768edo: Difference between revisions
Royalmilktea (talk | contribs) 768edo/6mu, infobox et, harmonics, commas, other mu temps. |
Rework |
||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{EDO intro|768}} | ||
768edo is consistent in the 7-limit | |||
== Theory == | |||
768edo is [[consistent]] in the [[7-odd-limit]]. The equal temperament [[tempering out|tempers out]] the [[mutt comma]] {{monzo| -44 -3 21 }} and the 5-limit [[commatic]] comma {{monzo| -37 38 -10 }} in the 5-limit, and [[Horwell comma|65625/65536]], [[250047/250000]], [[Mitonismic temperaments|5250987/5242880]], {{monzo| -12 -5 11 -2 }}, {{monzo| 7 18 -2 -11 }}, and {{monzo| -36 8 4 5 }} in the 7-limit. | |||
=== As a tuning standard === | |||
Its adjacent step is known as '''hexamu''' (sixth MIDI-resolution unit, 6mu, 2<sup>6</sup> = 64 equal divisions of the [[12edo]] semitone). The internal data structure of the 6mu requires one byte, with the first two bits reserved as flags, one to indicate the byte's status as data, and one to indicate the sign (+ or -) showing the direction of the pitch-bend up or down, and all six of the remaining bits used for the tuning data | |||
=== Odd harmonics === | |||
{{Harmonics in equal|768|intervals=prime}} | |||
=== Subsets and supersets === | |||
Since 768 factors into {{factorization|768}}, 768edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, and 384 }}. | |||
== See also == | == See also == | ||
| Line 19: | Line 30: | ||
** [[98304edo]] (13mu tuning) | ** [[98304edo]] (13mu tuning) | ||
** [[196608edo]] (14mu tuning) | ** [[196608edo]] (14mu tuning) | ||
== External links == | |||
* [http://tonalsoft.com/enc/number/6mu.aspx Tonalsoft Encyclopedia | ''6mu / hexamu / 768-edo''] | |||
Revision as of 07:51, 7 November 2023
| ← 767edo | 768edo | 769edo → |
Theory
768edo is consistent in the 7-odd-limit. The equal temperament tempers out the mutt comma [-44 -3 21⟩ and the 5-limit commatic comma [-37 38 -10⟩ in the 5-limit, and 65625/65536, 250047/250000, 5250987/5242880, [-12 -5 11 -2⟩, [7 18 -2 -11⟩, and [-36 8 4 5⟩ in the 7-limit.
As a tuning standard
Its adjacent step is known as hexamu (sixth MIDI-resolution unit, 6mu, 26 = 64 equal divisions of the 12edo semitone). The internal data structure of the 6mu requires one byte, with the first two bits reserved as flags, one to indicate the byte's status as data, and one to indicate the sign (+ or -) showing the direction of the pitch-bend up or down, and all six of the remaining bits used for the tuning data
Odd harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.393 | -0.376 | -0.076 | +0.245 | +0.097 | -0.268 | -0.638 | -0.149 | +0.110 | +0.277 |
| Relative (%) | +0.0 | -25.1 | -24.1 | -4.9 | +15.7 | +6.2 | -17.1 | -40.8 | -9.6 | +7.1 | +17.7 | |
| Steps (reduced) |
768 (0) |
1217 (449) |
1783 (247) |
2156 (620) |
2657 (353) |
2842 (538) |
3139 (67) |
3262 (190) |
3474 (402) |
3731 (659) |
3805 (733) | |
Subsets and supersets
Since 768 factors into 28 × 3, 768edo has subset edos 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, and 384.
See also
- Equal multiplications of MIDI-resolution units