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]]

Latest revision as of 00:31, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The octagar family of temperaments are rank-3 temperaments 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 temperaments that temper out the octagar comma can be found in octagar temperaments.

Octagar

This temperament is also known as octagari.

Subgroup: 2.3.5.7

Comma list: 4000/3969

Mapping: [⟨1 0 1 4], ⟨0 1 0 -2], ⟨0 0 2 3]]

Mapping generators: ~2, ~3, ~63/40

Mapping to lattice: [⟨0 -1 -2 -1], ⟨0 -1 0 2]]

Lattice basis:

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

Optimal tuning (POTE): ~3/2 = 703.6224, ~21/20 = 89.3227

Minimax tuning:

[[1 0 0 0⟩, [5/6 1/3 1/2 -1/3⟩, [0 0 1 0⟩, [5/6 -2/3 1/2 2/3⟩]
unchanged-interval (eigenmonzo) basis: 2.5.7/3
[[1 0 0 0⟩, [5/8 1/2 3/8 -1/4⟩, [0 0 1 0⟩, [5/4 -1 3/4 1/2⟩]
unchanged-interval (eigenmonzo) basis: 2.5.9/7

Optimal ET sequence: 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d

Badness: 0.216 × 10-3

Projection pairs: 5 - 3969/800, 7 - 27783/4000 to 2.3.7/5

Hobbit bases

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: [⟨1 0 1 4 4], ⟨0 1 0 -2 -2], ⟨0 0 2 3 4]]

Mapping to lattice: [⟨0 1 2 1 2], ⟨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: 12, 15, 26, 27e, 41, 109e

Badness: 0.539 × 10-3

Projection pairs: 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: [⟨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: 12f, 14cf, 15, 26, 29, 41

Octasand

Subgroup: 2.3.5.7.11

Comma list: 540/539, 2200/2187

Mapping: [⟨1 0 1 4 -5], ⟨0 1 0 -2 7], ⟨0 0 2 3 -4]]

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

Optimal ET sequence: 12e, 14c, 27e, 41, 53, 80, 94, 121

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 540/539

Mapping: [⟨1 0 1 4 -5 0], ⟨0 1 0 -2 7 4], ⟨0 0 2 3 -4 -4]]

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

Optimal ET sequence: 12e, 14c, 27e, 41, 53, 80, 94, 121

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