Octagar family: Difference between revisions

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 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]].

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.

This temperament is catalogued as ''octagari'' in [[Graham Breed]]'s [https://x31eq.com/temper/ Temperament Finder].  

[[Subgroup]]: 2.3.5.7

[[Comma list]]: [[4000/3969]]

{{Mapping|legend=1| 1 0 1 4 | 0 1 0 -2 | 0 0 2 3 }}

: Mapping generators: ~2, ~3, ~63/40

[[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

 

* [[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 }}

 

* [[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

 

{{Optimal ET sequence|legend=1| 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d }}

 

[[Badness]] (Sintel): 0.951

 

[[Projection pair]]s: <code>5 3969/800 7 27783/4000</code> to 2.3.7/5

 

{{Databox|[[Hobbit|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 to lattice: [{{val| 0 1 2 1 2 }}, {{val| 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{{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 }}

 

 

[[Badness]] (Sintel): 0.647

 

[[Projection pair]]s: <code>5 242/49 7 21296/3087 11 234256/21609</code> to 2.3.11/7

 

 

 

Subgroup: 2.3.5.7.11.13

 

Comma list: 100/99, 105/104, 245/242

 

Mapping: {{mapping| 1 0 1 4 4 2 | 0 1 0 -2 -2 -1 | 0 0 2 3 4 5 }}

 

* 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

 

 

[[Comma list]]: 540/539, 2200/2187

 

{{Mapping|legend=1| 1 0 1 4 -5 | 0 1 0 -2 7 | 0 0 2 3 -4 }}

 

* [[WE]]: ~2 = 1199.5255{{c}}, ~3/2 = 703.2719{{c}}, ~63/40 = 792.6321{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.2662{{c}}, ~63/40 = 792.6212{{c}}

 

 

[[Badness]] (Sintel): 1.20

 

=== 13-limit ===

 

Comma list: 325/324, 352/351, 540/539

 

 

* 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}}

 

 

 

[[Category:Octagar family| ]] <!-- main article -->