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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.
 
[[Linear temperament]]s that temper out the octagar comma can be found in [[octagar temperaments]].


== Octagar ==
== Octagar ==
[[Comma]]: 4000/3969
This temperament is also known as ''octagari''.
 
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 0 1 4}}, {{val|0 1 0 -2}}, {{val|0 0 2 3}}]
[[Comma list]]: [[4000/3969]]


Mapping generators: 2, ~3, ~63/40
{{Mapping|legend=1| 1 0 1 4 | 0 1 0 -2 | 0 0 2 3 }}


Minimax tuning:
: Mapping generators: ~2, ~3, ~63/40
* 7-limit minimax:  
: [{{monzo|1 0 0 0}}, {{monzo|5/6 1/3 1/2 -1/3}}, {{monzo|0 0 1 0}}, {{monzo|5/6 -2/3 1/2 2/3}}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 7/6, 5/4


* 9-limit minimax:
[[Mapping to lattice]]: [{{val| 0 -1 -2 -1 }}, {{val| 0 -1 0 2 }}]
: [{{monzo|1 0 0 0}}, {{monzo|5/8 1/2 3/8 -1/4}}, {{monzo|0 0 1 0}}, {{monzo|5/4 -1 3/4 1/2}}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 5/4, 9/7


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: [{{val|0 -1 -2 -1}}, {{val|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


{{Val list|legend=1| 12, 26, 27, 41, 53, 94, 121, 162, 189, 215, 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
}}
== Nakika ==
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 100/99, 245/242
{{Mapping|legend=1| 1 0 1 4 4 | 0 1 0 -2 -2 | 0 0 2 3 4 }}
Mapping to lattice: [{{val| 0 1 2 1 2 }}, {{val| 0 -1 0 2 2 }}]
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>


12: 50/49, 256000/250047
[[Projection pair]]s: 5 - 242/49, 7 - 21296/3087, 11 - 234256/21609 to 2.3.11/7


15: 256000/250047, 1029/1000
[[Associated temperament]]: [[octacot]]


23: 12800000/12252303, 107163/102400
Scales: [[nakika12]]     


=== Nakika ===
=== 13-limit ===
[[Comma list]]: 100/99, 245/242
Subgroup: 2.3.5.7.11.13


Mapping: [{{val|1 0 1 4 4}}, {{val|0 1 0 -2 -2}}, {{val|0 0 2 3 4}}]
Comma list: 100/99, 105/104, 245/242


Mapping generators: 2, ~3, ~11/7
Mapping: {{mapping| 1 0 1 4 4 2 | 0 1 0 -2 -2 -1 | 0 0 2 3 4 5 }}


Associated linear temperament: [[Tetracot family|octacot]]
Optimal tuning (POTE): ~3/2 = 701.8881, ~21/20 = 87.4143


Lattice basis: 11/7 length = 0.798, 22/21 length = 0.906
Optimal ET sequence: {{Optimal ET sequence| 12f, 14cf, 15, 26, 29, 41 }}


Angle(11/7, 22/21) = 97.747 degrees
== Octasand ==
[[Subgroup]]: 2.3.5.7.11


Map to lattice: [{{val|0 1 2 1 2}}, {{val|0 -1 0 2 2}}]
[[Comma list]]: 540/539, 2200/2187


Vals: {{Val list| 12, 15, 26, 29, 41 }}
{{Mapping|legend=1| 1 0 1 4 -5 | 0 1 0 -2 7 | 0 0 2 3 -4 }}


Badness: 0.000539
[[Optimal tuning]] (POTE): ~3/2 = 703.5501, ~21/20 = 89.3956


Projection pairs: 5 - 242/49, 7 - 21296/3087, 11 - 234256/21609 to 2.3.11/7
{{Optimal ET sequence|legend=1| 12e, 14c, 27e, 41, 53, 80, 94, 121 }}


Scales: [[nakika12]]     
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==== 13-limit ====
Comma list: 325/324, 352/351, 540/539
Comma list: 100/99, 105/104, 245/242


Mapping: [{{val|1 0 1 4 4 2}}, {{val|0 1 0 -2 -2 -1}}, {{val|0 0 2 3 4 5}}]
Mapping: {{mapping| 1 0 1 4 -5 0 | 0 1 0 -2 7 4 | 0 0 2 3 -4 -4 }}


Mapping generators: 2, ~3, ~11/7
Optimal tuning (POTE): ~3/2 = 703.5688, ~21/20 = 89.4032


Vals: {{Val list| 12f, 15, 26, 29, 41 }}
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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