Octagar family: Difference between revisions

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m Units & misc. cleanup. - redundant category
Switch to Sintel's badness, WE & CWE tunings
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: Angle (63/50, 21/20) = 97.743 degrees
: Angle (63/50, 21/20) = 97.743 degrees


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 703.6224{{c}}, ~21/20 = 89.3227{{c}}
[[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 }}
<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6224{{c}}, ~21/20 = 89.3227{{c}} -->


[[Minimax tuning]]:
[[Minimax tuning]]:
Line 32: Line 37:
{{Optimal ET sequence|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]] (Smith): 0.216 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.951


[[Projection pair]]s: <code>5 3969/800 7 27783/4000</code> to 2.3.7/5
[[Projection pair]]s: <code>5 3969/800 7 27783/4000</code> to 2.3.7/5
Line 56: Line 61:
: Angle (11/7, 22/21) = 97.747 degrees
: Angle (11/7, 22/21) = 97.747 degrees


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 703.8837{{c}}, ~21/20 = 87.8919{{c}}
[[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 }}
<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.8837{{c}}, ~21/20 = 87.8919{{c}} -->


{{Optimal ET sequence|legend=1| 12, 15, 26, 27e, 41, 109e }}
{{Optimal ET sequence|legend=1| 12, 15, 26, 27e, 41, 109e }}


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


[[Projection pair]]s: <code>5 242/49 7 21296/3087 11 234256/21609</code> to 2.3.11/7
[[Projection pair]]s: <code>5 242/49 7 21296/3087 11 234256/21609</code> to 2.3.11/7
Line 75: Line 85:
Mapping: {{mapping| 1 0 1 4 4 2 | 0 1 0 -2 -2 -1 | 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): ~2 = 1200.0000{{c}}, ~3/2 = 701.8881{{c}}, ~21/20 = 87.4143{{c}}
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}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8881{{c}}, ~21/20 = 87.4143{{c}} -->


{{Optimal ET sequence|legend=0| 12f, 14cf, 15, 26, 29, 41 }}
{{Optimal ET sequence|legend=0| 12f, 14cf, 15, 26, 29, 41 }}
Badness (Sintel): 0.832


== Octasand ==
== Octasand ==
Line 86: Line 101:
{{Mapping|legend=1| 1 0 1 4 -5 | 0 1 0 -2 7 | 0 0 2 3 -4 }}
{{Mapping|legend=1| 1 0 1 4 -5 | 0 1 0 -2 7 | 0 0 2 3 -4 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 703.5501{{c}}, ~21/20 = 89.3956{{c}}
[[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 }}
<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.5501{{c}}, ~21/20 = 89.3956{{c}} -->


{{Optimal ET sequence|legend=1| 12e, 14c, 27e, 41, 53, 80, 94, 121 }}
{{Optimal ET sequence|legend=1| 27e, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d }}
 
[[Badness]] (Sintel): 1.20


=== 13-limit ===
=== 13-limit ===
Line 97: Line 119:
Mapping: {{mapping| 1 0 1 4 -5 0 | 0 1 0 -2 7 4 | 0 0 2 3 -4 -4 }}
Mapping: {{mapping| 1 0 1 4 -5 0 | 0 1 0 -2 7 4 | 0 0 2 3 -4 -4 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 703.5688{{c}}, ~21/20 = 89.4032{{c}}
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}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 703.5688{{c}}, ~21/20 = 89.4032{{c}} -->
 
{{Optimal ET sequence|legend=0| 27e, 41, 53, 80, 94, 121, 174d, 215, 295d, 336def }}


{{Optimal ET sequence|legend=0| 12e, 14c, 27e, 41, 53, 80, 94, 121 }}
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]]

Revision as of 14:08, 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 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 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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