3072edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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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. | 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 === | ||
3072edo | 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 === | === Prime harmonics === | ||
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== 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]] | ||
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| {{monzo| 37 25 -33 }}, {{monzo| 161 -84 -12 }} | | {{monzo| 37 25 -33 }}, {{monzo| 161 -84 -12 }} | ||
| {{mapping| 3072 4869 7133 }} | | {{mapping| 3072 4869 7133 }} | ||
| | | −0.002 | ||
| 0.003 | | 0.003 | ||
| | | | ||
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| 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 }} | | {{mapping| 3072 4869 7133 8624 }} | ||
| 0.006 | | +0.006 | ||
| 0.013 | | 0.013 | ||
| | | | ||
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| 9801/9800, 151263/151250, 184549376/184528125, 73525096183/73466403840 | | 9801/9800, 151263/151250, 184549376/184528125, 73525096183/73466403840 | ||
| {{mapping| 3072 4869 7133 8624 10627 }} | | {{mapping| 3072 4869 7133 8624 10627 }} | ||
| 0.013 | | +0.013 | ||
| 0.019 | | 0.019 | ||
| | | | ||
| Line 51: | Line 52: | ||
| 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 }} | | {{mapping| 3072 4869 7133 8624 10627 11638 }} | ||
| 0.006 | | +0.006 | ||
| 0.022 | | 0.022 | ||
| | | | ||
|} | |} | ||
== Music == | |||
; [[Eliora]] | |||
* [https://www.youtube.com/watch?v=sksIgNTJ-XY ''Etude for Celtic Harp in Whoosh''] (2023) | |||
== See also == | == See also == | ||
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** [[196608edo]] (14mu tuning) | ** [[196608edo]] (14mu tuning) | ||
== | == External links == | ||
* [http://tonalsoft.com/enc/number/8mu.aspx 8mu / oktamu] on [[Tonalsoft Encyclopedia]] | |||
* [ | |||
[[Category:Listen]] | [[Category:Listen]] | ||