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1012edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1012}}
{{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.   
=== 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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=== Subsets and supersets ===
=== Subsets and supersets ===
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.
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>Ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
| 1
| 1
| 361\1012
| 361\1012
| 428.066
| 428.06
| 2800/2187  
| 2800/2187
| [[Osiris]]
| [[Osiris]]
|-
|-
| 2
| 2
| 491\1012
| 145\1012
| 498.023
| 171.94
| 243/220
| [[Semiosiris]]
|-
| 2
| 420\1012
| 498.02
| 7/5
| 7/5
| [[Quarvish]]
| [[Quarvish]]
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| 44
| 44
| 420\1012<br>(6\1012)
| 420\1012<br>(6\1012)
| 498.023<br>(7.115)
| 498.02<br>(7.11)
| 4/3<br>(18375/18304)
| 4/3<br>(18375/18304)
| [[Ruthenium]]
| [[Ruthenium]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]
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