|-
| | 90328
| 90328
| | 551.318359375
| 551.318359375000
| | [[11/8]]
| [[11/8]]
| | 551.317942364757
| 551.317942364757
| | +0.068323
| +0.068323
| | +0.000417010243
| +0.000417010243
|-
|-
| | 92305
| 92305
| | 563.385009765625
| 563.385009765625
| | [[18/13]]
| [[18/13]]
| | 563.382339961464
| 563.382339961464
| | +0.437421
| +0.437421
| | +0.002669804161
| +0.002669804161
|-
|-
| | 95439
| 95439
| | 582.513427734375
| 582.513427734375
| | [[7/5]]
| [[7/5]]
| | 582.512192604290
| 582.512192604290
| | +0.202364
| +0.202364
| | +0.001235130085
| +0.001235130085
|-
|-
| | 101169
| 101169
| | 617.486572265625
| 617.486572265625
| | [[10/7]]
| [[10/7]]
| | 617.487807395710
| 617.487807395710
| | &minus;0.202364
| −0.202364
| | &minus;0.001235130085
| −0.001235130085
|-
|-
| | 106280
| 106280
| | 648.681640625
| 648.681640625000
| | [[16/11]]
| [[16/11]]
| | 648.682057635243
| 648.682057635243
| | &minus;0.068323
| −0.068323
| | &minus;0.000417010243
| −0.000417010243
|-
|-
| | 115008
| 115008
| | 701.953125
| 701.953125000000
| | [[3/2]]
| [[3/2]]
| | 701.955000865387
| 701.955000865387
| | &minus;0.307342
| −0.307342
| | &minus;0.001875865387
| −0.001875865387
|-
|-
| | 125324
| 125324
| | 764.9169921875
| 764.916992187500
| | [[14/9]]
| [[14/9]]
| | 764.915904738350
| 764.915904738350
| | +0.178168
| +0.178168
| | +0.001087449150
| +0.001087449150
|-
|-
| | 128203
| 128203
| | 782.489013671875
| 782.489013671875
| | [[11/7]]
| [[11/7]]
| | 782.492035895632
| 782.492035895632
| | &minus;0.495161
| −0.495161
| | &minus;0.003022223757
| −0.003022223757
|-
|-
| | 133314
| 133314
| | 813.68408203125
| 813.684082031250
| | [[8/5]]
| [[8/5]]
| | 813.686286135165
| 813.686286135165
| | &minus;0.361120
| −0.361120
| | &minus;0.002204103915
| −0.002204103915
|-
|-
| | 139689
| 139689
| | 852.593994140625
| 852.593994140625
| | [[18/11]]
| [[18/11]]
| | 852.592059366018
| 852.592059366018
| | +0.316993
| +0.316993
| | +0.001934774607
| +0.001934774607
|-
|-
| | 144893
| 144893
| | 884.356689453125
| 884.356689453125
| | [[5/3]]
| [[5/3]]
| | 884.358712999447
| 884.358712999447
| | &minus;0.331538
| −0.331538
| | &minus;0.002023546322
| −0.002023546322
|-
|-
| | 152884
| 152884
| | 933.1298828125
| 933.129882812500
| | [[12/7]]
| [[12/7]]
| | 933.129094396262
| 933.129094396262
| | +0.129174
| +0.129174
| | +0.000788416238
| +0.000788416238
|-
|-
| | 158732
| 158732
| | 968.8232421875
| 968.823242187500
| | [[7/4]]
| [[7/4]]
| | 968.825906469125
| 968.825906469125
| | &minus;0.436516
| −0.436516
| | &minus;0.002664281625
| −0.002664281625
|-
|-
| | 163199
| 163199
| | 996.087646484375
| 996.087646484375
| | [[16/9]]
| [[16/9]]
| | 996.089998269225
| 996.089998269225
| | &minus;0.385316
| −0.385316
| | &minus;0.002351784850
| −0.002351784850
|-
|-
| | 166723
| 166723
| | 1017.596435546875
| 1017.596435546875
| | [[9/5]]
| [[9/5]]
| | 1017.596287865940
| 1017.596287865940
| | +0.024196
| +0.024196
| | +0.000147680935
| +0.000147680935
|-
|-
| | 169574
| 169574
| | 1034.99755859375
| 1034.997558593750
| | [[20/11]]
| [[20/11]]
| | 1034.995771500078
| 1034.995771500078
| | +0.292797
| +0.292797
| | +0.001787093672
| +0.001787093672
|-
|-
| | 171928
| 171928
| | 1049.365234375
| 1049.365234375000
| | [[11/6]]
| [[11/6]]
| | 1049.362941499369
| 1049.362941499369
| | +0.375665
| +0.375665
| | +0.002292875631
| +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
* [[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)
**[[98304edo]] (13mu tuning)
** [[98304edo]] (13mu tuning)


[[Category:Equal divisions of the octave|######]] <!-- 6-digit number -->
== External links ==
* [http://tonalsoft.com/enc/number/14mu.aspx 14mu / tetradekamu] on [[Tonalsoft Encyclopedia]]

Latest revision as of 18:02, 8 September 2025

This page presents a primarily software-related tuning.

This tuning is more notable for use in software rather than for its musical properties.

← 196607edo 196608edo 196609edo →
Prime factorization 216 × 3
Step size 0.00610352 ¢ 
Fifth 115008\196608 (701.953 ¢) (→ 599\1024)
Semitones (A1:m2) 18624:14784 (113.7 ¢ : 90.23 ¢)
Consistency limit 3
Distinct consistency limit 3

196608 equal divisions of the octave (abbreviated 196608edo or 196608ed2), also called 196608-tone equal temperament (196608tet) or 196608 equal temperament (196608et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 196608 equal parts of about 0.0061 ¢ each. Each step represents a frequency ratio of 21/196608, or the 196608th root of 2.

Theory

196608edo is 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.

As a tuning standard

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, 214 = 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

Approximation of odd harmonics in 196608edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.00188 +0.00220 -0.00266 +0.00235 +0.00042 -0.00032 +0.00033 +0.00065 +0.00285 +0.00156 -0.00286
Relative (%) -30.7 +36.1 -43.7 +38.5 +6.8 -5.2 +5.4 +10.6 +46.7 +25.6 -46.9
Steps
(reduced) 		311616
(115008) 		456510
(63294) 		551948
(158732) 		623233
(33409) 		680152
(90328) 		727536
(137712) 		768126
(178302) 		803628
(17196) 		835177
(48745) 		863565
(77133) 		889368
(102936)

Selected intervals

Below is a list for just intervals.

# Cents JI Interval Error
(Steps) 		Error
(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
43724 266.870117187500 7/6 266.870905603738 −0.129174 −0.000788416238
47384 289.208984375000 13/11 289.209719404554 −0.120427 −0.000735029554
51715 315.643310546875 6/5 315.641287000553 +0.331538 +0.002023546322
56919 347.406005859375 11/9 347.407940633982 −0.316993 −0.001934774607
58896 359.472656250000 16/13 359.472338230689 +0.052104 +0.000318019311
63294 386.315917968750 5/4 386.313713864835 +0.361120 +0.002204103915
68405 417.510986328125 14/11 417.507964104368 +0.495161 +0.003022223757
71284 435.083007812500 9/7 435.084095261650 −0.178168 −0.001087449150
74418 454.211425781250 13/10 454.213947904476 −0.413225 −0.002522123226
81600 498.046875000000 4/3 498.044999134613 +0.307342 +0.001875865387
87974 536.950683593750 15/11 536.950772365466 −0.014544 −0.000088771716
90328 551.318359375000 11/8 551.317942364757 +0.068323 +0.000417010243
92305 563.385009765625 18/13 563.382339961464 +0.437421 +0.002669804161
95439 582.513427734375 7/5 582.512192604290 +0.202364 +0.001235130085
101169 617.486572265625 10/7 617.487807395710 −0.202364 −0.001235130085
106280 648.681640625000 16/11 648.682057635243 −0.068323 −0.000417010243
115008 701.953125000000 3/2 701.955000865387 −0.307342 −0.001875865387
125324 764.916992187500 14/9 764.915904738350 +0.178168 +0.001087449150
128203 782.489013671875 11/7 782.492035895632 −0.495161 −0.003022223757
133314 813.684082031250 8/5 813.686286135165 −0.361120 −0.002204103915
139689 852.593994140625 18/11 852.592059366018 +0.316993 +0.001934774607
144893 884.356689453125 5/3 884.358712999447 −0.331538 −0.002023546322
152884 933.129882812500 12/7 933.129094396262 +0.129174 +0.000788416238
158732 968.823242187500 7/4 968.825906469125 −0.436516 −0.002664281625
163199 996.087646484375 16/9 996.089998269225 −0.385316 −0.002351784850
166723 1017.596435546875 9/5 1017.596287865940 +0.024196 +0.000147680935
169574 1034.997558593750 20/11 1034.995771500078 +0.292797 +0.001787093672
171928 1049.365234375000 11/6 1049.362941499369 +0.375665 +0.002292875631
196608 1200.000000000000 2/1 1200.000000000000 ±0.000000 ±0.000000000000

See also

External links

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