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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]]
* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Augene|Augmented family]]
* ''[[Waage]]'', {225/224, 250047/250000} → [[Compton family #Compton]]
* ''[[Compton]]'' (+225/224) → [[Compton family #Compton|Compton family]]
* ''[[Tritikleismic]]'', {1029/1024, 15625/15552} → [[Kleismic family #Tritikleismic]]
* ''[[Tritikleismic]]'' (+1029/1024) → [[Kleismic family #Tritikleismic|Kleismic family]]
* [[Ennealimmal]], {2401/2400, 4375/4374} → [[Ragismic microtemperaments #Ennealimmal]]
* ''[[Trisensory]]'' (+1728/1715) → [[Sensipent family #Trisensory|Sensipent family]]
* ''[[Misty]]'', {3136/3125, 5120/5103} → [[Misty family]]
* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Nessafof]]'', {6144/6125, 250047/250000} → [[Porwell temperaments #Nessafof]]
* ''[[Misty]]'' (+3136/3125 or 5120/5103) → [[Misty family]]
* ''[[Chromat]]'', {10976/10935, 235298/234375} → [[Hemimage temperaments #Chromat]]
* ''[[Nessafof]]'' (+6144/6125) → [[Porwell temperaments #Nessafof|Porwell temperaments]]
* ''[[Term]]'', {32805/32768, 250047/250000} → [[Schismatic family #Term]]
* ''[[Chromat]]'' (10976/10935) → [[Hemimage temperaments #Chromat|Hemimage temperaments]]
* ''[[Caleb]]'', {33075/32768, 250047/250000} → [[Mirwomo temperaments #Caleb]]
* ''[[Term]]'' (+32805/32768) → [[Schismatic family #Term|Schismatic family]]
* [[Mutt temperament|Mutt]], {65625/65536, 250047/250000} → [[Horwell temperaments #Mutt]]
* ''[[Caleb]]'' (+33075/32768) → [[Mirwomo temperaments #Caleb|Mirwomo temperaments]]
* ''[[Stearnscape]]'', {118098/117649, 250047/250000} → [[Stearnsmic clan #Stearnscape]]
* [[Mutt temperament|Mutt]] (+65625/65536) → [[Horwell temperaments #Mutt|Horwell temperaments]]
* ''[[Tritricot]]'', {250047/250000, 11785390260224/11767897353375} → [[Tricot family #Tritricot]]
* ''[[Triquart]]'' (+117649/116640) → [[Quartonic family #Triquart|Quartonic 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 35: 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
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 5632/5625, 10648/10647
Mapping: {{mapping| 6 0 71 150 230 279 | 0 1 -6 -14 -22 -27 }}
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 }}
Badness: 0.0191
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 936/935, 1701/1700, 1716/1715, 5632/5625, 7744/7735
Mapping: {{mapping| 6 0 71 150 230 279 -4 | 0 1 -6 -14 -22 -27 3 }}
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 }}
Badness: 0.0209
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 936/935, 1216/1215, 1701/1700, 1716/1715, 2376/2375, 4200/4199
Mapping: {{mapping| 6 0 71 150 230 279 -4 35 | 0 1 -6 -14 -22 -27 3 -1 }}
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 }}
Badness: 0.0156
=== Sextilia ===
Subgroup: 2.3.5.7.11.13
Comma list: 1001/1000, 4096/4095, 4459/4455, 20449/20412
Mapping: {{mapping| 6 0 71 150 230 -149 | 0 1 -6 -14 -22 18 }}
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 }}
Badness: 0.0391
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 715/714, 1001/1000, 1701/1700, 4096/4095, 4459/4455
Mapping: {{mapping| 6 0 71 150 230 -149 -4 | 0 1 -6 -14 -22 18 3 }}
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 }}
Badness: 0.0384
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 715/714, 1001/1000, 1216/1215, 1701/1700, 1729/1728, 2912/2907
Mapping: {{mapping| 6 0 71 150 230 -149 -4 35 | 0 1 -6 -14 -22 18 3 -1 }}
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 }}
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


{{Multival|legend=1| 15 51 72 46 72 24 }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13


[[POTE generator]]: ~243/224 = 140.367
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 61965/61952


{{Val list|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}
Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 | 0 5 17 24 31 53 15 }}


[[Badness]]: 0.016814
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 }}


[[POTE generator]]: ~192/175 = 161.399
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~192/175 = 161.399


{{Val list|legend=1| 15, 141, 156, 171, 2409cd, 2580cd, …, 4461bccddd, 4632bccddd }}
{{Optimal ET sequence|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 }}


[[POTE generator]]: ~1024/945 = 137.768
[[Optimal tuning]]s:
* [[CTE]]: ~63/50 = 1\3, ~1024/945 = 137.772
* [[POTE]]: ~63/50 = 1\3, ~1024/945 = 137.768


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


[[Badness]]: 0.062187
[[Badness]]: 0.062187
Line 110: 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 123: 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 151: 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 164: 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 177: 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 190: 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 203: 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


[[Category:Regular temperament theory]]
== Slendscape ==
[[Category:Temperament collection]]
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 ==
{{See also| 21st-octave temperaments }}
 
Akjayland tempers out the [[akjaysma]] in addition to landscape comma, and thereby features a period of 1＼21.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 250047/250000, {{monzo| 43 -1 -13 -4 }}
 
{{Mapping|legend=1| 21 1 38 102 | 0 3 1 -4 }}
 
: mapping generators: ~1323/1280, ~131072/91875
 
[[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 }}
 
[[Badness]]: 0.0309
 
=== 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.
 
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 102487/102400, {{monzo| 39 -4 -11 -5 2 }}
 
Mapping: {{mapping| 21 4 39 98 58 | 0 6 2 -8 3 3 }}
 
: mapping generators: ~1323/1280, ~6615/5632
 
Optimal tuning (CTE): ~1323/1280 = 1＼21, ~6615/5632 = 278.8998
 
{{Optimal ET sequence|legend=1| 168, 357, 525, 882, 1407, 2289e }}
 
Badness: 0.0949
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 3025/3024, 4096/4095, 14641/14625, 85750/85683
 
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 ET sequence|legend=1| 168, 357, 525, 882 }}
 
Badness: 0.0551
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 2601/2600, 3025/3024, 4096/4095, 8624/8619, 14641/14625
 
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 ET sequence|legend=1| 168, 357, 525, 882 }}
 
Badness: 0.0319
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 2376/2375, 2601/2600, 2926/2925, 3025/3024, 3213/3211, 4096/4095
 
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 ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
 
Badness: 0.0270
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211
 
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 ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
 
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 ==
: ''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
 
[[Comma list]]: 250047/250000, 49589822592/49433168575
 
{{Mapping|legend=1| 24 1 -6 18 | 0 3 5 4 }}
 
: mapping generators: ~250/243, ~10/7
 
[[Optimal tuning]] ([[CTE]]): ~250/243 = 1\24, ~10/7 = 617.2710
 
{{Optimal ET sequence|legend=1| 72, …, 480, 552, 624, 1320, 1944d, 3264d }}
 
[[Badness]]: 0.139
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 46656/46585, 250047/250000
 
Mapping: {{mapping| 24 1 -6 18 46 | 0 3 5 4 3 }}
 
Optimal tuning (CTE): ~250/243 = 1\24, ~10/7 = 617.2597
 
{{Optimal ET sequence|legend=1| 72, …, 480, 552, 624 }}
 
Badness: 0.0398
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 34398/34375, 39366/39325
 
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 ET sequence|legend=1| 72, …, 480f, 552, 624, 1176de, 1800cdee }}
 
Badness: 0.0293
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 11016/11011
 
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 ET sequence|legend=1| 72, …, 480fgg, 552g, 624 }}
 
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)
 
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[[Category:Pages with mostly numerical content]]
[[Category:Landscape microtemperaments| ]] <!-- main article -->
[[Category:Landscape microtemperaments| ]] <!-- main article -->
[[Category:Microtemperament]]
[[Category:Landscape| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Microtemperaments]]
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