Octagar family: Difference between revisions
Switch to Sintel's badness, WE & CWE tunings |
- POTE tunings |
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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.3764{{c}}, ~63/40 = 792.6299{{c}} | * [[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 }} | : error map: {{val| 0.000 +1.421 -1.054 +2.311 }} | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6783{{c}}, ~11/7 = 791.5168{{c}} | * [[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 }} | : error map: {{val| 0.000 +1.723 -3.280 -1.632 +7.393 }} | ||
{{Optimal ET sequence|legend=1| 12, 15, 26, 27e, 41, 109e }} | {{Optimal ET sequence|legend=1| 12, 15, 26, 27e, 41, 109e }} | ||
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* WE: ~2 = 1200.5397{{c}}, ~3/2 = 702.2038{{c}}, ~11/7 = 789.6574{{c}} | * 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}} | * 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 }} | {{Optimal ET sequence|legend=0| 12f, 14cf, 15, 26, 29, 41 }} | ||
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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.2662{{c}}, ~63/40 = 792.6212{{c}} | * [[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 }} | : 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 }} | {{Optimal ET sequence|legend=1| 27e, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d }} | ||
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* WE: ~2 = 1199.5112{{c}}, ~3/2 = 703.2823{{c}}, ~63/40 = 792.6491{{c}} | * 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}} | * 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 }} | {{Optimal ET sequence|legend=0| 27e, 41, 53, 80, 94, 121, 174d, 215, 295d, 336def }} | ||
Revision as of 14:09, 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
- 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]
- [[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
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
- 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]]
- 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