783edo: Difference between revisions
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== Theory == | |||
783edo provides solid approximations of the 5-limit, the 7-limit and to some extent the 11-limit, and is [[consistent]] up to the [[11-odd-limit]], making it a flexible and versatile tuning. The equal temperament [[tempering out|tempers out]] the [[ennealimma]] and the [[counterschisma]] in the 5-limit; [[2401/2400]], and [[4375/4374]] in the 7-limit. It [[support]]s [[counterschismic]], [[dodifo]], [[ennealimmal]], [[raider]], [[vavoom]], and [[Very high accuracy temperaments #Francium|francium]]. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|783}} | |||
=== Subsets and supersets === | |||
Since 783 factors into {{factorization|783}}, 783edo has subset edos {{EDOs| 3, 9, 27, 29, 87, and 261 }}. | |||
{ | == Regular temperament properties == | ||
=== Rank-2 temperaments === | |||
[[ | {| class="wikitable center-all left-5" | ||
[[ | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
[[ | |- | ||
[[ | ! Periods<br />per 8ve | ||
[[ | ! Generator* | ||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 73\783 | |||
| 111.877 | |||
| 16/15 | |||
| [[Vavoom]] | |||
|- | |||
| 1 | |||
| 224\783 | |||
| 343.295 | |||
| 8000/6561 | |||
| [[Raider]] | |||
|- | |||
| 1 | |||
| 233\783 | |||
| 357.088 | |||
| 768/625 | |||
| [[Dodifo]] (5-limit) | |||
|- | |||
| 1 | |||
| 325\783 | |||
| 498.084 | |||
| 4/3 | |||
| [[Counterschismic]] | |||
|- | |||
| 9 | |||
| 206\783<br />(32\783) | |||
| 315.709<br />(49.042) | |||
| 6/5<br />(36/35) | |||
| [[Ennealimmal]] | |||
|- | |||
| 29 | |||
| 325\783<br />(1\723) | |||
| 498.084<br />(1.533) | |||
| 4/3<br />(32805/32768) | |||
| [[Copper]] | |||
|- | |||
| 87 | |||
| 308\783<br />(2\783) | |||
| 472.031<br />(3.065) | |||
| {{monzo| 35 1 -6 -3 -4 }}<br />(?) | |||
| [[87th-octave temperaments #Francium|Francium]] | |||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
Latest revision as of 06:16, 21 February 2025
| ← 782edo | 783edo | 784edo → |
783 equal divisions of the octave (abbreviated 783edo or 783ed2), also called 783-tone equal temperament (783tet) or 783 equal temperament (783et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 783 equal parts of about 1.53 ¢ each. Each step represents a frequency ratio of 21/783, or the 783rd root of 2.
Theory
783edo provides solid approximations of the 5-limit, the 7-limit and to some extent the 11-limit, and is consistent up to the 11-odd-limit, making it a flexible and versatile tuning. The equal temperament tempers out the ennealimma and the counterschisma in the 5-limit; 2401/2400, and 4375/4374 in the 7-limit. It supports counterschismic, dodifo, ennealimmal, raider, vavoom, and francium.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.039 | -0.107 | -0.244 | +0.406 | -0.681 | -0.741 | -0.195 | +0.078 | +0.308 | -0.208 |
| Relative (%) | +0.0 | -2.6 | -7.0 | -15.9 | +26.5 | -44.4 | -48.3 | -12.7 | +5.1 | +20.1 | -13.6 | |
| Steps (reduced) |
783 (0) |
1241 (458) |
1818 (252) |
2198 (632) |
2709 (360) |
2897 (548) |
3200 (68) |
3326 (194) |
3542 (410) |
3804 (672) |
3879 (747) | |
Subsets and supersets
Since 783 factors into 33 × 29, 783edo has subset edos 3, 9, 27, 29, 87, and 261.
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 73\783 | 111.877 | 16/15 | Vavoom |
| 1 | 224\783 | 343.295 | 8000/6561 | Raider |
| 1 | 233\783 | 357.088 | 768/625 | Dodifo (5-limit) |
| 1 | 325\783 | 498.084 | 4/3 | Counterschismic |
| 9 | 206\783 (32\783) |
315.709 (49.042) |
6/5 (36/35) |
Ennealimmal |
| 29 | 325\783 (1\723) |
498.084 (1.533) |
4/3 (32805/32768) |
Copper |
| 87 | 308\783 (2\783) |
472.031 (3.065) |
[35 1 -6 -3 -4⟩ (?) |
Francium |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct