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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 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]]
* ''[[Noletaland]]'', → [[The Flashmob]]
* ''[[Tritricot]]'' (+{{monzo| 35 -23 -3 3 }}) → [[Alphatricot family #Tritricot|Alphatricot family]]
Discussed below are sextile, septichrome, pnict, domain, avicenna, terture, akjayland and terture.
* ''[[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 ==
Sextile tempers out the [[garischisma]] with a 1/6-octave period and is the 12 & 270 temperament.
[[Subgroup]]: 2.3.5.7
[[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 }}
 
Mapping generators: ~4096/3645, ~3


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


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


{{Optimal ET sequence|legend=1| 12, 234d, 246d, 258, 270, 1362c, 1632c, 1902c, 2172c, 2442bc, 2712bc }}
{{Optimal ET sequence|legend=1| 12, 234d, 246d, 258, 270, 1362c, 1632c, 1902c, 2172c, 2442bc, 2712bc }}
Line 40: 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 }}


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


{{Optimal ET sequence|legend=1| 12, 246dee, 258e, 270, 822, 1092, 1362c }}
{{Optimal ET sequence|legend=1| 12, 246dee, 258e, 270, 822, 1092, 1362c }}
Line 53: Line 60:
Comma list: 1716/1715, 2080/2079, 5632/5625, 10648/10647
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 }}


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 702.2141
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 702.2141
* CWE: ~55/49 = 1\6, ~3/2 = 702.2001


{{Optimal ET sequence|legend=1| 12f, 258ef, 270, 552, 822, 1092, 1914cde }}
{{Optimal ET sequence|legend=1| 12f, 258ef, 270, 552, 822, 1092, 1914cde }}
Line 66: Line 75:
Comma list: 936/935, 1701/1700, 1716/1715, 5632/5625, 7744/7735
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 }}


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 702.2117
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 702.2117
* CWE: ~55/49 = 1\6, ~3/2 = 702.1869


{{Optimal ET sequence|legend=1| 12f, 270, 552g }}
{{Optimal ET sequence|legend=1| 12f, 270, 552g }}
Line 79: Line 90:
Comma list: 936/935, 1216/1215, 1701/1700, 1716/1715, 2376/2375, 4200/4199
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 }}


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 702.2118
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 702.2118
* CWE: ~55/49 = 1\6, ~3/2 = 702.1890


{{Optimal ET sequence|legend=1| 12f, 270, 552g, 822gg }}
{{Optimal ET sequence|legend=1| 12f, 270, 552g, 822gg }}
Line 92: Line 105:
Comma list: 1001/1000, 4096/4095, 4459/4455, 20449/20412
Comma list: 1001/1000, 4096/4095, 4459/4455, 20449/20412


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


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 702.2296
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 702.2296
* CWE: ~55/49 = 1\6, ~3/2 = 702.2285


{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
Line 105: Line 120:
Comma list: 715/714, 1001/1000, 1701/1700, 4096/4095, 4459/4455
Comma list: 715/714, 1001/1000, 1701/1700, 4096/4095, 4459/4455


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


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 702.2264
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 702.2264
* CWE: ~55/49 = 1\6, ~3/2 = 702.2207


{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
Line 118: Line 135:
Comma list: 715/714, 1001/1000, 1216/1215, 1701/1700, 1729/1728, 2912/2907
Comma list: 715/714, 1001/1000, 1216/1215, 1701/1700, 1729/1728, 2912/2907


Mapping: [{{val| 6 0 71 150 230 -149 -4 35 }}, {{val| 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 }}


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 702.2266
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 702.2266
* CWE: ~55/49 = 1\6, ~3/2 = 702.2208


{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
Line 131: Line 150:
[[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 }}
 
{{Multival|legend=1| 15 51 72 46 72 24 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~243/224 = 140.367
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~243/224 = 140.367
Line 140: Line 157:


[[Badness]]: 0.016814
[[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
Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 -12 | 0 5 17 24 31 53 15 -5 23 }}
Optimal tuning (CTE): ~55/49 = 1\6, ~162/133 = 340.3757 (~88/85 = 59.6243)
Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}
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
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~192/175 = 161.399
Line 157: Line 239:


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 163: Line 245:
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467
Line 178: Line 258:
[[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
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~1024/945 = 137.768
* [[POTE]]: ~63/50 = 1\3, ~1024/945 = 137.768


{{Optimal ET sequence|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}
{{Optimal ET sequence|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}
Line 193: 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 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~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 ET sequence|legend=1| 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 206: 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 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~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 ET sequence|legend=1| 87, 183, 270 }}
Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270 }}


Badness: 0.015557
Badness: 0.015557
Line 219: Line 303:
Comma list: 676/675, 715/714, 1001/1000, 3025/3024, 4096/4095
Comma list: 676/675, 715/714, 1001/1000, 3025/3024, 4096/4095


Mapping: [{{val|3 2 8 16 9 8 4}}, {{val|0 8 -3 -22 4 9 24}}]
Mapping: {{mapping| 3 2 8 16 9 8 4 | 0 8 -3 -22 4 9 24 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~13/12 = 137.777
Optimal tunings:
* CTE: ~63/50 = 1\3, ~13/12 = 137.758
* POTE: ~63/50 = 1\3, ~13/12 = 137.761


{{Optimal ET sequence|legend=1| 87, 183, 270 }}
Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270, 453 }}


Badness: 0.015557
Badness: 0.0171


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


Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 3025/3024, 4096/4095
Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 1729/1728, 3025/3024


Mapping: [{{val|3 2 8 16 9 8 4 0}}, {{val|0 8 -3 -22 4 9 24 37}}]
Mapping: {{mapping| 3 2 8 16 9 8 4 0 | 0 8 -3 -22 4 9 24 37 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~13/12 = 137.777
Optimal tunings:
* CTE: ~63/50 = 1\3, ~13/12 = 137.763
* POTE: ~63/50 = 1\3, ~13/12 = 137.767


{{Optimal ET sequence|legend=1| 87, 183, 270 }}
Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270 }}


Badness: 0.015557
Badness: 0.0153


== Terture ==
== Terture ==
Line 247: Line 335:
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~392/375 = 75.555
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~392/375 = 75.555
Line 260: 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 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~392/375 = 75.550
Optimal tuning (POTE): ~63/50 = 1\3, ~392/375 = 75.550
Line 273: 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 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~117/112 = 75.553
Optimal tuning (POTE): ~63/50 = 1\3, ~117/112 = 75.553
Line 286: 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 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~117/112 = 75.560
Optimal tuning (POTE): ~63/50 = 1\3, ~117/112 = 75.560
Line 299: 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 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~95/91 = 75.560
Optimal tuning (POTE): ~63/50 = 1\3, ~95/91 = 75.560
Line 312: 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 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~24/23 = 75.548
Optimal tuning (POTE): ~63/50 = 1\3, ~24/23 = 75.548
Line 319: Line 407:


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 329: 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]]): ~1323/1280 = 1\21, ~131072/91875 = 614.9354
[[Optimal tuning]] ([[CTE]]): ~1323/1280 = 1＼21, ~131072/91875 = 614.9354


{{Optimal ET sequence|legend=1| 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 }}
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=== 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 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.  
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
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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): ~1323/1280 = 1\21, ~6615/5632 = 278.8998
Optimal tuning (CTE): ~1323/1280 = 1＼21, ~6615/5632 = 278.8998


{{Optimal ET sequence|legend=1| 168, 357, 525, 882, 1407, 2289e }}
{{Optimal ET sequence|legend=1| 168, 357, 525, 882, 1407, 2289e }}
Line 361: 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): ~336/325 = 1\21, ~168/143 = 278.9058
Optimal tuning (CTE): ~336/325 = 1＼21, ~168/143 = 278.9058


{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
Line 374: 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): ~336/325 = 1\21, ~168/143 = 278.9036
Optimal tuning (CTE): ~336/325 = 1＼21, ~168/143 = 278.9036


{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
Line 387: 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): ~336/325 = 1\21, ~168/143 = 278.9036
Optimal tuning (CTE): ~336/325 = 1＼21, ~168/143 = 278.9036


{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
Line 400: 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 4 39 98 58 107 120 16 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): ~336/325 = 1\21, ~168/143 = 278.8971
Optimal tuning (CTE): ~336/325 = 1＼21, ~168/143 = 278.8971


{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
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== Magnesium ==
== Magnesium ==
: ''For the 5-limit version, see [[12th-octave temperaments#Magnesium (5-limit)]].
: ''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
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 }}


Comma list: 250047/250000, 9234096523681640625/9223372036854775808
{{Mapping|legend=1| 12 2 23 58 | 0 7 2 -10 }}


Mapping: [{{val|12 2 23 58}}, {{val|0 7 2 -10}}]
: mapping generators: ~138915/131072, 3145728/2734375


Mapping generators: ~138915/131072 = 1\12, 3145728/2734375 = 243.130
Optimal tuning (CTE): ~138915/131072 = 1\12, ~3145728/2734375 = 243.130


Optimal tuning (CTE): 3145728/2734375 = 243.130
{{Optimal ET sequence|legend=1| 84, 360d, 444, 528, 612, 1920, 2532, 10740cd, 13272bcdd, 15804bcdd, 18336bcddd }}


{{Optimal ET sequence|legend=1|84, 528, 612, 696, 1308, 1920, 2532}}, ...
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
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[[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]]): ~250/243 = 1\24, ~10/7 = 617.2710
[[Optimal tuning]] ([[CTE]]): ~250/243 = 1\24, ~10/7 = 617.2710
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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): ~250/243 = 1\24, ~10/7 = 617.2597
Optimal tuning (CTE): ~250/243 = 1\24, ~10/7 = 617.2597
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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): ~250/243 = 1\24, ~10/7 = 617.2869
Optimal tuning (CTE): ~250/243 = 1\24, ~10/7 = 617.2869
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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): ~35/34 = 1\24, ~10/7 = 617.2732
Optimal tuning (CTE): ~35/34 = 1\24, ~10/7 = 617.2732
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== Zinc ==
== Zinc ==
Zinc is described as the 270 & 1920 temperament, and named after the 30th element due to featuring a period of 1/30 octave.
{{See also| 10th-octave temperaments #Neon }}
{{See also| 60th-octave temperaments #Neodymium }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Comma list: 250047/250000, {{monzo|-45 -24 14 18}}
[[Comma list]]: 250047/250000, {{monzo| -53 -12 2 24 }}


Mapping: [{{val|30 2 15 66}}, {{val|0 5 6 2}}]
{{Mapping|legend=1| 30 2 15 66 | 0 5 6 2 }}


Optimal tuning (CTE): ~53747712/52521875 = 1\30, ~216/175 = 364.389
[[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, 1110, 1380, 1650, 1920}}, ...
{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650 }}


=== 11-limit ===
[[Badness]]: 0.0742
 
=== 11-limit===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 9801/9800, 250047/250000, 97869261875/97844723712
Comma list: 9801/9800, 151263/151250, {{monzo| -27 -6 4 6 3 }}


Mapping: [{{val|30 2 15 66 122}}, {{val|0 5 6 2 -2}}]
Mapping: {{mapping| 30 2 15 66 122 | 0 5 6 2 -2 }}


Optimal tuning (CTE): ~18865/18432 = 1\30, ~216/175 = 364.389
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|legend=1|270, 840, 1110, 1380, 1650, 1920}}, ...
Optimal ET sequence: {{Optimal ET sequence| 270, 1110, 1380, 1650, 1920, 2190 }}
 
Badness: 0.0220


=== 13-limit ===
=== 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
Subgroup: 2.3.5.7.11


Comma list: 9801/9800, 123201/123200, 250047/250000, 1990656/1990625
Comma list: 9801/9800, 19712/19683, 250047/250000


Mapping: [{{val|30 2 15 66 122 193}}, {{val|0 5 6 2 -2 -9}}]
Mapping: {{mapping| 30 0 -73 -106 -134 | 0 1 3 4 5 }}


Optimal tuning (CTE): ~351/343 = 1\30, ~216/175 = 364.389
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)


{{Optimal ET sequence|legend=1|270, 1380, 1650, 1920}}, ...
[[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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