3072edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|3072}} | {{EDO intro|3072}} | ||
Although consistent to the 11-limit, it makes more sense to actually see 3072edo as a 2.3.5.7.13 subgroup tuning, due to lower relative error. There it tempers out 140625/140608 and 1990656/1990625. Overall in the 13-limit, the patent val still has smaller errors than any other val despite incosistency. In higher limits, it is not as impressive, with only [[53/32]] being 17% off and 2.3.5.67.71 subgroup having less than 4% error. | == Theory == | ||
3072edo is [[consistent]] to the [[11-odd-limit]] and it is an extremely accurate 5-limit tuning, tempering out {{monzo| 37 25 -33 }} (whoosh) and {{monzo| 161 -84 -12 }} ([[Kirnberger's atom|atom]]) in the 5-limit; 250047/250000 ([[landscape comma]]), {{monzo| -2 -25 1 14 }}, and {{monzo| -53 -1 9 12 }}; in the 7-limit; [[9801/9800]], 151263/151250, 184549376/184528125, and 73525096183/73466403840 in the 11-limit. | |||
Although consistent to the 11-odd-limit, it makes more sense to actually see 3072edo as a 2.3.5.7.13 [[subgroup]] tuning, due to lower relative error. There it tempers out 140625/140608 and 1990656/1990625. Overall in the 13-limit, the [[patent val]] still has smaller errors than any other val despite incosistency. In higher limits, it is not as impressive, with only [[53/32]] being 17% off and 2.3.5.67.71 subgroup having less than 4% error. | |||
=== Significance in digital audio software === | === Significance in digital audio software === | ||
3072edo's step is known as '''Octamu''' (eighth MIDI-resolution unit, 8mu, 2<sup>8</sup> = 256 equal divisions of the [[12edo]] semitone). The internal data structure of the 8mu 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 −) showing the direction of the pitch-bend up or down, and 5 other bits which are not 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. | 3072edo's step is known as '''Octamu''' (eighth MIDI-resolution unit, 8mu, 2<sup>8</sup> = 256 equal divisions of the [[12edo]] semitone). The internal data structure of the 8mu 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 −) showing the direction of the pitch-bend up or down, and 5 other bits which are not 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 === | === Prime harmonics === | ||
{{ | {{Harmonics in equal|3072}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
3072 factors as | 3072 factors as {{factorization|3072}}, with subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, and 1536 }}. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" |[[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
8ve Stretch (¢) | ! colspan="2" | Tuning Error | ||
! colspan="2" |Tuning Error | |||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo|37 25 -33}}, {{monzo|161 -84 -12}} | | {{monzo| 37 25 -33 }}, {{monzo| 161 -84 -12 }} | ||
| | | {{mapping| 3072 4869 7133 }} | ||
| -0.002 | | -0.002 | ||
|0.003 | | 0.003 | ||
| | | | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|250047/250000, {{monzo|-2 -25 1 14}}, {{monzo|-53 -1 9 12}} | | 250047/250000, {{monzo| -2 -25 1 14 }}, {{monzo| -53 -1 9 12 }} | ||
| | | {{mapping| 3072 4869 7133 8624 }} | ||
|0.006 | | 0.006 | ||
|0.013 | | 0.013 | ||
| | | | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|9801/9800, 151263/151250, 184549376/184528125, 73525096183/73466403840 | | 9801/9800, 151263/151250, 184549376/184528125, 73525096183/73466403840 | ||
| | | {{mapping| 3072 4869 7133 8624 10627 }} | ||
|0.013 | | 0.013 | ||
|0.019 | | 0.019 | ||
| | | | ||
|- | |- | ||
|2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
|9801/9800, 140625/140608, 151263/151250, 196625/196608, 3327500/3326427 | | 9801/9800, 140625/140608, 151263/151250, 196625/196608, 3327500/3326427 | ||
| | | {{mapping| 3072 4869 7133 8624 10627 11638 }} | ||
|0.006 | | 0.006 | ||
|0.022 | | 0.022 | ||
| | | | ||
|} | |} | ||
==See also== | |||
*[[Equal-step tuning|Equal multiplications]] of MIDI-resolution units | == See also == | ||
**[[24edo]] (1mu tuning) | * [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units | ||
**[[48edo]] (2mu tuning) | ** [[24edo]] (1mu tuning) | ||
**[[96edo]] (3mu tuning) | ** [[48edo]] (2mu tuning) | ||
**[[192edo]] (4mu tuning) | ** [[96edo]] (3mu tuning) | ||
**[[384edo]] (5mu tuning) | ** [[192edo]] (4mu tuning) | ||
**[[768edo]] (6mu tuning) | ** [[384edo]] (5mu tuning) | ||
**[[1536edo]] (7mu tuning) | ** [[768edo]] (6mu tuning) | ||
**[[6144edo]] (9mu tuning) | ** [[1536edo]] (7mu tuning) | ||
**[[12288edo]] (10mu tuning) | ** [[6144edo]] (9mu tuning) | ||
**[[24576edo]] (11mu tuning) | ** [[12288edo]] (10mu tuning) | ||
**[[49152edo]] (12mu tuning) | ** [[24576edo]] (11mu tuning) | ||
**[[98304edo]] (13mu tuning) | ** [[49152edo]] (12mu tuning) | ||
**[[196608edo]] (14mu tuning) | ** [[98304edo]] (13mu tuning) | ||
** [[196608edo]] (14mu tuning) | |||
== Music == | == Music == | ||
; [[Eliora]] | ; [[Eliora]] | ||
* | * [https://www.youtube.com/watch?v=sksIgNTJ-XY ''Etude for Celtic Harp in Whoosh''] (2023) | ||
[[Category:Listen]] | [[Category:Listen]] | ||
Revision as of 09:58, 30 October 2023
| ← 3071edo | 3072edo | 3073edo → |
Theory
3072edo is consistent to the 11-odd-limit and it is an extremely accurate 5-limit tuning, tempering out [37 25 -33⟩ (whoosh) and [161 -84 -12⟩ (atom) in the 5-limit; 250047/250000 (landscape comma), [-2 -25 1 14⟩, and [-53 -1 9 12⟩; in the 7-limit; 9801/9800, 151263/151250, 184549376/184528125, and 73525096183/73466403840 in the 11-limit.
Although consistent to the 11-odd-limit, it makes more sense to actually see 3072edo as a 2.3.5.7.13 subgroup tuning, due to lower relative error. There it tempers out 140625/140608 and 1990656/1990625. Overall in the 13-limit, the patent val still has smaller errors than any other val despite incosistency. In higher limits, it is not as impressive, with only 53/32 being 17% off and 2.3.5.67.71 subgroup having less than 4% error.
Significance in digital audio software
3072edo's step is known as Octamu (eighth MIDI-resolution unit, 8mu, 28 = 256 equal divisions of the 12edo semitone). The internal data structure of the 8mu 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 −) showing the direction of the pitch-bend up or down, and 5 other bits which are not 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
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.002 | +0.014 | -0.076 | -0.146 | +0.097 | +0.123 | +0.143 | -0.149 | +0.110 | -0.114 |
| Relative (%) | +0.0 | -0.5 | +3.7 | -19.4 | -37.4 | +24.9 | +31.4 | +36.7 | -38.2 | +28.2 | -29.1 | |
| Steps (reduced) |
3072 (0) |
4869 (1797) |
7133 (989) |
8624 (2480) |
10627 (1411) |
11368 (2152) |
12557 (269) |
13050 (762) |
13896 (1608) |
14924 (2636) |
15219 (2931) | |
Subsets and supersets
3072 factors as 210 × 3, with subset edos 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, and 1536.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [37 25 -33⟩, [161 -84 -12⟩ | [⟨3072 4869 7133]] | -0.002 | 0.003 | |
| 2.3.5.7 | 250047/250000, [-2 -25 1 14⟩, [-53 -1 9 12⟩ | [⟨3072 4869 7133 8624]] | 0.006 | 0.013 | |
| 2.3.5.7.11 | 9801/9800, 151263/151250, 184549376/184528125, 73525096183/73466403840 | [⟨3072 4869 7133 8624 10627]] | 0.013 | 0.019 | |
| 2.3.5.7.11.13 | 9801/9800, 140625/140608, 151263/151250, 196625/196608, 3327500/3326427 | [⟨3072 4869 7133 8624 10627 11638]] | 0.006 | 0.022 | |
See also
- Equal multiplications of MIDI-resolution units