Octagar family: Difference between revisions
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{{Mapping|legend=1| 1 0 1 4 | 0 1 0 -2 | 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 | ||
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: 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 | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 703.6224{{c}}, ~21/20 = 89.3227{{c}} | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit]] | * [[7-odd-limit]] | ||
: {{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 }} | : {{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 basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3 | ||
* [[9-odd-limit]] | * [[9-odd-limit]] | ||
: {{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 }} | : {{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 basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7 | ||
{{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]]: 0.216 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.216 × 10<sup>-3</sup> | ||
[[Projection pair]]s: 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]]| | ||
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: Angle (11/7, 22/21) = 97.747 degrees | : Angle (11/7, 22/21) = 97.747 degrees | ||
[[Optimal tuning]] (POTE): ~3/2 = 703.8837, ~21/20 = 87.8919 | [[Optimal tuning]] ([[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]]: 0.539 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.539 × 10<sup>-3</sup> | ||
[[Projection pair]]s: 5 | [[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]] | ||
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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): ~3/2 = 701.8881, ~21/20 = 87.4143 | Optimal tuning (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 }} | |||
== Octasand == | == Octasand == | ||
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{{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): ~3/2 = 703.5501, ~21/20 = 89.3956 | [[Optimal tuning]] ([[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| 12e, 14c, 27e, 41, 53, 80, 94, 121 }} | ||
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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): ~3/2 = 703.5688, ~21/20 = 89.4032 | Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 703.5688{{c}}, ~21/20 = 89.4032{{c}} | ||
{{Optimal ET sequence|legend=0| 12e, 14c, 27e, 41, 53, 80, 94, 121 }} | |||
[[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 10:48, 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
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): ~2 = 1200.0000 ¢, ~3/2 = 703.6224 ¢, ~21/20 = 89.3227 ¢
- [[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 (Smith): 0.216 × 10-3
Projection pairs: 5 3969/800 7 27783/4000 to 2.3.7/5
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): ~2 = 1200.0000 ¢, ~3/2 = 703.8837 ¢, ~21/20 = 87.8919 ¢
Optimal ET sequence: 12, 15, 26, 27e, 41, 109e
Badness (Smith): 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): ~2 = 1200.0000 ¢, ~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): ~2 = 1200.0000 ¢, ~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): ~2 = 1200.0000 ¢, ~3/2 = 703.5688 ¢, ~21/20 = 89.4032 ¢