1012edo: Difference between revisions
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{{EDO intro|1012}} | {{EDO intro|1012}} | ||
== Theory == | == Theory == | ||
1012edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit. | 1012edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}. | ||
In the 5-limit, it is enfactored, with the same mapping as [[506edo]], providing a tuning for [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it tempers out the [[breedsma]], 2401/2400. Furthermore, it is a [[septiruthenia]]<nowiki/>n system, tempering 64/63 comma to 23 steps, 1/44th of the octave.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, {{monzo| 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 === | === Prime harmonics === | ||
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[[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 == | ||
Revision as of 14:22, 7 June 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 is 2401/2400, 4096/4095, 6656/6655, 9801/9800 and [2 6 -1 2 0 4⟩.
In the 5-limit, it is enfactored, with the same mapping as 506edo, providing a tuning for vishnu, monzismic, and lafa. In the 7-limit, it tempers out the breedsma, 2401/2400. Furthermore, it is a septiruthenian system, tempering 64/63 comma to 23 steps, 1/44th of the octave.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
1012 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 (Reduced) |
Cents (Reduced) |
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 |