Octagar family: Difference between revisions
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{{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. | 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. | ||
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* [[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| | : [[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| | : [[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 }} | ||
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[[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: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
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
- [[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
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