3072edo: Difference between revisions
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[[ | == 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. | |||
=== As a tuning standard === | |||
A step of 3072edo is known as an '''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 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 === | |||
{{Harmonics in equal|3072}} | |||
=== Subsets and supersets === | |||
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 == | |||
{| class="wikitable center-4 center-5 center-6" | |||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve Stretch (¢) | |||
! colspan="2" | Tuning Error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3.5 | |||
| {{monzo| 37 25 -33 }}, {{monzo| 161 -84 -12 }} | |||
| {{mapping| 3072 4869 7133 }} | |||
| −0.002 | |||
| 0.003 | |||
| | |||
|- | |||
| 2.3.5.7 | |||
| 250047/250000, {{monzo| -2 -25 1 14 }}, {{monzo| -53 -1 9 12 }} | |||
| {{mapping| 3072 4869 7133 8624 }} | |||
| +0.006 | |||
| 0.013 | |||
| | |||
|- | |||
| 2.3.5.7.11 | |||
| 9801/9800, 151263/151250, 184549376/184528125, 73525096183/73466403840 | |||
| {{mapping| 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 | |||
| {{mapping| 3072 4869 7133 8624 10627 11638 }} | |||
| +0.006 | |||
| 0.022 | |||
| | |||
|} | |||
== Music == | |||
; [[Eliora]] | |||
* [https://www.youtube.com/watch?v=sksIgNTJ-XY ''Etude for Celtic Harp in Whoosh''] (2023) | |||
== See also == | |||
* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units | |||
** [[24edo]] (1mu tuning) | |||
** [[48edo]] (2mu tuning) | |||
** [[96edo]] (3mu tuning) | |||
** [[192edo]] (4mu tuning) | |||
** [[384edo]] (5mu tuning) | |||
** [[768edo]] (6mu tuning) | |||
** [[1536edo]] (7mu tuning) | |||
** [[6144edo]] (9mu tuning) | |||
** [[12288edo]] (10mu tuning) | |||
** [[24576edo]] (11mu tuning) | |||
** [[49152edo]] (12mu tuning) | |||
** [[98304edo]] (13mu tuning) | |||
** [[196608edo]] (14mu tuning) | |||
== External links == | |||
* [http://tonalsoft.com/enc/number/8mu.aspx 8mu / oktamu] on [[Tonalsoft Encyclopedia]] | |||
[[Category:Listen]] | |||