Landscape microtemperaments: Difference between revisions

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- sextile (more properly addressed in the garischismic clan)
 
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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]]
* ''[[Sextile]]'' (+33554432/33480783) → [[Garischismic clan #Sextile|Garischismic clan]]
* ''[[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]]
* ''[[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, terture, slendscape, akjayland, magnesium, chromium, zinc, and poe.
Considered below are septichrome, akjayland, pnict, atomic, slendscape, avicenna, 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 265: Line 240:


Atomic tempers out the [[atom]], {{monzo| 161 -84 -12 }}, and in the [[7-limit]] the [[nommisma]], {{monzo| -55 30 2 1 }}, so that a stack of two [[schisma]]s gives the [[garischisma]], from which intervals of [[7/1|7]] can be derived. It may be described as the {{nowrap| 12 & 612 }} temperament, with a [[ploidacot]] signature of dodecaploid monocot.  
Atomic tempers out the [[atom]], {{monzo| 161 -84 -12 }}, and in the [[7-limit]] the [[nommisma]], {{monzo| -55 30 2 1 }}, so that a stack of two [[schisma]]s gives the [[garischisma]], from which intervals of [[7/1|7]] can be derived. It may be described as the {{nowrap| 12 & 612 }} temperament, with a [[ploidacot]] signature of dodecaploid monocot.  
Atomic extensions discussed elsewhere include [[72nd-octave temperaments #Hafnium|hafnium]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 347: Line 320:
Badness (Sintel): 1.50
Badness (Sintel): 1.50


== Avicenna ==
==== Hafnium ====
[[Subgroup]]: 2.3.5.7
Hafnium (1224 & 4320), named after the 72nd element, splits the 1/12-octave period into six 1/72-octave parts. Since 4320edo and 5544edo have good 31st and 37th harmonics, addition of these primes are also prescribed. In the add-37 version, [[37/22]] is mapped to exact 3\4.  
 
[[Comma list]]: 250047/250000, 29360128/29296875
 
{{Mapping|legend=1| 3 2 8 16 | 0 8 -3 -22 }}
: mapping generators: ~63/50, ~1024/945
 
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 399.9681{{c}}, ~1024/945 = 137.7570{{c}}
: [[error map]]: {{val| -0.096 +0.037 +0.160 +0.010 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~1024/945 = 137.7689{{c}}
: error map: {{val| 0.000 +0.196 +0.380 +0.259 }}
 
{{Optimal ET sequence|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}
 
[[Badness]] (Sintel): 1.57
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 5632/5625, 102487/102400
 
Mapping: {{mapping| 3 2 8 16 9 | 0 8 -3 -22 4 }}
 
Optimal tunings:
* WE: ~63/50 = 399.9798{{c}}, ~693/640 = 137.7643{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~693/640 = 137.7716{{c}}
 
{{Optimal ET sequence|legend=0| 87, 183, 270, 1263, 1533, 1803c }}
 
Badness (Sintel): 0.763
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 3025/3024, 4096/4095
 
Mapping: {{mapping| 3 2 8 16 9 8 | 0 8 -3 -22 4 9 }}
 
Optimal tunings:
* WE: ~63/50 = 399.9921{{c}}, ~13/12 = 137.7743{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~13/12 = 137.7770{{c}}
 
{{Optimal ET sequence|legend=0| 87, 183, 270 }}


Badness (Sintel): 0.643
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 676/675, 715/714, 936/935, 1001/1000, 4096/4095
Mapping: {{mapping| 3 2 8 16 9 8 4 | 0 8 -3 -22 4 9 24 }}
Optimal tunings:
* WE: ~34/27 = 399.9776{{c}}, ~13/12 = 137.7535{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~13/12 = 137.7608{{c}}
{{Optimal ET sequence|legend=0| 87, 183, 270, 453 }}
Badness (Sintel): 0.869
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1729/1728
Mapping: {{mapping| 3 2 8 16 9 8 4 0 | 0 8 -3 -22 4 9 24 37 }}
Optimal tunings:
* WE: ~34/27 = 399.9804{{c}}, ~13/12 = 137.7602{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~13/12 = 137.7664{{c}}
{{Optimal ET sequence|legend=0| 87, 183, 270 }}
Badness (Sintel): 0.928
== Terture ==
{{See also| Vulture family }}
Named by [[Xenllium]] in 2021, terture may be described as the {{nowrap| 111 & 159 }} temperament.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 250047/250000, 359661568/358722675
{{Mapping|legend=1| 3 0 -18 -32 | 0 4 21 34 }}
: mapping generators: ~63/50, ~320/243
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 399.9723{{c}}, ~320/243 = 475.5221{{c}} (~392/375 = 75.5499{{c}})
: [[error map]]: {{val| -0.083 +0.134 +0.151 -0.185 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~320/243 = 475.5519{{c}} (~392/375 = 75.5519{{c}})
: error map: {{val| 0.000 +0.253 +0.276 -0.061 }}
{{Optimal ET sequence|legend=1| 111, 159, 270 }}
[[Badness]] (Sintel): 2.21
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 19712/19683, 102487/102400
Mapping: {{mapping| 3 0 -18 -32 8 | 0 4 21 34 2 }}
Optimal tunings:
* WE: ~63/50 = 399.9902{{c}}, ~320/243 = 475.5383{{c}} (~392/375 = 75.5481{{c}})
* CWE: ~63/50 = 400.0000{{c}}, ~320/243 = 475.5490{{c}} (~392/375 = 75.5490{{c}})
{{Optimal ET sequence|legend=0| 111, 159, 270, 1239, 1509, 1779, 2049, 2319 }}
Badness (Sintel): 0.969
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 676/675, 1001/1000, 3025/3024, 10985/10976
Comma list: 9801/9800, 151263/151250, 184549376/184528125, 308915776/308828625


Mapping: {{mapping| 3 0 -18 -32 8 -21 | 0 4 21 34 2 27 }}
Mapping: {{mapping| 72 0 966 2028 3102 2777 | 0 1 -7 -16 -25 -22 }}
: mapping generators: ~105/104, ~3


Optimal tunings:  
Optimal tunings:  
* WE: ~63/50 = 399.9958{{c}}, ~154/117 = 475.5485{{c}} (~117/112 = 75.5527{{c}})
* WE: ~105/104 = 16.666629{{c}}, ~3/2 = 701.945668{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~154/117 = 475.5531{{c}} (~117/112 = 75.5531{{c}})
* CWE: ~105/104 = 16.666667{{c}}, ~3/2 = 701.947388{{c}}


{{Optimal ET sequence|legend=0| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 1224, 3096e, 4320, 5544, 9864c }}


Badness (Sintel): 0.771
Badness (Sintel): 4.76


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


Comma list: 676/675, 715/714, 936/935, 1001/1000, 4928/4913
Comma list: 9801/9800, 12376/12375, 151263/151250, 1713660/1713481, 97144749/97140736


Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 | 0 4 21 34 2 27 12 }}
Mapping: {{mapping| 72 0 966 2028 3102 2777 979 | 0 1 -7 -16 -25 -22 -6 }}
: mapping generators: ~105/104, ~3


Optimal tunings:  
Optimal tunings:  
* WE: ~34/27 = 399.9664{{c}}, ~112/85 = 475.5198{{c}} (~117/112 = 75.5534{{c}})
* WE: ~105/104 = 16.666615{{c}}, ~3/2 = 701.944968{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5568{{c}} (~117/112 = 75.5568{{c}})
* CWE: ~105/104 = 16.666667{{c}}, ~3/2 = 701.947293{{c}}


{{Optimal ET sequence|legend=0| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 1224, 3096e, 4320, 5544, 9864c }}


Badness (Sintel): 0.953
Badness (Sintel): 2.73


=== 19-limit ===
===== 2.3.5.7.11.13.17.31 subgroup =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.31


Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1617/1615
Comma list: 9801/9800, 10881/10880, 12376/12375, 57629/57624, 179712/179707, 61456384/61448625


Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 -36 | 0 4 21 34 2 27 12 41 }}
Subgroup-val mapping: {{mapping| 72 0 966 2028 3102 2777 979 -328 | 0 1 -7 -16 -25 -22 -6 6 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~34/27 = 399.9665{{c}}, ~112/85 = 475.5198{{c}} (~95/91 = 75.5533{{c}})
* WE: ~105/104 = 16.666643{{c}}, ~3/2 = 701.946542{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5568{{c}} (~95/91 = 75.5568{{c}})
* CWE: ~105/104 = 16.666667{{c}}, ~3/2 = 701.947581{{c}}


{{Optimal ET sequence|legend=0| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 1224, 3096e, 4320, 5544 }}


Badness (Sintel): 0.846
Badness (Sintel): 2.36


=== 23-limit ===
===== 2.3.5.7.11.13.17.31.37 subgroup =====
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.31.37


Comma list: 460/459, 529/528, 676/675, 715/714, 936/935, 1001/1000, 1216/1215
Comma list: 9801/9800, 10881/10880, 12376/12375, 16576/16575, 57629/57624, 93093/93092, 179712/179707


Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 -36 10 | 0 4 21 34 2 27 12 41 3 }}
Subgroup-val mapping: {{mapping| 72 0 966 2028 3102 2777 979 -328 3228 | 0 1 -7 -16 -25 -22 -6 6 -25 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~34/27 = 400.0026{{c}}, ~112/85 = 475.5510{{c}} (~24/23 = 75.5485{{c}})
* WE: ~105/104 = 16.666648{{c}}, ~3/2 = 701.946549{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5482{{c}} (~24/23 = 75.5482{{c}})
* CWE: ~105/104 = 16.666667{{c}}, ~3/2 = 701.947386{{c}}


{{Optimal ET sequence|legend=0| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 1224, 3096el, 4320, 5544 }}


Badness (Sintel): 1.07
Badness (Sintel): 1.98


== Slendscape ==
== Slendscape ==
Line 572: Line 434:
Badness (Sintel): 0.877
Badness (Sintel): 0.877


== Akjayland ==
== Avicenna ==
{{See also| 21st-octave temperaments }}
 
Named by [[Eliora]] in 2022, akjayland tempers out the [[akjaysma]] in addition to landscape comma, and thereby features a period of 1\21.
 
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, {{monzo| 43 -1 -13 -4 }}
[[Comma list]]: 250047/250000, 29360128/29296875


{{Mapping|legend=1| 21 1 38 102 | 0 3 1 -4 }}
{{Mapping|legend=1| 3 2 8 16 | 0 8 -3 -22 }}
: mapping generators: ~1323/1280, ~131072/91875
: mapping generators: ~63/50, ~1024/945


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~1323/1280 = 57.1426{{c}}, ~131072/91875 = 614.9336{{c}}
* [[WE]]: ~63/50 = 399.9681{{c}}, ~1024/945 = 137.7570{{c}}
: [[error map]]: {{val| -0.005 -0.012 +0.039 -0.013 }}
: [[error map]]: {{val| -0.096 +0.037 +0.160 +0.010 }}
* [[CWE]]: ~1323/1280 = 57.1429{{c}}, ~131072/91875 = 614.9360{{c}}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~1024/945 = 137.7689{{c}}
: error map: {{val| 0.000 -0.004 +0.051 +0.002 }}
: error map: {{val| 0.000 +0.196 +0.380 +0.259 }}


{{Optimal ET sequence|legend=1| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
{{Optimal ET sequence|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}


[[Badness]] (Sintel): 0.838
[[Badness]] (Sintel): 1.57
 
=== 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.  


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


Comma list: 3025/3024, 102487/102400, {{monzo| 39 -4 -11 -5 2 }}
Comma list: 3025/3024, 5632/5625, 102487/102400


Mapping: {{mapping| 21 4 39 98 58 | 0 6 2 -8 3 3 }}
Mapping: {{mapping| 3 2 8 16 9 | 0 8 -3 -22 4 }}
: mapping generators: ~1323/1280, ~6615/5632


Optimal tunings:  
Optimal tunings:  
* WE: ~1323/1280 = 57.1436{{c}}, ~6615/5632 = 278.9017{{c}}
* WE: ~63/50 = 399.9798{{c}}, ~693/640 = 137.7643{{c}}
* CWE: ~1323/1280 = 57.1429{{c}}, ~6615/5632 = 278.8985{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~693/640 = 137.7716{{c}}


{{Optimal ET sequence|legend=0| 168, 357, 525, 882, 1407, 2289e }}
{{Optimal ET sequence|legend=0| 87, 183, 270, 1263, 1533, 1803c }}


Badness (Sintel): 3.14
Badness (Sintel): 0.763


==== 13-limit ====
=== 13-limit ===
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: 676/675, 1001/1000, 3025/3024, 4096/4095


Mapping: {{mapping| 21 4 39 98 58 107 | 0 6 2 -8 3 -6 }}
Mapping: {{mapping| 3 2 8 16 9 8 | 0 8 -3 -22 4 9 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~336/325 = 57.1426{{c}}, ~168/143 = 278.9047{{c}}
* WE: ~63/50 = 399.9921{{c}}, ~13/12 = 137.7743{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9060{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~13/12 = 137.7770{{c}}


{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=0| 87, 183, 270 }}


Badness (Sintel): 2.28
Badness (Sintel): 0.643


==== 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: 676/675, 715/714, 936/935, 1001/1000, 4096/4095


Mapping: {{mapping| 21 4 39 98 58 107 120 | 0 6 2 -8 3 -6 -7 }}
Mapping: {{mapping| 3 2 8 16 9 8 4 | 0 8 -3 -22 4 9 24 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9037{{c}}
* WE: ~34/27 = 399.9776{{c}}, ~13/12 = 137.7535{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9036{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~13/12 = 137.7608{{c}}


{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=0| 87, 183, 270, 453 }}


Badness (Sintel): 1.62
Badness (Sintel): 0.869


==== 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: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1729/1728


Mapping: {{mapping| 21 4 39 98 58 107 120 16 | 0 6 2 -8 3 -6 -7 15 }}
Mapping: {{mapping| 3 2 8 16 9 8 4 0 | 0 8 -3 -22 4 9 24 37 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~336/325 = 57.1425{{c}}, ~168/143 = 278.8960{{c}}
* WE: ~34/27 = 399.9804{{c}}, ~13/12 = 137.7602{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.8976{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~13/12 = 137.7664{{c}}


{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}
{{Optimal ET sequence|legend=0| 87, 183, 270 }}


Badness (Sintel): 1.64
Badness (Sintel): 0.928
 
==== 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 ==
{{See also| 10th-octave temperaments #Neon | 60th-octave temperaments #Neodymium }}
{{See also| 10th-octave temperaments #Neon }}


Zinc maybe described as the {{nowrap| 270 & 2190 }} temperament. It was named by [[Eliora]] in 2023 after the 30th element for having a 30th-octave period.
Zinc maybe described as the {{nowrap| 270 & 2190 }} temperament. It was named by [[Eliora]] in 2023 after the 30th element for having a 30th-octave period.
Line 860: Line 611:


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 910: Line 683:


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]]

Latest revision as of 09:45, 16 May 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of rank-2 landscape microtemperaments, which temper out the landscape comma (monzo: [-4 6 -6 3⟩, ratio: 250047/250000). For the rank-3 temperament, see Landscape family #Landscape.

Temperaments discussed elsewhere are:

Considered below are septichrome, akjayland, pnict, atomic, slendscape, avicenna, zinc, magnesium, poe, and chromium in the order of increasing badness.

Septichrome

Subgroup: 2.3.5.7

Comma list: 250047/250000, 2460375/2458624

Mapping: [⟨3 3 1 0], ⟨0 5 17 24]]

mapping generators: ~63/50, ~243/224

Optimal tunings:

  • WE: ~63/50 = 400.0100 ¢, ~243/224 = 140.3702 ¢
error map: ⟨+0.030 -0.074 -0.010 +0.059]
  • CWE: ~63/50 = 400.0000 ¢, ~243/224 = 140.3685 ¢
error map: ⟨0.000 -0.113 -0.050 +0.017]

Optimal ET sequence: 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc

Badness (Sintel): 0.426

Semiseptichrome

Subgroup: 2.3.5.7.11

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

Mapping: [⟨6 1 -15 -24 -32], ⟨0 5 17 24 31]]

mapping generators: ~55/49, ~375/308

Optimal tunings:

  • WE: ~55/49 = 200.0058 ¢, ~375/308 = 340.3742 ¢ (~1760/1701 = 59.6375 ¢)
  • CWE: ~55/49 = 200.0000 ¢, ~375/308 = 340.3661 ¢ (~1760/1701 = 59.6339 ¢)

Optimal ET sequence: 60e, 222cdee, 282, 342, 966, 1308, 1650, 4608b, 6258bc

Badness (Sintel): 0.642

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 34398/34375, 85293/85184

Mapping: [⟨6 1 -15 -24 -32 -68], ⟨0 5 17 24 31 53]]

Optimal tunings:

  • WE: ~55/49 = 199.9936 ¢, ~375/308 = 340.3707 ¢ (~121/117 = 59.6165 ¢)
  • CWE: ~55/49 = 200.0000 ¢, ~375/308 = 340.3802 ¢ (~121/117 = 59.6198 ¢)

Optimal ET sequence: 282, 342f, 624

Badness (Sintel): 1.64

17-limit

Subgroup: 2.3.5.7.11.13

Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 61965/61952

Mapping: [⟨6 1 -15 -24 -32 -68 -1], ⟨0 5 17 24 31 53 15]]

Optimal tunings:

  • WE: ~55/49 = 199.9865 ¢, ~375/308 = 340.3619 ¢ (~88/85 = 59.6111 ¢)
  • CWE: ~55/49 = 200.0000 ¢, ~375/308 = 340.3821 ¢ (~88/85 = 59.6179 ¢)

Optimal ET sequence: 282, 342f, 624

Badness (Sintel): 1.39

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨6 1 -15 -24 -32 -68 -1 34], ⟨0 5 17 24 31 53 15 -5]]

Optimal tunings:

  • WE: ~55/49 = 199.9837 ¢, ~162/133 = 340.3589 ¢ (~88/85 = 59.6084 ¢)
  • CWE: ~55/49 = 200.0000 ¢, ~162/133 = 340.3844 ¢ (~88/85 = 59.6156 ¢)

Optimal ET sequence: 282, 342f, 624

Badness (Sintel): 1.35

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: [⟨6 1 -15 -24 -32 -68 -1 34 -12], ⟨0 5 17 24 31 53 15 -5 23]]

Optimal tunings:

  • WE: ~55/49 = 199.9829 ¢, ~162/133 = 340.3576 ¢ (~88/85 = 59.6081 ¢)
  • CWE: ~55/49 = 200.0000 ¢, ~162/133 = 340.3844 ¢ (~88/85 = 59.6156 ¢)

Optimal ET sequence: 282, 342f, 624

Badness (Sintel): 1.14

Akjayland

See also: 21st-octave temperaments

Named by Eliora in 2022, 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 tunings:

  • WE: ~1323/1280 = 57.1426 ¢, ~131072/91875 = 614.9336 ¢
error map: ⟨-0.005 -0.012 +0.039 -0.013]
  • CWE: ~1323/1280 = 57.1429 ¢, ~131072/91875 = 614.9360 ¢
error map: ⟨0.000 -0.004 +0.051 +0.002]

Optimal ET sequence: 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c

Badness (Sintel): 0.838

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 tunings:

  • WE: ~1323/1280 = 57.1436 ¢, ~6615/5632 = 278.9017 ¢
  • CWE: ~1323/1280 = 57.1429 ¢, ~6615/5632 = 278.8985 ¢

Optimal ET sequence: 168, 357, 525, 882, 1407, 2289e

Badness (Sintel): 3.14

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 tunings:

  • WE: ~336/325 = 57.1426 ¢, ~168/143 = 278.9047 ¢
  • CWE: ~336/325 = 57.1429 ¢, ~168/143 = 278.9060 ¢

Optimal ET sequence: 168, 357, 525, 882

Badness (Sintel): 2.28

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 tunings:

  • WE: ~336/325 = 57.1429 ¢, ~168/143 = 278.9037 ¢
  • CWE: ~336/325 = 57.1429 ¢, ~168/143 = 278.9036 ¢

Optimal ET sequence: 168, 357, 525, 882

Badness (Sintel): 1.62

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 tunings:

  • WE: ~336/325 = 57.1425 ¢, ~168/143 = 278.8960 ¢
  • CWE: ~336/325 = 57.1429 ¢, ~168/143 = 278.8976 ¢

Optimal ET sequence: 168h, 357, 525, 882, 1407

Badness (Sintel): 1.64

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 tunings:

  • WE: ~336/325 = 57.1422 ¢, ~168/143 = 278.8949 ¢
  • CWE: ~336/325 = 57.1429 ¢, ~168/143 = 278.8980 ¢

Optimal ET sequence: 168h, 357, 525, 882, 1407

Badness (Sintel): 1.43

Pnict

Subgroup: 2.3.5.7

Comma list: 250047/250000, 2100875/2097152

Mapping: [⟨3 -3 1 12], ⟨0 13 10 -6]]

mapping generators: ~63/50, ~147/128

Optimal tunings:

  • WE: ~63/50 = 400.0312 ¢, ~147/128 = 238.6196 ¢ (~192/175 = 161.4116 ¢)
error map: ⟨+0.094 +0.006 -0.087 -0.169]
  • CWE: ~63/50 = 400.0000 ¢, ~147/128 = 238.6038 ¢ (~192/175 = 161.3962 ¢)
error map: ⟨0.000 -0.106 -0.276 -0.449]

Optimal ET sequence: 15, 141, 156, 171, 2409cd, 2580cd, …, 4461bccddd, 4632bccddd

Badness (Sintel): 1.16

Atomic

For the 5-limit version, see Very high accuracy temperaments #Atomic.

Atomic tempers out the atom, [161 -84 -12⟩, and in the 7-limit the nommisma, [-55 30 2 1⟩, so that a stack of two schismas gives the garischisma, from which intervals of 7 can be derived. It may be described as the 12 & 612 temperament, with a ploidacot signature of dodecaploid monocot.

Subgroup: 2.3.5.7

Comma list: 250047/250000, [-55 30 2 1⟩

Mapping: [⟨12 0 161 338], ⟨0 1 -7 -16]]

Optimal tunings:

  • WE: ~30375/28672 = 99.999866 ¢, ~3/2 = 701.948670 ¢ (~32805/32768 = 1.949605 ¢)
error map: ⟨-0.0016 -0.0079 +0.0353 -0.0241]
  • CWE: ~30375/28672 = 100.000000 ¢, ~3/2 = 701.949698 ¢ (~32805/32768 = 1.949698 ¢)
error map: ⟨0.0000 -0.0053 +0.0384 -0.0211]

Optimal ET sequence: 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd

Badness (Sintel): 1.16

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 151263/151250, 184549376/184528125

Mapping: [⟨12 0 161 338 517], ⟨0 1 -7 -16 -25]]

Optimal tunings:

  • WE: ~30375/28672 = 99.999760 ¢, ~3/2 = 701.946301 ¢ (~32805/32768 = 1.947983 ¢)
  • CWE: ~30375/28672 = 100.000000 ¢, ~3/2 = 701.948121 ¢ (~32805/32768 = 1.948121 ¢)

Optimal ET sequence: 12, …, 600e, 612, 1236, 1848

Badness (Sintel): 0.530

Minutes

Minutes (600e & 2460) splits the 1/12-octave period into five 1/60-octave parts.

Subgroup: 2.3.5.7.11.13

Comma list: 9801/9800, 151263/151250, 371293/371250, 184549376/184528125

Mapping: [⟨60 0 805 1690 2585 1173], ⟨0 1 7 -16 -25 -10]]

mapping generators: ~2704/2673, ~3

Optimal tunings:

  • WE: ~2704/2673 = 19.999967 ¢, ~3/2 = 701.946866 ¢
  • CWE: ~2704/2673 = 20.000000 ¢, ~3/2 = 701.948111 ¢

Optimal ET sequence: 600e, …, 1860, 2460, 6780, 9240

Badness (Sintel): 2.82

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 9801/9800, 12376/12375, 28561/28560, 151263/151250, 11275335/11275264

Mapping: [⟨60 0 805 1690 2585 1173 1957], ⟨0 1 7 -16 -25 -10 -18]]

Optimal tunings:

  • WE: ~2704/2673 = 19.999974 ¢, ~3/2 = 701.946956 ¢
  • CWE: ~2704/2673 = 20.000000 ¢, ~3/2 = 701.947926 ¢

Optimal ET sequence: 600e, …, 1860, 2460, 6780, 9240

Badness (Sintel): 1.67

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 9801/9800, 12376/12375, 12636/12635, 23409/23408, 28561/28560, 151263/151250

Mapping: [⟨60 60 385 730 1085 573 877 -61], ⟨0 1 7 -16 -25 -10 -18 9]]

Optimal tunings:

  • WE: ~2704/2673 = 19.999962 ¢, ~3/2 = 701.946354 ¢
  • CWE: ~2704/2673 = 20.000000 ¢, ~3/2 = 701.947751 ¢

Optimal ET sequence: 600e, 1860, 2460, 4320, 6780, 9240

Badness (Sintel): 1.50

Hafnium

Hafnium (1224 & 4320), named after the 72nd element, splits the 1/12-octave period into six 1/72-octave parts. Since 4320edo and 5544edo have good 31st and 37th harmonics, addition of these primes are also prescribed. In the add-37 version, 37/22 is mapped to exact 3\4.

Subgroup: 2.3.5.7.11.13

Comma list: 9801/9800, 151263/151250, 184549376/184528125, 308915776/308828625

Mapping: [⟨72 0 966 2028 3102 2777], ⟨0 1 -7 -16 -25 -22]]

mapping generators: ~105/104, ~3

Optimal tunings:

  • WE: ~105/104 = 16.666629 ¢, ~3/2 = 701.945668 ¢
  • CWE: ~105/104 = 16.666667 ¢, ~3/2 = 701.947388 ¢

Optimal ET sequence: 1224, 3096e, 4320, 5544, 9864c

Badness (Sintel): 4.76

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 9801/9800, 12376/12375, 151263/151250, 1713660/1713481, 97144749/97140736

Mapping: [⟨72 0 966 2028 3102 2777 979], ⟨0 1 -7 -16 -25 -22 -6]]

mapping generators: ~105/104, ~3

Optimal tunings:

  • WE: ~105/104 = 16.666615 ¢, ~3/2 = 701.944968 ¢
  • CWE: ~105/104 = 16.666667 ¢, ~3/2 = 701.947293 ¢

Optimal ET sequence: 1224, 3096e, 4320, 5544, 9864c

Badness (Sintel): 2.73

2.3.5.7.11.13.17.31 subgroup

Subgroup: 2.3.5.7.11.13.17.31

Comma list: 9801/9800, 10881/10880, 12376/12375, 57629/57624, 179712/179707, 61456384/61448625

Subgroup-val mapping: [⟨72 0 966 2028 3102 2777 979 -328], ⟨0 1 -7 -16 -25 -22 -6 6]]

Optimal tunings:

  • WE: ~105/104 = 16.666643 ¢, ~3/2 = 701.946542 ¢
  • CWE: ~105/104 = 16.666667 ¢, ~3/2 = 701.947581 ¢

Optimal ET sequence: 1224, 3096e, 4320, 5544

Badness (Sintel): 2.36

2.3.5.7.11.13.17.31.37 subgroup

Subgroup: 2.3.5.7.11.13.17.31.37

Comma list: 9801/9800, 10881/10880, 12376/12375, 16576/16575, 57629/57624, 93093/93092, 179712/179707

Subgroup-val mapping: [⟨72 0 966 2028 3102 2777 979 -328 3228], ⟨0 1 -7 -16 -25 -22 -6 6 -25]]

Optimal tunings:

  • WE: ~105/104 = 16.666648 ¢, ~3/2 = 701.946549 ¢
  • CWE: ~105/104 = 16.666667 ¢, ~3/2 = 701.947386 ¢

Optimal ET sequence: 1224, 3096el, 4320, 5544

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: [⟨15 0 17 54], ⟨0 4 3 -2]]

mapping generators: ~8575/8192, ~1152/875

Optimal tunings:

  • WE: ~8575/8192 = 79.9771 ¢, ~1152/875 = 475.4832 ¢
error map: ⟨-0.043 -0.022 +0.087 +0.053]
  • CWE: ~8575/8192 = 80.0000 ¢, ~1152/875 = 475.4962 ¢
error map: ⟨0.000 +0.030 +0.175 +0.182]

Optimal ET sequence: 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: [⟨15 0 17 54 40], ⟨0 4 3 -2 2]]

Optimal tunings:

  • WE: ~22/21 = 79.9991 ¢, ~968/735 = 475.4915 ¢
  • CWE: ~22/21 = 80.0000 ¢, ~968/735 = 475.4955 ¢

Optimal ET sequence: 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: [⟨15 0 17 54 40 109], ⟨0 4 3 -2 2 -9]]

Optimal tunings:

  • WE: ~22/21 = 79.9993 ¢, ~154/117 = 475.4902 ¢
  • CWE: ~22/21 = 80.0000 ¢, ~154/117 = 475.4943 ¢

Optimal ET sequence: 255, 270, 795, 1065

Badness (Sintel): 0.877

Avicenna

Subgroup: 2.3.5.7

Comma list: 250047/250000, 29360128/29296875

Mapping: [⟨3 2 8 16], ⟨0 8 -3 -22]]

mapping generators: ~63/50, ~1024/945

Optimal tunings:

  • WE: ~63/50 = 399.9681 ¢, ~1024/945 = 137.7570 ¢
error map: ⟨-0.096 +0.037 +0.160 +0.010]
  • CWE: ~63/50 = 400.0000 ¢, ~1024/945 = 137.7689 ¢
error map: ⟨0.000 +0.196 +0.380 +0.259]

Optimal ET sequence: 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd

Badness (Sintel): 1.57

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 tunings:

  • WE: ~63/50 = 399.9798 ¢, ~693/640 = 137.7643 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~693/640 = 137.7716 ¢

Optimal ET sequence: 87, 183, 270, 1263, 1533, 1803c

Badness (Sintel): 0.763

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 tunings:

  • WE: ~63/50 = 399.9921 ¢, ~13/12 = 137.7743 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~13/12 = 137.7770 ¢

Optimal ET sequence: 87, 183, 270

Badness (Sintel): 0.643

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 936/935, 1001/1000, 4096/4095

Mapping: [⟨3 2 8 16 9 8 4], ⟨0 8 -3 -22 4 9 24]]

Optimal tunings:

  • WE: ~34/27 = 399.9776 ¢, ~13/12 = 137.7535 ¢
  • CWE: ~34/27 = 400.0000 ¢, ~13/12 = 137.7608 ¢

Optimal ET sequence: 87, 183, 270, 453

Badness (Sintel): 0.869

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1729/1728

Mapping: [⟨3 2 8 16 9 8 4 0], ⟨0 8 -3 -22 4 9 24 37]]

Optimal tunings:

  • WE: ~34/27 = 399.9804 ¢, ~13/12 = 137.7602 ¢
  • CWE: ~34/27 = 400.0000 ¢, ~13/12 = 137.7664 ¢

Optimal ET sequence: 87, 183, 270

Badness (Sintel): 0.928

Zinc

See also: 10th-octave temperaments § Neon

Zinc maybe described as the 270 & 2190 temperament. It was named by Eliora in 2023 after the 30th element for having a 30th-octave period.

Subgroup: 2.3.5.7

Comma list: 250047/250000, [-53 -12 2 24⟩

Mapping: [⟨30 2 15 66], ⟨0 5 6 2]]

mapping generators: ~53747712/52521875, ~216/175

Optimal tunings:

  • WE: ~53747712/52521875 = 40.0002 ¢, ~216/175 = 364.3890 ¢ (~[21 3 1 -10⟩ = 4.3869 ¢)
error map: ⟨+0.007 -0.009 +0.024 -0.032]
  • CWE: ~53747712/52521875 = 40.0000 ¢, ~216/175 = 364.3879 ¢ (~[21 3 1 -10⟩ = 4.3879 ¢)
error map: ⟨0.000 -0.015 +0.014 -0.050]

Optimal ET sequence: 270, 1380, 1650, 1920, 2190, 4650

Badness (Sintel): 1.88

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 151263/151250, [-27 -6 4 6 3⟩

Mapping: [⟨30 2 15 66 122], ⟨0 5 6 2 -2]]

Optimal tunings:

  • WE: ~18865/18432 = 40.0005 ¢, ~216/175 = 364.3881 ¢ (~385/384 = 4.3837 ¢)
  • CWE: ~18865/18432 = 40.0000 ¢, ~216/175 = 364.3849 ¢ (~385/384 = 4.3849 ¢)

Optimal ET sequence: 270, 1110, 1380, 1650, 1920, 2190

Badness (Sintel): 0.727

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 9801/9800, 10648/10647, 105644/105625, 196625/196608

Mapping: [⟨30 2 15 66 122 193], ⟨0 5 6 2 -2 -9]]

Optimal tunings:

  • WE: ~351/343 = 40.0003 ¢, ~216/175 = 364.3894 ¢ (~385/384 = 4.3865 ¢)
  • CWE: ~351/343 = 40.0000 ¢, ~216/175 = 364.3867 ¢ (~385/384 = 4.3867 ¢)

Optimal ET sequence: 270, 1380, 1650, 1920, 2190, 4650, 6840e, 11490de

Badness (Sintel): 0.640

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: [⟨30 2 15 66 122 193 91], ⟨0 5 6 2 -2 -9 4]]

Optimal tunings:

  • WE: ~175/171 = 40.0002 ¢, ~216/175 = 364.3885 ¢ (~400/399 = 4.3862 ¢)
  • CWE: ~175/171 = 40.0000 ¢, ~216/175 = 364.3864 ¢ (~400/399 = 4.3864 ¢)

Optimal ET sequence: 270, 1380, 1650, 1920, 2190, 4650, 6840e

Badness (Sintel): 0.477

Neodymium

Neodymium (540 & 1920) splits the period into 1/60-octave halves for prime 17.

Subgroup: 2.3.5.7.11.13.17

Comma list: 9801/9800, 10648/10647, 31213/31212, 105644/105625, 196625/196608

Mapping: [⟨60 4 30 132 244 386 391], ⟨0 5 6 2 -2 -9 -8]]

mapping generators: ~612/605, ~216/175

Optimal tunings:

  • WE: ~612/605 = 20.0002 ¢, ~216/175 = 364.3894 ¢
  • CWE: ~612/605 = 20.0000 ¢, ~216/175 = 364.3864 ¢

Optimal ET sequence: 540, 1380, 1920, 2460, 4380, 6840e

Badness (Sintel): 1.23

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 5929/5928, 6860/6859, 9801/9800, 10241/10240, 23409/23408, 89376/89375

Mapping: [⟨60 4 30 132 244 386 391 182], ⟨0 5 6 2 -2 -9 -8 4]]

Optimal tunings:

  • WE: ~612/605 = 20.0001 ¢, ~216/175 = 364.3883 ¢
  • CWE: ~612/605 = 20.0000 ¢, ~216/175 = 364.3860 ¢

Optimal ET sequence: 540, 1380, 1920, 2460, 4380, 6840e

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 [-157 -24 84⟩ in the 5-limit, and 7 generator steps together with two 12edo semitones reach the 3rd harmonic. It may be described as the 84 & 612 temperament, with a ploidacot signature of dodecaploid gamma-heptacot.

Subgroup: 2.3.5.7

Comma list: 250047/250000, [-59 2 18 5⟩

Mapping: [⟨12 2 23 58], ⟨0 7 2 -10]]

mapping generators: ~138915/131072, ~3145728/2734375

Optimal tunings:

  • WE: ~138915/131072 = 100.0021 ¢, ~3145728/2734375 = 243.1333 ¢
error map: ⟨+0.025 -0.018 +0.000 -0.039]
  • CWE: ~138915/131072 = 100.0000 ¢, ~3145728/2734375 = 243.1285 ¢
error map: ⟨0.000 -0.055 -0.057 -0.111]

Optimal ET sequence: 84, 360d, 444, 528, 612, 1920, 2532, 10740cd, 13272bcdd, 15804bcdd, 18336bcddd

Badness (Sintel): 2.44

Poe

Named by Tristan Bay in 2025, poe may be described as the 60 & 270 temperament.

Subgroup: 2.3.5.7

Comma list: 250047/250000, [15 -16 -4 7⟩

Mapping: [⟨30 0 -73 -106], ⟨0 1 3 4]]

mapping generators: ~2240/2187, ~3

Optimal tunings:

  • WE: ~2240/2187 = 39.9982 ¢, ~3/2 = 702.1533 ¢
error map: ⟨-0.055 +0.143 +0.115 -0.238]
  • CWE: ~2240/2187 = 40.0000 ¢, ~3/2 = 702.1656 ¢
error map: ⟨0.000 +0.211 +0.183 -0.163]

Optimal ET sequence: 60, 150cd, 210, 270, 1950, 2220, 2490, 2760b, 3030bc, 3300bc

Badness (Sintel): 2.90

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 19712/19683, 151263/151250

Mapping: [⟨30 0 -73 -106 -134], ⟨0 1 3 4 5]]

Optimal tunings:

  • WE: ~45/44 = 39.9976 ¢, ~3/2 = 702.1955 ¢
  • CWE: ~45/44 = 40.0000 ¢, ~3/2 = 702.2129 ¢

Optimal ET sequence: 60e, …, 210e, 270

Badness (Sintel): 1.31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 4225/4224, 4459/4455, 19712/19683

Mapping: [⟨30 0 -73 -106 -134 111], ⟨0 1 3 4 5 0]]

Optimal tunings:

  • WE: ~45/44 = 40.0008 ¢, ~3/2 = 702.1778 ¢
  • CWE: ~45/44 = 40.0000 ¢, ~3/2 = 702.1699 ¢

Optimal ET sequence: 60e, 210e, 270, 1410ef, 1680ef

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 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: [⟨24 1 -6 18], ⟨0 3 5 4]]

mapping generators: ~250/243, ~10/7

Optimal tunings:

  • WE: ~250/243 = 49.9992 ¢, ~10/7 = 617.2714 ¢
error map: ⟨-0.019 -0.142 +0.048 +0.246]
  • CWE: ~250/243 = 50.0000 ¢, ~10/7 = 617.2762 ¢
error map: ⟨0.000 -0.126 +0.067 +0.279]

Optimal ET sequence: 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: [⟨24 1 -6 18 46], ⟨0 3 5 4 3]]

Optimal tunings:

  • WE: ~250/243 = 49.9972 ¢, ~10/7 = 617.2639 ¢
  • CWE: ~250/243 = 50.0000 ¢, ~10/7 = 617.2823 ¢

Optimal ET sequence: 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: [⟨24 1 -6 18 46 -47 -13], ⟨0 3 5 4 3 11]]

Optimal tunings:

  • WE: ~250/243 = 49.9958 ¢, ~10/7 = 617.2824 ¢
  • CWE: ~250/243 = 50.0000 ¢, ~10/7 = 617.3161 ¢

Optimal ET sequence: 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: [⟨24 1 -6 18 46 -47 -13], ⟨0 3 5 4 3 11 9]]

Optimal tunings:

  • WE: ~35/34 = 49.9959 ¢, ~10/7 = 617.2685 ¢
  • CWE: ~35/34 = 50.0000 ¢, ~10/7 = 617.3015 ¢

Optimal ET sequence: 72, …, 480fgg, 552g, 624

Badness (Sintel): 1.06

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