@@ 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:
@@ Line 5: / Line 6: @@
 * ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]]
 * ''[[Opossum]]'' (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]]
-* [[Diminished (temperament)|Diminished]] (+36/35) → [[Dimipent family #Diminished|Dimipent family]]
+* [[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]]
@@ Line 11: / Line 12: @@
 * [[Mavila]] (+135/128) → [[Pelogic family #Mavila|Pelogic family]]
 * [[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 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 {{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.
+-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 {{nowrap|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 ==
@@ 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 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 ==
@@ 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 989: / Line 700: @@
 [[Category:Temperament collections]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Starling temperaments| ]] <!-- main article -->
 [[Category:Myna]]
 [[Category:Rank 2]]

Starling temperaments: Difference between revisions