Canou family: Difference between revisions

The '''canou family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[canousma]] ({{monzo|legend=1| 4 -14 3 4 }}, [[ratio]]: 4802000/4782969), a 7-limit comma measuring about 6.9 [[cent]]s.

{{Main| Canou }}

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 wide, three of which make [[14/9]], and four make [[9/5]]. It therefore splits the large septimal diesis, [[49/48]], into three equal parts, guaranteeing the existence of two [[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]], associating it with the [[amicable]] temperament.  

[[Comma list]]: 4802000/4782969

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9597{{c}}, ~3/2 = 702.3492{{c}}, ~81/70 = 254.6168{{c}}

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

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

[[Badness]] (Sintel): 4.95

== Undecimal canou ==

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]], while the other 11-limit comma, [[42875/42768]] (S34⋅S35²), suggests 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]]s:

* [[WE]]: ~2 = 1200.0568{{c}}, ~3/2 = 702.2009{{c}}, ~81/70 = 254.6291{{c}}

[[Badness]] (Sintel): 2.45

Optimal tunings:  

* WE: ~2 = 1200.0501{{c}}, ~3/2 = 702.2100{{c}}, ~81/70 = 254.6345{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1889{{c}}, ~81/70 = 254.6222{{c}}

{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}

Badness (Sintel): 2.39

Optimal tunings:  

* WE: ~2 = 1200.0630{{c}}, ~3/2 = 702.2317{{c}}, ~51/44 = 254.6224{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2055{{c}}, ~51/44 = 254.6066{{c}}

{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}

Badness (Sintel): 1.41

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 }}

 

 

 

 

 

 

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

Optimal tunings:  

* WE: ~2 = 1200.0004{{c}}, ~3/2 = 702.2361{{c}}, ~22/19 = 254.6225{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2359{{c}}, ~22/19 = 254.6223{{c}}

{{Optimal ET sequence|legend=0| 94, 193f, 212gh, 217, 311 }}

Badness (Sintel): 1.09

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 {{nowrap| 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]]s:

* [[WE]]: ~2 = 1199.0708{{c}}, ~3/2 = 703.1969{{c}}, ~64/55 = 254.4161{{c}}

: [[error map]]: {{val| -0.929 +0.313 +0.557 -0.113 +1.820 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.5249{{c}}, ~64/55 = 254.5492{{c}}

{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e, 457bcddeeee }}

[[Badness]] (Sintel): 5.43

Optimal tunings:  

* WE: ~2 = 1199.0093{{c}}, ~3/2 = 703.2884{{c}}, ~64/55 = 254.4219{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8323{{c}}, ~64/55 = 254.5887{{c}}

{{Optimal ET sequence|legend=0| 75e, 80, 99ef, 179ef, 462bccddeeeff }}

Badness (Sintel): 4.47

Semicanou adds 9801/9800, the [[kalisma]], to the comma list, and may be described as {{nowrap| 80 & 94 & 118 }}. It splits the octave into two equal parts, each representing 99/70~140/99. This takes advantage of the fact that {{nowrap| 99/70 {{=}} (81/70)⋅(11/9) }}.  

The other comma necessary to define it is 14641/14580, the [[semicanousma]], which is the difference between [[121/120]] and [[243/242]]. By flattening the 11th harmonic by about one cent, it identifies [[20/11]] by three [[11/9]]'s stacked, so an octave can be divided into 11/9, 11/9, 11/9, and 11/10.  

[[Optimal tuning]]s:

* [[WE]]: ~99/70 = 600.0142{{c}}, ~3/2 = 702.4017{{c}}, ~81/70 = 254.6228{{c}}

: [[error map]]: {{val| +0.028 +0.475 +0.055 -0.126 -1.066 }}

* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 702.4048{{c}}, ~81/70 = 254.6179{{c}}

[[Badness]] (Sintel): 2.64