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12edo: Difference between revisions

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| 2.3
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| {{monzo| -19 12 }}
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* 12et (using the 12f val, where 9 steps - a major sixth - is used as the approximation of [[13/8]] instead of 8 steps) is lower in relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], [[11-limit|11-]], [[13-limit|13-]], and [[19-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], 19/19e, and 19egh, respectively.
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this.
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].


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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]
{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-
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| 1\12
| 1\12
| (P8, P4/5)
| (P8, P4/5)
| [[Ripple]] / [[passion]]
| [[Ripple]], [[passion]]
|-
|-
| 1
| 1
| 5\12
| 5\12
| (P8, P5)
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
|-
| 2
| 2
| 5\12 (1\12)
| 5\12 (1\12)
| (P8/2, P5)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
| [[Pajara]], [[injera]]
|-
|-
| 3
| 3
| 5\12 (1\12)
| 5\12 (1\12)
| (P8/3, P5)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
|-
| 4
| 4
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<nowiki>*</nowiki> [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki>*</nowiki> [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


Rank-2 temperaments to which 12et can be [[Detempering|detempered]]:
Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 & 72), [[schismic]]/[[garibaldi]] (41 & 53), and [[diaschismic]] (46 & 58). Rank-3 temperaments to which 12et can be detempered include [[marvel]], [[starling]]/[[thrush]], [[aberschismic]], [[orthoschismic]], and [[cassaschismic]]. For more comprehensive lists, see:  
 
* [[List of 12et rank two temperaments by badness]]
* [[Schismic]] / [[Schismatic family#Schismic, schismatic, a.k.a. helmholtz|helmholtz]] (12 & 53)
* [[List of 12et rank two temperaments by complexity]]
* [[Compton]] (12 & 72)
* [[List of edo-distinct 12f rank two temperaments]]
* [[Atomic]] (12 & 612)
* [[Schismic–commatic equivalence continuum]]
* [[Garibaldi]] / [[cassandra]] (41 & 53)*
* [[Heptacot]] (311 & 323)*
* [[Misty]] / murky** (87 & 99ef)
* [[Septimal diaschismic]] (46 & 58)**
 
Rank-3 temperaments to which 12et can be detempered:
 
* [[Odin]] (12 & 84 & 270)
* [[Marvel]] (31 & 41 & 53)*
* [[Hemifamity|Aberschismic]] / [[Pele]] (41 & 46 & 53)*
* [[Cassaschismic]] (41 & 53 & 270)*
 
<nowiki>*</nowiki>Through the 12e val
 
<nowiki>**</nowiki>Through the 12f val


== Octave stretch or compression ==
== Octave stretch or compression ==
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