196608edo: Difference between revisions
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Clarify the context |
Move the 14mu info to the theory section; misc. cleanup |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|196608}} | {{EDO intro|196608}} | ||
== 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 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. | |||
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 | 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 === | ||
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==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) | ||
== External links == | |||
* [http://tonalsoft.com/enc/number/14mu.aspx Tonalsoft Encyclopedia | ''14mu / tetradekamu''] | |||
Revision as of 08:09, 7 November 2023
| ← 196607edo | 196608edo | 196609edo → |
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
A step of 196608edo is known as the MIDI Tuning Standard unit (MTSU) or 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
| 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 | 111.73095703125 | 16/15 | 111.731285269778 | −0.053779 | −0.000328238528 |
| 19570 | 119.44580078125 | 15/14 | 119.442808261097 | +0.490295 | +0.002992520153 |
| 21020 | 128.2958984375 | 14/13 | 128.298244699814 | −0.384412 | −0.002346262314 |
| 22704 | 138.57421875 | 13/12 | 138.572660903923 | +0.255238 | +0.001557846077 |
| 24680 | 150.634765625 | 12/11 | 150.637058500631 | −0.375665 | −0.002292875631 |
| 27034 | 165.00244140625 | 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 | 231.1767578125 | 8/7 | 231.174093530875 | +0.436516 | +0.002664281625 |
| 40590 | 247.74169921875 | 15/13 | 247.741052960912 | +0.105883 | +0.000646257838 |
| 43724 | 266.8701171875 | 7/6 | 266.870905603738 | −0.129174 | −0.000788416238 |
| 47384 | 289.208984375 | 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 | 359.47265625 | 16/13 | 359.472338230689 | +0.052104 | +0.000318019311 |
| 63294 | 386.31591796875 | 5/4 | 386.313713864835 | +0.361120 | +0.002204103915 |
| 68405 | 417.510986328125 | 14/11 | 417.507964104368 | +0.495161 | +0.003022223757 |
| 71284 | 435.0830078125 | 9/7 | 435.084095261650 | −0.178168 | −0.001087449150 |
| 74418 | 454.21142578125 | 13/10 | 454.213947904476 | −0.413225 | −0.002522123226 |
| 81600 | 498.046875 | 4/3 | 498.044999134613 | +0.307342 | +0.001875865387 |
| 87974 | 536.95068359375 | 15/11 | 536.950772365466 | −0.014544 | −0.000088771716 |
| 90328 | 551.318359375 | 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 | 648.681640625 | 16/11 | 648.682057635243 | −0.068323 | −0.000417010243 |
| 115008 | 701.953125 | 3/2 | 701.955000865387 | −0.307342 | −0.001875865387 |
| 125324 | 764.9169921875 | 14/9 | 764.915904738350 | +0.178168 | +0.001087449150 |
| 128203 | 782.489013671875 | 11/7 | 782.492035895632 | −0.495161 | −0.003022223757 |
| 133314 | 813.68408203125 | 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 | 933.1298828125 | 12/7 | 933.129094396262 | +0.129174 | +0.000788416238 |
| 158732 | 968.8232421875 | 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 | 1034.99755859375 | 20/11 | 1034.995771500078 | +0.292797 | +0.001787093672 |
| 171928 | 1049.365234375 | 11/6 | 1049.362941499369 | +0.375665 | +0.002292875631 |
| 196608 | 1200.000000000000 | 2/1 | 1200.000000000000 | ±0.000000 | ±0.000000000000 |
See also
- Interval size measure
- Equal multiplications of MIDI-resolution units