Cathartic 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.

== Canou ==

{{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.  

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

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

 

 

 

 

 

 

=== 17-limit ===

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 }}

* 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 ===

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 }}

* 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

=== 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 }}

* 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

== Semicanou ==

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.  

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