Starling temperaments: Difference between revisions

* [[Diaschismic]] (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]

* [[Wollemia]] (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]

 

Since {{nowrap|(6/5)³ {{=}} (126/125)⋅(12/7)}}, these temperaments tend to have a relatively small complexity for 6/5. They also possess the [[starling tetrad]], the 6/5–6/5–6/5–7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before [[12edo]] established itself as the standard tuning, it is actually three stacked minor thirds and an augmented second, contrary to the popular belief that it is four stacked minor thirds.  

7-limit myna is naturally found by establishing a structure of thirds, by making [[7/6]]–[[6/5]]–[[49/40]]–[[5/4]]–[[9/7]] all equidistant (the distances between which are [[36/35]], [[49/48]], and [[50/49]]). [[11-limit]] myna then arises from equating this neutral third to [[11/9]]. Myna's characteristic feature is that the pental thirds are tuned outwards so that the chroma between them ([[25/24]]) is twice the size of the interval between the pental and septimal thirds ([[36/35]]). In that sense, it is opposed to [[keemic temperaments]], in particular [[quasitemp]], where the distance between the pental and septimal thirds is the same as the chroma between the pental thirds and different from the septimal dieses.  

In terms of vanishing commas, in addition to 126/125, myna adds [[1728/1715]], the orwell comma, and [[2401/2400]], the breedsma. It can also be described as the {{nowrap| 27 & 31 }} temperament, and has a [[ploidacot]] signature of beta-decacot. It has [[~]][[6/5]] as a generator.

 

[[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round cent values may like [[120edo]]. It is also possible to tune myna with pure fifths by taking 6^1/10 as the generator. Myna extends naturally but with much increased complexity to the 11- and 13-limit.

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3

== Paraguay ==

: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic]].''

 

[[Comma list]]: 126/125, 12005/11664

{{Mapping|legend=1| 1 -8 -8 -9 | 0 13 14 16 }}

* [[WE]]: ~2 = 1200.6421{{c}}, ~5/3 = 885.3232{{c}}

: [[error map]]: {{val| +0.642 +2.110 +3.074 -9.434 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8949{{c}}

: error map: {{val| 0.000 +1.678 +2.214 -10.508 }}

{{Optimal ET sequence|legend=1| 19, 61, 80d, 99d }}

[[Badness]] (Sintel): 2.47

Comma list: 56/55, 100/99, 12005/11664

 

 

 

 

 

 

Comma list: 56/55, 91/90, 100/99, 343/338

Mapping: {{mapping| 1 -8 -8 -9 2 -14 | 0 13 14 16 2 24 }}

* WE: ~2 = 1197.7848{{c}}, ~5/3 = 883.6431{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1623{{c}}

{{Optimal ET sequence|legend=0| 19, 42ef, 61e }}

Badness (Sintel): 1.86

==== Uruguay ====

Comma list: 56/55, 78/77, 100/99, 1183/1152

Mapping: {{mapping| 1 -8 -8 -9 2 0 | 0 13 14 16 2 5 }}

* WE: ~2 = 1199.6132{{c}}, ~5/3 = 884.7325{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.0005{{c}}

{{Optimal ET sequence|legend=0| 19, 42e }}

Badness (Sintel): 2.51

Aside from 126/125, casablanca tempers out the no-threes comma [[823543/819200]] and also [[589824/588245]], and may be described as {{nowrap| 31 & 73 }} with a [[ploidacot]] signature of eta-19-cot. 61\135 or 75\166 supply good tunings for the generator, and 20- and 31-note [[mos scale]]s are available.

Named by [[Xenllium]] in 2022, supersensi tempers out the no-fives comma [[17496/16807]], and may be described as {{nowrap| 8d & 43 }}. It has a ultramajor third generator, which is sharper than the generator for [[sensi]], hence the name. Its [[ploidacot]] is epsilon-15-cot.  

Cobalt was named by [[Xenllium]] in 2022 after the 27th element.