3072edo: Difference between revisions

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rtp table, i'll fill in the relative error a bit later
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3072edo is consistent to the [[11-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, {{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.
3072edo is consistent to the [[11-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, {{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-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. 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.
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.
=== 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 &minus;) 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 &minus;) 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.
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3072 factors as 2<sup>10</sup> x 3, with subset edos {{EDOs|1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536}}.
3072 factors as 2<sup>10</sup> x 3, with subset edos {{EDOs|1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536}}.


== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal
8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3.5
|{{monzo|37 25 -33}}, {{monzo|161 -84 -12}}
|[{{val|3072 4869 7133}}]
| -0.002
|0.003
|
|-
|2.3.5.7
|250047/250000, {{monzo|-2 -25 1 14}}, {{monzo|-53 -1 9 12}}
|[{{val|3072 4869 7133 8624}}]
|0.006
|0.013
|
|-
|2.3.5.7.11
|9801/9800, 151263/151250, 184549376/184528125, 73525096183/73466403840
|[{{val|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
|[{{val|3072 4869 7133 8624 10627 11638}}]
|0.006
|0.022
|
|}
==See also==
==See also==
*[[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
*[[Equal-step tuning|Equal multiplications]] of MIDI-resolution units

Revision as of 21:40, 12 July 2023

← 3071edo 3072edo 3073edo →
Prime factorization 210 × 3
Step size 0.390625 ¢ 
Fifth 1797\3072 (701.953 ¢) (→ 599\1024)
Semitones (A1:m2) 291:231 (113.7 ¢ : 90.23 ¢)
Consistency limit 11
Distinct consistency limit 11

Template:EDO intro

Theory

3072edo is consistent to the 11-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, [-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-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

Approximation of prime harmonics in 3072edo
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 x 3, with subset edos 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 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

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