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]]
* ''[[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]]
* ''[[Tritricot]]'' (+{{monzo| 35 -23 -3 3 }}) → [[Alphatricot family #Tritricot|Alphatricot family]]
* ''[[Domain (temperament)|Domain]]'' (+645700815/645657712) → [[Minortonic family #Domain|Minortonic 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, domain, avicenna, akjayland and terture.
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–Pythagorean equivalence continuum #Sextile (5-limit)]].''
 
Sextile tempers out the [[garischisma]] with a 1/6-octave period and is the 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: ~63/50, ~243/224
: mapping generators: ~4096/3645, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~4096/3645 = 1\6, ~3/2 = 702.2347
* [[WE]]: ~63/50 = 400.0100{{c}}, ~243/224 = 140.3702{{c}}
* [[POTE]]: ~4096/3645 = 1\6, ~3/2 = 702.212
: [[error map]]: {{val| +0.030 -0.074 -0.010 +0.059 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~243/224 = 140.3685{{c}}
: error map: {{val| 0.000 -0.113 -0.050 +0.017 }}


{{Optimal ET sequence|legend=1| 12, 234d, 246d, 258, 270, 1362c, 1632c, 1902c, 2172c, 2442bc, 2712bc }}
{{Optimal ET sequence|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}


[[Badness]]: 0.070097
[[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 = 1\6, ~3/2 = 702.2225
* WE: ~55/49 = 200.0058{{c}}, ~375/308 = 340.3742{{c}} (~1760/1701 = 59.6375{{c}})
* POTE: ~55/49 = 1\6, ~3/2 = 702.202
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3661{{c}} (~1760/1701 = 59.6339{{c}})


{{Optimal ET sequence|legend=1| 12, 246dee, 258e, 270, 822, 1092, 1362c }}
{{Optimal ET sequence|legend=0| 60e, 222cdee, 282, 342, 966, 1308, 1650, 4608b, 6258bc }}


Badness: 0.029677
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:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2141
* WE: ~55/49 = 199.9936{{c}}, ~375/308 = 340.3707{{c}} (~121/117 = 59.6165{{c}})
* CWE: ~55/49 = 1\6, ~3/2 = 702.2001
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3802{{c}} (~121/117 = 59.6198{{c}})


{{Optimal ET sequence|legend=1| 12f, 258ef, 270, 552, 822, 1092, 1914cde }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness: 0.0191
Badness (Sintel): 1.64


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


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:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2117
* WE: ~55/49 = 199.9865{{c}}, ~375/308 = 340.3619{{c}} (~88/85 = 59.6111{{c}})
* CWE: ~55/49 = 1\6, ~3/2 = 702.1869
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3821{{c}} (~88/85 = 59.6179{{c}})


{{Optimal ET sequence|legend=1| 12f, 270, 552g }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness: 0.0209
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 = 1\6, ~3/2 = 702.2118
* WE: ~55/49 = 199.9837{{c}}, ~162/133 = 340.3589{{c}} (~88/85 = 59.6084{{c}})
* CWE: ~55/49 = 1\6, ~3/2 = 702.1890
* CWE: ~55/49 = 200.0000{{c}}, ~162/133 = 340.3844{{c}} (~88/85 = 59.6156{{c}})


{{Optimal ET sequence|legend=1| 12f, 270, 552g, 822gg }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness: 0.0156
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:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2296
* WE: ~55/49 = 199.9829{{c}}, ~162/133 = 340.3576{{c}} (~88/85 = 59.6081{{c}})
* CWE: ~55/49 = 1\6, ~3/2 = 702.2285
* CWE: ~55/49 = 200.0000{{c}}, ~162/133 = 340.3844{{c}} (~88/85 = 59.6156{{c}})


{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness: 0.0391
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 }}
[[Subgroup]]: 2.3.5.7


Optimal tunings:  
[[Comma list]]: 250047/250000, {{monzo| 43 -1 -13 -4 }}
* CTE: ~55/49 = 1\6, ~3/2 = 702.2264
* CWE: ~55/49 = 1\6, ~3/2 = 702.2207


{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
{{Mapping|legend=1| 21 1 38 102 | 0 3 1 -4 }}
: mapping generators: ~1323/1280, ~131072/91875


Badness: 0.0384
[[Optimal tuning]]s:  
* [[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 }}


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


Comma list: 715/714, 1001/1000, 1216/1215, 1701/1700, 1729/1728, 2912/2907
[[Badness]] (Sintel): 0.838


Mapping: {{mapping| 6 0 71 150 230 -149 -4 35 | 0 1 -6 -14 -22 18 3 -1 }}
=== 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.


Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 702.2266
* CWE: ~55/49 = 1\6, ~3/2 = 702.2208
{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
Badness: 0.0252
== Septichrome ==
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 250047/250000, 2460375/2458624
{{Mapping|legend=1| 3 3 1 0 | 0 5 17 24 }}
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~243/224 = 140.367
{{Optimal ET sequence|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}
[[Badness]]: 0.016814
=== Semiseptichrome ===
Subgroup: 2.3.5.7.11
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: ~1323/1280, ~6615/5632


: mapping generators: ~55/49, ~375/308
Optimal tunings:  
 
* WE: ~1323/1280 = 57.1436{{c}}, ~6615/5632 = 278.9017{{c}}
Optimal tuning (CTE): ~55/49 = 1\6, ~375/308 = 340.3687 (~1760/1701 = 59.6313)
* CWE: ~1323/1280 = 57.1429{{c}}, ~6615/5632 = 278.8985{{c}}


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


Badness: 0.0194
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 tuning (CTE): ~55/49 = 1\6, ~375/308 = 340.3778 (~121/117 = 59.6222)
Optimal tunings:
* WE: ~336/325 = 57.1426{{c}}, ~168/143 = 278.9047{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9060{{c}}


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


Badness: 0.0397
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 tuning (CTE): ~55/49 = 1\6, ~375/308 = 340.3763 (~88/85 = 59.6237)
Optimal tunings:
* WE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9037{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9036{{c}}


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


Badness: 0.0272
Badness (Sintel): 1.62


==== 19-limit ====
==== 19-limit ====
Subgroup: 2.3.5.7.11.13
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 tuning (CTE): ~55/49 = 1\6, ~162/133 = 340.3767 (~88/85 = 59.6233)
Optimal tunings:
* WE: ~336/325 = 57.1425{{c}}, ~168/143 = 278.8960{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.8976{{c}}


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


Badness: 0.0222
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 tuning (CTE): ~55/49 = 1\6, ~162/133 = 340.3757 (~88/85 = 59.6243)
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: {{Optimal ET sequence| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}


Badness: 0.0160
Badness (Sintel): 1.43


== Pnict ==
== Pnict ==
Line 232: Line 223:
[[Comma list]]: 250047/250000, 2100875/2097152
[[Comma list]]: 250047/250000, 2100875/2097152


{{Mapping|legend=1| 3 10 11 6 | 0 -13 -10 6 }}
{{Mapping|legend=1| 3 -3 1 12 | 0 13 10 -6 }}
: mapping generators: ~63/50, ~147/128


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~192/175 = 161.399
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0312{{c}}, ~147/128 = 238.6196{{c}} (~192/175 = 161.4116{{c}})
: [[error map]]: {{val| +0.094 +0.006 -0.087 -0.169 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~147/128 = 238.6038{{c}} (~192/175 = 161.3962{{c}})
: error map: {{val| 0.000 -0.106 -0.276 -0.449 }}


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


[[Badness]]: 0.045660
[[Badness]] (Sintel): 1.16
 
== Domain ==
{{See also| Minortonic family }}


[[Subgroup]]: 2.3.5.7
== Atomic ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Atomic]].''


[[Comma list]]: 250047/250000, 645700815/645657712
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.


{{Mapping|legend=1| 3 -3 -9 -8 | 0 17 35 36 }}
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467
{{Optimal ET sequence|legend=1| 171, 1164, 1335, 1506, 1677, 1848, 2019, 2190, 11943, 13962, 15981, 18000, 20019, 22038 }}
[[Badness]]: 0.013979
== Avicenna ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 29360128/29296875
[[Comma list]]: 250047/250000, {{monzo| -55 30 2 1 }}


{{Mapping|legend=1| 3 2 8 16 | 0 8 -3 -22 }}
{{Mapping|legend=1| 12 0 161 338 | 0 1 -7 -16 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~63/50 = 1\3, ~1024/945 = 137.772
* [[WE]]: ~30375/28672 = 99.999866{{c}}, ~3/2 = 701.948670{{c}} (~32805/32768 = 1.949605{{c}})
* [[POTE]]: ~63/50 = 1\3, ~1024/945 = 137.768
: [[error map]]: {{val| -0.0016 -0.0079  +0.0353 -0.0241 }}
* [[CWE]]: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.949698{{c}} (~32805/32768 = 1.949698{{c}})
: error map: {{val| 0.0000 -0.0053 +0.0384 -0.0211 }}


{{Optimal ET sequence|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}
{{Optimal ET sequence|legend=1| 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}


[[Badness]]: 0.062187
[[Badness]] (Sintel): 1.16


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


Comma list: 3025/3024, 5632/5625, 102487/102400
Comma list: 9801/9800, 151263/151250, 184549376/184528125


Mapping: {{mapping| 3 2 8 16 9 | 0 8 -3 -22 4 }}
Mapping: {{mapping| 12 0 161 338 517 | 0 1 -7 -16 -25 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~63/50 = 1\3, ~693/640 = 137.773
* WE: ~30375/28672 = 99.999760{{c}}, ~3/2 = 701.946301{{c}} (~32805/32768 = 1.947983{{c}})
* POTE: ~63/50 = 1\3, ~693/640 = 137.771
* CWE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.948121{{c}} (~32805/32768 = 1.948121{{c}})
 
{{Optimal ET sequence|legend=0| 12, …, 600e, 612, 1236, 1848 }}


Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270, 1263, 1533, 1803c }}
Badness (Sintel): 0.530


Badness: 0.023085
==== Minutes ====
Minutes (600e & 2460) splits the 1/12-octave period into five 1/60-octave parts.  


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


Comma list: 676/675, 1001/1000, 3025/3024, 4096/4095
Comma list: 9801/9800, 151263/151250, 371293/371250, 184549376/184528125


Mapping: {{mapping| 3 2 8 16 9 8 | 0 8 -3 -22 4 9 }}
Mapping: {{mapping| 60 0 805 1690 2585 1173 | 0 1 7 -16 -25 -10 }}
: mapping generators: ~2704/2673, ~3


Optimal tunings:  
Optimal tunings:  
* CTE: ~63/50 = 1\3, ~13/12 = 137.777
* WE: ~2704/2673 = 19.999967{{c}}, ~3/2 = 701.946866{{c}}
* POTE: ~63/50 = 1\3, ~13/12 = 137.777
* CWE: ~2704/2673 = 20.000000{{c}}, ~3/2 = 701.948111{{c}}


Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270 }}
{{Optimal ET sequence|legend=0| 600e, …, 1860, 2460, 6780, 9240 }}


Badness: 0.015557
Badness (Sintel): 2.82


=== 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, 1001/1000, 3025/3024, 4096/4095
Comma list: 9801/9800, 12376/12375, 28561/28560, 151263/151250, 11275335/11275264


Mapping: {{mapping| 3 2 8 16 9 8 4 | 0 8 -3 -22 4 9 24 }}
Mapping: {{mapping| 60 0 805 1690 2585 1173 1957 | 0 1 7 -16 -25 -10 -18 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~63/50 = 1\3, ~13/12 = 137.758
* WE: ~2704/2673 = 19.999974{{c}}, ~3/2 = 701.946956{{c}}
* POTE: ~63/50 = 1\3, ~13/12 = 137.761
* CWE: ~2704/2673 = 20.000000{{c}}, ~3/2 = 701.947926{{c}}


Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270, 453 }}
{{Optimal ET sequence|legend=0| 600e, …, 1860, 2460, 6780, 9240 }}


Badness: 0.0171
Badness (Sintel): 1.67


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


Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 1729/1728, 3025/3024
Comma list: 9801/9800, 12376/12375, 12636/12635, 23409/23408, 28561/28560, 151263/151250


Mapping: {{mapping| 3 2 8 16 9 8 4 0 | 0 8 -3 -22 4 9 24 37 }}
Mapping: {{mapping| 60 60 385 730 1085 573 877 -61 | 0 1 7 -16 -25 -10 -18 9 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~63/50 = 1\3, ~13/12 = 137.763
* WE: ~2704/2673 = 19.999962{{c}}, ~3/2 = 701.946354{{c}}
* POTE: ~63/50 = 1\3, ~13/12 = 137.767
* CWE: ~2704/2673 = 20.000000{{c}}, ~3/2 = 701.947751{{c}}


Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270 }}
{{Optimal ET sequence|legend=0| 600e, 1860, 2460, 4320, 6780, 9240 }}


Badness: 0.0153
Badness (Sintel): 1.50


== Terture ==
==== Hafnium ====
{{See also| Vulture family }}
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
[[Comma list]]: 250047/250000, 359661568/358722675
{{Mapping|legend=1| 3 4 3 2 | 0 4 21 34 }}
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~392/375 = 75.555
{{Optimal ET sequence|legend=1| 111, 159, 270 }}
[[Badness]]: 0.087156
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 19712/19683, 102487/102400
Mapping: {{mapping| 3 4 3 2 10 | 0 4 21 34 2 }}
Optimal tuning (POTE): ~63/50 = 1\3, ~392/375 = 75.550
{{Optimal ET sequence|legend=1| 111, 159, 270, 1239, 1509, 1779, 2049, 2319 }}
Badness: 0.029326
=== 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 4 3 2 10 6 | 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 tuning (POTE): ~63/50 = 1\3, ~117/112 = 75.553
Optimal tunings:
* WE: ~105/104 = 16.666629{{c}}, ~3/2 = 701.945668{{c}}
* CWE: ~105/104 = 16.666667{{c}}, ~3/2 = 701.947388{{c}}


{{Optimal ET sequence|legend=1| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 1224, 3096e, 4320, 5544, 9864c }}


Badness: 0.018647
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 4 3 2 10 6 10 | 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 tuning (POTE): ~63/50 = 1\3, ~117/112 = 75.560
Optimal tunings:
* WE: ~105/104 = 16.666615{{c}}, ~3/2 = 701.944968{{c}}
* CWE: ~105/104 = 16.666667{{c}}, ~3/2 = 701.947293{{c}}


{{Optimal ET sequence|legend=1| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 1224, 3096e, 4320, 5544, 9864c }}


Badness: 0.018705
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 4 3 2 10 6 10 5 | 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 tuning (POTE): ~63/50 = 1\3, ~95/91 = 75.560
Optimal tunings:
* WE: ~105/104 = 16.666643{{c}}, ~3/2 = 701.946542{{c}}
* CWE: ~105/104 = 16.666667{{c}}, ~3/2 = 701.947581{{c}}


{{Optimal ET sequence|legend=1| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 1224, 3096e, 4320, 5544 }}


Badness: 0.013902
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 4 3 2 10 6 10 5 13 | 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 tuning (POTE): ~63/50 = 1\3, ~24/23 = 75.548
Optimal tunings:
* WE: ~105/104 = 16.666648{{c}}, ~3/2 = 701.946549{{c}}
* CWE: ~105/104 = 16.666667{{c}}, ~3/2 = 701.947386{{c}}


{{Optimal ET sequence|legend=1| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 1224, 3096el, 4320, 5544 }}


Badness: 0.014915
Badness (Sintel): 1.98


== Slendscape ==
== Slendscape ==
Slendscape tempers out the [[slendroschisma]] (68719476736/68641485507) in addition to landscape comma, and thereby features a period of 1\15.
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
[[Subgroup]]: 2.3.5.7
Line 418: Line 392:


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


: 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 tuning]] ([[CTE]]): ~8575/8192 = 80.0000, ~1152/875 = 475.4847 (~6/5 = 315.4847 or ~21/20 = 84.5153)
{{Optimal ET sequence|legend=1| 15, 240, 255, 270, 795, 1065, 1335, 2400 }}


{{Optimal ET sequence|legend=1| 255, 270, 525, 795, 1065 }}
[[Badness]] (Sintel): 1.47
 
[[Badness]]: 0.058002


=== 11-limit ===
=== 11-limit ===
Line 432: Line 409:
Comma list: 3025/3024, 102487/102400, 180224/180075
Comma list: 3025/3024, 102487/102400, 180224/180075


Mapping: {{Mapping| 15 0 17 54 40 | 0 4 3 -2 2 }}
Mapping: {{mapping| 15 0 17 54 40 | 0 4 3 -2 2 }}


Optimal tuning (CTE): ~22/21 = 80.0000, ~968/735 = 475.4912 (~6/5 = 315.4912 or ~21/20 = 84.5088)
Optimal 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: {{EDOs| 255, 270, 525, 795, 1065 }}
{{Optimal ET sequence|legend=0| 15, 240, 255, 270, 795, 1065, 2400e }}


Badness: 0.026246
Badness (Sintel): 0.868


=== 13-limit ===
=== 13-limit ===
Line 445: Line 424:
Comma list: 1716/1715, 3025/3024, 4096/4095, 14641/14625
Comma list: 1716/1715, 3025/3024, 4096/4095, 14641/14625


Mapping: {{Mapping| 15 0 17 54 40 109 | 0 4 3 -2 2 -9 }}
Mapping: {{mapping| 15 0 17 54 40 109 | 0 4 3 -2 2 -9 }}


Optimal tuning (CTE): ~22/21 = 80.0000, ~154/117 = 475.4935 (~6/5 = 315.4935 or ~21/20 = 84.5065)
Optimal 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: {{EDOs| 255, 270, 795, 1065 }}
{{Optimal ET sequence|legend=0| 255, 270, 795, 1065 }}


Badness: 0.021230
Badness (Sintel): 0.877


== Akjayland ==
== Avicenna ==
{{See also| 21st-octave temperaments }}
[[Subgroup]]: 2.3.5.7


Akjayland tempers out the [[akjaysma]] in addition to landscape comma, and thereby features a period of 1＼21.
[[Comma list]]: 250047/250000, 29360128/29296875


[[Subgroup]]: 2.3.5.7
{{Mapping|legend=1| 3 2 8 16 | 0 8 -3 -22 }}
: mapping generators: ~63/50, ~1024/945


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


{{Mapping|legend=1| 21 1 38 102 | 0 3 1 -4 }}
{{Optimal ET sequence|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}


: mapping generators: ~1323/1280, ~131072/91875
[[Badness]] (Sintel): 1.57


[[Optimal tuning]] ([[CTE]]): ~1323/1280 = 1＼21, ~131072/91875 = 614.9354
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Optimal ET sequence|legend=1| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
Comma list: 3025/3024, 5632/5625, 102487/102400


[[Badness]]: 0.0309
Mapping: {{mapping| 3 2 8 16 9 | 0 8 -3 -22 4 }}


=== Vasca ===
Optimal tunings:
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.  
* WE: ~63/50 = 399.9798{{c}}, ~693/640 = 137.7643{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~693/640 = 137.7716{{c}}


Subgroup: 2.3.5.7.11
{{Optimal ET sequence|legend=0| 87, 183, 270, 1263, 1533, 1803c }}


Comma list: 3025/3024, 102487/102400, {{monzo| 39 -4 -11 -5 2 }}
Badness (Sintel): 0.763


Mapping: {{mapping| 21 4 39 98 58 | 0 6 2 -8 3 3 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


: mapping generators: ~1323/1280, ~6615/5632
Comma list: 676/675, 1001/1000, 3025/3024, 4096/4095


Optimal tuning (CTE): ~1323/1280 = 1＼21, ~6615/5632 = 278.8998
Mapping: {{mapping| 3 2 8 16 9 8 | 0 8 -3 -22 4 9 }}


{{Optimal ET sequence|legend=1| 168, 357, 525, 882, 1407, 2289e }}
Optimal tunings:
* WE: ~63/50 = 399.9921{{c}}, ~13/12 = 137.7743{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~13/12 = 137.7770{{c}}


Badness: 0.0949
{{Optimal ET sequence|legend=0| 87, 183, 270 }}


==== 13-limit ====
Badness (Sintel): 0.643
Subgroup: 2.3.5.7.11.13


Comma list: 3025/3024, 4096/4095, 14641/14625, 85750/85683
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Mapping: {{mapping| 21 4 39 98 58 107 | 0 6 2 -8 3 -6 }}
Comma list: 676/675, 715/714, 936/935, 1001/1000, 4096/4095


Optimal tuning (CTE): ~336/325 = 1＼21, ~168/143 = 278.9058
Mapping: {{mapping| 3 2 8 16 9 8 4 | 0 8 -3 -22 4 9 24 }}


{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
Optimal tunings:
* WE: ~34/27 = 399.9776{{c}}, ~13/12 = 137.7535{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~13/12 = 137.7608{{c}}


Badness: 0.0551
{{Optimal ET sequence|legend=0| 87, 183, 270, 453 }}


==== 17-limit ====
Badness (Sintel): 0.869
Subgroup: 2.3.5.7.11.13.17


Comma list: 2601/2600, 3025/3024, 4096/4095, 8624/8619, 14641/14625
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Mapping: {{mapping| 21 4 39 98 58 107 120 | 0 6 2 -8 3 -6 -7 }}
Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1729/1728


Optimal tuning (CTE): ~336/325 = 1＼21, ~168/143 = 278.9036
Mapping: {{mapping| 3 2 8 16 9 8 4 0 | 0 8 -3 -22 4 9 24 37 }}


{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
Optimal tunings:
* WE: ~34/27 = 399.9804{{c}}, ~13/12 = 137.7602{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~13/12 = 137.7664{{c}}


Badness: 0.0319
{{Optimal ET sequence|legend=0| 87, 183, 270 }}


==== 19-limit ====
Badness (Sintel): 0.928
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 2376/2375, 2601/2600, 2926/2925, 3025/3024, 3213/3211, 4096/4095
== Zinc ==
{{See also| 10th-octave temperaments #Neon }}


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


Optimal tuning (CTE): ~336/325 = 1＼21, ~168/143 = 278.9036
[[Subgroup]]: 2.3.5.7


{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
[[Comma list]]: 250047/250000, {{monzo| -53 -12 2 24 }}


Badness: 0.0270
{{Mapping|legend=1| 30 2 15 66 | 0 5 6 2 }}
: mapping generators: ~53747712/52521875, ~216/175


==== 23-limit ====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13.17.19.23
* [[WE]]: ~53747712/52521875 = 40.0002{{c}}, ~216/175 = 364.3890{{c}} (~{{monzo| 21 3 1 -10 }} = 4.3869{{c}})
: [[error map]]: {{val| +0.007 -0.009 +0.024 -0.032 }}
* [[CWE]]: ~53747712/52521875 = 40.0000{{c}}, ~216/175 = 364.3879{{c}} (~{{monzo| 21 3 1 -10 }} = 4.3879{{c}})
: error map: {{val| 0.000 -0.015 +0.014 -0.050 }}


Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211
{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650 }}


Mapping: {{mapping| 21 4 39 98 58 107 120 16 95 | 0 -6 -2 8 -3 6 7 -15 0 }}
[[Badness]] (Sintel): 1.88


Optimal tuning (CTE): ~336/325 = 1＼21, ~168/143 = 278.8971
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
Comma list: 9801/9800, 151263/151250, {{monzo| -27 -6 4 6 3 }}


Badness: 0.0199
Mapping: {{mapping| 30 2 15 66 122 | 0 5 6 2 -2 }}


== Magnesium ==
Optimal tunings:
: ''For the 5-limit version, see [[12th-octave temperaments#Magnesium (5-limit)]].
* WE: ~18865/18432 = 40.0005{{c}}, ~216/175 = 364.3881{{c}} (~385/384 = 4.3837{{c}})
* CWE: ~18865/18432 = 40.0000{{c}}, ~216/175 = 364.3849{{c}} (~385/384 = 4.3849{{c}})


Magnesium is named after element 12 for being period 12, however, it's not an extension of the [[atomic]] – the associated comma is {{monzo| -157 -24 84 }} in the 5-limit and the 7 generators together with [[12edo]] major second reach the just perfect fifth, [[3/2]].
{{Optimal ET sequence|legend=0| 270, 1110, 1380, 1650, 1920, 2190 }}


[[Subgroup]]: 2.3.5.7
Badness (Sintel): 0.727


[[Comma list]]: 250047/250000, {{monzo| -59 2 18 5 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Mapping|legend=1| 12 2 23 58 | 0 7 2 -10 }}
Comma list: 9801/9800, 10648/10647, 105644/105625, 196625/196608


: mapping generators: ~138915/131072, 3145728/2734375
Mapping: {{mapping| 30 2 15 66 122 193 | 0 5 6 2 -2 -9 }}


Optimal tuning (CTE): ~138915/131072 = 1\12, ~3145728/2734375 = 243.130
Optimal tunings:
* WE: ~351/343 = 40.0003{{c}}, ~216/175 = 364.3894{{c}} (~385/384 = 4.3865{{c}})
* CWE: ~351/343 = 40.0000{{c}}, ~216/175 = 364.3867{{c}} (~385/384 = 4.3867{{c}})


{{Optimal ET sequence|legend=1| 84, 360d, 444, 528, 612, 1920, 2532, 10740cd, 13272bcdd, 15804bcdd, 18336bcddd }}
{{Optimal ET sequence|legend=0| 270, 1380, 1650, 1920, 2190, 4650, 6840e, 11490de }}


Badness: 0.0964
Badness (Sintel): 0.640


== Chromium ==
==== 2.3.5.7.11.13.19 subgroup (neozinc) ====
: ''For the 5-limit version, see [[24th-octave temperaments#Chromium (5-limit)]].
Subgroup: 2.3.5.7.11.13.19


[[Chromium]] is defined by associating the porcupine comma [[250/243]] to the 24th of an octave. Named after the 24th element for being period 24.
Comma list: 5929/5928, 6860/6859, 9801/9800, 10241/10240, 89376/89375


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 30 2 15 66 122 193 91 | 0 5 6 2 -2 -9 4 }}


[[Comma list]]: 250047/250000, 49589822592/49433168575
Optimal tunings:  
* WE: ~175/171 = 40.0002{{c}}, ~216/175 = 364.3885{{c}} (~400/399 = 4.3862{{c}})
* CWE: ~175/171 = 40.0000{{c}}, ~216/175 = 364.3864{{c}} (~400/399 = 4.3864{{c}})


{{Mapping|legend=1| 24 1 -6 18 | 0 3 5 4 }}
{{Optimal ET sequence|legend=0| 270, 1380, 1650, 1920, 2190, 4650, 6840e }}


: mapping generators: ~250/243, ~10/7
Badness (Sintel): 0.477


[[Optimal tuning]] ([[CTE]]): ~250/243 = 1\24, ~10/7 = 617.2710
=== Neodymium ===
Neodymium (540 & 1920) splits the period into 1/60-octave halves for prime 17.  


{{Optimal ET sequence|legend=1| 72, …, 480, 552, 624, 1320, 1944d, 3264d }}
Subgroup: 2.3.5.7.11.13.17


[[Badness]]: 0.139
Comma list: 9801/9800, 10648/10647, 31213/31212, 105644/105625, 196625/196608


=== 11-limit ===
Mapping: {{mapping| 60 4 30 132 244 386 391 | 0 5 6 2 -2 -9 -8 }}
Subgroup: 2.3.5.7.11
: mapping generators: ~612/605, ~216/175


Comma list: 9801/9800, 46656/46585, 250047/250000
Optimal tunings:  
* WE: ~612/605 = 20.0002{{c}}, ~216/175 = 364.3894{{c}}
* CWE: ~612/605 = 20.0000{{c}}, ~216/175 = 364.3864{{c}}


Mapping: {{mapping| 24 1 -6 18 46 | 0 3 5 4 3 }}
{{Optimal ET sequence|legend=0| 540, 1380, 1920, 2460, 4380, 6840e }}


Optimal tuning (CTE): ~250/243 = 1\24, ~10/7 = 617.2597
Badness (Sintel): 1.23


{{Optimal ET sequence|legend=1| 72, …, 480, 552, 624 }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0398
Comma list: 5929/5928, 6860/6859, 9801/9800, 10241/10240, 23409/23408, 89376/89375


=== 13-limit ===
Mapping: {{mapping| 60 4 30 132 244 386 391 182 | 0 5 6 2 -2 -9 -8 4 }}
Subgroup: 2.3.5.7.11.13


Comma list: 1716/1715, 2080/2079, 34398/34375, 39366/39325
Optimal tunings:  
* WE: ~612/605 = 20.0001{{c}}, ~216/175 = 364.3883{{c}}
* CWE: ~612/605 = 20.0000{{c}}, ~216/175 = 364.3860{{c}}


Mapping: {{mapping| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 }}
{{Optimal ET sequence|legend=0| 540, 1380, 1920, 2460, 4380, 6840e }}


Optimal tuning (CTE): ~250/243 = 1\24, ~10/7 = 617.2869
Badness (Sintel): 1.02


{{Optimal ET sequence|legend=1| 72, …, 480f, 552, 624, 1176de, 1800cdee }}
== Magnesium ==
: ''For the 5-limit version, see [[12th-octave temperaments #Magnesium (5-limit)]].


Badness: 0.0293
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.  


=== 17-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17


Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 11016/11011
[[Comma list]]: 250047/250000, {{monzo| -59 2 18 5 }}


Mapping: {{mapping| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 9 }}
{{Mapping|legend=1| 12 2 23 58 | 0 7 2 -10 }}
: mapping generators: ~138915/131072, ~3145728/2734375


Optimal tuning (CTE): ~35/34 = 1\24, ~10/7 = 617.2732
[[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| 72, …, 480fgg, 552g, 624 }}
{{Optimal ET sequence|legend=1| 84, 360d, 444, 528, 612, 1920, 2532, 10740cd, 13272bcdd, 15804bcdd, 18336bcddd }}


Badness: 0.0209
Badness (Sintel): 2.44


== Zinc ==
== Poe ==
{{See also| 10th-octave temperaments #Neon }}
Named by [[Tristan Bay]] in 2025, poe may be described as the {{nowrap| 60 & 270 }} temperament.
{{See also| 60th-octave temperaments #Neodymium }}


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


[[Comma list]]: 250047/250000, {{monzo| -53 -12 2 24 }}
[[Comma list]]: 250047/250000, {{monzo| 15 -16 -4 7 }}


{{Mapping|legend=1| 30 2 15 66 | 0 5 6 2 }}
{{Mapping|legend=1| 30 0 -73 -106 | 0 1 3 4 }}
: mapping generators: ~2240/2187, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~53747712/52521875 = 1\30, ~216/175 = 364.3896 (~{{monzo| 21 3 1 -10 }} = 4.3896)
* [[WE]]: ~2240/2187 = 39.9982{{c}}, ~3/2 = 702.1533{{c}}
* [[CWE]]: ~53747712/52521875 = 1\30, ~216/175 = 364.3879 (~{{monzo| 21 3 1 -10 }} = 4.3879)
: [[error map]]: {{val| -0.055 +0.143 +0.115 -0.238 }}
* [[CWE]]: ~2240/2187 = 40.0000{{c}}, ~3/2 = 702.1656{{c}}
: error map: {{val| 0.000 +0.211 +0.183 -0.163 }}


{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650 }}
{{Optimal ET sequence|legend=1| 60, 150cd, 210, 270, 1950, 2220, 2490, 2760b, 3030bc, 3300bc }}


[[Badness]]: 0.0742
[[Badness]] (Sintel): 2.90


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


Comma list: 9801/9800, 151263/151250, {{monzo| -27 -6 4 6 3 }}
Comma list: 9801/9800, 19712/19683, 151263/151250


Mapping: {{mapping| 30 2 15 66 122 | 0 5 6 2 -2 }}
Mapping: {{mapping| 30 0 -73 -106 -134 | 0 1 3 4 5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~18865/18432 = 1\30, ~216/175 = 364.3886 (~385/384 = 4.3886)
* WE: ~45/44 = 39.9976{{c}}, ~3/2 = 702.1955{{c}}
* CWE: ~18865/18432 = 1\30, ~216/175 = 364.3849 (~385/384 = 4.3849)
* CWE: ~45/44 = 40.0000{{c}}, ~3/2 = 702.2129{{c}}


Optimal ET sequence: {{Optimal ET sequence| 270, 1110, 1380, 1650, 1920, 2190 }}
{{Optimal ET sequence|legend=0| 60e, …, 210e, 270 }}


Badness: 0.0220
Badness (Sintel): 1.31


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


Comma list: 9801/9800, 10648/10647, 105644/105625, 196625/196608
Comma list: 1001/1000, 4225/4224, 4459/4455, 19712/19683


Mapping: {{mapping| 30 2 15 66 122 193 | 0 5 6 2 -2 -9 }}
Mapping: {{mapping| 30 0 -73 -106 -134 111 | 0 1 3 4 5 0 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~351/343 = 1\30, ~216/175 = 364.3879 (~385/384 = 4.3879)
* WE: ~45/44 = 40.0008{{c}}, ~3/2 = 702.1778{{c}}
* CWE: ~351/343 = 1\30, ~216/175 = 364.3867 (~385/384 = 4.3867)
* CWE: ~45/44 = 40.0000{{c}}, ~3/2 = 702.1699{{c}}


Optimal ET sequence: {{Optimal ET sequence| 270, 1380, 1650, 1920, 2190, 4650, 6840e, 11490de }}
{{Optimal ET sequence|legend=0| 60e, 210e, 270, 1410ef, 1680ef }}


Badness: 0.0155
Badness (Sintel): 1.19


=== 2.3.5.7.11.13.19 subgroup (neozinc) ===
== Chromium ==
Subgroup: 2.3.5.7.11.13.19
: ''For the 5-limit version, see [[24th-octave temperaments #Chromium (5-limit)]].


Comma list: 5929/5928, 6860/6859, 9801/9800, 10241/10240, 89376/89375
[[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.


Mapping: {{mapping| 30 2 15 66 122 193 91 | 0 5 6 2 -2 -9 4 }}
Optimal tunings:
* CTE: ~175/171 = 1\30, ~216/175 = 364.3876 (~400/399 = 4.3876)
* CWE: ~175/171 = 1\30, ~216/175 = 364.3864 (~400/399 = 4.3864)
Optimal ET sequence: {{Optimal ET sequence| 270, 1380, 1650, 1920, 2190, 4650, 6840e }}
Badness: 0.00921
== Poe ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, {{monzo| 15 -16 -4 7 }}
[[Comma list]]: 250047/250000, 49589822592/49433168575


{{Mapping|legend=1| 30 0 -73 -106 | 0 1 3 4 }}
{{Mapping|legend=1| 24 1 -6 18 | 0 3 5 4 }}
: mapping generators: ~250/243, ~10/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2240/2187 = 1\30, ~3/2 = 702.141 (~81/80 = 22.141)
* [[WE]]: ~250/243 = 49.9992{{c}}, ~10/7 = 617.2714{{c}}
* [[CWE]]: ~2240/2187 = 1\30, ~3/2 = 702.166 (~81/80 = 22.166)
: [[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| 60, 210, 270 }}
{{Optimal ET sequence|legend=1| 72, …, 480, 552, 624, 1320, 1944d, 3264d }}


[[Badness]] (Sintel): 2.900
[[Badness]] (Sintel): 3.52


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


Comma list: 9801/9800, 19712/19683, 250047/250000
Comma list: 9801/9800, 46656/46585, 151263/151250


Mapping: {{mapping| 30 0 -73 -106 -134 | 0 1 3 4 5 }}
Mapping: {{mapping| 24 1 -6 18 46 | 0 3 5 4 3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~45/44 = 1\30, ~3/2 = 702.182 (~81/80 = 22.182)
* WE: ~250/243 = 49.9972{{c}}, ~10/7 = 617.2639{{c}}
* CWE: ~45/44 = 1\30, ~3/2 = 702.213 (~81/80 = 22.213)
* CWE: ~250/243 = 50.0000{{c}}, ~10/7 = 617.2823{{c}}


{{Optimal ET sequence|legend=1| 60e, 210e, 270 }}
{{Optimal ET sequence|legend=0| 72, …, 480, 552, 624 }}


Badness (Sintel): 1.307
Badness (Sintel): 1.32


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


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


Mapping: {{mapping| 30 0 -73 -106 -134 111 | 0 1 3 4 5 0 }}
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:  
Optimal tunings:  
* CTE: ~45/44 = 1\30, ~3/2 = 702.182 (~81/80 = 22.182)
* WE: ~35/34 = 49.9959{{c}}, ~10/7 = 617.2685{{c}}
* CWE: ~45/44 = 1\30, ~3/2 = 702.170 (~81/80 = 22.170)
* CWE: ~35/34 = 50.0000{{c}}, ~10/7 = 617.3015{{c}}


{{Optimal ET sequence|legend=1| 60e, 210e, 270 }}
{{Optimal ET sequence|legend=0| 72, …, 480fgg, 552g, 624 }}


Badness (Sintel): 1.194
Badness (Sintel): 1.06


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Landscape microtemperaments| ]] <!-- main article -->
[[Category:Landscape microtemperaments| ]] <!-- main article -->
[[Category:Landscape| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Microtemperaments]]

Latest revision as of 17:41, 8 June 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: 2.3.5.7.11.13.17

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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