Starling temperaments: Difference between revisions

This page discusses miscellaneous [[rank-2 temperament]]s tempering out [[126/125]], the starling comma or septimal semicomma.  

* [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]]

* ''[[Mavling]]'' (+135/128) → [[Mavila family #Mavling|Mavila family]]

* ''[[Grackle]]'' (+32805/32768) → [[Schismatic family #Grackle|Schismatic family]]

* ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]]

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

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

* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]

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

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

* ''[[Ripple]]'' (+405/392) → [[Ripple family #Septimal ripple|Ripple family]]

* [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]]

* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]

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.  

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.

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

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.3410{{c}}, ~6/5 = 309.9756{{c}}

: [[error map]]: {{val| -0.659 -1.540 +3.467 +0.344 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 310.0880{{c}}

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

{{Optimal ET sequence|legend=1| 27, 31, 58, 89, 236cc }}

[[Badness]] (Sintel): 0.684

{{Mapping|legend=0| 1 -1 0 1 -3 | 0 10 9 7 25 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.0982{{c}}

{{Optimal ET sequence|legend=0| 27e, 31, 58, 89, 236cce }}

Badness (Sintel): 0.557

{{Mapping|legend=0| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.2218{{c}}

{{Optimal ET sequence|legend=0| 27e, 31, 58, 205cceff, 263ccdeefff }}

Badness (Sintel): 0.708

{{Mapping|legend=0| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}

Optimal tunings:

* WE: ~2 = 1199.1929{{c}}, ~6/5 = 310.1724{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.3251{{c}}

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

Badness (Sintel): 1.14

Comma list: 66/65, 105/104, 126/125, 243/242

{{Mapping|legend=0| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}

Optimal tunings:

* WE: ~2 = 1199.9109{{c}}, ~6/5 = 309.7815{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7987{{c}}

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

Badness (Sintel): 1.23

{{Mapping|legend=0| 1 -1 0 1 5 | 0 10 9 7 -6 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7812{{c}}

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

Badness (Sintel): 1.11

{{Mapping|legend=0| 1 -1 0 1 4 | 0 10 9 7 -2 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.6398{{c}}

{{Optimal ET sequence|legend=0| 4, 23bc, 27e }}

Badness (Sintel): 1.61

== Nusecond ==

Nusecond tempers out [[2430/2401]] and [[16875/16807]] in addition to 126/125, and may be described as the {{nowrap| 31 & 39d }} temperament. 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.  

[[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 -8 -9 -12 | 0 11 13 17 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.6138{{c}}, ~49/27 = 1045.0850{{c}}

: [[error map]]: {{val| -0.386 -2.931 +3.267 +2.253 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/27 = 1045.3909{{c}}

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

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

[[Badness]] (Sintel): 1.28

{{Mapping|legend=0| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}

Optimal tunings:

* WE: ~2 = 1200.3420{{c}}, ~11/6 = 1045.6528{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.3816{{c}}

* [[11-odd-limit]]: ~11/6 = {{monzo| 9/10 1/5 0 0 -1/10 }}

: unchanged-interval (eigenmonzo) basis: 2.11/9

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

Badness (Sintel): 0.847

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

Optimal tunings:

* WE: ~2 = 1198.9982{{c}}, ~11/6 = 1044.6488{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.4476{{c}}

{{Optimal ET sequence|legend=0| 8d, 23de, 31 }}

Badness (Sintel): 0.964

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

{{Mapping|legend=1| 1 -11 -11 -12 | 0 17 18 20 }}

* [[WE]]: ~2 = 1199.9188{{c}}, ~5/3 = 888.2606{{c}}

: [[error map]]: {{val| -0.081 -0.632 +3.269 -2.640 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 888.3163{{c}}

: error map: {{val| 0.000 -0.578 +3.379 -2.500 }}

{{Optimal ET sequence|legend=1| 23d, 27, 50, 77 }}

[[Badness]] (Sintel): 1.86

{{Mapping|legend=0| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}

Optimal tunings:

* WE: ~2 = 1198.9982{{c}}, ~5/3 = 888.0239{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3941{{c}}

{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}

Badness (Sintel): 1.88

{{Mapping|legend=0| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}

Optimal tunings:

* WE: ~2 = 1199.5177{{c}}, ~5/3 = 888.0521{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3959{{c}}

{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}

Badness (Sintel): 1.47

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

 

{{Mapping|legend=1| 2 -1 1 3 | 0 8 7 5 }}

[[Optimal tuning]]s:

* [[WE]]: ~343/240 = 600.2436{{c}}, ~6/5 = 312.7294{{c}}

* [[CWE]]: ~343/240 = 600.0000{{c}}, ~6/5 = 312.6547{{c}}

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

[[Badness]] (Sintel): 1.98

{{Mapping|legend=0| 2 -1 1 3 9 | 0 8 7 5 -4 }}

Optimal tunings:  

* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 312.6282{{c}}

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

Badness (Sintel): 1.47

{{Mapping|legend=0| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }}

Optimal tunings:  

* WE: ~55/39 = 600.3065{{c}}, ~6/5 = 312.7240{{c}}

* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 312.5836{{c}}

{{Optimal ET sequence|legend=0| 46, 96d }}

Badness (Sintel): 1.23

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

Named by [[User:Xenllium|Xenllium]] in 2026, xenial may be described as the {{nowrap| 19 & 70 }} temperament, splitting the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. Equivalently, a stack of nine [[9/5]]s is equated with the [[3/2|perfect fifth]] above 7 [[octave]]s, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name.

[[Comma list]]: 126/125, 177147/175616

{{Mapping|legend=1| 1 -6 -12 -25 | 0 9 17 33 }}

* [[WE]]: ~2 = 1200.0095{{c}}, ~9/5 = 1011.1532{{c}}

: [[error map]]: {{val| +0.010 -1.634 +3.176 -1.009 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.1456{{c}}

: error map: {{val| 0.000 -1.644 +3.162 -1.021 }}

{{Optimal ET sequence|legend=1| 19, 51cd, 70, 89 }}

[[Badness]] (Sintel): 2.13

Comma list: 126/125, 540/539, 16384/16335

{{Mapping|legend=0| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}

Optimal tunings:

* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.1915{{c}}

{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}

Badness (Sintel): 2.31

Comma list: 126/125, 169/168, 540/539, 729/728

{{Mapping|legend=0| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }}

Optimal tunings:

* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.2102{{c}}

{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}

Badness (Sintel): 1.98

Comma list: 126/125, 169/168, 221/220, 256/255, 540/539

{{Mapping|legend=0| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2323{{c}}

{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}

Badness (Sintel): 2.06

Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539

{{Mapping|legend=0| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2230{{c}}

{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}

Badness (Sintel): 2.03

Comma list: 126/125, 162/161, 169/168, 171/170, 208/207, 221/220, 231/230

{{Mapping|legend=0| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2245{{c}}

{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}

Badness (Sintel): 1.93

== Kumonga ==

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

[[Comma list]]: 126/125, 12288/12005

{{Mapping|legend=1| 1 -9 -5 2 | 0 13 9 1 }}

* [[WE]]: ~2 = 1198.0653{{c}}, ~7/4 = 975.6277{{c}}

: [[error map]]: {{val| -1.935 -1.382 +4.009 +2.932 }}

: error map: {{val| 0.000 +0.470 +7.673 +8.284 }}

{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 167ccdd }}

[[Badness]] (Sintel): 2.21

Comma list: 126/125, 176/175, 864/847

{{Mapping|legend=0| 1 -9 -5 2 -12 | 0 13 9 1 19 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9964{{c}}

{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e }}

Badness (Sintel): 1.43

Comma list: 78/77, 126/125, 144/143, 176/175

{{Mapping|legend=0| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9677{{c}}

{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e, 113cdee }}

Badness (Sintel): 1.19

 

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

{{Mapping|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}

: [[error map]]: {{val| +0.165 -3.634 +2.988 +2.272 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~196/135 = 658.1814{{c}}

: error map: {{val| 0.000 -3.779 +2.769 +2.071 }}

{{Optimal ET sequence|legend=1| 11cd, 20cd, 31 }}

[[Badness]] (Sintel): 2.53

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

{{Mapping|legend=0| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}

Optimal tunings:  

* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}

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

Badness (Sintel): 1.41

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

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

Optimal tunings:  

* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}

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

Badness (Sintel): 1.56

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

 

It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the [[~]][[48/35]] 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.  

If we add 385/384 to the list of commas, 48/35 is identified with [[11/8]], and casablanca is revealed as an [[11-limit]] temperament with a very low complexity for [[11/1|11]] and not too high a one for [[7/1|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 [[15/11]] as opposed to 11/8 in casablanca.  

{{Mapping|legend=1| 1 -7 -4 1 | 0 19 14 4 }}

* [[WE]]: ~2 = 1199.6286{{c}}, ~48/35 = 542.0141{{c}}

: [[error map]]: {{val| -0.371 -1.087 +3.370 -1.141 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 542.1684{{c}}

: error map: {{val| 0.000 -0.756 +4.044 -0.152 }}

[[Badness]] (Sintel): 2.56

{{Mapping|legend=0| 1 -7 -4 1 3 | 0 19 14 4 1 }}

Optimal tunings:

* WE: ~2 = 1200.6404{{c}}, ~11/8 = 542.3659{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.0945{{c}}

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

Badness (Sintel): 2.22

{{Mapping|legend=0| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }}

 

* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.1392{{c}}

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

Badness (Sintel): 2.31

{{Mapping|legend=0| 1 -7 -4 1 -11 | 0 19 14 4 32 }}

* WE: ~2 = 1199.6315{{c}}, ~15/11 = 542.0428{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.1958{{c}}

{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c }}

Badness (Sintel): 1.34

{{Mapping|legend=0| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }}

Optimal tunings:

* WE: ~2 = 1199.3741{{c}}, ~15/11 = 541.9613{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2361{{c}}

{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c, 239ccf }}

Badness (Sintel): 1.68

{{Mapping|legend=0| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }}

Optimal tunings:

* WE: ~2 = 1198.6578{{c}}, ~15/11 = 541.6930{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2577{{c}}

{{Optimal ET sequence|legend=0| 31, 73f, 104cff }}

Badness (Sintel): 1.71

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

* [[WE]]: ~2 = 1199.4354{{c}}, ~5/4 = 390.9104{{c}}

: [[error map]]: {{val| -0.565 -0.811 +3.467 -1.206 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 391.0937{{c}}

[[Badness]] (Sintel): 2.81

{{Mapping|legend=0| 1 -2 2 9 9 | 0 11 1 -19 -17 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0783{{c}}

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

Badness (Sintel): 1.44

{{Mapping|legend=0| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}

Optimal tunings:

* WE: ~2 = 1199.8174{{c}}, ~5/4 = 391.0130{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0737{{c}}

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

Badness (Sintel): 1.27

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.  

: [[error map]]: {{val| -0.859 -4.800 +3.337 +6.675 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/270 = 446.5163{{c}}

: error map: {{val| 0.000 -4.210 +4.464 +8.017 }}

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

[[Badness]] (Sintel): 3.76

{{Mapping|legend=0| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}

Optimal tunings:

* WE: ~2 = 1198.6099{{c}}, ~72/55 = 446.0983{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 446.5381{{c}}

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

Badness (Sintel): 1.97

{{Mapping|legend=0| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}

Optimal tunings:

* WE: ~2 = 1198.9947{{c}}, ~13/10 = 446.2243{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5420{{c}}

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

Badness (Sintel): 1.46

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

Optimal tunings:

* WE: ~2 = 1198.7070{{c}}, ~13/10 = 446.1493{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5645{{c}}

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