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Octagar family: Difference between revisions

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The '''octagar family''' of temperaments are [[planar temperament]]s tempering out [[4000/3969]]. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7 odd limit minimax tuning has fifths and 7s 2.245 cents sharp, with just major thirds.
{{Technical data page}}
The '''octagar family''' of [[temperament]]s are [[rank-3 temperament]]s [[tempering out]] the octagar comma, [[4000/3969]]. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7-odd-limit minimax tuning has fifths and 7's 2.245 cents sharp, with just major thirds.


= Octagar =
[[Linear temperament]]s that temper out the octagar comma can be found in [[octagar temperaments]].
[[Comma]] c = 4000/3969


7-limit minimax: 3 and 7 1/6c sharp, 5 just
== Octagar ==
This temperament is also known as ''octagari''.


[|1 0 0 0>, |5/6 1/3 1/2 -1/3>,
[[Subgroup]]: 2.3.5.7
|0 0 1 0>, |5/6 -2/3 1/2 2/3>]


[[Eigenmonzo]]s: 2, 7/6, 5/4
[[Comma list]]: [[4000/3969]]


9-limit minimax: 3 1/8c sharp, 5 just, 7 1/4c sharp
{{Mapping|legend=1| 1 0 1 4 | 0 1 0 -2 | 0 0 2 3 }}


[|1 0 0 0>, |5/8 1/2 3/8 -1/4>,  
: Mapping generators: ~2, ~3, ~63/40
|0 0 1 0>, |5/4 -1 3/4 1/2>]


[[Eigenmonzo]]s: 2, 5/4, 9/7
[[Mapping to lattice]]: [{{val| 0 -1 -2 -1 }}, {{val| 0 -1 0 2 }}]


Lattice basis: 63/50 length 0.8966, 21/20 length 1.0605
Lattice basis:
: 63/50 length = 0.8966, 21/20 length = 1.0605
: Angle (63/50, 21/20) = 97.743 degrees


Angle(63/50, 21/20) = 97.743 cents
[[Optimal tuning]] (POTE): ~3/2 = 703.6224, ~21/20 = 89.3227


Map to lattice: [<0 -1 -2 -1|, <0 -1 0 2|]
[[Minimax tuning]]:
* [[7-odd-limit]]
: {{monzo list| 1 0 0 0 | 5/6 1/3 1/2 -1/3 | 0 0 1 0 | 5/6 -2/3 1/2 2/3 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
* [[9-odd-limit]]
: {{monzo list| 1 0 0 0 | 5/8 1/2 3/8 -1/4 | 0 0 1 0 | 5/4 -1 3/4 1/2 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7


[[EDO|EDOs]]: [[12edo|12]], [[26edo|26]], [[27edo|27]], [[41edo|41]], [[53edo|53]], [[94edo|94]], [[121edo|121]], [[162edo|162]], [[189edo|189]], [[215edo|215]], [[230edo|230]]
{{Optimal ET sequence|legend=1| 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d }}


Badness: 0.000216
[[Badness]]: 0.216 × 10<sup>-3</sup>


[[Projection pair]]s: 5 3969/800 7 27783/4000 to 2.3.7/5
[[Projection pair]]s: 5 - 3969/800, 7 - 27783/4000 to 2.3.7/5


=== [[Hobbits|Hobbit bases]] ===
{{Databox|[[Hobbit|Hobbit bases]]|
{2, 3, 7/5} subgroup
2.3.7/5 subgroup
* 12: 50/49, 256000/250047
* 15: 256000/250047, 1029/1000
* 23: 12800000/12252303, 107163/102400
}}


12: 50/49, 256000/250047
== Nakika ==
[[Subgroup]]: 2.3.5.7.11


15: 256000/250047, 1029/1000
[[Comma list]]: 100/99, 245/242


23: 12800000/12252303, 107163/102400
{{Mapping|legend=1| 1 0 1 4 4 | 0 1 0 -2 -2 | 0 0 2 3 4 }}


== Nakika ==
Mapping to lattice: [{{val| 0 1 2 1 2 }}, {{val| 0 -1 0 2 2 }}]
Commas: 100/99, 245/242
 
Lattice basis:
: 11/7 length = 0.798, 22/21 length = 0.906
: Angle (11/7, 22/21) = 97.747 degrees
 
[[Optimal tuning]] (POTE): ~3/2 = 703.8837, ~21/20 = 87.8919
 
{{Optimal ET sequence|legend=1| 12, 15, 26, 27e, 41, 109e }}
 
[[Badness]]: 0.539 × 10<sup>-3</sup>
 
[[Projection pair]]s: 5 - 242/49, 7 - 21296/3087, 11 - 234256/21609 to 2.3.11/7
 
[[Associated temperament]]: [[octacot]]
 
Scales: [[nakika12]]     
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 105/104, 245/242
 
Mapping: {{mapping| 1 0 1 4 4 2 | 0 1 0 -2 -2 -1 | 0 0 2 3 4 5 }}
 
Optimal tuning (POTE): ~3/2 = 701.8881, ~21/20 = 87.4143
 
Optimal ET sequence: {{Optimal ET sequence| 12f, 14cf, 15, 26, 29, 41 }}
 
== Octasand ==
[[Subgroup]]: 2.3.5.7.11


Associated linear temperament: [[Tetracot family|octacot]]
[[Comma list]]: 540/539, 2200/2187


Lattice basis: 11/7 0.798 22/21 0.906
{{Mapping|legend=1| 1 0 1 4 -5 | 0 1 0 -2 7 | 0 0 2 3 -4 }}


Angle(11/7, 22/21) = 97.747 degrees
[[Optimal tuning]] (POTE): ~3/2 = 703.5501, ~21/20 = 89.3956


Map to lattice: [&lt;0 1 2 1 2|, &lt;0 -1 0 2 2|]
{{Optimal ET sequence|legend=1| 12e, 14c, 27e, 41, 53, 80, 94, 121 }}


Map: [&lt;1 0 1 4 4|, &lt;0 1 0 -2 -2|, &lt;0 0 2 3 4|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 12, 15, 26, 29, 41
Comma list: 325/324, 352/351, 540/539


Badness: 0.000539
Mapping: {{mapping| 1 0 1 4 -5 0 | 0 1 0 -2 7 4 | 0 0 2 3 -4 -4 }}


Projection pairs: 5 242/49 7 21296/3087 11 234256/21609 to 2.3.11/7
Optimal tuning (POTE): ~3/2 = 703.5688, ~21/20 = 89.4032


Scales: [[nakika12]]     
Optimal ET sequence: {{Optimal ET sequence| 12e, 14c, 27e, 41, 53, 80, 94, 121 }}


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Octagar]]
[[Category:Octagar family| ]] <!-- main article -->
[[Category:Octagar| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
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