1012edo: Difference between revisions
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== Theory == | == Theory == | ||
1012edo is a strong 13-limit system, | 1012edo is a strong 13-limit system, [[consistency|distinctly consistent]] through the [[15-odd-limit]]. It is a [[zeta peak edo]], though not [[zeta integral edo|zeta integral]] nor [[zeta gap edo|zeta gap]]. A basis for the 13-limit [[comma]]s consists of [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}. | ||
In the 5-limit, 1012edo is enfactored, with the same | In the 5-limit, 1012edo is [[enfactoring|enfactored]], with the same tuning as [[506edo]], [[support]]ing [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it [[tempering out|tempers out]] the [[breedsma]], 2401/2400, and tunes the [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 [[subgroup]], it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit. | ||
=== Other techniques === | === Other techniques === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
1012 has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. | Since 1012 factors into {{factorization|1012}}, 1012edo has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. [[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic. | ||
[[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
Revision as of 10:59, 31 October 2023
| ← 1011edo | 1012edo | 1013edo → |
Theory
1012edo is a strong 13-limit system, distinctly consistent through the 15-odd-limit. It is a zeta peak edo, though not zeta integral nor zeta gap. A basis for the 13-limit commas consists of 2401/2400, 4096/4095, 6656/6655, 9801/9800 and [2 6 -1 2 0 4⟩.
In the 5-limit, 1012edo is enfactored, with the same tuning as 506edo, supporting vishnu, monzismic, and lafa. In the 7-limit, it tempers out the breedsma, 2401/2400, and tunes the osiris temperament. Furthermore, noting its exceptional strength in the 2.3.7 subgroup, it is a septiruthenian system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the optimal patent val for quarvish temperament in the 7-limit and also in the 11-limit.
Other techniques
In addition to containing 22edo and 23edo, it contains a 22L 1s scale produced by generator of 45\1012 associated with 33/32, and is associated with the 45 & 1012 temperament, making it concoctic. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, [18 15 -12 -1 0 -3⟩.
In the 2.3.7.11.101, it tempers out 7777/7776 and is a tuning for the neutron star temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.021 | +0.248 | -0.051 | +0.065 | +0.184 | +0.578 | +0.115 | +0.184 | -0.328 | +0.419 |
| Relative (%) | +0.0 | +1.8 | +20.9 | -4.3 | +5.5 | +15.5 | +48.8 | +9.7 | +15.5 | -27.7 | +35.3 | |
| Steps (reduced) |
1012 (0) |
1604 (592) |
2350 (326) |
2841 (817) |
3501 (465) |
3745 (709) |
4137 (89) |
4299 (251) |
4578 (530) |
4916 (868) |
5014 (966) | |
Subsets and supersets
Since 1012 factors into 22 × 11 × 23, 1012edo has subset edos 2, 4, 11, 22, 23, 44, 46, 92, 253, 506. 2024edo, which divides the edostep in two, provides a good correction for the 17th harmonic.
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 361\1012 | 428.066 | 2800/2187 | Osiris |
| 2 | 491\1012 | 498.023 | 7/5 | Quarvish |
| 44 | 420\1012 (6\1012) |
498.023 (7.115) |
4/3 (18375/18304) |
Ruthenium |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct