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.
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.


== Octagar ==
== Octagar ==
[[Comma]] c = 4000/3969
[[Comma]]: 4000/3969


7-limit minimax: 3 and 7 1/6c sharp, 5 just
[[Mapping]]: [{{val|1 0 1 4}}, {{val|0 1 0 -2}}, {{val|0 0 2 3}}]


[|1 0 0 0>, |5/6 1/3 1/2 -1/3>,
Mapping generators: 2, ~3, ~63/40
|0 0 1 0>, |5/6 -2/3 1/2 2/3>]


[[Eigenmonzo]]s (unchanged intervals): 2, 7/6, 5/4
Minimax tuning:
* 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: 3 1/8c sharp, 5 just, 7 1/4c sharp
* 9-limit minimax:  
: [{{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


[|1 0 0 0>, |5/8 1/2 3/8 -1/4>,
Lattice basis: 63/50 length = 0.8966, 21/20 length = 1.0605
|0 0 1 0>, |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


Angle(63/50, 21/20) = 97.743 cents
Angle(63/50, 21/20) = 97.743 cents


Map to lattice: [<0 -1 -2 -1|, <0 -1 0 2|]
Map to lattice: [{{val|0 -1 -2 -1}}, {{val|0 -1 0 2}}]


[[EDO|EDOs]]: [[12edo|12]], [[26edo|26]], [[27edo|27]], [[41edo|41]], [[53edo|53]], [[94edo|94]], [[121edo|121]], [[162edo|162]], [[189edo|189]], [[215edo|215]], [[230edo|230]]
{{Val list|legend=1| 12, 26, 27, 41, 53, 94, 121, 162, 189, 215, 230 }}


Badness: 0.000216
[[Badness]]: 0.000216


[[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]] ====
==== [[Hobbits|Hobbit bases]] ====
{2, 3, 7/5} subgroup
{2, 3, 7/5} subgroup


Line 39: Line 38:
23: 12800000/12252303, 107163/102400
23: 12800000/12252303, 107163/102400


=== Nakika ===
=== Nakika ===
Commas: 100/99, 245/242
[[Comma list]]: 100/99, 245/242
 
Mapping: [{{val|1 0 1 4 4}}, {{val|0 1 0 -2 -2}}, {{val|0 0 2 3 4}}]
 
Mapping generators: 2, ~3, ~11/7


Associated linear temperament: [[Tetracot family|octacot]]
Associated linear temperament: [[Tetracot family|octacot]]


Lattice basis: 11/7 0.798 22/21 0.906
Lattice basis: 11/7 length = 0.798, 22/21 length = 0.906


Angle(11/7, 22/21) = 97.747 degrees
Angle(11/7, 22/21) = 97.747 degrees


Map to lattice: [<0 1 2 1 2|, <0 -1 0 2 2|]
Map to lattice: [{{val|0 1 2 1 2}}, {{val|0 -1 0 2 2}}]


Map: [<1 0 1 4 4|, <0 1 0 -2 -2|, <0 0 2 3 4|]
Vals: {{Val list| 12, 15, 26, 29, 41 }}
 
EDOs: 12, 15, 26, 29, 41


Badness: 0.000539
Badness: 0.000539


Projection pairs: 5 242/49 7 21296/3087 11 234256/21609 to 2.3.11/7
Projection pairs: 5 - 242/49, 7 - 21296/3087, 11 - 234256/21609 to 2.3.11/7


Scales: [[nakika12]]       
Scales: [[nakika12]]       
==== 13-limit ====
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 generators: 2, ~3, ~11/7
Vals: {{Val list| 12f, 15, 26, 29, 41 }}


[[Category:Theory]]
[[Category:Theory]]

Revision as of 12:49, 17 July 2021

The octagar family of temperaments are planar temperaments 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.

Octagar

Comma: 4000/3969

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

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

Minimax tuning:

  • 7-limit minimax:
[[1 0 0 0⟩, [5/6 1/3 1/2 -1/3⟩, [0 0 1 0⟩, [5/6 -2/3 1/2 2/3⟩]
Eigenmonzos (unchanged intervals): 2, 7/6, 5/4
  • 9-limit minimax:
[[1 0 0 0⟩, [5/8 1/2 3/8 -1/4⟩, [0 0 1 0⟩, [5/4 -1 3/4 1/2⟩]
Eigenmonzos (unchanged intervals): 2, 5/4, 9/7

Lattice basis: 63/50 length = 0.8966, 21/20 length = 1.0605

Angle(63/50, 21/20) = 97.743 cents

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

Template:Val list

Badness: 0.000216

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

Comma list: 100/99, 245/242

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

Mapping generators: 2, ~3, ~11/7

Associated linear temperament: octacot

Lattice basis: 11/7 length = 0.798, 22/21 length = 0.906

Angle(11/7, 22/21) = 97.747 degrees

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

Vals: Template:Val list

Badness: 0.000539

Projection pairs: 5 - 242/49, 7 - 21296/3087, 11 - 234256/21609 to 2.3.11/7

Scales: nakika12

13-limit

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]]

Mapping generators: 2, ~3, ~11/7

Vals: Template:Val list

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