Starling temperaments: Difference between revisions

(17 intermediate revisions by 6 users not shown)

Line 1:

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

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

Temperaments discussed in families and clans are:

* ''[[Pater]]'' (+16/15) → [[Father family #Pater|Father family]]

* ''[[~~Flat (temperament)|Flat~~]]'' (+21/20) → [[Dicot family #~~Flat~~|Dicot family]]

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

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

* ''[[Diminished]]'' (+36/35) → [[~~Dimipent~~ family #Diminished~~|Dimipent~~ family~~]] / [[Jubilismic clan #Diminished|jubilismic clan~~]]

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

* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]

* [[Augene]] (+64/63) → [[Augmented family #Augene|Augmented family]]

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

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

* [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent ~~family]] / [[Sensamagic clan #Sensi|sensamagic clan]]~~

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

* ''[[Gilead]]'' (+343/324) → [[Shibboleth family #Gilead|Shibboleth family]]

* [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]

* ''[[Diaschismic]]'' (+2048/2025) → [[Diaschismic family #~~Diaschismic~~|Diaschismic family]]

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

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

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

* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]]

Line 19:

Line 20:

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

* ''[[Worschmidt]]'' (+33075/32768) → [[Würschmidt family #Worschmidt|Würschmidt family]]

* ''[[Thuja]]'' (+65536/64827) → [[Buzzardsmic clan #Thuja|Buzzardsmic clan]]

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

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

Line 24:

Line 26:

* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila family]]

Since (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.

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.

== Myna ==

Line 30:

Line 32:

In addition to 126/125, myna ~~tempers out~~ [[1728/1715]], the orwell comma, and [[2401/2400]], the breedsma. It can also be described as the 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 61/10 as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.

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 61/10 as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.

[[Subgroup]]: 2.3.5.7

Line 39:

Line 43:

: mapping generators: ~2, ~5/3

~~{{Multival|legend=1| 10 9 7 -9 -17 -9 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 310.146

Line 47:

Line 49:

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

: {{monzo list| 1 0 0 0 | 0 1 0 0 | 9/10 9/10 0 0 | 17/10 7/10 0 0 }}

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

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

Line 59:

Line 61:

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

~~{{Multival|legend=1| 10 9 7 25 -9 -17 5 -9 27 46 }}~~

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.144

Line 132:

Badness: 0.048687

~~== Valentine ==~~

~~{{Main| Valentine }}~~

Valentine tempers out [[1029/1024]] and [[6144/6125]] as well as 126/125, so it also fits under the heading of the gamelismic clan. It has a generator of 21/20, which can be stripped of its 2 and taken as 3×7/5. In this respect it resembles miracle, with a generator of 3×5/7, and casablanca, with a generator of 5×7/3. These three generators are the simplest in terms of the relationship of tetrads in the [[The Seven Limit Symmetrical Lattices|lattice of 7-limit tetrads]]. Valentine can also be described as the 31 & 46 temperament, and [[77edo]], [[108edo]] or [[185edo]] make for excellent tunings, which also happen to be excellent tunings for starling temperament, the 126/125 planar temperament. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)1/9 as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as {{multival| 9 5 -3 7 … }}, tempering out 121/120 and 441/440; 46edo has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, (11/7)1/10.

Valentine is very closely related to [[Carlos Alpha]], the rank-1 non-octave temperament of Wendy Carlos, as the generator chain of valentine is the same thing as Carlos Alpha. Indeed, the way Carlos uses Alpha in ''Beauty in the Beast'' suggests that she really intended Alpha to be the same thing as valentine, and that it is misdescribed as a rank-1 temperament. Carlos tells us that "[t]he melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before", and since Alpha lives inside valentine this comment carries over and applies to it if you stick close melodically to generator steps, which is almost impossible not to do since the generator step is so small. MOSes of 15, 16, 31 and 46 notes are available to explore these exotic and fresh melodies, or the less exotic ones you might cook up otherwise.

~~[[Subgroup]]: 2.3.5~~

~~[[Comma list]]: 1990656/1953125~~

~~{{Mapping|legend=1| 1 1 2 | 0 9 5 }}~~

~~[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/24 = 78.039~~

~~{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 123 }}~~

~~[[Badness]]: 0.122765~~

~~=== 7-limit ===~~

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 126/125, 1029/1024~~

~~{{Mapping|legend=1| 1 1 2 3 | 0 9 5 -3 }}~~

~~: mapping generators: ~2, ~21/20~~

~~Wedgie: {{multival| 9 5 -3 -13 -30 -21 }}~~

~~[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 77.864~~

~~[[Minimax tuning]]:~~

* [[7-odd-limit]]: ~21/20 = {{monzo| 1/6 1/12 0 -1/12 }}

~~: [{{monzo| 1 0 0 0 }}, {{monzo| 5/2 3/4 0 -3/4 }}, {{monzo| 17/6 5/12 0 -5/12 }}, {{monzo| 5/2 -1/4 0 1/4 }}]~~

~~: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/3~~

* [[9-odd-limit]]: ~21/20 = {{monzo| 1/21 2/21 0 -1/21}}

~~: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 47/21 10/21 0 -5/21 }}, {{monzo| 20/7 -2/7 0 1/7 }}]~~

~~: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7~~

~~[[Algebraic generator]]: smaller root of ''x''2 - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.~~

~~{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 185, 262cd }}~~

~~[[Badness]]: 0.031056~~

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 121/120, 126/125, 176/175~~

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

~~: mapping generators: ~2, ~21/20~~

~~Wedgie: {{multival| 9 5 -3 7 -13 -30 -20 -21 -1 30 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.881~~

~~Minimax tuning:~~

* [[11-odd-limit]]: ~21/20 = {{monzo| 0 0 0 -1/10 1/10 }}

~~: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 -9/10 9/10 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 0 0 3/10 -3/10 }}, {{monzo| 3 0 0 -7/10 7/10 }}]~~

~~: eigenmonzo (unchanged-interval) basis: 2.11/7~~

~~Algebraic generator: positive root of 4''x''3 + 15''x''2 - 21, or else Gontrand2, the smallest positive root of 4''x''7 - 8''x''6 + 5.~~

~~{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 262cdee, 339cdeee }}~~

~~Badness: 0.016687~~

~~==== Dwynwen ====~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 91/90, 121/120, 126/125, 176/175~~

~~Mapping: {{mapping| 1 1 2 3 3 2 | 0 9 5 -3 7 26 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 78.219~~

~~{{Optimal ET sequence|legend=1| 15, 31f, 46 }}~~

~~Badness: 0.023461~~

~~==== Lupercalia ====~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 66/65, 105/104, 121/120, 126/125~~

~~Mapping: {{mapping| 1 1 2 3 3 3 | 0 9 5 -3 7 11 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.709~~

~~{{Optimal ET sequence|legend=1| 15, 31, 77ff, 108eff, 139efff }}~~

~~Badness: 0.021328~~

~~==== Valentino ====~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 121/120, 126/125, 176/175, 196/195~~

~~Mapping: {{mapping| 1 1 2 3 3 5 | 0 9 5 -3 7 -20 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.958~~

~~{{Optimal ET sequence|legend=1| 15f, 31, 46, 77 }}~~

~~Badness: 0.020665~~

~~===== 17-limit =====~~

~~Subgroup: 2.3.5.7.11.13.17~~

~~Comma list: 121/120, 126/125, 154/153, 176/175, 196/195~~

~~Mapping: {{mapping| 1 1 2 3 3 5 5 | 0 9 5 -3 7 -20 -14 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 78.003~~

~~{{Optimal ET sequence|legend=1| 15f, 31, 46, 77, 123e, 200ceg }}~~

~~Badness: 0.016768~~

~~==== Semivalentine ====~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 121/120, 126/125, 169/168, 176/175~~

~~Mapping: {{mapping| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.839~~

~~{{Optimal ET sequence|legend=1| 16, 30, 46, 62, 108ef }}~~

~~Badness: 0.032749~~

~~==== Hemivalentine ====~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 121/120, 126/125, 176/175, 343/338~~

~~Mapping: {{mapping| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~40/39 = 39.044~~

~~{{Optimal ET sequence|legend=1| 30, 31, 61, 92f, 123f }}~~

~~Badness: 0.047059~~

~~=== Hemivalentino ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 126/125, 243/242, 1029/1024~~

~~Mapping: {{mapping| 1 1 2 3 2 | 0 18 10 -6 45 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.921~~

~~{{Optimal ET sequence|legend=1| 31, 92e, 123, 154, 185 }}~~

~~Badness: 0.061275~~

~~==== 13-limit ====~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 126/125, 196/195, 243/242, 1029/1024~~

~~Mapping: {{mapping| 1 1 2 3 2 5 | 0 18 10 -6 45 -40 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.948~~

~~{{Optimal ET sequence|legend=1| 31, 92e, 123, 154 }}~~

~~Badness: 0.057919~~

~~==== Hemivalentoid ====~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 126/125, 144/143, 243/242, 343/338~~

~~Mapping: {{mapping| 1 1 2 3 2 4 | 0 18 10 -6 45 -9 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~40/39 = 38.993~~

~~{{Optimal ET sequence|legend=1| 31, 92ef, 123f }}~~

~~Badness: 0.057931~~

== Nusecond ==

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Nusecond]].''

Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 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.

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.

[[Subgroup]]: 2.3.5.7

Line 332:

Line 145:

: mapping generators: ~2, ~49/45

~~{{Multival|legend=1| 11 13 17 -5 -4 3 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/45 = 154.579

Line 340:

Line 151:

* [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}

: {{monzo list| 1 0 0 0 | -5/13 0 11/13 0 | 0 0 1 0 | -3/13 0 17/13 0 }}

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

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

* [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}

: {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/11 13/11 0 0 | 4/11 17/11 0 0 }}

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

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

Line 361:

Line 172:

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

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]

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

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

Algebraic generator: positive root of 15''x''2 - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.

Line 391:

Line 202:

~~{{Multival|legend=1| 17 18 20 -11 -16 -4 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 311.679

Line 475:

Line 284:

~~{{Multival|legend=1| 13 9 1 -16 -35 -23 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 222.797

Line 509:

Line 316:

Badness: 0.028920

~~== Thuja ==~~

~~: ''For the 5-limit version of this temperament, see [[High badness temperaments #Thuja]].''~~

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 126/125, 65536/64827~~

~~{{Mapping|legend=1| 1 -4 0 7 | 0 12 5 -9 }}~~

~~{{Multival|legend=1| 12 5 -9 -20 -48 -35 }}~~

~~[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 558.605~~

~~{{Optimal ET sequence|legend=1| 15, 43, 58 }}~~

~~[[Badness]]: 0.088441~~

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 126/125, 176/175, 1344/1331~~

~~Mapping: {{mapping| 1 -4 0 7 3 | 0 12 5 -9 1 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.620~~

~~{{Optimal ET sequence|legend=1| 15, 43, 58 }}~~

~~Badness: 0.033078~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 126/125, 144/143, 176/175, 364/363~~

~~Mapping: {{mapping| 1 -4 0 7 3 -7 | 0 12 5 -9 1 23 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.589~~

~~{{Optimal ET sequence|legend=1| 15, 43, 58 }}~~

~~Badness: 0.022838~~

~~=== 17-limit ===~~

~~Subgroup: 2.3.5.7.11.13.17~~

~~Comma list: 126/125, 144/143, 176/175, 221/220, 256/255~~

~~Mapping: {{mapping| 1 -4 0 7 3 -7 12 | 0 12 5 -9 1 23 -17 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.509~~

~~{{Optimal ET sequence|legend=1| 15, 43, 58 }}~~

~~Badness: 0.022293~~

~~=== 19-limit ===~~

~~Subgroup: 2.3.5.7.11.13.17.19~~

~~Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220~~

~~Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 | 0 12 5 -9 1 23 -17 7 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.504~~

~~{{Optimal ET sequence|legend=1| 15, 43, 58h }}~~

~~Badness: 0.018938~~

~~=== 23-limit ===~~

~~Subgroup: 2.3.5.7.11.13.17.19.23~~

~~Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230~~

~~Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 5 | 0 12 5 -9 1 23 -17 7 -1 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.522~~

~~{{Optimal ET sequence|legend=1| 15, 43, 58hi }}~~

~~Badness: 0.016581~~

~~=== 29-limit ===~~

~~The ''raison d'etre'' of this entry is the simple and accurate approximation of factor twenty-nine, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.~~

~~Subgroup: 2.3.5.7.11.13.17.19.23.29~~

~~Comma list: 96/95, 116/115, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230~~

~~Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 5 3 | 0 12 5 -9 1 23 -17 7 -1 4 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.520~~

~~{{Optimal ET sequence|legend=1| 15, 43, 58hi }}~~

~~Badness: 0.013762~~

== Cypress ==

Line 615:

Line 325:

~~{{Multival|legend=1| 12 17 27 -1 9 15 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~135/98 = 541.828

Line 656:

Line 364:

~~{{Multival|legend=1| 18 22 30 -7 -3 8 }}~~

[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~35/27 = 455.445

Line 694:

Line 400:

: ''For the 5-limit version of this temperament, see [[High badness 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 31 & 73. 74\135 or 91\166 supply good tunings for the generator, and 20- and 31-note mosses are available.

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.

Line 705:

Line 411:

~~{{Multival|legend=1| 19 14 4 -22 -47 -30 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/24 = 657.818

Line 720:

Line 424:

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

~~{{Multival|legend=1| 19 14 4 1 -22 -47 -64 -30 -46 -11 }}~~

Optimal tuning (POTE): ~2 = 1\1, ~16/11 = 657.923

Line 746:

Line 448:

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

~~{{Multival|legend=1| 19 14 4 32 -22 -47 -15 -30 26 76 }}~~

Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 657.791

Line 789:

Line 489:

~~{{Multival|legend=1| 11 1 -19 -24 -61 -47 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 391.094

Line 823:

Line 521:

Badness: 0.030666

== Gilead ==

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 343/324

[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1\1, ~6/5 = 321.109

* [[POTE]]: ~2 = 1\1, ~6/5 = 321.423

[[Badness]]: 0.115292

== Supersensi ==

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

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.

[[Subgroup]]: 2.3.5.7

Line 832:

Line 545:

~~{{Multival|legend=1| 15 17 21 -8 -9 1 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/270 = 446.568

Line 883:

Line 594:

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

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

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

[[Subgroup]]: 2.3.5.7

Line 930:

Line 641:

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

Optimal tuning (POTE): ~~~11/10 =~~, ~3/2 = 700.397

Optimal tuning (POTE): 1\27, ~3/2 = 700.397

Line 987:

Line 698:

Badness: 0.052732

~~== Nutmeg ==~~

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 126/125, 99/98, 540/539~~

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

~~Optimal tuning (POTE): 2 = 2/1, ~11/10 = 154.618~~

~~{{Optimal ET sequence|legend=1| 31, 23de, 70 }}~~

~~Badness: 0.847~~

[[Category:Temperament collections]]

[[Category:Pages with mostly numerical content]]

[[Category:Starling temperaments| ]]

[[Category:Myna]]

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
-This page discusses miscellaneous rank-2 temperaments tempering out [[126/125]], the starling comma or septimal semicomma.
+{{Technical data page}}
+This page discusses miscellaneous [[rank-2 temperament]]s tempering out [[126/125]], the starling comma or septimal semicomma.
 Temperaments discussed in families and clans are:
 * ''[[Pater]]'' (+16/15) → [[Father family #Pater|Father family]]
-* ''[[Flat (temperament)|Flat]]'' (+21/20) → [[Dicot family #Flat|Dicot family]]
+* ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]]
 * ''[[Opossum]]'' (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]]
-* ''[[Diminished]]'' (+36/35) → [[Dimipent family #Diminished|Dimipent family]] / [[Jubilismic clan #Diminished|jubilismic clan]]
+* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]
 * [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
 * [[Augene]] (+64/63) → [[Augmented family #Augene|Augmented family]]
 * [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]]
 * [[Mavila]] (+135/128) → [[Pelogic family #Mavila|Pelogic family]]
-* [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]] / [[Sensamagic clan #Sensi|sensamagic clan]]
+* [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]]
-* ''[[Gilead]]'' (+343/324) → [[Shibboleth family #Gilead|Shibboleth family]]
 * [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]
-* ''[[Diaschismic]]'' (+2048/2025) → [[Diaschismic family #Diaschismic|Diaschismic family]]
+* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]
+* ''[[Diaschismic]]'' (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
 * ''[[Wollemia]]'' (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]
 * ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]]
@@ Line 19: / Line 20: @@
 * ''[[Grackle]]'' (+32805/32768) → [[Schismatic family #Grackle|Schismatic family]]
 * ''[[Worschmidt]]'' (+33075/32768) → [[Würschmidt family #Worschmidt|Würschmidt family]]
+* ''[[Thuja]]'' (+65536/64827) → [[Buzzardsmic clan #Thuja|Buzzardsmic clan]]
 * ''[[Passionate]]'' (+131072/127575) → [[Passion family #Passionate|Passion family]]
 * ''[[Vishnean]]'' (+540225/524288) → [[Vishnuzmic family #Vishnean|Vishnuzmic family]]
@@ Line 24: / Line 26: @@
 * ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila family]]
-Since (6/5)<sup>3</sup> = 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.
+Since {{nowrap|(6/5)<sup>3</sup> {{=}} 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.
 == Myna ==
@@ Line 30: / Line 32: @@
 {{Main| Myna }}
-In addition to 126/125, myna tempers out [[1728/1715]], the orwell comma, and [[2401/2400]], the breedsma. It can also be described as the 27 &amp; 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<sup>1/10</sup> as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.
+-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 &amp; 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<sup>1/10</sup> as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.
 [[Subgroup]]: 2.3.5.7
@@ Line 39: / Line 43: @@
 : mapping generators: ~2, ~5/3
-{{Multival|legend=1| 10 9 7 -9 -17 -9 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 310.146
@@ Line 47: / Line 49: @@
 * 7- and [[9-odd-limit]]: ~6/5 = {{monzo| 1/10 1/10 0 0}}
 : {{monzo list| 1 0 0 0 | 0 1 0 0 | 9/10 9/10 0 0 | 17/10 7/10 0 0 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
 {{Optimal ET sequence|legend=1| 27, 31, 58, 89 }}
@@ Line 59: / Line 61: @@
 Mapping: {{mapping| 1 9 9 8 22 | 0 -10 -9 -7 -25 }}
-{{Multival|legend=1| 10 9 7 25 -9 -17 5 -9 27 46 }}
 Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.144
@@ Line 132: / Line 132: @@
 Badness: 0.048687
-== Valentine ==
-{{Main| Valentine }}
-Valentine tempers out [[1029/1024]] and [[6144/6125]] as well as 126/125, so it also fits under the heading of the gamelismic clan. It has a generator of 21/20, which can be stripped of its 2 and taken as 3×7/5. In this respect it resembles miracle, with a generator of 3×5/7, and casablanca, with a generator of 5×7/3. These three generators are the simplest in terms of the relationship of tetrads in the [[The Seven Limit Symmetrical Lattices|lattice of 7-limit tetrads]]. Valentine can also be described as the 31 &amp; 46 temperament, and [[77edo]], [[108edo]] or [[185edo]] make for excellent tunings, which also happen to be excellent tunings for starling temperament, the 126/125 planar temperament. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)<sup>1/9</sup> as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as {{multival| 9 5 -3 7 … }}, tempering out 121/120 and 441/440; 46edo has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, (11/7)<sup>1/10</sup>.
-Valentine is very closely related to [[Carlos Alpha]], the rank-1 non-octave temperament of Wendy Carlos, as the generator chain of valentine is the same thing as Carlos Alpha. Indeed, the way Carlos uses Alpha in ''Beauty in the Beast'' suggests that she really intended Alpha to be the same thing as valentine, and that it is misdescribed as a rank-1 temperament. Carlos tells us that "[t]he melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before", and since Alpha lives inside valentine this comment carries over and applies to it if you stick close melodically to generator steps, which is almost impossible not to do since the generator step is so small. MOSes of 15, 16, 31 and 46 notes are available to explore these exotic and fresh melodies, or the less exotic ones you might cook up otherwise.
-[[Subgroup]]: 2.3.5
-[[Comma list]]: 1990656/1953125
-{{Mapping|legend=1| 1 1 2 | 0 9 5 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/24 = 78.039
-{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 123 }}
-[[Badness]]: 0.122765
-=== 7-limit ===
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 126/125, 1029/1024
-{{Mapping|legend=1| 1 1 2 3 | 0 9 5 -3 }}
-: mapping generators: ~2, ~21/20
-Wedgie: {{multival| 9 5 -3 -13 -30 -21 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 77.864
-[[Minimax tuning]]:
-* [[7-odd-limit]]: ~21/20 = {{monzo| 1/6 1/12 0 -1/12 }}
-: [{{monzo| 1 0 0 0 }}, {{monzo| 5/2 3/4 0 -3/4 }}, {{monzo| 17/6 5/12 0 -5/12 }}, {{monzo| 5/2 -1/4 0 1/4 }}]
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/3
-* [[9-odd-limit]]: ~21/20 = {{monzo| 1/21 2/21 0 -1/21}}
-: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 47/21 10/21 0 -5/21 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7
-[[Algebraic generator]]: smaller root of ''x''<sup>2</sup> - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.
-{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 185, 262cd }}
-[[Badness]]: 0.031056
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 121/120, 126/125, 176/175
-Mapping: {{mapping| 1 1 2 3 3 | 0 9 5 -3 7 }}
-: mapping generators: ~2, ~21/20
-Wedgie: {{multival| 9 5 -3 7 -13 -30 -20 -21 -1 30 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.881
-Minimax tuning:
-* [[11-odd-limit]]: ~21/20 = {{monzo| 0 0 0 -1/10 1/10 }}
-: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 -9/10 9/10 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 0 0 3/10 -3/10 }}, {{monzo| 3 0 0 -7/10 7/10 }}]
-: eigenmonzo (unchanged-interval) basis: 2.11/7
-Algebraic generator: positive root of 4''x''<sup>3</sup> + 15''x''<sup>2</sup> - 21, or else Gontrand2, the smallest positive root of 4''x''<sup>7</sup> - 8''x''<sup>6</sup> + 5.
-{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 262cdee, 339cdeee }}
-Badness: 0.016687
-==== Dwynwen ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 121/120, 126/125, 176/175
-Mapping: {{mapping| 1 1 2 3 3 2 | 0 9 5 -3 7 26 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 78.219
-{{Optimal ET sequence|legend=1| 15, 31f, 46 }}
-Badness: 0.023461
-==== Lupercalia ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 66/65, 105/104, 121/120, 126/125
-Mapping: {{mapping| 1 1 2 3 3 3 | 0 9 5 -3 7 11 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.709
-{{Optimal ET sequence|legend=1| 15, 31, 77ff, 108eff, 139efff }}
-Badness: 0.021328
-==== Valentino ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 121/120, 126/125, 176/175, 196/195
-Mapping: {{mapping| 1 1 2 3 3 5 | 0 9 5 -3 7 -20 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.958
-{{Optimal ET sequence|legend=1| 15f, 31, 46, 77 }}
-Badness: 0.020665
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 121/120, 126/125, 154/153, 176/175, 196/195
-Mapping: {{mapping| 1 1 2 3 3 5 5 | 0 9 5 -3 7 -20 -14 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 78.003
-{{Optimal ET sequence|legend=1| 15f, 31, 46, 77, 123e, 200ceg }}
-Badness: 0.016768
-==== Semivalentine ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 121/120, 126/125, 169/168, 176/175
-Mapping: {{mapping| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.839
-{{Optimal ET sequence|legend=1| 16, 30, 46, 62, 108ef }}
-Badness: 0.032749
-==== Hemivalentine ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 121/120, 126/125, 176/175, 343/338
-Mapping: {{mapping| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }}
-Optimal tuning (POTE): ~2 = 1\1, ~40/39 = 39.044
-{{Optimal ET sequence|legend=1| 30, 31, 61, 92f, 123f }}
-Badness: 0.047059
-=== Hemivalentino ===
-Subgroup: 2.3.5.7.11
-Comma list: 126/125, 243/242, 1029/1024
-Mapping: {{mapping| 1 1 2 3 2 | 0 18 10 -6 45 }}
-Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.921
-{{Optimal ET sequence|legend=1| 31, 92e, 123, 154, 185 }}
-Badness: 0.061275
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 126/125, 196/195, 243/242, 1029/1024
-Mapping: {{mapping| 1 1 2 3 2 5 | 0 18 10 -6 45 -40 }}
-Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.948
-{{Optimal ET sequence|legend=1| 31, 92e, 123, 154 }}
-Badness: 0.057919
-==== Hemivalentoid ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 126/125, 144/143, 243/242, 343/338
-Mapping: {{mapping| 1 1 2 3 2 4 | 0 18 10 -6 45 -9 }}
-Optimal tuning (POTE): ~2 = 1\1, ~40/39 = 38.993
-{{Optimal ET sequence|legend=1| 31, 92ef, 123f }}
-Badness: 0.057931
 == Nusecond ==
 : ''For the 5-limit version of this temperament, see [[High badness temperaments #Nusecond]].''
-Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31 &amp; 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.
+Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as {{nowrap|31 &amp; 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.
 [[Subgroup]]: 2.3.5.7
@@ Line 332: / Line 145: @@
 : mapping generators: ~2, ~49/45
-{{Multival|legend=1| 11 13 17 -5 -4 3 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/45 = 154.579
@@ Line 340: / Line 151: @@
 * [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
 : {{monzo list| 1 0 0 0 | -5/13 0 11/13 0 | 0 0 1 0 | -3/13 0 17/13 0 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 * [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}
 : {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/11 13/11 0 0 | 4/11 17/11 0 0 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
 {{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}
@@ Line 361: / Line 172: @@
 * [[11-odd-limit]]: ~11/10 = {{monzo| 1/10 -1/5 0 0 1/10 }}
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]
-: eigenmonzo (unchanged-interval) basis: 2.11/9
+: unchanged-interval (eigenmonzo) basis: 2.11/9
 Algebraic generator: positive root of 15''x''<sup>2</sup> - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.
@@ Line 391: / Line 202: @@
 {{Mapping|legend=1| 1 6 7 8 | 0 -17 -18 -20 }}
-{{Multival|legend=1| 17 18 20 -11 -16 -4 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 311.679
@@ Line 475: / Line 284: @@
 {{Mapping|legend=1| 1 4 4 3 | 0 -13 -9 -1 }}
-{{Multival|legend=1| 13 9 1 -16 -35 -23 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 222.797
@@ Line 509: / Line 316: @@
 Badness: 0.028920
-== Thuja ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Thuja]].''
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 126/125, 65536/64827
-{{Mapping|legend=1| 1 -4 0 7 | 0 12 5 -9 }}
-{{Multival|legend=1| 12 5 -9 -20 -48 -35 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 558.605
-{{Optimal ET sequence|legend=1| 15, 43, 58 }}
-[[Badness]]: 0.088441
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 126/125, 176/175, 1344/1331
-Mapping: {{mapping| 1 -4 0 7 3 | 0 12 5 -9 1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.620
-{{Optimal ET sequence|legend=1| 15, 43, 58 }}
-Badness: 0.033078
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 126/125, 144/143, 176/175, 364/363
-Mapping: {{mapping| 1 -4 0 7 3 -7 | 0 12 5 -9 1 23 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.589
-{{Optimal ET sequence|legend=1| 15, 43, 58 }}
-Badness: 0.022838
-=== 17-limit ===
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 126/125, 144/143, 176/175, 221/220, 256/255
-Mapping: {{mapping| 1 -4 0 7 3 -7 12 | 0 12 5 -9 1 23 -17 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.509
-{{Optimal ET sequence|legend=1| 15, 43, 58 }}
-Badness: 0.022293
-=== 19-limit ===
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220
-Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 | 0 12 5 -9 1 23 -17 7 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.504
-{{Optimal ET sequence|legend=1| 15, 43, 58h }}
-Badness: 0.018938
-=== 23-limit ===
-Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230
-Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 5 | 0 12 5 -9 1 23 -17 7 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.522
-{{Optimal ET sequence|legend=1| 15, 43, 58hi }}
-Badness: 0.016581
-=== 29-limit ===
-The ''raison d'etre'' of this entry is the simple and accurate approximation of factor twenty-nine, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.
-Subgroup: 2.3.5.7.11.13.17.19.23.29
-Comma list: 96/95, 116/115, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230
-Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 5 3 | 0 12 5 -9 1 23 -17 7 -1 4 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.520
-{{Optimal ET sequence|legend=1| 15, 43, 58hi }}
-Badness: 0.013762
 == Cypress ==
@@ Line 615: / Line 325: @@
 {{Mapping|legend=1| 1 7 10 15 | 0 -12 -17 -27 }}
-{{Multival|legend=1| 12 17 27 -1 9 15 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~135/98 = 541.828
@@ Line 656: / Line 364: @@
 {{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
-{{Multival|legend=1| 18 22 30 -7 -3 8 }}
 [[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~35/27 = 455.445
@@ Line 694: / Line 400: @@
 : ''For the 5-limit version of this temperament, see [[High badness 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 31 &amp; 73. 74\135 or 91\166 supply good tunings for the generator, and 20- and 31-note mosses are available.
+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 &amp; 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.
@@ Line 705: / Line 411: @@
 {{Mapping|legend=1| 1 12 10 5 | 0 -19 -14 -4 }}
-{{Multival|legend=1| 19 14 4 -22 -47 -30 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/24 = 657.818
@@ Line 720: / Line 424: @@
 Mapping: {{mapping| 1 12 10 5 4 | 0 -19 -14 -4 -1 }}
-{{Multival|legend=1| 19 14 4 1 -22 -47 -64 -30 -46 -11 }}
 Optimal tuning (POTE): ~2 = 1\1, ~16/11 = 657.923
@@ Line 746: / Line 448: @@
 Mapping: {{mapping| 1 12 10 5 21 | 0 -19 -14 -4 -32 }}
-{{Multival|legend=1| 19 14 4 32 -22 -47 -15 -30 26 76 }}
 Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 657.791
@@ Line 789: / Line 489: @@
 {{Mapping|legend=1| 1 -2 2 9 | 0 11 1 -19 }}
-{{Multival|legend=1| 11 1 -19 -24 -61 -47 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 391.094
@@ Line 823: / Line 521: @@
 Badness: 0.030666
+== Gilead ==
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 126/125, 343/324
+{{Mapping|legend=1| 1 4 5 6 | 0 -9 -10 -12 }}
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1\1, ~6/5 = 321.109
+* [[POTE]]: ~2 = 1\1, ~6/5 = 321.423
+{{Optimal ET sequence|legend=1| 11cd, 15, 41dd, 56dd }}
+[[Badness]]: 0.115292
 == Supersensi ==
-Supersensi (8d &amp; 43) has supermajor third as a generator like [[sensi]], but the no-fives comma 17496/16807 rather than 245/243 tempered out.
+Supersensi ({{nowrap|8d &amp; 43}}) has supermajor third as a generator like [[sensi]], but the no-fives comma 17496/16807 rather than 245/243 tempered out.
 [[Subgroup]]: 2.3.5.7
@@ Line 832: / Line 545: @@
 {{Mapping|legend=1| 1 -4 -4 -5 | 0 15 17 21 }}
-{{Multival|legend=1| 15 17 21 -8 -9 1 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/270 = 446.568
@@ Line 883: / Line 594: @@
 The name of the cobalt temperament comes from the 27th element.
-Cobalt (27 &amp; 81) has a period of 1/27 octave and tempers out 126/125 and 540/539, as well as the [[Starling family #Aplonis|aplonis temperament]].
+Cobalt ({{nowrap|27 &amp; 81}}) has a period of 1/27 octave and tempers out 126/125 and 540/539, as well as the [[Starling family #Aplonis|aplonis temperament]].
 [[Subgroup]]: 2.3.5.7
@@ Line 930: / Line 641: @@
 Mapping: {{mapping| 27 43 63 76 94 100 111 | 0 -1 -1 -1 -2 0 -2 }}
-Optimal tuning (POTE): 	~11/10 =, ~3/2 = 700.397
+Optimal tuning (POTE): 1\27, ~3/2 = 700.397
 {{Optimal ET sequence|legend=1| 27eg, 81, 108g }}
@@ Line 987: / Line 698: @@
 Badness: 0.052732
-== Nutmeg ==
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list:  126/125, 99/98, 540/539
-Mapping: {{mapping| 1 3 4 5 5 | 0 -11 -13 -17 -12 }}
-Optimal tuning (POTE): 2 = 2/1, ~11/10 = 154.618
-{{Optimal ET sequence|legend=1| 31, 23de, 70 }}
-Badness: 0.847
 [[Category:Temperament collections]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Starling temperaments| ]] <!-- main article -->
 [[Category:Myna]]
 [[Category:Rank 2]]