Landscape microtemperaments: Difference between revisions

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~~'''Landscape microtemperaments''' are~~ [[~~Rank~~-2 temperament|rank-2]] ~~[[temperament]]s~~ [[tempering out]] the [[landscape comma]] ({{monzo| -4 6 -6 3 }} = 250047/250000).

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~~ discussed elsewhere are:

Temperaments discussed elsewhere are:

* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Augene|Augmented family]]

* ''[[Compton]]'' (+225/224) → [[Compton family #Compton|Compton family]]

* ''[[Tritikleismic]]'' (+1029/1024) → [[Kleismic family #Tritikleismic|Kleismic family]]

* ''[[Trisensory]]'' (+1728/1715) → [[Sensipent family #Trisensory|Sensipent family]]

* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[~~Ragismic microtemperaments~~ #Ennealimmal|~~Ragismic microtemperaments~~]]

* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]

* ''[[Misty]]'' (+3136/3125 or 5120/5103) → [[Misty family]]

* ''[[Nessafof]]'' (+6144/6125) → [[Porwell temperaments #Nessafof|Porwell temperaments]]

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* ''[[Triquart]]'' (+117649/116640) → [[Quartonic family #Triquart|Quartonic family]]

* ''[[Stearnscape]]'' (+118098/117649) → [[Stearnsmic clan #Stearnscape|Stearnsmic clan]]

* ''[[Tritricot]]'' (+{{monzo| 35 -23 -3 3 }}) → [[~~Tricot~~ family #Tritricot|~~Tricot~~ family]]

* ''[[Tritricot]]'' (+{{monzo| 35 -23 -3 3 }}) → [[Alphatricot family #Tritricot|Alphatricot family]]

* ''[[Aemilic]]'' (+{{monzo|-84 53}}) → [[159th-octave temperaments#Aemilic|159th-octave temperaments]]

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== Sextile ==

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

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~4096/3645, ~3

~~{{Multival|legend=1| 6 -36 -84 -71 -150 -94 }}~~

[[Optimal tuning]]s:

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~~{{Multival|legend=1| 15 51 72 46 72 24 }}~~

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

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~~{{Multival|legend=1| 39 30 -18 -43 -138 -126 }}~~

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

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~~{{Multival|legend=1| 51 105 108 48 28 -44 }}~~

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

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~~{{Multival|legend=1| 24 -9 -66 -70 -172 -128 }}~~

[[Optimal tuning]]s:

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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 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)

[[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 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

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: 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 }}

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=== 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

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: 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 }}

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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

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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

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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

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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

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Badness: 0.00921

== Poe ==

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 250047/250000, {{monzo| 15 -16 -4 7 }}

[[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:Pages with mostly numerical content]]

[[Category:Landscape microtemperaments| ]]

[[Category:Landscape| ]]

[[Category:Rank 2]]

[[Category:Microtemperaments]]

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-'''Landscape microtemperaments''' are [[Rank-2 temperament|rank-2]] [[temperament]]s [[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 discussed elsewhere are:
+Temperaments discussed elsewhere are:
 * [[Augene]] (+64/63 or 126/125) → [[Augmented family #Augene|Augmented family]]
 * ''[[Compton]]'' (+225/224) → [[Compton family #Compton|Compton family]]
 * ''[[Tritikleismic]]'' (+1029/1024) → [[Kleismic family #Tritikleismic|Kleismic family]]
 * ''[[Trisensory]]'' (+1728/1715) → [[Sensipent family #Trisensory|Sensipent family]]
-* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Ragismic microtemperaments #Ennealimmal|Ragismic microtemperaments]]
+* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
 * ''[[Misty]]'' (+3136/3125 or 5120/5103) → [[Misty family]]
 * ''[[Nessafof]]'' (+6144/6125) → [[Porwell temperaments #Nessafof|Porwell temperaments]]
@@ Line 15: / Line 16: @@
 * ''[[Triquart]]'' (+117649/116640) → [[Quartonic family #Triquart|Quartonic family]]
 * ''[[Stearnscape]]'' (+118098/117649) → [[Stearnsmic clan #Stearnscape|Stearnsmic clan]]
-* ''[[Tritricot]]'' (+{{monzo| 35 -23 -3 3 }}) → [[Tricot family #Tritricot|Tricot family]]
+* ''[[Tritricot]]'' (+{{monzo| 35 -23 -3 3 }}) → [[Alphatricot family #Tritricot|Alphatricot family]]
 * ''[[Aemilic]]'' (+{{monzo|-84 53}}) → [[159th-octave temperaments#Aemilic|159th-octave temperaments]]
@@ Line 21: / Line 22: @@
 == Sextile ==
-Sextile tempers out the [[garischisma]] with a 1/6-octave period and is the 12 & 270 temperament.
+Sextile tempers out the [[garischisma]] with a 1/6-octave period and is the 12 &amp; 270 temperament.
 [[Subgroup]]: 2.3.5.7
@@ Line 30: / Line 31: @@
 : mapping generators: ~4096/3645, ~3
-{{Multival|legend=1| 6 -36 -84 -71 -150 -94 }}
 [[Optimal tuning]]s:
@@ Line 152: / Line 151: @@
 {{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
@@ Line 234: / Line 231: @@
 {{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
@@ Line 251: / Line 246: @@
 {{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
@@ Line 266: / Line 259: @@
 {{Mapping|legend=1| 3 2 8 16 | 0 8 -3 -22 }}
-{{Multival|legend=1| 24 -9 -66 -70 -172 -128 }}
 [[Optimal tuning]]s:
@@ Line 416: / Line 407: @@
 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 ==
 {{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
@@ Line 430: / Line 464: @@
 : 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 }}
@@ Line 437: / Line 471: @@
 === 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 &amp; 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
@@ Line 447: / Line 481: @@
 : 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 }}
@@ Line 460: / Line 494: @@
 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 }}
@@ Line 473: / Line 507: @@
 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 }}
@@ Line 486: / Line 520: @@
 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 }}
@@ Line 499: / Line 533: @@
 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 }}
@@ Line 644: / Line 678: @@
 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:Pages with mostly numerical content]]
 [[Category:Landscape microtemperaments| ]] <!-- main article -->
 [[Category:Landscape| ]] <!-- key article -->
 [[Category:Rank 2]]
 [[Category:Microtemperaments]]