Octagar family: Difference between revisions

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== Octagar ==
== Octagar ==
[[Comma]]: 4000/3969
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]]: [{{val| 1 0 1 4 }}, {{val| 0 1 0 -2 }}, {{val| 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
[[Minimax tuning]]:
 
* [[7-odd-limit]]
[[Mapping to lattice]]: [{{val|0 -1 -2 -1}}, {{val|0 -1 0 2}}]
: [{{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-odd-limit]]
: [{{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


{{Val list|legend=1| 12, 26, 27, 41, 53, 94, 121, 162, 189, 215, 230 }}
{{Val list|legend=1| 12, 26, 27, 41, 53, 94, 121, 162, 189, 215, 230 }}


[[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
23: 12800000/12252303, 107163/102400
=== Nakika ===
[[Comma list]]: 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]]: [{{val| 1 0 1 4 4 }}, {{val| 0 1 0 -2 -2 }}, {{val| 0 0 2 3 4 }}]


Mapping generators: 2, ~3, ~11/7
Mapping to lattice: [{{val| 0 1 2 1 2 }}, {{val| 0 -1 0 2 2 }}]


Associated linear temperament: [[Tetracot family|octacot]]
Lattice basis:
: 11/7 length = 0.798, 22/21 length = 0.906
: Angle (11/7, 22/21) = 97.747 degrees


Lattice basis: 11/7 length = 0.798, 22/21 length = 0.906
{{Val list|legend=1| 12, 15, 26, 29, 41 }}


Angle(11/7, 22/21) = 97.747 degrees
[[Badness]]: 0.539 × 10<sup>-3</sup>


Mapping to lattice: [{{val|0 1 2 1 2}}, {{val|0 -1 0 2 2}}]
[[Projection pair]]s: 5 - 242/49, 7 - 21296/3087, 11 - 234256/21609 to 2.3.11/7


Vals: {{Val list| 12, 15, 26, 29, 41 }}
[[Associated temperament]]: [[octacot]]


Badness: 0.000539
Scales: [[nakika12]]     


Projection pairs: 5 - 242/49, 7 - 21296/3087, 11 - 234256/21609 to 2.3.11/7
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Scales: [[nakika12]]     
==== 13-limit ====
Comma list: 100/99, 105/104, 245/242
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: [{{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 }}
Vals: {{Val list| 12f, 15, 26, 29, 41 }}


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Octagar]]
[[Category:Octagar family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 12:33, 6 November 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

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

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⟩]
Eigenmonzos (unchanged intervals): 2, 7/6, 5/4
[[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

Template:Val list

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

Template:Val list

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]]

Vals: Template:Val list

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