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'''Landscape microtemperaments''' are rank-2 temperaments tempering out the [[landscape comma]] ({{Monzo| -4 6 -6 3 }} = 250047/250000).
{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] '''landscape microtemperaments''', which [[tempering out|temper out]] the [[landscape comma]] ({{monzo|legend=1| -4 6 -6 3 }}, [[ratio]]: 250047/250000). For the [[rank-3 temperament]], see [[Landscape family #Landscape]].  


Landscape rank-2 temperaments include sextile, septichrome, pnict, and domain. Considered elsewhere are:  
Temperaments discussed elsewhere are:  
* [[Augene]], {64/63, 126/125} → [[Augmented family #Augene|Augmented family]]
* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Augene|Augmented family]]
* ''[[Waage]]'', {225/224, 250047/250000} → [[Compton family #Compton|Compton family]]
* ''[[Compton]]'' (+225/224) → [[Compton family #Compton|Compton family]]
* ''[[Tritikleismic]]'', {1029/1024, 15625/15552} → [[Kleismic family #Tritikleismic|Kleismic family]]
* ''[[Tritikleismic]]'' (+1029/1024) → [[Kleismic family #Tritikleismic|Kleismic family]]
* ''[[Trisensory]]'', {1728/1715, 78732/78125} → [[Sensipent family #Trisensory|Sensipent family]]
* ''[[Trisensory]]'' (+1728/1715) → [[Sensipent family #Trisensory|Sensipent family]]
* [[Ennealimmal]], {2401/2400, 4375/4374} → [[Ragismic microtemperaments #Ennealimmal|Ragismic microtemperaments]]
* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Misty]]'', {3136/3125, 5120/5103} → [[Misty family]]
* ''[[Misty]]'' (+3136/3125 or 5120/5103) → [[Misty family]]
* ''[[Nessafof]]'', {6144/6125, 250047/250000} → [[Porwell temperaments #Nessafof|Porwell temperaments]]
* ''[[Nessafof]]'' (+6144/6125) → [[Porwell temperaments #Nessafof|Porwell temperaments]]
* ''[[Chromat]]'', {10976/10935, 235298/234375} → [[Hemimage temperaments #Chromat|Hemimage temperaments]]
* ''[[Chromat]]'' (10976/10935) → [[Hemimage temperaments #Chromat|Hemimage temperaments]]
* ''[[Term]]'', {32805/32768, 250047/250000} → [[Schismatic family #Term|Schismatic family]]
* ''[[Term]]'' (+32805/32768) → [[Schismatic family #Term|Schismatic family]]
* ''[[Caleb]]'', {33075/32768, 250047/250000} → [[Mirwomo temperaments #Caleb|Mirwomo temperaments]]
* ''[[Caleb]]'' (+33075/32768) → [[Mirwomo temperaments #Caleb|Mirwomo temperaments]]
* [[Mutt temperament|Mutt]], {65625/65536, 250047/250000} → [[Horwell temperaments #Mutt|Horwell temperaments]]
* [[Mutt temperament|Mutt]] (+65625/65536) → [[Horwell temperaments #Mutt|Horwell temperaments]]
* ''[[Stearnscape]]'', {118098/117649, 250047/250000} → [[Stearnsmic clan #Stearnscape|Stearnsmic clan]]
* ''[[Triquart]]'' (+117649/116640) → [[Quartonic family #Triquart|Quartonic family]]
* ''[[Tritricot]]'', {250047/250000, 11785390260224/11767897353375} → [[Tricot family #Tritricot|Tricot family]]
* ''[[Stearnscape]]'' (+118098/117649) → [[Stearnsmic clan #Stearnscape|Stearnsmic clan]]
* ''[[Tritricot]]'' (+{{monzo| 35 -23 -3 3 }}) → [[Alphatricot family #Tritricot|Alphatricot family]]
* ''[[Aemilic]]'' (+{{monzo|-84 53}}) → [[159th-octave temperaments#Aemilic|159th-octave temperaments]]
 
Considered below are sextile, septichrome, pnict, domain, avicenna, terture, akjayland and terture.


== Sextile ==
== Sextile ==
Subgroup: 2.3.5.7
Sextile tempers out the [[garischisma]] with a 1/6-octave period and is the 12 & 270 temperament.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 33554432/33480783
[[Comma list]]: 250047/250000, 33554432/33480783


[[Mapping]]: [{{val| 6 0 71 150 }}, {{val| 0 1 -6 -14 }}]
{{Mapping|legend=1| 6 0 71 150 | 0 1 -6 -14 }}


{{Multival|legend=1| 6 -36 -84 -71 -150 -94 }}
: mapping generators: ~4096/3645, ~3


[[POTE generator]]: ~3/2 = 702.212
[[Optimal tuning]]s:
* [[CTE]]: ~4096/3645 = 1\6, ~3/2 = 702.2347
* [[POTE]]: ~4096/3645 = 1\6, ~3/2 = 702.212


{{Val list|legend=1| 12, 234d, 246d, 258, 270, 1362c, 1632c, 1902c, 2172c }}
{{Optimal ET sequence|legend=1| 12, 234d, 246d, 258, 270, 1362c, 1632c, 1902c, 2172c, 2442bc, 2712bc }}


[[Badness]]: 0.070097
[[Badness]]: 0.070097
Line 36: Line 45:
Comma list: 5632/5625, 9801/9800, 151263/151250
Comma list: 5632/5625, 9801/9800, 151263/151250


Mapping: [{{val|6 0 71 150 230}}, {{val|0 1 -6 -14 -22}}]
Mapping: {{mapping| 6 0 71 150 230 | 0 1 -6 -14 -22 }}


POTE generator: ~3/2 = 702.202
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 702.2225
* POTE: ~55/49 = 1\6, ~3/2 = 702.202


Optimal GPV sequence: {{Val list| 12, 246dee, 258e, 270, 822, 1092 }}
{{Optimal ET sequence|legend=1| 12, 246dee, 258e, 270, 822, 1092, 1362c }}


Badness: 0.029677
Badness: 0.029677
Line 47: Line 58:
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1716/1715, 5632/5625, 9801/9800, 151263/151250
Comma list: 1716/1715, 2080/2079, 5632/5625, 10648/10647


Mapping: [{{val|6 0 71 150 230 279}}, {{val|0 1 -6 -14 -22 -27}}]
Mapping: {{mapping| 6 0 71 150 230 279 | 0 1 -6 -14 -22 -27 }}


POTE generator: ~3/2 = 702.202
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2141
* CWE: ~55/49 = 1\6, ~3/2 = 702.2001


Optimal GPV sequence: {{Val list| 12f, 258ef, 270, 552, 822, 1092 }}
{{Optimal ET sequence|legend=1| 12f, 258ef, 270, 552, 822, 1092, 1914cde }}


Badness: 0.029677
Badness: 0.0191


==== 17-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 936/935, 1716/1715, 5632/5625, 9801/9800, 151263/151250
Comma list: 936/935, 1701/1700, 1716/1715, 5632/5625, 7744/7735


Mapping: [{{val|6 0 71 150 230 279 -4}}, {{val|0 1 -6 -14 -22 -27 3}}]
Mapping: {{mapping| 6 0 71 150 230 279 -4 | 0 1 -6 -14 -22 -27 3 }}


POTE generator: ~3/2 = 702.202
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2117
* CWE: ~55/49 = 1\6, ~3/2 = 702.1869


Optimal GPV sequence: {{Val list| 12f, 258ef, 270, 552, 822, 1092 }}
{{Optimal ET sequence|legend=1| 12f, 270, 552g }}


Badness: 0.029677
Badness: 0.0209


==== 19-limit ====
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 936/935, 1216/1215, 1716/1715, 5632/5625, 9801/9800, 151263/151250
Comma list: 936/935, 1216/1215, 1701/1700, 1716/1715, 2376/2375, 4200/4199


Mapping: [{{val|6 0 71 150 230 279 -4 35}}, {{val|0 1 -6 -14 -22 -27 3 -1}}]
Mapping: {{mapping| 6 0 71 150 230 279 -4 35 | 0 1 -6 -14 -22 -27 3 -1 }}


POTE generator: ~3/2 = 702.202
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2118
* CWE: ~55/49 = 1\6, ~3/2 = 702.1890


Optimal GPV sequence: {{Val list| 12f, 258ef, 270, 552, 822, 1092 }}
{{Optimal ET sequence|legend=1| 12f, 270, 552g, 822gg }}


Badness: 0.029677
Badness: 0.0156


=== Sextilia ===
=== Sextilia ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1001/1000, 5632/5625, 9801/9800, 151263/151250
Comma list: 1001/1000, 4096/4095, 4459/4455, 20449/20412


Mapping: [⟨6 0 71 150 230 -149], ⟨0 1 -6 -14 -22 18]]
Mapping: {{mapping| 6 0 71 150 230 -149 | 0 1 -6 -14 -22 18 }}


POTE generator: ~3/2 = 702.202
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2296
* CWE: ~55/49 = 1\6, ~3/2 = 702.2285


Optimal GPV sequence: 12, 246deef, 258e, 270
{{Optimal ET sequence|legend=1| 12, 258e, 270 }}


Badness: 0.029677
Badness: 0.0391


==== 17-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 715/714, 1001/1000, 5632/5625, 9801/9800, 151263/151250
Comma list: 715/714, 1001/1000, 1701/1700, 4096/4095, 4459/4455


Mapping: [⟨6 0 71 150 230 -149 -4], ⟨0 1 -6 -14 -22 18 3]]
Mapping: {{mapping| 6 0 71 150 230 -149 -4 | 0 1 -6 -14 -22 18 3 }}


POTE generator: ~3/2 = 702.202
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2264
* CWE: ~55/49 = 1\6, ~3/2 = 702.2207


Optimal GPV sequence: 12, 246deef, 258e, 270
{{Optimal ET sequence|legend=1| 12, 258e, 270 }}


Badness: 0.029677
Badness: 0.0384


==== 19-limit ====
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 715/714, 1001/1000, 1216/1215, 5632/5625, 9801/9800, 151263/151250
Comma list: 715/714, 1001/1000, 1216/1215, 1701/1700, 1729/1728, 2912/2907


Mapping: [⟨6 0 71 150 230], ⟨0 1 -6 -14 -22 18 3 -1]]
Mapping: {{mapping| 6 0 71 150 230 -149 -4 35 | 0 1 -6 -14 -22 18 3 -1 }}


POTE generator: ~3/2 = 702.202
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2266
* CWE: ~55/49 = 1\6, ~3/2 = 702.2208


Optimal GPV sequence: 12, 246deef, 258e, 270
{{Optimal ET sequence|legend=1| 12, 258e, 270 }}


Badness: 0.029677
Badness: 0.0252


== Septichrome ==
== Septichrome ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 2460375/2458624
[[Comma list]]: 250047/250000, 2460375/2458624


[[Mapping]]: [{{val| 3 3 1 0 }}, {{val| 0 5 17 24 }}]
{{Mapping|legend=1| 3 3 1 0 | 0 5 17 24 }}
 
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~243/224 = 140.367
 
{{Optimal ET sequence|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}
 
[[Badness]]: 0.016814
 
=== Semiseptichrome ===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 151263/151250, 234375/234256
 
Mapping: {{mapping| 6 1 -15 -24 -32 | 0 5 17 24 31 }}
 
: mapping generators: ~55/49, ~375/308
 
Optimal tuning (CTE): ~55/49 = 1\6, ~375/308 = 340.3687 (~1760/1701 = 59.6313)
 
Optimal ET sequence: {{Optimal ET sequence| 60e, 222cdee, 282, 342, 966, 1308, 1650, 4608b, 6258bc }}
 
Badness: 0.0194
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 34398/34375, 85293/85184
 
Mapping: {{mapping| 6 1 -15 -24 -32 -68 | 0 5 17 24 31 53 }}
 
Optimal tuning (CTE): ~55/49 = 1\6, ~375/308 = 340.3778 (~121/117 = 59.6222)
 
Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}
 
Badness: 0.0397
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 61965/61952
 
Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 | 0 5 17 24 31 53 15 }}
 
Optimal tuning (CTE): ~55/49 = 1\6, ~375/308 = 340.3763 (~88/85 = 59.6237)
 
Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}
 
Badness: 0.0272
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 23409/23408
 
Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 | 0 5 17 24 31 53 15 -5 }}
 
Optimal tuning (CTE): ~55/49 = 1\6, ~162/133 = 340.3767 (~88/85 = 59.6233)
 
Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}
 
Badness: 0.0222
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 936/935, 1701/1700, 1716/1715, 1863/1862, 2024/2023, 2025/2023, 2376/2375


{{Multival|legend=1| 15 51 72 46 72 24 }}
Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 -12 | 0 5 17 24 31 53 15 -5 23 }}


[[POTE generator]]: ~243/224 = 140.367
Optimal tuning (CTE): ~55/49 = 1\6, ~162/133 = 340.3757 (~88/85 = 59.6243)


{{Val list|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}
Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}


[[Badness]]: 0.016814
Badness: 0.0160


== Pnict ==
== Pnict ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 2100875/2097152
[[Comma list]]: 250047/250000, 2100875/2097152


[[Mapping]]: [{{val| 3 10 11 6 }}, {{val| 0 -13 -10 6 }}]
{{Mapping|legend=1| 3 10 11 6 | 0 -13 -10 6 }}


{{Multival|legend=1| 39 30 -18 -43 -138 -126 }}
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~192/175 = 161.399


[[POTE generator]]: ~192/175 = 161.399
{{Optimal ET sequence|legend=1| 15, 141, 156, 171, 2409cd, 2580cd, …, 4461bccddd, 4632bccddd }}
 
{{Val list|legend=1| 15, 141, 156, 171, 2409cd, 2580cd, …, 4461bccddd, 4632bccddd }}


[[Badness]]: 0.045660
[[Badness]]: 0.045660


== Domain ==
== Domain ==
{{see also| Minortonic family }}
{{See also| Minortonic family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 645700815/645657712
[[Comma list]]: 250047/250000, 645700815/645657712


[[Mapping]]: [{{val| 3 -3 -9 -8 }}, {{val| 0 17 35 36 }}]
{{Mapping|legend=1| 3 -3 -9 -8 | 0 17 35 36 }}
 
{{Multival|legend=1| 51 105 108 48 28 -44 }}


[[POTE generator]]: ~10/9 = 182.467
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467


{{Val list|legend=1| 171, 1164, 1335, 1506, 1677, 1848, 2019, 2190, 11943, 13962, 15981, 18000, 20019, 22038 }}
{{Optimal ET sequence|legend=1| 171, 1164, 1335, 1506, 1677, 1848, 2019, 2190, 11943, 13962, 15981, 18000, 20019, 22038 }}


[[Badness]]: 0.013979
[[Badness]]: 0.013979


== Avicenna ==
== Avicenna ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 29360128/29296875
[[Comma list]]: 250047/250000, 29360128/29296875


[[Mapping]]: [{{val|3 2 8 16}}, {{val|0 8 -3 -22}}]
{{Mapping|legend=1| 3 2 8 16 | 0 8 -3 -22 }}


{{Multival|legend=1| 24 -9 -66 -70 -172 -128 }}
[[Optimal tuning]]s:
* [[CTE]]: ~63/50 = 1\3, ~1024/945 = 137.772
* [[POTE]]: ~63/50 = 1\3, ~1024/945 = 137.768


[[POTE generator]]: ~1024/945 = 137.768
{{Optimal ET sequence|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}
 
{{Val list|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}


[[Badness]]: 0.062187
[[Badness]]: 0.062187
Line 189: Line 273:
Comma list: 3025/3024, 5632/5625, 102487/102400
Comma list: 3025/3024, 5632/5625, 102487/102400


Mapping: [{{val|3 2 8 16 9}}, {{val|0 8 -3 -22 4}}]
Mapping: {{mapping| 3 2 8 16 9 | 0 8 -3 -22 4 }}


POTE generator: ~693/640 = 137.771
Optimal tunings:
* CTE: ~63/50 = 1\3, ~693/640 = 137.773
* POTE: ~63/50 = 1\3, ~693/640 = 137.771


Optimal GPV sequence: {{Val list| 87, 183, 270, 1263, 1533, 1803c, 2073c }}
Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270, 1263, 1533, 1803c }}


Badness: 0.023085
Badness: 0.023085
Line 202: Line 288:
Comma list: 676/675, 1001/1000, 3025/3024, 4096/4095
Comma list: 676/675, 1001/1000, 3025/3024, 4096/4095


Mapping: [{{val|3 2 8 16 9 8}}, {{val|0 8 -3 -22 4 9}}]
Mapping: {{mapping| 3 2 8 16 9 8 | 0 8 -3 -22 4 9 }}


POTE generator: ~13/12 = 137.777
Optimal tunings:
* CTE: ~63/50 = 1\3, ~13/12 = 137.777
* POTE: ~63/50 = 1\3, ~13/12 = 137.777


Optimal GPV sequence: {{Val list| 87, 183, 270 }}
Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270 }}


Badness: 0.015557
Badness: 0.015557
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 676/675, 715/714, 1001/1000, 3025/3024, 4096/4095
Mapping: {{mapping| 3 2 8 16 9 8 4 | 0 8 -3 -22 4 9 24 }}
Optimal tunings:
* CTE: ~63/50 = 1\3, ~13/12 = 137.758
* POTE: ~63/50 = 1\3, ~13/12 = 137.761
Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270, 453 }}
Badness: 0.0171
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 1729/1728, 3025/3024
Mapping: {{mapping| 3 2 8 16 9 8 4 0 | 0 8 -3 -22 4 9 24 37 }}
Optimal tunings:
* CTE: ~63/50 = 1\3, ~13/12 = 137.763
* POTE: ~63/50 = 1\3, ~13/12 = 137.767
Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270 }}
Badness: 0.0153


== Terture ==
== Terture ==
{{see also| Vulture family }}
{{See also| Vulture family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 359661568/358722675
[[Comma list]]: 250047/250000, 359661568/358722675


[[Mapping]]: [{{val| 3 4 3 2 }}, {{val| 0 4 21 34 }}]
{{Mapping|legend=1| 3 4 3 2 | 0 4 21 34 }}


[[POTE generator]]: ~392/375 = 75.555
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~392/375 = 75.555


{{Val list|legend=1| 111, 159, 270 }}
{{Optimal ET sequence|legend=1| 111, 159, 270 }}


[[Badness]]: 0.087156
[[Badness]]: 0.087156
Line 230: Line 348:
Comma list: 3025/3024, 19712/19683, 102487/102400
Comma list: 3025/3024, 19712/19683, 102487/102400


Mapping: [{{val| 3 4 3 2 10 }}, {{val| 0 4 21 34 2 }}]
Mapping: {{mapping| 3 4 3 2 10 | 0 4 21 34 2 }}


POTE generator: ~392/375 = 75.550
Optimal tuning (POTE): ~63/50 = 1\3, ~392/375 = 75.550


Optimal GPV sequence: {{Val list| 111, 159, 270, 1239, 1509, 1779, 2049, 2319 }}
{{Optimal ET sequence|legend=1| 111, 159, 270, 1239, 1509, 1779, 2049, 2319 }}


Badness: 0.029326
Badness: 0.029326
Line 243: Line 361:
Comma list: 676/675, 1001/1000, 3025/3024, 10985/10976
Comma list: 676/675, 1001/1000, 3025/3024, 10985/10976


Mapping: [{{val| 3 4 3 2 10 6 }}, {{val| 0 4 21 34 2 27 }}]
Mapping: {{mapping| 3 4 3 2 10 6 | 0 4 21 34 2 27 }}


POTE generator: ~117/112 = 75.553
Optimal tuning (POTE): ~63/50 = 1\3, ~117/112 = 75.553


Optimal GPV sequence: {{Val list| 111, 159, 270 }}
{{Optimal ET sequence|legend=1| 111, 159, 270 }}


Badness: 0.018647
Badness: 0.018647
Line 256: Line 374:
Comma list: 676/675, 715/714, 936/935, 1001/1000, 4928/4913
Comma list: 676/675, 715/714, 936/935, 1001/1000, 4928/4913


Mapping: [{{val| 3 4 3 2 10 6 10 }}, {{val| 0 4 21 34 2 27 12 }}]
Mapping: {{mapping| 3 4 3 2 10 6 10 | 0 4 21 34 2 27 12 }}


POTE generator: ~117/112 = 75.560
Optimal tuning (POTE): ~63/50 = 1\3, ~117/112 = 75.560


Optimal GPV sequence: {{Val list| 111, 159, 270 }}
{{Optimal ET sequence|legend=1| 111, 159, 270 }}


Badness: 0.018705
Badness: 0.018705
Line 269: Line 387:
Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1617/1615
Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1617/1615


Mapping: [{{val| 3 4 3 2 10 6 10 5 }}, {{val| 0 4 21 34 2 27 12 41 }}]
Mapping: {{mapping| 3 4 3 2 10 6 10 5 | 0 4 21 34 2 27 12 41 }}


POTE generator: ~95/91 = 75.560
Optimal tuning (POTE): ~63/50 = 1\3, ~95/91 = 75.560


Optimal GPV sequence: {{Val list| 111, 159, 270 }}
{{Optimal ET sequence|legend=1| 111, 159, 270 }}


Badness: 0.013902
Badness: 0.013902
Line 282: Line 400:
Comma list: 460/459, 529/528, 676/675, 715/714, 936/935, 1001/1000, 1216/1215
Comma list: 460/459, 529/528, 676/675, 715/714, 936/935, 1001/1000, 1216/1215


Mapping: [{{val| 3 4 3 2 10 6 10 5 13 }}, {{val| 0 4 21 34 2 27 12 41 3 }}]
Mapping: {{mapping| 3 4 3 2 10 6 10 5 13 | 0 4 21 34 2 27 12 41 3 }}


POTE generator: ~24/23 = 75.548
Optimal tuning (POTE): ~63/50 = 1\3, ~24/23 = 75.548


Optimal GPV sequence: {{Val list| 111, 159, 270 }}
{{Optimal ET sequence|legend=1| 111, 159, 270 }}


Badness: 0.014915
Badness: 0.014915
== Slendscape ==
Slendscape tempers out the [[slendroschisma]] (68719476736/68641485507) in addition to landscape comma, and thereby features a period of 1\15.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 250047/250000, 12884901888/12867859375
{{Mapping|legend=1| 15 0 17 54 | 0 4 3 -2 }}
: Mapping generators: ~8575/8192, ~1152/875
[[Optimal tuning]] ([[CTE]]): ~8575/8192 = 80.0000, ~1152/875 = 475.4847 (~6/5 = 315.4847 or ~21/20 = 84.5153)
{{Optimal ET sequence|legend=1| 255, 270, 525, 795, 1065 }}
[[Badness]]: 0.058002
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 102487/102400, 180224/180075
Mapping: {{Mapping| 15 0 17 54 40 | 0 4 3 -2 2 }}
Optimal tuning (CTE): ~22/21 = 80.0000, ~968/735 = 475.4912 (~6/5 = 315.4912 or ~21/20 = 84.5088)
Optimal ET sequence: {{EDOs| 255, 270, 525, 795, 1065 }}
Badness: 0.026246
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 3025/3024, 4096/4095, 14641/14625
Mapping: {{Mapping| 15 0 17 54 40 109 | 0 4 3 -2 2 -9 }}
Optimal tuning (CTE): ~22/21 = 80.0000, ~154/117 = 475.4935 (~6/5 = 315.4935 or ~21/20 = 84.5065)
Optimal ET sequence: {{EDOs| 255, 270, 795, 1065 }}
Badness: 0.021230


== Akjayland ==
== Akjayland ==
{{See also| 21st-octave temperaments }}
{{See also| 21st-octave temperaments }}


''Akjayland'' tempers out the [[akjaysma]] in addition to landscape comma, and thereby features a period of 1\21.  
Akjayland tempers out the [[akjaysma]] in addition to landscape comma, and thereby features a period of 1＼21.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 299: Line 460:
[[Comma list]]: 250047/250000, {{monzo| 43 -1 -13 -4 }}
[[Comma list]]: 250047/250000, {{monzo| 43 -1 -13 -4 }}


[[Mapping]]: [{{val| 21 1 38 102 }}, {{val| 0 3 1 -4 }}]
{{Mapping|legend=1| 21 1 38 102 | 0 3 1 -4 }}


Mapping generators: ~1323/1280, ~131072/91875
: mapping generators: ~1323/1280, ~131072/91875


[[Optimal tuning]] ([[CTE]]): ~131072/91875 = 614.9354
[[Optimal tuning]] ([[CTE]]): ~1323/1280 = 1＼21, ~131072/91875 = 614.9354


[[Optimal GPV sequence]]: {{val list| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
{{Optimal ET sequence|legend=1| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}


[[Badness]]: 0.0309
[[Badness]]: 0.0309


=== Vasca ===
=== Vasca ===
Vasca can be described as the 357 & 525 temperament, extended as high as the 23-limit. It tempers out the {{monzo| 95 0 0 0 0 0 0 0 -21 }}, and sets a stack of 21 [[23/16]]'s equal with 11 octaves. The name derives from elements vanadium (23) and scandium (21), since this uses the 23rd harmonic, which itself is extremely well represented in 21edo.  
Vasca can be described as the 357 & 525 temperament, extended as high as the 23-limit. It tempers out the {{monzo| 95 0 0 0 0 0 0 0 -21 }}, and sets a stack of twenty-one [[23/16]]'s equal with eleven octaves. The name derives from elements vanadium (23) and scandium (21), since this uses the 23rd harmonic, which itself is extremely well represented in 21edo.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 316: Line 477:
Comma list: 3025/3024, 102487/102400, {{monzo| 39 -4 -11 -5 2 }}
Comma list: 3025/3024, 102487/102400, {{monzo| 39 -4 -11 -5 2 }}


Mapping: [{{val| 21 4 39 98 58 }}, {{val| 0 6 2 -8 3 3 }}]
Mapping: {{mapping| 21 4 39 98 58 | 0 6 2 -8 3 3 }}


Mapping generators: ~1323/1280, ~6615/5632
: mapping generators: ~1323/1280, ~6615/5632


Optimal tuning (CTE): ~6615/5632 = 278.8998
Optimal tuning (CTE): ~1323/1280 = 1＼21, ~6615/5632 = 278.8998


Optimal GPV sequence: {{val list| 168, 357, 525, 882, 1407, 2289e }}
{{Optimal ET sequence|legend=1| 168, 357, 525, 882, 1407, 2289e }}


Badness: 0.0949
Badness: 0.0949
Line 331: Line 492:
Comma list: 3025/3024, 4096/4095, 14641/14625, 85750/85683
Comma list: 3025/3024, 4096/4095, 14641/14625, 85750/85683


Mapping: [{{val| 21 4 39 98 58 107 }}, {{val| 0 6 2 -8 3 -6 }}]
Mapping: {{mapping| 21 4 39 98 58 107 | 0 6 2 -8 3 -6 }}


Optimal tuning (CTE): ~168/143 = 278.9058
Optimal tuning (CTE): ~336/325 = 1＼21, ~168/143 = 278.9058


Optimal GPV sequence: {{val list| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}


Badness: 0.0551
Badness: 0.0551
Line 344: Line 505:
Comma list: 2601/2600, 3025/3024, 4096/4095, 8624/8619, 14641/14625
Comma list: 2601/2600, 3025/3024, 4096/4095, 8624/8619, 14641/14625


Mapping: [{{val| 21 4 39 98 58 107 120 }}, {{val| 0 6 2 -8 3 -6 -7 }}]
Mapping: {{mapping| 21 4 39 98 58 107 120 | 0 6 2 -8 3 -6 -7 }}


Optimal tuning (CTE): ~168/143 = 278.9036
Optimal tuning (CTE): ~336/325 = 1＼21, ~168/143 = 278.9036


Optimal GPV sequence: {{val list| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}


Badness: 0.0319
Badness: 0.0319
Line 357: Line 518:
Comma list: 2376/2375, 2601/2600, 2926/2925, 3025/3024, 3213/3211, 4096/4095
Comma list: 2376/2375, 2601/2600, 2926/2925, 3025/3024, 3213/3211, 4096/4095


Mapping: [{{val| 21 4 39 98 58 107 120 16 }}, {{val| 0 6 2 -8 3 -6 -7 15 }}]
Mapping: {{mapping| 21 4 39 98 58 107 120 16 | 0 6 2 -8 3 -6 -7 15 }}


Optimal tuning (CTE): ~168/143 = 278.9036
Optimal tuning (CTE): ~336/325 = 1＼21, ~168/143 = 278.9036


Optimal GPV sequence: {{val list| 168h, 357, 525, 882, 1407 }}
{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}


Badness: 0.0270
Badness: 0.0270
Line 370: Line 531:
Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211
Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211


Mapping: [{{val| 21 34 49 58 73 77 85 91 95 }}, {{val| 0 -6 -2 8 -3 6 7 -15 0 }}]
Mapping: {{mapping| 21 4 39 98 58 107 120 16 95 | 0 -6 -2 8 -3 6 7 -15 0 }}


Optimal tuning (CTE): ~168/143 = 278.8971
Optimal tuning (CTE): ~336/325 = 1＼21, ~168/143 = 278.8971


Optimal GPV sequence: {{val list| 168h, 357, 525, 882, 1407 }}
{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}


Badness: 0.0199
Badness: 0.0199
== Magnesium ==
: ''For the 5-limit version, see [[12th-octave temperaments#Magnesium (5-limit)]].
Magnesium is named after element 12 for being period 12, however, it's not an extension of the [[atomic]] – the associated comma is {{monzo| -157 -24 84 }} in the 5-limit and the 7 generators together with [[12edo]] major second reach the just perfect fifth, [[3/2]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 250047/250000, {{monzo| -59 2 18 5 }}
{{Mapping|legend=1| 12 2 23 58 | 0 7 2 -10 }}
: mapping generators: ~138915/131072, 3145728/2734375
Optimal tuning (CTE): ~138915/131072 = 1\12, ~3145728/2734375 = 243.130
{{Optimal ET sequence|legend=1| 84, 360d, 444, 528, 612, 1920, 2532, 10740cd, 13272bcdd, 15804bcdd, 18336bcddd }}
Badness: 0.0964


== Chromium ==
== Chromium ==
Chromium is described as the 72 & 624 temperament, and named after the 24th element for being period 24.
: ''For the 5-limit version, see [[24th-octave temperaments#Chromium (5-limit)]].
 
[[Chromium]] is defined by associating the porcupine comma [[250/243]] to the 24th of an octave. Named after the 24th element for being period 24.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 385: Line 567:
[[Comma list]]: 250047/250000, 49589822592/49433168575
[[Comma list]]: 250047/250000, 49589822592/49433168575


[[Mapping]]: [{{val| 24 1 -6 18 }}, {{val| 0 3 5 4 }}]
{{Mapping|legend=1| 24 1 -6 18 | 0 3 5 4 }}


Mapping generators: ~250/243, ~10/7
: mapping generators: ~250/243, ~10/7


[[Optimal tuning]] ([[CTE]]): ~10/7 = 617.2710
[[Optimal tuning]] ([[CTE]]): ~250/243 = 1\24, ~10/7 = 617.2710


{{Val list|legend=1| 72, …, 480, 552, 624, 1320, 1944d, 3264d }}
{{Optimal ET sequence|legend=1| 72, …, 480, 552, 624, 1320, 1944d, 3264d }}


[[Badness]]: 0.139
[[Badness]]: 0.139
Line 400: Line 582:
Comma list: 9801/9800, 46656/46585, 250047/250000
Comma list: 9801/9800, 46656/46585, 250047/250000


Mapping: [{{val| 24 1 -6 18 46 }}, {{val| 0 3 5 4 3 }}]
Mapping: {{mapping| 24 1 -6 18 46 | 0 3 5 4 3 }}


Optimal tuning (CTE): ~10/7 = 617.2597
Optimal tuning (CTE): ~250/243 = 1\24, ~10/7 = 617.2597


Optimal GPV sequence: {{Val list| 72, …, 480, 552, 624 }}
{{Optimal ET sequence|legend=1| 72, …, 480, 552, 624 }}


Badness: 0.0398
Badness: 0.0398
Line 413: Line 595:
Comma list: 1716/1715, 2080/2079, 34398/34375, 39366/39325
Comma list: 1716/1715, 2080/2079, 34398/34375, 39366/39325


Mapping: [{{val| 24 1 -6 18 46 -47 -13 }}, {{val| 0 3 5 4 3 11 }}]
Mapping: {{mapping| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 }}


Optimal tuning (CTE): ~10/7 = 617.2869
Optimal tuning (CTE): ~250/243 = 1\24, ~10/7 = 617.2869


Optimal GPV sequence: {{Val list| 72, …, 480f, 552, 624, 1176de, 1800cdee }}
{{Optimal ET sequence|legend=1| 72, …, 480f, 552, 624, 1176de, 1800cdee }}


Badness: 0.0293
Badness: 0.0293
Line 426: Line 608:
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 11016/11011
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 11016/11011


Mapping: [{{val| 24 1 -6 18 46 -47 -13 }}, {{val| 0 3 5 4 3 11 9 }}]
Mapping: {{mapping| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 9 }}


Optimal tuning (CTE): ~10/7 = 617.2732
Optimal tuning (CTE): ~35/34 = 1\24, ~10/7 = 617.2732


Optimal GPV sequence: {{Val list| 72, …, 480fgg, 552g, 624 }}
{{Optimal ET sequence|legend=1| 72, …, 480fgg, 552g, 624 }}


Badness: 0.0209
Badness: 0.0209
== Zinc ==
{{See also| 10th-octave temperaments #Neon }}
{{See also| 60th-octave temperaments #Neodymium }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 250047/250000, {{monzo| -53 -12 2 24 }}
{{Mapping|legend=1| 30 2 15 66 | 0 5 6 2 }}
[[Optimal tuning]]s:
* [[CTE]]: ~53747712/52521875 = 1\30, ~216/175 = 364.3896 (~{{monzo| 21 3 1 -10 }} = 4.3896)
* [[CWE]]: ~53747712/52521875 = 1\30, ~216/175 = 364.3879 (~{{monzo| 21 3 1 -10 }} = 4.3879)
{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650 }}
[[Badness]]: 0.0742
=== 11-limit===
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 151263/151250, {{monzo| -27 -6 4 6 3 }}
Mapping: {{mapping| 30 2 15 66 122 | 0 5 6 2 -2 }}
Optimal tunings:
* CTE: ~18865/18432 = 1\30, ~216/175 = 364.3886 (~385/384 = 4.3886)
* CWE: ~18865/18432 = 1\30, ~216/175 = 364.3849 (~385/384 = 4.3849)
Optimal ET sequence: {{Optimal ET sequence| 270, 1110, 1380, 1650, 1920, 2190 }}
Badness: 0.0220
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 9801/9800, 10648/10647, 105644/105625, 196625/196608
Mapping: {{mapping| 30 2 15 66 122 193 | 0 5 6 2 -2 -9 }}
Optimal tunings:
* CTE: ~351/343 = 1\30, ~216/175 = 364.3879 (~385/384 = 4.3879)
* CWE: ~351/343 = 1\30, ~216/175 = 364.3867 (~385/384 = 4.3867)
Optimal ET sequence: {{Optimal ET sequence| 270, 1380, 1650, 1920, 2190, 4650, 6840e, 11490de }}
Badness: 0.0155
=== 2.3.5.7.11.13.19 subgroup (neozinc) ===
Subgroup: 2.3.5.7.11.13.19
Comma list: 5929/5928, 6860/6859, 9801/9800, 10241/10240, 89376/89375
Mapping: {{mapping| 30 2 15 66 122 193 91 | 0 5 6 2 -2 -9 4 }}
Optimal tunings:
* CTE: ~175/171 = 1\30, ~216/175 = 364.3876 (~400/399 = 4.3876)
* CWE: ~175/171 = 1\30, ~216/175 = 364.3864 (~400/399 = 4.3864)
Optimal ET sequence: {{Optimal ET sequence| 270, 1380, 1650, 1920, 2190, 4650, 6840e }}
Badness: 0.00921
== Poe ==
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 250047/250000, {{monzo| 15 -16 -4 7 }}
{{Mapping|legend=1| 30 0 -73 -106 | 0 1 3 4 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2240/2187 = 1\30, ~3/2 = 702.141 (~81/80 = 22.141)
* [[CWE]]: ~2240/2187 = 1\30, ~3/2 = 702.166 (~81/80 = 22.166)
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 19712/19683, 250047/250000
Mapping: {{mapping| 30 0 -73 -106 -134 | 0 1 3 4 5 }}
Optimal tunings:
* CTE: ~45/44 = 1\30, ~3/2 = 702.182 (~81/80 = 22.182)
* CWE: ~45/44 = 1\30, ~3/2 = 702.213 (~81/80 = 22.213)
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 19712/19683, 1001/1000, 4459/4455, 4225/4224
Mapping: {{mapping| 30 0 -73 -106 -134 111 | 0 1 3 4 5 0 }}
Optimal tunings:
* CTE: ~45/44 = 1\30, ~3/2 = 702.182 (~81/80 = 22.182)
* CWE: ~45/44 = 1\30, ~3/2 = 702.170 (~81/80 = 22.170)


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Landscape microtemperaments| ]] <!-- main article -->
[[Category:Landscape microtemperaments| ]] <!-- main article -->
[[Category:Landscape| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Microtemperaments]]
[[Category:Microtemperaments]]
[[Category:Rank 2]]
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