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.

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.  

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4802000/4782969

: mapping generators: ~2, ~3, ~81/70

Lattice basis:  

: 3/2 length = 0.8110, 81/70 length = 0.5135

: Angle (3/2, 81/70) = 73.88 deg

[[Optimal tuning]]s:  

: [[error map]]: {{val| -0.040 +0.354 -0.163 -0.317 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.3455{{c}}, ~81/70 = 254.6237{{c}}

: error map: {{val| 0.000 +0.390 -0.118 -0.264 }}

* [[7-odd-limit]]: 3 +c/14, 5 and 7 just

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

* [[9-odd-limit]]: 3 just, 5 and 7 -c/7 to 3 +c/14, 5 and 7 just

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

{{Optimal ET sequence|legend=1| 19, 56d, 61d, 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b, 1659b }}

[[Badness]] (Sintel): 4.95

 

== Undecimal canou ==

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.  

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 19712/19683, 42875/42768

[[Optimal tuning]]s:  

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

: [[error map]]: {{val| +0.057 +0.303 -0.314 -0.480 +0.201 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1829{{c}}, ~81/70 = 254.6186{{c}}

: error map: {{val| 0.0000 +0.228 -0.422 -0.604 +0.107 }}

{{Optimal ET sequence|legend=1| 94, 99e, 118, 193, 212, 311, 740, 1051d }}

[[Badness]] (Sintel): 2.45

[[Complexity spectrum]]: 4/3, 9/8, 9/7, 7/6, 5/4, 6/5, 10/9, 11/9, 8/7, 12/11, 11/10, 14/11, 11/8, 7/5

Subgroup: 2.3.5.7.11.13.17

Comma list: 595/594, 833/832, 1156/1155, 19712/19683

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

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

=== 19-limit ===

Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215

Optimal tunings:  

* WE: ~2 = 1200.0624{{c}}, ~3/2 = 702.2377{{c}}, ~22/19 = 254.6139{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2117{{c}}, ~22/19 = 254.5983{{c}}

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

Badness (Sintel): 1.03

Subgroup: 2.3.5.7.11.13.17.19.23

 

 

* 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}}

 

 

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.  

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 896/891, 472392/471625

{{Mapping|legend=1| 1 0 0 -1 6 | 0 1 2 2 -2 | 0 0 4 -3 -3 }}

: [[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}}

: error map: {{val| 0.000 +1.570 +2.539 +1.871 +5.280 }}

 

[[Badness]] (Sintel): 5.43

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 472392/471625

Mapping: {{mapping| 1 0 0 -1 6 11 | 0 1 2 2 -2 -5 | 0 0 4 -3 -3 -3 }}

Optimal tunings:  

* 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 }}

 

 

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.

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 9801/9800, 14641/14580

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

: mapping generators: ~99/70, ~3, ~81/70

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

{{Optimal ET sequence|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}

[[Badness]] (Sintel): 2.64

[[Category:Temperament families]]

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

[[Category:Rank 3]]