Starling temperaments: Difference between revisions

* ''[[Opossum]]'' (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]]

* ''[[Passionate]]'' (+131072/127575) → [[Passion family #Passionate|Passion family]]

* ''[[Vishnean]]'' (+540225/524288) → [[Vishnuzmic family #Vishnean|Vishnuzmic family]]

* ''[[Ditonic]]'' (+8751645/8388608) → [[Ditonmic family #Ditonic|Ditonmic family]]

* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila 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]]), leaving space for a neutral third in between. In that sense, it is opposed to [[keemic temperaments]], where the chroma between the pental thirds is the same as the distance between the pental and septimal thirds.

In terms of commas tempered, 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. It has 6/5 as a generator, and [[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round amounts in cents 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 limits.

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

Mapping: {{mapping| 1 -1 0 1 -3 | 0 10 9 7 25 }}

Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}

Mapping: {{mapping| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}

Mapping: {{mapping| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}

Mapping: {{mapping| 1 -1 0 1 5 | 0 10 9 7 -6 }}

Mapping: {{mapping| 1 -1 0 1 4 | 0 10 9 7 -2 }}

Nusecond tempers out [[2430/2401]] and [[16875/16807]] in addition to 126/125, and may be described as {{nowrap| 31 & 70 }}. It has a neutral second generator of [[49/45]], two of which make up a 6/5 minor third since 2430/2401 is tempered out. Note that in the data below, the generator is its [[octave complement]] since eleven such generators [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus theta-hendecacot.  

Mapping: {{mapping| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}

Mapping: {{mapping| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}

Mapping: {{mapping| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}

Mapping: {{mapping| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}

Mapping: {{mapping| 2 -1 1 3 9 | 0 8 7 5 -4 }}

Mapping: {{mapping| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }}

Mapping: {{mapping| 1 -9 -5 2 -12 | 0 13 9 1 19 }}

Mapping: {{mapping| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }}

Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}

Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}

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 }}. 61\135 or 75\166 supply good tunings for the generator, and 20- and 31-note [[mos scale]]s are available.

Mapping: {{mapping| 1 -7 -4 1 3 | 0 19 14 4 1 }}

Mapping: {{mapping| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }}

Mapping: {{mapping| 1 -7 -4 1 -11 | 0 19 14 4 32 }}

Mapping: {{mapping| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }}

Mapping: {{mapping| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }}

Mapping: {{mapping| 1 -2 2 9 9 | 0 11 1 -19 -17 }}

Mapping: {{mapping| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}

Supersensi may be described as {{nowrap| 8d & 43 }}. It has supermajor third as a generator like [[sensi]], but the no-fives comma [[17496/16807]] rather than [[245/243]] tempered out.

Mapping: {{mapping| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}

Mapping: {{mapping| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}

Mapping: {{mapping| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}

The name of the cobalt temperament comes from the 27th element.

Mapping: {{mapping| 27 0 20 33 8 | 0 1 1 1 2 }}

Mapping: {{mapping| 27 0 20 33 8 100 | 0 1 1 1 2 0 }}

Mapping: {{mapping| 27 0 20 33 8 100 79 | 0 1 1 1 2 0 2 }}

Mapping: {{mapping| 27 0 20 33 8 100 79 99 | 0 1 1 1 2 0 2 1 }}

Mapping: {{mapping| 27 0 20 33 8 100 -18 | 0 1 1 1 2 0 3 }}

Mapping: {{mapping| 27 0 20 33 8 100 -18 72 | 0 1 1 1 2 0 3 1 }}

Mapping: {{mapping| 27 0 20 33 8 57 | 0 1 1 1 2 1 }}

[[Category:Starling temperaments| ]] <!-- main article -->

[[Category:Rank 2]]