Landscape microtemperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
→Sextile: OGPV sequence
Improve intro; update keys; normalize mappings and generators
Line 1: Line 1:
'''Landscape microtemperaments''' are rank-2 temperaments tempering out the [[landscape comma]] ({{Monzo| -4 6 -6 3 }} = 250047/250000).
'''Landscape microtemperaments''' are rank-2 temperaments tempering out the [[landscape comma]] ({{Monzo| -4 6 -6 3 }} = 250047/250000).


Landscape rank-2 temperaments include sextile, septichrome, pnict, and domain. Considered elsewhere are:  
Landscape rank-2 temperaments considered elsewhere are:  
* [[Augene]], {64/63, 126/125} → [[Augmented family #Augene|Augmented family]]
* [[Augene]], {64/63, 126/125} → [[Augmented family #Augene|Augmented family]]
* ''[[Waage]]'', {225/224, 250047/250000} → [[Compton family #Compton|Compton family]]
* ''[[Waage]]'', {225/224, 250047/250000} → [[Compton family #Compton|Compton family]]
Line 15: Line 15:
* ''[[Stearnscape]]'', {118098/117649, 250047/250000} → [[Stearnsmic clan #Stearnscape|Stearnsmic clan]]
* ''[[Stearnscape]]'', {118098/117649, 250047/250000} → [[Stearnsmic clan #Stearnscape|Stearnsmic clan]]
* ''[[Tritricot]]'', {250047/250000, 11785390260224/11767897353375} → [[Tricot family #Tritricot|Tricot family]]
* ''[[Tritricot]]'', {250047/250000, 11785390260224/11767897353375} → [[Tricot family #Tritricot|Tricot family]]
Discussed below are sextile, septichrome, pnict, domain, avicenna, terture, akjayland and terture.


== Sextile ==
== Sextile ==
Line 125: Line 126:


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


[[Comma list]]: 250047/250000, 2460375/2458624
[[Comma list]]: 250047/250000, 2460375/2458624
Line 133: Line 134:
{{Multival|legend=1| 15 51 72 46 72 24 }}
{{Multival|legend=1| 15 51 72 46 72 24 }}


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


{{Val list|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}
{{Val list|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}
Line 148: Line 149:
{{Multival|legend=1| 39 30 -18 -43 -138 -126 }}
{{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 }}
{{Val list|legend=1| 15, 141, 156, 171, 2409cd, 2580cd, …, 4461bccddd, 4632bccddd }}
Line 157: Line 158:
{{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
Line 165: Line 166:
{{Multival|legend=1| 51 105 108 48 28 -44 }}
{{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 }}
{{Val list|legend=1| 171, 1164, 1335, 1506, 1677, 1848, 2019, 2190, 11943, 13962, 15981, 18000, 20019, 22038 }}
Line 172: Line 173:


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


[[Comma list]]: 250047/250000, 29360128/29296875
[[Comma list]]: 250047/250000, 29360128/29296875
Line 180: Line 181:
{{Multival|legend=1| 24 -9 -66 -70 -172 -128 }}
{{Multival|legend=1| 24 -9 -66 -70 -172 -128 }}


[[POTE generator]]: ~1024/945 = 137.768
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~1024/945 = 137.768


{{Val list|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}
{{Val list|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}
Line 193: Line 194:
Mapping: [{{val|3 2 8 16 9}}, {{val|0 8 -3 -22 4}}]
Mapping: [{{val|3 2 8 16 9}}, {{val|0 8 -3 -22 4}}]


POTE generator: ~693/640 = 137.771
Optimal tuning (POTE): ~63/50 = 1\3, ~693/640 = 137.771


Optimal GPV sequence: {{Val list| 87, 183, 270, 1263, 1533, 1803c, 2073c }}
Optimal GPV sequence: {{Val list| 87, 183, 270, 1263, 1533, 1803c, 2073c }}
Line 206: Line 207:
Mapping: [{{val|3 2 8 16 9 8}}, {{val|0 8 -3 -22 4 9}}]
Mapping: [{{val|3 2 8 16 9 8}}, {{val|0 8 -3 -22 4 9}}]


POTE generator: ~13/12 = 137.777
Optimal tuning (POTE): ~63/50 = 1\3, ~13/12 = 137.777


Optimal GPV sequence: {{Val list| 87, 183, 270 }}
Optimal GPV sequence: {{Val list| 87, 183, 270 }}
Line 213: Line 214:


== 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
Line 221: Line 222:
[[Mapping]]: [{{val| 3 4 3 2 }}, {{val| 0 4 21 34 }}]
[[Mapping]]: [{{val| 3 4 3 2 }}, {{val| 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 }}
{{Val list|legend=1| 111, 159, 270 }}
Line 234: Line 235:
Mapping: [{{val| 3 4 3 2 10 }}, {{val| 0 4 21 34 2 }}]
Mapping: [{{val| 3 4 3 2 10 }}, {{val| 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 GPV sequence: {{Val list| 111, 159, 270, 1239, 1509, 1779, 2049, 2319 }}
Line 247: Line 248:
Mapping: [{{val| 3 4 3 2 10 6 }}, {{val| 0 4 21 34 2 27 }}]
Mapping: [{{val| 3 4 3 2 10 6 }}, {{val| 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 GPV sequence: {{Val list| 111, 159, 270 }}
Line 260: Line 261:
Mapping: [{{val| 3 4 3 2 10 6 10 }}, {{val| 0 4 21 34 2 27 12 }}]
Mapping: [{{val| 3 4 3 2 10 6 10 }}, {{val| 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 GPV sequence: {{Val list| 111, 159, 270 }}
Line 273: Line 274:
Mapping: [{{val| 3 4 3 2 10 6 10 5 }}, {{val| 0 4 21 34 2 27 12 41 }}]
Mapping: [{{val| 3 4 3 2 10 6 10 5 }}, {{val| 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 GPV sequence: {{Val list| 111, 159, 270 }}
Line 286: Line 287:
Mapping: [{{val| 3 4 3 2 10 6 10 5 13 }}, {{val| 0 4 21 34 2 27 12 41 3 }}]
Mapping: [{{val| 3 4 3 2 10 6 10 5 13 }}, {{val| 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 GPV sequence: {{Val list| 111, 159, 270 }}
Line 305: Line 306:
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 GPV sequence]]: {{val list| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
Line 312: Line 313:


=== 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 322: Line 323:
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 GPV sequence: {{val list| 168, 357, 525, 882, 1407, 2289e }}
Line 335: Line 336:
Mapping: [{{val| 21 4 39 98 58 107 }}, {{val| 0 6 2 -8 3 -6 }}]
Mapping: [{{val| 21 4 39 98 58 107 }}, {{val| 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 GPV sequence: {{val list| 168, 357, 525, 882 }}
Line 348: Line 349:
Mapping: [{{val| 21 4 39 98 58 107 120 }}, {{val| 0 6 2 -8 3 -6 -7 }}]
Mapping: [{{val| 21 4 39 98 58 107 120 }}, {{val| 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 GPV sequence: {{val list| 168, 357, 525, 882 }}
Line 361: Line 362:
Mapping: [{{val| 21 4 39 98 58 107 120 16 }}, {{val| 0 6 2 -8 3 -6 -7 15 }}]
Mapping: [{{val| 21 4 39 98 58 107 120 16 }}, {{val| 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 GPV sequence: {{val list| 168h, 357, 525, 882, 1407 }}
Line 372: Line 373:
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: [{{val| 21 4 39 98 58 107 120 16 95 }}, {{val| 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 GPV sequence: {{val list| 168h, 357, 525, 882, 1407 }}
Line 391: Line 392:
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 }}
{{Val list|legend=1| 72, …, 480, 552, 624, 1320, 1944d, 3264d }}
Line 404: Line 405:
Mapping: [{{val| 24 1 -6 18 46 }}, {{val| 0 3 5 4 3 }}]
Mapping: [{{val| 24 1 -6 18 46 }}, {{val| 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 GPV sequence: {{Val list| 72, …, 480, 552, 624 }}
Line 417: Line 418:
Mapping: [{{val| 24 1 -6 18 46 -47 -13 }}, {{val| 0 3 5 4 3 11 }}]
Mapping: [{{val| 24 1 -6 18 46 -47 -13 }}, {{val| 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 GPV sequence: {{Val list| 72, …, 480f, 552, 624, 1176de, 1800cdee }}
Line 430: Line 431:
Mapping: [{{val| 24 1 -6 18 46 -47 -13 }}, {{val| 0 3 5 4 3 11 9 }}]
Mapping: [{{val| 24 1 -6 18 46 -47 -13 }}, {{val| 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 GPV sequence: {{Val list| 72, …, 480fgg, 552g, 624 }}

Revision as of 09:51, 17 December 2022

Landscape microtemperaments are rank-2 temperaments tempering out the landscape comma ([-4 6 -6 3⟩ = 250047/250000).

Landscape rank-2 temperaments considered elsewhere are:

Discussed below are sextile, septichrome, pnict, domain, avicenna, terture, akjayland and terture.

Sextile

Subgroup: 2.3.5.7

Comma list: 250047/250000, 33554432/33480783

Mapping: [⟨6 0 71 150], ⟨0 1 -6 -14]]

Mapping generators: ~4096/3645, ~3

Wedgie: ⟨⟨ 6 -36 -84 -71 -150 -94 ]]

Optimal tuning (CTE): ~4096/3645 = 1\6, ~3/2 = 702.2347

Template:Val list

Badness: 0.070097

11-limit

Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 9801/9800, 151263/151250

Mapping: [⟨6 0 71 150 230], ⟨0 1 -6 -14 -22]]

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

Optimal GPV sequence: Template:Val list

Badness: 0.029677

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 5632/5625, 10648/10647

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 GPV sequence: Template:Val list

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: [⟨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 GPV sequence: Template:Val list

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: [⟨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 GPV sequence: Template:Val list

Badness: 0.0156

Sextilia

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 4096/4095, 4459/4455, 20449/20412

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 GPV sequence: Template:Val list

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: [⟨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 GPV sequence: Template:Val list

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: [⟨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 GPV sequence: Template:Val list

Badness: 0.0252

Septichrome

Subgroup: 2.3.5.7

Comma list: 250047/250000, 2460375/2458624

Mapping: [⟨3 3 1 0], ⟨0 5 17 24]]

Wedgie: ⟨⟨ 15 51 72 46 72 24 ]]

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

Template:Val list

Badness: 0.016814

Pnict

Subgroup: 2.3.5.7

Comma list: 250047/250000, 2100875/2097152

Mapping: [⟨3 10 11 6], ⟨0 -13 -10 6]]

Wedgie: ⟨⟨ 39 30 -18 -43 -138 -126 ]]

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

Template:Val list

Badness: 0.045660

Domain

See also: Minortonic family

Subgroup: 2.3.5.7

Comma list: 250047/250000, 645700815/645657712

Mapping: [⟨3 -3 -9 -8], ⟨0 17 35 36]]

Wedgie: ⟨⟨ 51 105 108 48 28 -44 ]]

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

Template:Val list

Badness: 0.013979

Avicenna

Subgroup: 2.3.5.7

Comma list: 250047/250000, 29360128/29296875

Mapping: [⟨3 2 8 16], ⟨0 8 -3 -22]]

Wedgie: ⟨⟨ 24 -9 -66 -70 -172 -128 ]]

Optimal tuning (POTE): ~63/50 = 1\3, ~1024/945 = 137.768

Template:Val list

Badness: 0.062187

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5632/5625, 102487/102400

Mapping: [⟨3 2 8 16 9], ⟨0 8 -3 -22 4]]

Optimal tuning (POTE): ~63/50 = 1\3, ~693/640 = 137.771

Optimal GPV sequence: Template:Val list

Badness: 0.023085

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3025/3024, 4096/4095

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 GPV sequence: Template:Val list

Badness: 0.015557

Terture

See also: Vulture family

Subgroup: 2.3.5.7

Comma list: 250047/250000, 359661568/358722675

Mapping: [⟨3 4 3 2], ⟨0 4 21 34]]

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

Template:Val list

Badness: 0.087156

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 19712/19683, 102487/102400

Mapping: [⟨3 4 3 2 10], ⟨0 4 21 34 2]]

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

Optimal GPV sequence: Template:Val list

Badness: 0.029326

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3025/3024, 10985/10976

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 GPV sequence: Template:Val list

Badness: 0.018647

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 936/935, 1001/1000, 4928/4913

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 GPV sequence: Template:Val list

Badness: 0.018705

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1617/1615

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 GPV sequence: Template:Val list

Badness: 0.013902

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 460/459, 529/528, 676/675, 715/714, 936/935, 1001/1000, 1216/1215

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 GPV sequence: Template:Val list

Badness: 0.014915

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, [43 -1 -13 -4⟩

Mapping: [⟨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 GPV sequence: Template:Val list

Badness: 0.0309

Vasca

Vasca can be described as the 357 & 525 temperament, extended as high as the 23-limit. It tempers out the [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, [39 -4 -11 -5 2⟩

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 GPV sequence: Template:Val list

Badness: 0.0949

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4096/4095, 14641/14625, 85750/85683

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 GPV sequence: Template:Val list

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: [⟨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 GPV sequence: Template:Val list

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: [⟨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 GPV sequence: Template:Val list

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: [⟨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 GPV sequence: Template:Val list

Badness: 0.0199

Chromium

Chromium is described as the 72 & 624 temperament, and named after the 24th element for being period 24.

Subgroup: 2.3.5.7

Comma list: 250047/250000, 49589822592/49433168575

Mapping: [⟨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

Template:Val list

Badness: 0.139

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 46656/46585, 250047/250000

Mapping: [⟨24 1 -6 18 46], ⟨0 3 5 4 3]]

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

Optimal GPV sequence: Template:Val list

Badness: 0.0398

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 34398/34375, 39366/39325

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 GPV sequence: Template:Val list

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: [⟨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 GPV sequence: Template:Val list

Badness: 0.0209

Retrieved from "https://en.xen.wiki/index.php?title=Landscape_microtemperaments&oldid=100148"