Cathartic family: Difference between revisions

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

== Cathartic ==

{{See also| Cathartic scales }}

Cathartic (formerly ''canou'') 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.  

9-, 14- and 19-note scales are highly characteristic for the temperament, with the abundance of [[28/27]] subminor seconds making it melodically active.  

[[Comma list]]: 4802000/4782969

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

[[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

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

[[Badness]] (Sintel): 4.95

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]].

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]]. These extensions add little additional error, and [[311edo]] remains an excellent tuning in all cases.  

{{Mapping|legend=1| 1 0 0 -1 -7 | 0 1 2 2 7 | 0 0 -4 3 -3 }}

[[Optimal tuning]]s:

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

{{Optimal ET sequence|legend=1| 24, …, 75e, 94, 99e, 118, 193, 212, 311, 740, 1051d }}

[[Badness]] (Sintel): 2.45

Subgroup: 2.3.5.7.11.17

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

Subgroup-val mapping: {{mapping| 1 0 0 -1 -7 -5 | 0 1 2 2 7 6 | 0 0 -4 3 -3 -2 }}

Optimal tunings:  

* WE: ~2 = 1200.0485{{c}}, ~3/2 = 702.2481{{c}}, ~51/44 = 254.6343{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2325{{c}}, ~51/44 = 254.6253{{c}}

{{Optimal ET sequence|legend=0| 24, …, 75e, 94, 99e, 118, 193, 212g, 217, 311, 1051dg }}

Badness (Sintel): 1.01

Subgroup: 2.3.5.7.11.17.19

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

Subgroup-val mapping: {{mapping| 1 0 0 -1 -7 -5 -6 | 0 1 2 2 7 6 7 | 0 0 -4 3 -3 -2 -4 }}

Optimal tunings:  

* WE: ~2 = 1200.0444{{c}}, ~3/2 = 702.2569{{c}}, ~22/19 = 254.6305{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2421{{c}}, ~22/19 = 254.6224{{c}}

{{Optimal ET sequence|legend=0| 24, …, 75e, 94, 99e, 118, 193, 217, 311, 1051dgh }}

Badness (Sintel): 0.641

Subgroup: 2.3.5.7.11.17.19.23

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

Optimal tunings:  

* WE: ~2 = 1199.9778{{c}}, ~3/2 = 702.2794{{c}}, ~22/19 = 254.6572{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2883{{c}}, ~22/19 = 254.6625{{c}}

{{Optimal ET sequence|legend=0| 24, 75e, 94, 99e, 118, 193, 217, 311 }}

Badness (Sintel): 0.703

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.  

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

[[Optimal tuning]]s:

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

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

[[Badness]] (Sintel): 5.43

{{Mapping|legend=0| 1 0 0 -1 6 11 | 0 1 2 2 -2 -5 | 0 0 4 -3 -3 -3 }}

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}}

Badness (Sintel): 4.47

Semicathart (formerly ''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 [[semicathartisma]], 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.  

{{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

[[Optimal tuning]]s:  

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

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

[[Badness]] (Sintel): 2.64

[[Category:Catalogs of rank-3 temperaments]]