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Landscape microtemperaments: Difference between revisions

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Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]
* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]
* [[Misty]] (+3136/3125 or 5120/5103) → [[Misty family]]
* ''[[Term (temperament)|Term]]'' (+32805/32768) → [[Schismatic family #Term|Schismatic family]]
* ''[[Tritricot]]'' (+{{monzo| 35 -23 -3 3 }}) → [[Alphatricot family #Tritricot|Alphatricot family]]
* [[Compton]] (+225/224) → [[Compton family #Compton|Compton family]]
* [[Compton]] (+225/224) → [[Compton family #Compton|Compton family]]
* ''[[Terture]]'' (+359661568/358722675) → [[Vulture family #Domain|Vulture family]]
* ''[[Chromat]]'' (10976/10935) → [[Hemimage temperaments #Chromat|Hemimage temperaments]]
* ''[[Tritikleismic]]'' (+1029/1024) → [[Kleismic family #Tritikleismic|Kleismic family]]
* ''[[Tritikleismic]]'' (+1029/1024) → [[Kleismic family #Tritikleismic|Kleismic family]]
* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Caleb]]'' (+33075/32768) → [[Mirwomo temperaments #Caleb|Mirwomo temperaments]]
* ''[[Trisensory]]'' (+1728/1715) → [[Sensipent family #Trisensory|Sensipent family]]
* ''[[Trisensory]]'' (+1728/1715) → [[Sensipent family #Trisensory|Sensipent family]]
* [[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]]
* ''[[Nessafof]]'' (+6144/6125) → [[Porwell temperaments #Nessafof|Porwell temperaments]]
* ''[[Chromat]]'' (10976/10935) → [[Hemimage temperaments #Chromat|Hemimage temperaments]]
* ''[[Term (temperament)|Term]]'' (+32805/32768) → [[Schismatic family #Term|Schismatic family]]
* ''[[Caleb]]'' (+33075/32768) → [[Mirwomo temperaments #Caleb|Mirwomo temperaments]]
* [[Mutt]] (+65625/65536) → [[Horwell temperaments #Mutt|Horwell temperaments]]
* [[Mutt]] (+65625/65536) → [[Horwell temperaments #Mutt|Horwell temperaments]]
* ''[[Triquart]]'' (+117649/116640) → [[Quartonic family #Triquart|Quartonic family]]
* ''[[Triquart]]'' (+117649/116640) → [[Quartonic family #Triquart|Quartonic family]]
* ''[[Stearnscape]]'' (+118098/117649) → [[Stearnsmic clan #Stearnscape|Stearnsmic clan]]
* ''[[Stearnscape]]'' (+118098/117649) → [[Stearnsmic clan #Stearnscape|Stearnsmic clan]]
* ''[[Terture]]'' (+359661568/358722675) → [[Vulture family #Domain|Vulture family]]
* ''[[Domain (temperament)|Domain]]'' (+645700815/645657712) → [[Minortonic family #Domain|Minortonic family]]
* ''[[Domain (temperament)|Domain]]'' (+645700815/645657712) → [[Minortonic family #Domain|Minortonic family]]
* ''[[Tritricot]]'' (+{{monzo| 35 -23 -3 3 }}) → [[Alphatricot family #Tritricot|Alphatricot family]]
* ''[[Aemilic]]'' (+{{monzo|-84 53}}) → [[159th-octave temperaments #Aemilic|159th-octave temperaments]]
* ''[[Aemilic]]'' (+{{monzo|-84 53}}) → [[159th-octave temperaments #Aemilic|159th-octave temperaments]]


Considered below are sextile, septichrome, pnict, atomic, avicenna, slendscape, akjayland, magnesium, chromium, zinc, and poe.
Considered below are septichrome, akjayland, pnict, atomic, slendscape, avicenna, sextile, zinc, magnesium, poe, and chromium in the order of increasing [[badness]].
 
== Sextile ==
: ''For the 5-limit version, see [[Schismic–commatic equivalence continuum #Sextile (5-limit)]].''
 
Sextile tempers out the [[garischisma]] with a 1/6-octave period and is the {{nowrap| 12 & 270 }} temperament.  


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


[[Comma list]]: 250047/250000, 33554432/33480783
[[Comma list]]: 250047/250000, 2460375/2458624


{{Mapping|legend=1| 6 0 71 150 | 0 1 -6 -14 }}
{{Mapping|legend=1| 3 3 1 0 | 0 5 17 24 }}
: mapping generators: ~4096/3645, ~3
: mapping generators: ~63/50, ~243/224


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~4096/3645 = 199.9828{{c}}, ~3/2 = 702.1521{{c}}
* [[WE]]: ~63/50 = 400.0100{{c}}, ~243/224 = 140.3702{{c}}
: [[error map]]: {{val| -0.103 +0.094 +0.173 -0.088 }}
: [[error map]]: {{val| +0.030 -0.074 -0.010 +0.059 }}
* [[CWE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2187{{c}}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~243/224 = 140.3685{{c}}
: error map: {{val| 0.000 +0.264 +0.374 +0.112 }}
: error map: {{val| 0.000 -0.113 -0.050 +0.017 }}


{{Optimal ET sequence|legend=1| 12, …, 258, 270, 1362c, 1632c, …, 2442bc, 2712bc }}
{{Optimal ET sequence|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}


[[Badness]] (Sintel): 1.77
[[Badness]] (Sintel): 0.426


=== 11-limit ===
=== Semiseptichrome ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 5632/5625, 9801/9800, 151263/151250
Comma list: 9801/9800, 151263/151250, 234375/234256


Mapping: {{mapping| 6 0 71 150 230 | 0 1 -6 -14 -22 }}
Mapping: {{mapping| 6 1 -15 -24 -32 | 0 5 17 24 31 }}
: mapping generators: ~55/49, ~375/308


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 199.9817{{c}}, ~3/2 = 702.1383{{c}}
* WE: ~55/49 = 200.0058{{c}}, ~375/308 = 340.3742{{c}} (~1760/1701 = 59.6375{{c}})
* POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2080{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3661{{c}} (~1760/1701 = 59.6339{{c}})


{{Optimal ET sequence|legend=0| 12, …, 258e, 270, 822, 1092, 1362c }}
{{Optimal ET sequence|legend=0| 60e, 222cdee, 282, 342, 966, 1308, 1650, 4608b, 6258bc }}


Badness (Sintel): 0.981
Badness (Sintel): 0.642


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1716/1715, 2080/2079, 5632/5625, 10648/10647
Comma list: 1716/1715, 2080/2079, 34398/34375, 85293/85184


Mapping: {{mapping| 6 0 71 150 230 279 | 0 1 -6 -14 -22 -27 }}
Mapping: {{mapping| 6 1 -15 -24 -32 -68 | 0 5 17 24 31 53 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 199.9804{{c}}, ~3/2 = 702.1260{{c}}
* WE: ~55/49 = 199.9936{{c}}, ~375/308 = 340.3707{{c}} (~121/117 = 59.6165{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2001{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3802{{c}} (~121/117 = 59.6198{{c}})


{{Optimal ET sequence|legend=0| 12f, …, 258ef, 270, 552, 822, 1092, 1914cde }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness (Sintel): 0.788
Badness (Sintel): 1.64


==== 17-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 936/935, 1701/1700, 1716/1715, 5632/5625, 7744/7735
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 61965/61952


Mapping: {{mapping| 6 0 71 150 230 279 -4 | 0 1 -6 -14 -22 -27 3 }}
Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 | 0 5 17 24 31 53 15 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 199.9669{{c}}, ~3/2 = 702.0643{{c}}
* WE: ~55/49 = 199.9865{{c}}, ~375/308 = 340.3619{{c}} (~88/85 = 59.6111{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.1869{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3821{{c}} (~88/85 = 59.6179{{c}})


{{Optimal ET sequence|legend=0| 12f, 270, 552g }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness (Sintel): 1.06
Badness (Sintel): 1.39


==== 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, 1701/1700, 1716/1715, 2376/2375, 4200/4199
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 23409/23408


Mapping: {{mapping| 6 0 71 150 230 279 -4 35 | 0 1 -6 -14 -22 -27 3 -1 }}
Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 | 0 5 17 24 31 53 15 -5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 199.9711{{c}}, ~3/2 = 702.0829{{c}}
* WE: ~55/49 = 199.9837{{c}}, ~162/133 = 340.3589{{c}} (~88/85 = 59.6084{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.1890{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~162/133 = 340.3844{{c}} (~88/85 = 59.6156{{c}})


{{Optimal ET sequence|legend=0| 12f, 270, 552g, 822gg }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness (Sintel): 0.948
Badness (Sintel): 1.35


=== Sextilia ===
==== 23-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 1001/1000, 4096/4095, 4459/4455, 20449/20412
Comma list: 936/935, 1701/1700, 1716/1715, 1863/1862, 2024/2023, 2025/2023, 2376/2375


Mapping: {{mapping| 6 0 71 150 230 -149 | 0 1 -6 -14 -22 18 }}
Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 -12 | 0 5 17 24 31 53 15 -5 23 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 199.9975{{c}}, ~3/2 = 702.2196{{c}}
* WE: ~55/49 = 199.9829{{c}}, ~162/133 = 340.3576{{c}} (~88/85 = 59.6081{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2285{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~162/133 = 340.3844{{c}} (~88/85 = 59.6156{{c}})


{{Optimal ET sequence|legend=0| 12, 258e, 270 }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness (Sintel): 1.62
Badness (Sintel): 1.14


==== 17-limit ====
== Akjayland ==
Subgroup: 2.3.5.7.11.13.17
{{See also| 21st-octave temperaments }}


Comma list: 715/714, 1001/1000, 1701/1700, 4096/4095, 4459/4455
Named by [[Eliora]] in 2022, akjayland tempers out the [[akjaysma]] in addition to landscape comma, and thereby features a period of 1\21.  
 
Mapping: {{mapping| 6 0 71 150 230 -149 -4 | 0 1 -6 -14 -22 18 3 }}
 
Optimal tunings:
* WE: ~55/49 = 199.9862{{c}}, ~3/2 = 702.1714{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2207{{c}}


{{Optimal ET sequence|legend=0| 12, 258e, 270 }}
Badness (Sintel): 1.95
==== 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:
* WE: ~55/49 = 199.9866{{c}}, ~3/2 = 702.1731{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2208{{c}}
{{Optimal ET sequence|legend=0| 12, 258e, 270 }}
Badness (Sintel): 1.53
== Septichrome ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 2460375/2458624
[[Comma list]]: 250047/250000, {{monzo| 43 -1 -13 -4 }}


{{Mapping|legend=1| 3 3 1 0 | 0 5 17 24 }}
{{Mapping|legend=1| 21 1 38 102 | 0 3 1 -4 }}
: mapping generators: ~63/50, ~243/224
: mapping generators: ~1323/1280, ~131072/91875


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~63/50 = 400.0100{{c}}, ~243/224 = 140.3702{{c}}
* [[WE]]: ~1323/1280 = 57.1426{{c}}, ~131072/91875 = 614.9336{{c}}
: [[error map]]: {{val| +0.030 -0.074 -0.010 +0.059 }}
: [[error map]]: {{val| -0.005 -0.012 +0.039 -0.013 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~243/224 = 140.3685{{c}}
* [[CWE]]: ~1323/1280 = 57.1429{{c}}, ~131072/91875 = 614.9360{{c}}
: error map: {{val| 0.000 -0.113 -0.050 +0.017 }}
: error map: {{val| 0.000 -0.004 +0.051 +0.002 }}
 
{{Optimal ET sequence|legend=1| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}


{{Optimal ET sequence|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}
[[Badness]] (Sintel): 0.838


[[Badness]] (Sintel): 0.426
=== Vasca ===
Vasca can be described as the {{nowrap| 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.  


=== Semiseptichrome ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 9801/9800, 151263/151250, 234375/234256
Comma list: 3025/3024, 102487/102400, {{monzo| 39 -4 -11 -5 2 }}


Mapping: {{mapping| 6 1 -15 -24 -32 | 0 5 17 24 31 }}
Mapping: {{mapping| 21 4 39 98 58 | 0 6 2 -8 3 3 }}
: mapping generators: ~55/49, ~375/308
: mapping generators: ~1323/1280, ~6615/5632


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 200.0058{{c}}, ~375/308 = 340.3742{{c}} (~1760/1701 = 59.6375{{c}})
* WE: ~1323/1280 = 57.1436{{c}}, ~6615/5632 = 278.9017{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3661{{c}} (~1760/1701 = 59.6339{{c}})
* CWE: ~1323/1280 = 57.1429{{c}}, ~6615/5632 = 278.8985{{c}}


{{Optimal ET sequence|legend=0| 60e, 222cdee, 282, 342, 966, 1308, 1650, 4608b, 6258bc }}
{{Optimal ET sequence|legend=0| 168, 357, 525, 882, 1407, 2289e }}


Badness (Sintel): 0.642
Badness (Sintel): 3.14


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1716/1715, 2080/2079, 34398/34375, 85293/85184
Comma list: 3025/3024, 4096/4095, 14641/14625, 85750/85683


Mapping: {{mapping| 6 1 -15 -24 -32 -68 | 0 5 17 24 31 53 }}
Mapping: {{mapping| 21 4 39 98 58 107 | 0 6 2 -8 3 -6 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 199.9936{{c}}, ~375/308 = 340.3707{{c}} (~121/117 = 59.6165{{c}})
* WE: ~336/325 = 57.1426{{c}}, ~168/143 = 278.9047{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3802{{c}} (~121/117 = 59.6198{{c}})
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9060{{c}}


{{Optimal ET sequence|legend=0| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}


Badness (Sintel): 1.64
Badness (Sintel): 2.28


==== 17-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17


Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 61965/61952
Comma list: 2601/2600, 3025/3024, 4096/4095, 8624/8619, 14641/14625


Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 | 0 5 17 24 31 53 15 }}
Mapping: {{mapping| 21 4 39 98 58 107 120 | 0 6 2 -8 3 -6 -7 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 199.9865{{c}}, ~375/308 = 340.3619{{c}} (~88/85 = 59.6111{{c}})
* WE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9037{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3821{{c}} (~88/85 = 59.6179{{c}})
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9036{{c}}


{{Optimal ET sequence|legend=0| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}


Badness (Sintel): 1.39
Badness (Sintel): 1.62


==== 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, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 23409/23408
Comma list: 2376/2375, 2601/2600, 2926/2925, 3025/3024, 3213/3211, 4096/4095


Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 | 0 5 17 24 31 53 15 -5 }}
Mapping: {{mapping| 21 4 39 98 58 107 120 16 | 0 6 2 -8 3 -6 -7 15 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 199.9837{{c}}, ~162/133 = 340.3589{{c}} (~88/85 = 59.6084{{c}})
* WE: ~336/325 = 57.1425{{c}}, ~168/143 = 278.8960{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~162/133 = 340.3844{{c}} (~88/85 = 59.6156{{c}})
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.8976{{c}}


{{Optimal ET sequence|legend=0| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}


Badness (Sintel): 1.35
Badness (Sintel): 1.64


==== 23-limit ====
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
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
Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211


Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 -12 | 0 5 17 24 31 53 15 -5 23 }}
Mapping: {{mapping| 21 4 39 98 58 107 120 16 95 | 0 6 2 -8 3 -6 -7 15 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 199.9829{{c}}, ~162/133 = 340.3576{{c}} (~88/85 = 59.6081{{c}})
* WE: ~336/325 = 57.1422{{c}}, ~168/143 = 278.8949{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~162/133 = 340.3844{{c}} (~88/85 = 59.6156{{c}})
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.8980{{c}}


{{Optimal ET sequence|legend=0| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}


Badness (Sintel): 1.14
Badness (Sintel): 1.43


== Pnict ==
== Pnict ==
Line 409: Line 382:


Badness (Sintel): 1.98
Badness (Sintel): 1.98
== Slendscape ==
Named by [[Xenllium]] in 2025, 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]]s:
* [[WE]]: ~8575/8192 = 79.9771{{c}}, ~1152/875 = 475.4832{{c}}
: [[error map]]: {{val| -0.043 -0.022 +0.087 +0.053 }}
* [[CWE]]: ~8575/8192 = 80.0000{{c}}, ~1152/875 = 475.4962{{c}}
: error map: {{val| 0.000 +0.030 +0.175 +0.182 }}
{{Optimal ET sequence|legend=1| 15, 240, 255, 270, 795, 1065, 1335, 2400 }}
[[Badness]] (Sintel): 1.47
=== 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 tunings:
* WE: ~22/21 = 79.9991{{c}}, ~968/735 = 475.4915{{c}}
* CWE: ~22/21 = 80.0000{{c}}, ~968/735 = 475.4955{{c}}
{{Optimal ET sequence|legend=0| 15, 240, 255, 270, 795, 1065, 2400e }}
Badness (Sintel): 0.868
=== 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 tunings:
* WE: ~22/21 = 79.9993{{c}}, ~154/117 = 475.4902{{c}}
* CWE: ~22/21 = 80.0000{{c}}, ~154/117 = 475.4943{{c}}
{{Optimal ET sequence|legend=0| 255, 270, 795, 1065 }}
Badness (Sintel): 0.877


== Avicenna ==
== Avicenna ==
Line 488: Line 511:
Badness (Sintel): 0.928
Badness (Sintel): 0.928


== Slendscape ==
== Sextile ==
Named by [[Xenllium]] in 2025, slendscape tempers out the [[slendroschisma]] (68719476736/68641485507) in addition to landscape comma, and thereby features a period of 1\15.
: ''For the 5-limit version, see [[Schismic–commatic equivalence continuum #Sextile (5-limit)]].''
 
Sextile tempers out the [[garischisma]] with a 1/6-octave period and is the {{nowrap| 12 & 270 }} temperament.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 12884901888/12867859375
[[Comma list]]: 250047/250000, 33554432/33480783


{{Mapping|legend=1| 15 0 17 54 | 0 4 3 -2 }}
{{Mapping|legend=1| 6 0 71 150 | 0 1 -6 -14 }}
: mapping generators: ~8575/8192, ~1152/875
: mapping generators: ~4096/3645, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~8575/8192 = 79.9771{{c}}, ~1152/875 = 475.4832{{c}}
* [[WE]]: ~4096/3645 = 199.9828{{c}}, ~3/2 = 702.1521{{c}}
: [[error map]]: {{val| -0.043 -0.022 +0.087 +0.053 }}
: [[error map]]: {{val| -0.103 +0.094 +0.173 -0.088 }}
* [[CWE]]: ~8575/8192 = 80.0000{{c}}, ~1152/875 = 475.4962{{c}}
* [[CWE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2187{{c}}
: error map: {{val| 0.000 +0.030 +0.175 +0.182 }}
: error map: {{val| 0.000 +0.264 +0.374 +0.112 }}


{{Optimal ET sequence|legend=1| 15, 240, 255, 270, 795, 1065, 1335, 2400 }}
{{Optimal ET sequence|legend=1| 12, …, 258, 270, 1362c, 1632c, …, 2442bc, 2712bc }}


[[Badness]] (Sintel): 1.47
[[Badness]] (Sintel): 1.77


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 102487/102400, 180224/180075
Comma list: 5632/5625, 9801/9800, 151263/151250


Mapping: {{mapping| 15 0 17 54 40 | 0 4 3 -2 2 }}
Mapping: {{mapping| 6 0 71 150 230 | 0 1 -6 -14 -22 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~22/21 = 79.9991{{c}}, ~968/735 = 475.4915{{c}}
* CTE: ~55/49 = 199.9817{{c}}, ~3/2 = 702.1383{{c}}
* CWE: ~22/21 = 80.0000{{c}}, ~968/735 = 475.4955{{c}}
* POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2080{{c}}


{{Optimal ET sequence|legend=0| 15, 240, 255, 270, 795, 1065, 2400e }}
{{Optimal ET sequence|legend=0| 12, …, 258e, 270, 822, 1092, 1362c }}


Badness (Sintel): 0.868
Badness (Sintel): 0.981


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1716/1715, 3025/3024, 4096/4095, 14641/14625
Comma list: 1716/1715, 2080/2079, 5632/5625, 10648/10647


Mapping: {{mapping| 15 0 17 54 40 109 | 0 4 3 -2 2 -9 }}
Mapping: {{mapping| 6 0 71 150 230 279 | 0 1 -6 -14 -22 -27 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~22/21 = 79.9993{{c}}, ~154/117 = 475.4902{{c}}
* WE: ~55/49 = 199.9804{{c}}, ~3/2 = 702.1260{{c}}
* CWE: ~22/21 = 80.0000{{c}}, ~154/117 = 475.4943{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2001{{c}}


{{Optimal ET sequence|legend=0| 255, 270, 795, 1065 }}
{{Optimal ET sequence|legend=0| 12f, …, 258ef, 270, 552, 822, 1092, 1914cde }}


Badness (Sintel): 0.877
Badness (Sintel): 0.788


== Akjayland ==
==== 17-limit ====
{{See also| 21st-octave temperaments }}
Subgroup: 2.3.5.7.11.13.17


Named by [[Eliora]] in 2022, akjayland tempers out the [[akjaysma]] in addition to landscape comma, and thereby features a period of 1\21.
Comma list: 936/935, 1701/1700, 1716/1715, 5632/5625, 7744/7735


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 6 0 71 150 230 279 -4 | 0 1 -6 -14 -22 -27 3 }}


[[Comma list]]: 250047/250000, {{monzo| 43 -1 -13 -4 }}
Optimal tunings:  
* WE: ~55/49 = 199.9669{{c}}, ~3/2 = 702.0643{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.1869{{c}}


{{Mapping|legend=1| 21 1 38 102 | 0 3 1 -4 }}
{{Optimal ET sequence|legend=0| 12f, 270, 552g }}
: mapping generators: ~1323/1280, ~131072/91875


[[Optimal tuning]]s:  
Badness (Sintel): 1.06
* [[WE]]: ~1323/1280 = 57.1426{{c}}, ~131072/91875 = 614.9336{{c}}
: [[error map]]: {{val| -0.005 -0.012 +0.039 -0.013 }}
* [[CWE]]: ~1323/1280 = 57.1429{{c}}, ~131072/91875 = 614.9360{{c}}
: error map: {{val| 0.000 -0.004 +0.051 +0.002 }}


{{Optimal ET sequence|legend=1| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


[[Badness]] (Sintel): 0.838
Comma list: 936/935, 1216/1215, 1701/1700, 1716/1715, 2376/2375, 4200/4199


=== Vasca ===
Mapping: {{mapping| 6 0 71 150 230 279 -4 35 | 0 1 -6 -14 -22 -27 3 -1 }}
Vasca can be described as the {{nowrap| 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 tunings:  
Optimal tunings:  
* WE: ~1323/1280 = 57.1436{{c}}, ~6615/5632 = 278.9017{{c}}
* CTE: ~55/49 = 199.9711{{c}}, ~3/2 = 702.0829{{c}}
* CWE: ~1323/1280 = 57.1429{{c}}, ~6615/5632 = 278.8985{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.1890{{c}}


{{Optimal ET sequence|legend=0| 168, 357, 525, 882, 1407, 2289e }}
{{Optimal ET sequence|legend=0| 12f, 270, 552g, 822gg }}


Badness (Sintel): 3.14
Badness (Sintel): 0.948


==== 13-limit ====
=== Sextilia ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 3025/3024, 4096/4095, 14641/14625, 85750/85683
Comma list: 1001/1000, 4096/4095, 4459/4455, 20449/20412


Mapping: {{mapping| 21 4 39 98 58 107 | 0 6 2 -8 3 -6 }}
Mapping: {{mapping| 6 0 71 150 230 -149 | 0 1 -6 -14 -22 18 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~336/325 = 57.1426{{c}}, ~168/143 = 278.9047{{c}}
* WE: ~55/49 = 199.9975{{c}}, ~3/2 = 702.2196{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9060{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2285{{c}}


{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=0| 12, 258e, 270 }}


Badness (Sintel): 2.28
Badness (Sintel): 1.62


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


Comma list: 2601/2600, 3025/3024, 4096/4095, 8624/8619, 14641/14625
Comma list: 715/714, 1001/1000, 1701/1700, 4096/4095, 4459/4455


Mapping: {{mapping| 21 4 39 98 58 107 120 | 0 6 2 -8 3 -6 -7 }}
Mapping: {{mapping| 6 0 71 150 230 -149 -4 | 0 1 -6 -14 -22 18 3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9037{{c}}
* WE: ~55/49 = 199.9862{{c}}, ~3/2 = 702.1714{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9036{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2207{{c}}


{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=0| 12, 258e, 270 }}


Badness (Sintel): 1.62
Badness (Sintel): 1.95


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


Comma list: 2376/2375, 2601/2600, 2926/2925, 3025/3024, 3213/3211, 4096/4095
Comma list: 715/714, 1001/1000, 1216/1215, 1701/1700, 1729/1728, 2912/2907


Mapping: {{mapping| 21 4 39 98 58 107 120 16 | 0 6 2 -8 3 -6 -7 15 }}
Mapping: {{mapping| 6 0 71 150 230 -149 -4 35 | 0 1 -6 -14 -22 18 3 -1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~336/325 = 57.1425{{c}}, ~168/143 = 278.8960{{c}}
* WE: ~55/49 = 199.9866{{c}}, ~3/2 = 702.1731{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.8976{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2208{{c}}


{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}
{{Optimal ET sequence|legend=0| 12, 258e, 270 }}


Badness (Sintel): 1.64
Badness (Sintel): 1.53
 
==== 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 tunings:
* WE: ~336/325 = 57.1422{{c}}, ~168/143 = 278.8949{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.8980{{c}}
 
{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}
 
Badness (Sintel): 1.43
 
== Magnesium ==
: ''For the 5-limit version, see [[12th-octave temperaments #Magnesium (5-limit)]].
 
Magnesium is named by [[Eliora]] in 2023 after the 12th element for having a 1/12-octave period; however, it is not an extension of the [[atomic]] – the associated comma is {{monzo| -157 -24 84 }} in the 5-limit, and 7 generator steps together with two [[12edo]] semitones reach the [[3/1|3rd]] [[harmonic]]. It may be described as the {{nowrap| 84 & 612 }} temperament, with a [[ploidacot]] signature of dodecaploid gamma-heptacot.
 
[[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]]s:
* [[WE]]: ~138915/131072 = 100.0021{{c}}, ~3145728/2734375 = 243.1333{{c}}
: [[error map]]: {{val| +0.025 -0.018 +0.000 -0.039 }}
* [[CWE]]: ~138915/131072 = 100.0000{{c}}, ~3145728/2734375 = 243.1285{{c}}
: error map: {{val| 0.000 -0.055 -0.057 -0.111 }}
 
{{Optimal ET sequence|legend=1| 84, 360d, 444, 528, 612, 1920, 2532, 10740cd, 13272bcdd, 15804bcdd, 18336bcddd }}
 
Badness (Sintel): 2.44
 
== 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, and may be described as the {{nowrap| 72 & 624 }} temperament. It was named by [[Eliora]] in 2022 after the 24th element for having a 24th-octave period.
 
[[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]]s:
* [[WE]]: ~250/243 = 49.9992{{c}}, ~10/7 = 617.2714{{c}}
: [[error map]]: {{val| -0.019 -0.142 +0.048 +0.246 }}
* [[CWE]]: ~250/243 = 50.0000{{c}}, ~10/7 = 617.2762{{c}}
: error map: {{val| 0.000 -0.126 +0.067 +0.279 }}
 
{{Optimal ET sequence|legend=1| 72, …, 480, 552, 624, 1320, 1944d, 3264d }}
 
[[Badness]] (Sintel): 3.52
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 46656/46585, 151263/151250
 
Mapping: {{mapping| 24 1 -6 18 46 | 0 3 5 4 3 }}
 
Optimal tunings:
* WE: ~250/243 = 49.9972{{c}}, ~10/7 = 617.2639{{c}}
* CWE: ~250/243 = 50.0000{{c}}, ~10/7 = 617.2823{{c}}
 
{{Optimal ET sequence|legend=0| 72, …, 480, 552, 624 }}
 
Badness (Sintel): 1.32
 
=== 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 tunings:
* WE: ~250/243 = 49.9958{{c}}, ~10/7 = 617.2824{{c}}
* CWE: ~250/243 = 50.0000{{c}}, ~10/7 = 617.3161{{c}}
 
{{Optimal ET sequence|legend=0| 72, …, 480f, 552, 624, 1176de, 1800cdee }}
 
Badness (Sintel): 1.21
 
=== 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 tunings:
* WE: ~35/34 = 49.9959{{c}}, ~10/7 = 617.2685{{c}}
* CWE: ~35/34 = 50.0000{{c}}, ~10/7 = 617.3015{{c}}
 
{{Optimal ET sequence|legend=0| 72, …, 480fgg, 552g, 624 }}
 
Badness (Sintel): 1.06


== Zinc ==
== Zinc ==
Line 826: Line 737:


Badness (Sintel): 1.02
Badness (Sintel): 1.02
== Magnesium ==
: ''For the 5-limit version, see [[12th-octave temperaments #Magnesium (5-limit)]].
Magnesium is named by [[Eliora]] in 2023 after the 12th element for having a 1/12-octave period; however, it is not an extension of the [[atomic]] – the associated comma is {{monzo| -157 -24 84 }} in the 5-limit, and 7 generator steps together with two [[12edo]] semitones reach the [[3/1|3rd]] [[harmonic]]. It may be described as the {{nowrap| 84 & 612 }} temperament, with a [[ploidacot]] signature of dodecaploid gamma-heptacot.
[[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]]s:
* [[WE]]: ~138915/131072 = 100.0021{{c}}, ~3145728/2734375 = 243.1333{{c}}
: [[error map]]: {{val| +0.025 -0.018 +0.000 -0.039 }}
* [[CWE]]: ~138915/131072 = 100.0000{{c}}, ~3145728/2734375 = 243.1285{{c}}
: error map: {{val| 0.000 -0.055 -0.057 -0.111 }}
{{Optimal ET sequence|legend=1| 84, 360d, 444, 528, 612, 1920, 2532, 10740cd, 13272bcdd, 15804bcdd, 18336bcddd }}
Badness (Sintel): 2.44


== Poe ==
== Poe ==
Line 876: Line 809:


Badness (Sintel): 1.19
Badness (Sintel): 1.19
== 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, and may be described as the {{nowrap| 72 & 624 }} temperament. It was named by [[Eliora]] in 2022 after the 24th element for having a 24th-octave period.
[[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]]s:
* [[WE]]: ~250/243 = 49.9992{{c}}, ~10/7 = 617.2714{{c}}
: [[error map]]: {{val| -0.019 -0.142 +0.048 +0.246 }}
* [[CWE]]: ~250/243 = 50.0000{{c}}, ~10/7 = 617.2762{{c}}
: error map: {{val| 0.000 -0.126 +0.067 +0.279 }}
{{Optimal ET sequence|legend=1| 72, …, 480, 552, 624, 1320, 1944d, 3264d }}
[[Badness]] (Sintel): 3.52
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 46656/46585, 151263/151250
Mapping: {{mapping| 24 1 -6 18 46 | 0 3 5 4 3 }}
Optimal tunings:
* WE: ~250/243 = 49.9972{{c}}, ~10/7 = 617.2639{{c}}
* CWE: ~250/243 = 50.0000{{c}}, ~10/7 = 617.2823{{c}}
{{Optimal ET sequence|legend=0| 72, …, 480, 552, 624 }}
Badness (Sintel): 1.32
=== 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 tunings:
* WE: ~250/243 = 49.9958{{c}}, ~10/7 = 617.2824{{c}}
* CWE: ~250/243 = 50.0000{{c}}, ~10/7 = 617.3161{{c}}
{{Optimal ET sequence|legend=0| 72, …, 480f, 552, 624, 1176de, 1800cdee }}
Badness (Sintel): 1.21
=== 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 tunings:
* WE: ~35/34 = 49.9959{{c}}, ~10/7 = 617.2685{{c}}
* CWE: ~35/34 = 50.0000{{c}}, ~10/7 = 617.3015{{c}}
{{Optimal ET sequence|legend=0| 72, …, 480fgg, 552g, 624 }}
Badness (Sintel): 1.06


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Landscape microtemperaments| ]] <!-- main article -->
[[Category:Landscape microtemperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Landscape_microtemperaments"