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 [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the octagar comma, [[4000/3969]].  
 
For [[rank-2 temperament]]s that temper out the octagar comma, see [[Octagar temperaments]].


== Octagar ==
== Octagar ==
Subgroup: 2.3.5.7
Octagar is generated by a [[3/2|perfect fifth]] and a minor sixth of [[~]][[63/40]], two of which make ~[[5/2]], and three make an interval short of two octaves by a [[64/63|septimal comma]].
 
While many 7-limit rank-3 temperaments exhibit a tendency towards tuning flat, a slight sharp tendency is often preferred instead. Octagar provides this; for instance the [[7-odd-limit]] [[minimax tuning]] has [[3/1|3]]'s and [[7/1|7]]'s 2.245 cents sharp, with just [[5/1|5]]'s.
 
This temperament is catalogued as ''octagari'' in [[Graham Breed]]'s [https://x31eq.com/temper/ Temperament Finder].
 
[[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 23:
: 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]]s:
* [[WE]]: ~2 = 1199.5353{{c}}, ~3/2 = 703.3499{{c}}, ~63/40 = 792.6380{{c}}
: [[error map]]: {{val| -0.465 +0.930 -1.502 +1.459 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.3764{{c}}, ~63/40 = 792.6299{{c}}
: error map: {{val| 0.000 +1.421 -1.054 +2.311 }}


[[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]] (Sintel): 0.951


[[Projection pair]]s: 5 - 3969/800, 7 - 27783/4000 to 2.3.7/5
[[Projection pair]]s: <code>5 3969/800 7 27783/4000</code> to 2.3.7/5


{{Databox|[[Hobbit|Hobbit bases]]|
{{Databox|[[Hobbit|Hobbit bases]]|
Line 38: Line 52:


== 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 64:
: 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]]s:
* [[WE]]: ~2 = 1199.6137{{c}}, ~3/2 = 703.6571{{c}}, ~11/7 = 791.5207{{c}}
: [[error map]]: {{val| -0.386 +1.316 -3.659 -2.351 +6.678 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6783{{c}}, ~11/7 = 791.5168{{c}}
: error map: {{val| 0.000 +1.723 -3.280 -1.632 +7.393 }}
 
{{Optimal ET sequence|legend=1| 12, 15, 26, 27e, 41, 109e }}


[[Badness]]: 0.539 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.647


[[Projection pair]]s: 5 - 242/49, 7 - 21296/3087, 11 - 234256/21609 to 2.3.11/7
[[Projection pair]]s: <code>5 242/49 7 21296/3087 11 234256/21609</code> to 2.3.11/7


[[Associated temperament]]: [[octacot]]
[[Associated temperament]]: [[octacot]]
Line 65: Line 85:
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 tunings:
* WE: ~2 = 1200.5397{{c}}, ~3/2 = 702.2038{{c}}, ~11/7 = 789.6574{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0977{{c}}, ~11/7 = 789.5686{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 14cf, 15, 26, 29, 41 }}
 
Badness (Sintel): 0.832
 
== 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]]s:
* [[WE]]: ~2 = 1199.5255{{c}}, ~3/2 = 703.2719{{c}}, ~63/40 = 792.6321{{c}}
: [[error map]]: {{val| -0.475 +0.842 -1.524 +1.578 +0.108 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.2662{{c}}, ~63/40 = 792.6212{{c}}
: error map: {{val| 0.000 +1.311 -1.071 +2.505 +1.061 }}
 
{{Optimal ET sequence|legend=1| 27e, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d }}
 
[[Badness]] (Sintel): 1.20
 
=== 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 tunings:
* WE: ~2 = 1199.5112{{c}}, ~3/2 = 703.2823{{c}}, ~63/40 = 792.6491{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.3684{{c}}, ~63/40 = 792.7875{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 41, 53, 80, 94, 121, 174d, 215, 295d, 336def }}


Vals: {{Val list| 12f, 14cf, 15, 26, 29, 41 }}
Badness (Sintel): 1.03


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Octagar family| ]] <!-- main article -->
[[Category:Octagar family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 16:28, 13 March 2026

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 rank-3 temperaments tempers out the octagar comma, 4000/3969.

For rank-2 temperaments that temper out the octagar comma, see Octagar temperaments.

Octagar

Octagar is generated by a perfect fifth and a minor sixth of ~63/40, two of which make ~5/2, and three make an interval short of two octaves by a septimal comma.

While many 7-limit rank-3 temperaments exhibit a tendency towards tuning flat, a slight sharp tendency is often preferred instead. Octagar provides this; for instance the 7-odd-limit minimax tuning has 3's and 7's 2.245 cents sharp, with just 5's.

This temperament is catalogued as octagari in Graham Breed's Temperament Finder.

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 tunings:

  • WE: ~2 = 1199.5353 ¢, ~3/2 = 703.3499 ¢, ~63/40 = 792.6380 ¢
error map: ⟨-0.465 +0.930 -1.502 +1.459]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.3764 ¢, ~63/40 = 792.6299 ¢
error map: ⟨0.000 +1.421 -1.054 +2.311]

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 (Sintel): 0.951

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 tunings:

  • WE: ~2 = 1199.6137 ¢, ~3/2 = 703.6571 ¢, ~11/7 = 791.5207 ¢
error map: ⟨-0.386 +1.316 -3.659 -2.351 +6.678]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.6783 ¢, ~11/7 = 791.5168 ¢
error map: ⟨0.000 +1.723 -3.280 -1.632 +7.393]

Optimal ET sequence: 12, 15, 26, 27e, 41, 109e

Badness (Sintel): 0.647

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 tunings:

  • WE: ~2 = 1200.5397 ¢, ~3/2 = 702.2038 ¢, ~11/7 = 789.6574 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0977 ¢, ~11/7 = 789.5686 ¢

Optimal ET sequence: 12f, 14cf, 15, 26, 29, 41

Badness (Sintel): 0.832

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 tunings:

  • WE: ~2 = 1199.5255 ¢, ~3/2 = 703.2719 ¢, ~63/40 = 792.6321 ¢
error map: ⟨-0.475 +0.842 -1.524 +1.578 +0.108]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.2662 ¢, ~63/40 = 792.6212 ¢
error map: ⟨0.000 +1.311 -1.071 +2.505 +1.061]

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

Badness (Sintel): 1.20

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 tunings:

  • WE: ~2 = 1199.5112 ¢, ~3/2 = 703.2823 ¢, ~63/40 = 792.6491 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.3684 ¢, ~63/40 = 792.7875 ¢

Optimal ET sequence: 27e, 41, 53, 80, 94, 121, 174d, 215, 295d, 336def

Badness (Sintel): 1.03

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