3072edo: Difference between revisions

Line 4:

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 ===

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.

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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}}.

== 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==

*[[Equal-step tuning|Equal multiplications]] of MIDI-resolution units

@@ Line 4: / Line 4: @@
 edo 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 ===
 edo'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.
@@ Line 13: / Line 13: @@
 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
+ve 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==
 *[[Equal-step tuning|Equal multiplications]] of MIDI-resolution units