Xenharmonic Wiki

Octagar family: Difference between revisions

← Older edit
VisualWikitext
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,164 edits
link for mapping to lattice
+ more intro to octagar
 
(19 intermediate revisions by 7 users not shown)
Line 1: Line 1:
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 ==
[[Comma]]: 4000/3969
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]].


[[Mapping]]: [{{val|1 0 1 4}}, {{val|0 1 0 -2}}, {{val|0 0 2 3}}]
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.


Mapping generators: 2, ~3, ~63/40
This temperament is catalogued as ''octagari'' in [[Graham Breed]]'s [https://x31eq.com/temper/ Temperament Finder].


Minimax tuning:
[[Subgroup]]: 2.3.5.7
* 7-limit minimax:
: [{{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}}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 7/6, 5/4


* 9-limit minimax:
[[Comma list]]: [[4000/3969]]
: [{{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}}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 5/4, 9/7


Lattice basis: 63/50 length = 0.8966, 21/20 length = 1.0605
{{Mapping|legend=1| 1 0 1 4 | 0 1 0 -2 | 0 0 2 3 }}
: Mapping generators: ~2, ~3, ~63/40


Angle(63/50, 21/20) = 97.743 cents
[[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}}]
Lattice basis:  
: 63/50 length = 0.8966, 21/20 length = 1.0605
: Angle (63/50, 21/20) = 97.743 degrees


{{Val list|legend=1| 12, 26, 27, 41, 53, 94, 121, 162, 189, 215, 230 }}
[[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 }}


[[Badness]]: 0.000216
[[Minimax tuning]]:
* [[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 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
* [[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 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7


[[Projection pair]]s: 5 - 3969/800, 7 - 27783/4000 to 2.3.7/5
{{Optimal ET sequence|legend=1| 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d }}


;[[Hobbits|Hobbit bases]]
[[Badness]] (Sintel): 0.951
2.3.7/5 subgroup


12: 50/49, 256000/250047
[[Projection pair]]s: <code>5 3969/800 7 27783/4000</code> to 2.3.7/5


15: 256000/250047, 1029/1000
{{Databox|[[Hobbit|Hobbit bases]]|
2.3.7/5 subgroup
* 12: 50/49, 256000/250047
* 15: 256000/250047, 1029/1000
* 23: 12800000/12252303, 107163/102400
}}


23: 12800000/12252303, 107163/102400
== Nakika ==
[[Subgroup]]: 2.3.5.7.11


=== Nakika ===
[[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 generators: 2, ~3, ~11/7
Mapping to lattice: [{{val| 0 1 2 1 2 }}, {{val| 0 -1 0 2 2 }}]


Associated linear temperament: [[Tetracot family|octacot]]
Lattice basis:
: 11/7 length = 0.798, 22/21 length = 0.906
: Angle (11/7, 22/21) = 97.747 degrees


Lattice basis: 11/7 length = 0.798, 22/21 length = 0.906
[[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 }}


Angle(11/7, 22/21) = 97.747 degrees
{{Optimal ET sequence|legend=1| 12, 15, 26, 27e, 41, 109e }}


Mapping to lattice: [{{val|0 1 2 1 2}}, {{val|0 -1 0 2 2}}]
[[Badness]] (Sintel): 0.647


Vals: {{Val list| 12, 15, 26, 29, 41 }}
[[Projection pair]]s: <code>5 242/49 7 21296/3087 11 234256/21609</code> to 2.3.11/7


Badness: 0.000539
[[Associated temperament]]: [[octacot]]


Projection pairs: 5 - 242/49, 7 - 21296/3087, 11 - 234256/21609 to 2.3.11/7
Scales: [[nakika12]]     


Scales: [[nakika12]]     
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==== 13-limit ====
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}}


Mapping generators: 2, ~3, ~11/7
{{Optimal ET sequence|legend=0| 27e, 41, 53, 80, 94, 121, 174d, 215, 295d, 336def }}


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


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Octagar family| ]] <!-- main article -->
[[Category:Octagar]]
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Octagar_family"