Canou family: Difference between revisions

The '''canou family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperament]]s [[tempering out|tempers out]] the [[canousma]], 4802000/4782969 ({{monzo| 4 -14 3 4 }}), a 7-limit comma measuring about 6.9 [[cent]]s.

{{Main| Canou temperament }}

The canou temperament features a [[period]] of an [[octave]] and [[generator]]s of [[3/2]] and [[81/70]]. The ~81/70-generator is about 255 cents. Three make [[14/9]]; four make [[9/5]]. It therefore splits the large septimal diesis, [[49/48]], into three equal parts, making two distinct [[interseptimal interval]]s related to the 35th harmonic.  

A basic tuning option would be [[99edo]], although [[80edo]] is even simpler and distinctive. More intricate tunings are provided by [[311edo]] and [[410edo]], whereas the [[optimal patent val]] goes up to [[1131edo]], relating it to the [[amicable]] temperament.  

[[Comma list]]: [[4802000/4782969]]

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.3175, ~81/70 = 254.6220

: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7

: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5

[[Badness]] (Smith): 1.122 × 10^-3

The fifth is in the range where a stack of four (i.e. a major third) can serve as ~[[19/15]] and a stack of five (i.e. a major seventh) can serve as ~[[19/10]], tempering out [[1216/1215]]. Moreover, the last generator of ~81/70 is sharpened to slightly overshoot [[22/19]], so it only makes sense to temper out their difference, [[1540/1539]]. The implied 11-limit comma is the [[symbiotic comma]], which suggests the [[wilschisma]] should also be tempered out in the 13-limit.  

Since the syntonic comma has been split in two, it is natural to map [[19/17]] to the mean of [[9/8]] and [[10/9]], tempering out [[1445/1444]]. From a commatic point of view, notice the other 11-limit comma, [[42875/42768]], is {{nowrap| S34 × S35² }}, suggesting tempering out [[595/594]] (S34 × S35), [[1156/1155]] (S34), and [[1225/1224]] (S35), which coincides with above. Finally, we can map [[23/20]] to the fourth complement of 22/19 to make an equidistant sequence consisting of 7/6, 22/19, 23/20, and 8/7, tempering out [[760/759]]. 311edo remains an excellent tuning in all the limits.  

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.2115, ~81/70 = 254.6215

[[Badness]] (Smith): 2.04 × 10^-3

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2075, ~81/70 = 254.6183

Badness (Smith): 2.56 × 10^-3

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2296, ~51/44 = 254.6012

Badness (Smith): 1.49 × 10^-3

Mapping: {{mapping|| 1 0 0 -1 -7 -13 -5 -6 | 0 1 2 2 7 10 6 7 | 0 0 -4 3 -3 4 -2 -4 }}

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2355, ~22/19 = 254.5930

Badness (Smith): 1.00 × 10^-3

By adding [[896/891]], the pentacircle comma, [[33/32]] is equated with 28/27, so the scale is filled with this 33/32~28/27 mixture. This may be described as 75e & 80 & 99e, and 80edo makes the optimal. It has a natural extension to the 13-limit since 896/891 = (352/351)(364/363), named ''gentcanta'' in earlier materials.  

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8093, ~64/55 = 254.3378

{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e }}

[[Badness]]: 4.523 × 10^-3

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 703.6228, ~64/55 = 254.3447

{{Optimal ET sequence|legend=1| 75e, 80, 99ef, 179ef }}

Badness: 4.781 × 10^-3

Semicanou adds [[9801/9800]], the kalisma, to the comma list, and may be described as 80 & 94 & 118. It splits the octave into two equal parts, each representing 99/70~140/99. Note that {{nowrap| 99/70 {{=}} (81/70)(11/9) }}, this extension is more than natural.  

[[Optimal tuning]] ([[CTE]]): ~99/70 = 600.0000, ~3/2 = 702.4262, ~81/70 = 254.6191

[[Badness]]: 2.197 × 10^-3

 

 

 

 

Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.4802, ~81/70 = 254.6526

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Badness: 3.511 × 10^-3