1012edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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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. | 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. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 16: | Line 11: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 1012 factors into {{ | Since 1012 factors into {{nowrap| 2<sup>2</sup> × 11 × 23 }}, 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. | ||
== 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 | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| 1 | | 1 | ||
| 361\1012 | | 361\1012 | ||
| 428. | | 428.06 | ||
| 2800/2187 | | 2800/2187 | ||
| [[Osiris]] | | [[Osiris]] | ||
|- | |- | ||
| 2 | | 2 | ||
| | | 145\1012 | ||
| 498. | | 171.94 | ||
| 243/220 | |||
| [[Semiosiris]] | |||
|- | |||
| 2 | |||
| 420\1012 | |||
| 498.02 | |||
| 7/5 | | 7/5 | ||
| [[Quarvish]] | | [[Quarvish]] | ||
| Line 43: | Line 44: | ||
| 44 | | 44 | ||
| 420\1012<br>(6\1012) | | 420\1012<br>(6\1012) | ||
| 498. | | 498.02<br>(7.11) | ||
| 4/3<br>(18375/18304) | | 4/3<br>(18375/18304) | ||
| [[Ruthenium]] | | [[Ruthenium]] | ||
|} | |} | ||
<nowiki>* | <nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]] | ||
Latest revision as of 10:30, 12 July 2026
| ← 1011edo | 1012edo | 1013edo → |
1012 equal divisions of the octave (abbreviated 1012edo or 1012ed2), also called 1012-tone equal temperament (1012tet) or 1012 equal temperament (1012et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1012 equal parts of about 1.19 ¢ each. Each step represents a frequency ratio of 21/1012, or the 1012th root of 2.
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.
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 | 428.06 | 2800/2187 | Osiris |
| 2 | 145\1012 | 171.94 | 243/220 | Semiosiris |
| 2 | 420\1012 | 498.02 | 7/5 | Quarvish |
| 44 | 420\1012 (6\1012) |
498.02 (7.11) |
4/3 (18375/18304) |
Ruthenium |