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]]), 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.

{{Mapping|legend=1| 1 9 9 8 | 0 -10 -9 -7 }}

: mapping generators: ~2, ~5/3

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 310.146{{c}}

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

{{Optimal ET sequence|legend=1| 27, 31, 58, 89 }}

[[Badness]] (Smith): 0.027044

Mapping: {{mapping| 1 9 9 8 22 | 0 -10 -9 -7 -25 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 310.144{{c}}

{{Optimal ET sequence|legend=0| 27e, 31, 58, 89 }}

Badness (Smith): 0.016842

Mapping: {{mapping| 1 9 9 8 22 0 | 0 -10 -9 -7 -25 5 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 310.276{{c}}

{{Optimal ET sequence|legend=0| 27e, 31, 58 }}

Badness (Smith): 0.017125

Mapping: {{mapping| 1 9 9 8 22 20 | 0 -10 -9 -7 -25 -22 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 310.381{{c}}

Badness (Smith): 0.027568

Comma list: 66/65, 105/104, 126/125, 540/539

Mapping: {{mapping| 1 9 9 8 22 23 | 0 -10 -9 -7 -25 -26 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 309.804{{c}}

Badness (Smith): 0.029868

Mapping: {{mapping| 1 9 9 8 -1 | 0 -10 -9 -7 6 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 309.737{{c}}

Badness (Smith): 0.033434

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

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 310.853{{c}}

Badness (Smith): 0.048687

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. [[31edo]] can be used as a tuning, or [[132edo]] with a val which is the sum of the [[patent val]]s for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. Mosses of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note mos might also be considered from the melodic point of view.

{{Mapping|legend=1| 1 3 4 5 | 0 -11 -13 -17 }}

: mapping generators: ~2, ~49/45

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/45 = 154.579{{c}}

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

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

[[Badness]] (Smith): 0.050389

Mapping: {{mapping| 1 3 4 5 5 | 0 -11 -13 -17 -12 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 154.645{{c}}

* [[11-odd-limit]]: ~11/10 = {{monzo| 1/10 -1/5 0 0 1/10 }}

{{Optimal ET sequence|legend=0| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}

Badness (Smith): 0.025621

Mapping: {{mapping| 1 3 4 5 5 5 | 0 -11 -13 -17 -12 -10 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 154.478{{c}}

{{Optimal ET sequence|legend=0| 8d, 23de, 31, 70f, 101ff }}

Badness (Smith): 0.023323

{{Mapping|legend=1| 1 6 7 8 | 0 -17 -18 -20 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 311.679{{c}}

{{Optimal ET sequence|legend=1| 27, 50, 77 }}

[[Badness]] (Smith): 0.073509

Mapping: {{mapping| 1 6 7 8 18 | 0 -17 -18 -20 -56 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 311.587{{c}}

{{Optimal ET sequence|legend=0| 27e, 77, 104c, 181c }}

Badness (Smith): 0.056915

Mapping: {{mapping| 1 6 7 8 18 5 | 0 -17 -18 -20 -56 -5 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 311.591{{c}}

{{Optimal ET sequence|legend=0| 27e, 77, 104c, 181c }}

Badness (Smith): 0.035582

{{Mapping|legend=1| 2 7 8 8 | 0 -8 -7 -5 }}

[[Optimal tuning]] ([[POTE]]): ~343/240 = 600.000{{c}}, ~6/5 = 312.602{{c}}

{{Optimal ET sequence|legend=1| 42, 46, 96d, 142d, 238dd }}

[[Badness]] (Smith): 0.078049

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

Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~6/5 = 312.601{{c}}

{{Optimal ET sequence|legend=0| 42, 46, 96d, 142d, 238dd }}

Badness (Smith): 0.044499

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

Optimal tuning (POTE): ~55/39 = 600.000{{c}}, ~6/5 = 312.564{{c}}

Badness (Smith): 0.029693

{{Mapping|legend=1| 1 4 4 3 | 0 -13 -9 -1 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~8/7 = 222.797{{c}}

[[Badness]] (Smith): 0.087500

Mapping: {{mapping| 1 4 4 3 7 | 0 -13 -9 -1 -19 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~8/7 = 222.898{{c}}

Badness (Smith): 0.043336

Mapping: {{mapping| 1 4 4 3 7 5 | 0 -13 -9 -1 -19 -7 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~8/7 = 222.961{{c}}

Badness (Smith): 0.028920

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Cypress]].''

[[Comma list]]: 126/125, 19683/19208

{{Mapping|legend=1| 1 7 10 15 | 0 -12 -17 -27 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~135/98 = 541.828{{c}}

{{Optimal ET sequence|legend=1| 11cd, 20cd, 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bbcd }}

[[Badness]] (Smith): 0.099801

Comma list: 99/98, 126/125, 243/242

Mapping: {{mapping| 1 7 10 15 17 | 0 -12 -17 -27 -30 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/11 = 541.772{{c}}

{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde }}

Badness (Smith): 0.042719

=== 13-limit ===

Comma list: 66/65, 99/98, 126/125, 243/242

Mapping: {{mapping| 1 7 10 15 17 15 | 0 -12 -17 -27 -30 -25 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/11 = 541.778{{c}}

{{Optimal ET sequence|legend=0| 11cdeef, 20cdef, 31 }}

Badness (Smith): 0.037849

[[Optimal tuning]] ([[POTE]]): ~343/243 = 600.000{{c}}, ~35/27 = 455.445{{c}}

[[Badness]] (Smith): 0.097786

Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~35/27 = 455.373{{c}}

Badness (Smith): 0.041190

Optimal tuning (POTE): ~55/39 = 600.000{{c}}, ~13/10 = 455.347{{c}}

Badness (Smith): 0.023877

: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Casablanca]].''

Aside from 126/125, casablanca tempers out the no-threes comma [[823543/819200]] and also [[589824/588245]], and may also be described as {{nowrap| 31 & 73 }}. 74\135 or 91\166 supply good tunings for the generator, and 20- and 31-note mosses are available.

It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the ~35/24 generator is particularly interesting; like 15/14 and 21/20, it represents an interval between one vertex of a [[hexany]] and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads. For another, if we add 385/384 to the list of commas, 35/24 is identified with 16/11, and casablanca is revealed as an 11-limit temperament with a very low complexity for 11 and not too high a one for 7; we might compare 1, 4, 14, 19, the generator steps to 11, 7, 5 and 3 respectively, with 1, 4, 10, 18, the steps to 3, 5, 7 and 11 in 11-limit meantone.

Marrakesh, named by [[Herman Miller]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19166.html#19186 Yahoo! Tuning Group | ''A rose by any other name . . .'']</ref>, is a more accurate 11-limit extension where the generator is identified with 22/15 as opposed to 16/11 in casablanca.  

{{Mapping|legend=1| 1 12 10 5 | 0 -19 -14 -4 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~35/24 = 657.818{{c}}

[[Badness]] (Smith): 0.101191

Mapping: {{mapping| 1 12 10 5 4 | 0 -19 -14 -4 -1 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/11 = 657.923{{c}}

Badness (Smith): 0.067291

Mapping: {{mapping| 1 12 10 5 4 7 | 0 -19 -14 -4 -1 -6 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/11 = 657.854{{c}}

Mapping: {{mapping| 1 12 10 5 21 | 0 -19 -14 -4 -32 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~22/15 = 657.791{{c}}

Badness (Smith): 0.040539

Mapping: {{mapping| 1 12 10 5 21 -10 | 0 -19 -14 -4 -32 25 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~22/15 = 657.756{{c}}

Badness (Smith): 0.040774

Mapping: {{mapping| 1 12 10 5 21 7 | 0 -19 -14 -4 -32 -6 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~22/15 = 657.700{{c}}

{{Optimal ET sequence|legend=0| 31, 104cff, 135cff }}

Badness (Smith): 0.041395

{{See also| High badness temperaments #Magus }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~5/4 = 391.094{{c}}

[[Badness]] (Smith): 0.110873

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~5/4 = 391.075{{c}}

Badness (Smith): 0.043438

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~5/4 = 391.073{{c}}

{{Optimal ET sequence|legend=0| 43, 46, 89, 135cf, 224cf }}

Badness (Smith): 0.030666

{{Mapping|legend=1| 1 4 5 6 | 0 -9 -10 -12 }}

* [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 321.109{{c}}

* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 321.423{{c}}

{{Optimal ET sequence|legend=1| 11cd, 15, 41dd, 56dd }}

[[Badness]] (Smith): 0.115292

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~343/270 = 446.568{{c}}

{{Optimal ET sequence|legend=1| 8d, 35, 43 }}

[[Badness]] (Smith): 0.148531

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~72/55 = 446.616{{c}}

{{Optimal ET sequence|legend=0| 8d, 35, 43 }}

Badness (Smith): 0.059449

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/10 = 446.598{{c}}

{{Optimal ET sequence|legend=0| 8d, 35f, 43 }}

Badness (Smith): 0.035258

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/10 = 446.631{{c}}

{{Optimal ET sequence|legend=0| 8d, 35f, 43 }}

Badness (Smith): 0.025907

Cobalt ({{nowrap| 27 & 81 }}) has a period of 1/27 octave and tempers out 126/125 and 540/539, as well as the [[aplonis]] temperament.  

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

{{Mapping|legend=1| 27 43 63 76 | 0 -1 -1 -1 }}

[[Optimal tuning]] ([[POTE]]): ~36/35 = 44.444, ~3/2 = 701.244{{c}}

{{Optimal ET sequence|legend=1| 27, 81, 108, 135c, 243c }}

[[Badness]] (Smith): 0.173308

Mapping: {{mapping| 27 43 63 76 94 | 0 -1 -1 -1 -2 }}

Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 700.001{{c}}

Badness (Smith): 0.078060

Mapping: {{mapping| 27 43 63 76 94 100 | 0 -1 -1 -1 -2 0 }}

Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 700.867{{c}}

Badness (Smith): 0.057145

Mapping: {{mapping| 27 43 63 76 94 100 111 | 0 -1 -1 -1 -2 0 -2 }}

Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 700.397{{c}}

Badness (Smith): 0.042106

Mapping: {{mapping| 27 43 63 76 94 100 111 115 | 0 -1 -1 -1 -2 0 -2 -1 }}

Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 700.429{{c}}

Badness (Smith): 0.030415

Mapping: {{mapping| 27 43 63 76 94 100 111 | 0 -1 -1 -1 -2 0 -3 }}

Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 701.595{{c}}

{{Optimal ET sequence|legend=0| 27eg, 81gg, 108, 135ce }}

Badness (Smith): 0.047163

Mapping: {{mapping| 27 43 63 76 94 100 111 115 | 0 -1 -1 -1 -2 0 -3 -1 }}

Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 701.673{{c}}

{{Optimal ET sequence|legend=0| 27eg, 81gg, 108, 135ceh }}

Badness (Smith): 0.034176

Mapping: {{mapping| 27 43 63 76 94 100 | 0 -1 -1 -1 -2 -1 }}

Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 699.179{{c}}

{{Optimal ET sequence|legend=0| 27e, 54bdef, 81f, 108f }}

Badness (Smith): 0.052732