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 ==
Subgroup: 2.3.5.7
This temperament is also known as ''octagari''.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[4000/3969]]
[[Comma list]]: [[4000/3969]]


[[Mapping]]: [{{val| 1 0 1 4 }}, {{val| 0 1 0 -2 }}, {{val| 0 0 2 3 }}]
{{Mapping|legend=1| 1 0 1 4 | 0 1 0 -2 | 0 0 2 3 }}


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


[[Mapping to lattice]]: [{{val| 0 -1 -2 -1 }}, {{val| 0 -1 0 2 }}]
[[Mapping to lattice]]: [{{val| 0 -1 -2 -1 }}, {{val| 0 -1 0 2 }}]
Line 15: Line 20:
: 63/50 length = 0.8966, 21/20 length = 1.0605
: 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 degrees
[[Optimal tuning]] (POTE): ~3/2 = 703.6224, ~21/20 = 89.3227


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit]]
* [[7-odd-limit]]
: [{{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 }}]
: {{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]]s (unchanged intervals): 2, 7/6, 5/4
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
* [[9-odd-limit]]
* [[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 }}]
: {{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]]s (unchanged intervals): 2, 5/4, 9/7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7


{{Val list|legend=1| 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d }}
{{Optimal ET sequence|legend=1| 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d }}


[[Badness]]: 0.216 × 10<sup>-3</sup>
[[Badness]]: 0.216 × 10<sup>-3</sup>
Line 38: Line 45:


== Nakika ==
== Nakika ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[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|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 }}]
Mapping to lattice: [{{val| 0 1 2 1 2 }}, {{val| 0 -1 0 2 2 }}]
Line 50: Line 57:
: Angle (11/7, 22/21) = 97.747 degrees
: Angle (11/7, 22/21) = 97.747 degrees


{{Val list|legend=1| 12, 15, 26, 27e, 41, 109e }}
[[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>
[[Badness]]: 0.539 × 10<sup>-3</sup>
Line 65: Line 74:
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: {{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
 
[[Comma list]]: 540/539, 2200/2187
 
{{Mapping|legend=1| 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|legend=1| 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: {{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


Vals: {{Val list| 12f, 14cf, 15, 26, 29, 41 }}
Optimal ET sequence: {{Optimal ET sequence| 12e, 14c, 27e, 41, 53, 80, 94, 121 }}


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Octagar family| ]] <!-- main article -->
[[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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