Cathartic 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.

== Canou ==

{{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]]

* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.3175, ~81/70 = 254.6220

: [[error map]]: {{val| 0.0000 +0.3625 -0.1667 -0.3249 }}

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

: error map: {{val| 0.0000 +0.3904 -0.1175 -0.2640 }}

[[Badness]] (Smith): 1.122 × 10^-3

== 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]]. 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.  

* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.2115, ~81/70 = 254.6215

: [[error map]]: {{val| 0.0000 +0.2565 -0.3768 -0.5383 +0.2980 }}

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

: error map: {{val| 0.0000 +0.2279 -0.4221 -0.6043 +0.1069 }}

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

[[Badness]] (Smith): 2.04 × 10^-3

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

 

 

 

* CTE: ~2 = 1200.0000, ~3/2 = 702.2075, ~81/70 = 254.6183

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

 

 

Badness (Smith): 2.56 × 10^-3

Subgroup: 2.3.5.7.11.13.17

 

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

* CTE: ~2 = 1200.0000, ~3/2 = 702.2296, ~51/44 = 254.6012

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

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

Badness (Smith): 1.49 × 10^-3

=== 19-limit ===

Subgroup: 2.3.5.7.11.13.17.19

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

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 }}

* CTE: ~2 = 1200.0000, ~3/2 = 702.2355, ~22/19 = 254.5930

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

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

Badness (Smith): 1.00 × 10^-3

=== 23-limit ===

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1156/1155

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 }}

* CTE: ~2 = 1200.0000, ~3/2 = 702.2361, ~22/19 = 254.6222

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

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

Badness (Smith): 0.948 × 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.  

* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.8093, ~64/55 = 254.3378

: [[error map]]: {{val| 0.0000 +0.8543 +1.9537 -0.1940 +6.0769 }}

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

: error map: {{val| 0.0000 +1.5699 +2.5393 +1.8714 +5.2799 }}

[[Badness]] (Smith): 4.52 × 10^-3

* CTE: ~2 = 1200.0000, ~3/2 = 703.6228, ~64/55 = 254.3447

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

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

Badness (Smith): 4.78 × 10^-3

== Semicanou ==

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.  

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.  

* [[CTE]]: ~99/70 = 600.0000, ~3/2 = 702.4262, ~81/70 = 254.6191

: [[error map]]: {{val| 0.0000 +0.4712 +0.0625 -0.1163 -1.0846 }}

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

: error map: {{val| 0.0000 +0.4498 +0.0245 -0.1627 -1.1262 }}

[[Badness]] (Smith): 2.20 × 10^-3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

[[Category:Canou| ]] <!-- key article -->