Landscape microtemperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Zhenlige (talk | contribs)
autopatrolled
318 edits
→Chromium: fix typo
Tags: Mobile edit Mobile web edit
- sextile (more properly addressed in the garischismic clan)
 
(38 intermediate revisions by 7 users not shown)
Line 1: Line 1:
'''Landscape microtemperaments''' are [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[landscape comma]] ({{monzo| -4 6 -6 3 }} = 250047/250000).
{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] '''landscape microtemperaments''', which [[tempering out|temper out]] the [[landscape comma]] ({{monzo|legend=1| -4 6 -6 3 }}, [[ratio]]: 250047/250000). For the [[rank-3 temperament]], see [[Landscape family #Landscape]].  


Landscape rank-2 temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Augene|Augmented family]]
* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]
* ''[[Compton]]'' (+225/224) → [[Compton family #Compton|Compton 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]]
* ''[[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) → [[Ragismic microtemperaments #Ennealimmal|Ragismic microtemperaments]]
* ''[[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]]
* [[Mutt]] (+65625/65536) → [[Horwell temperaments #Mutt|Horwell temperaments]]
* ''[[Term]]'' (+32805/32768) → [[Schismatic family #Term|Schismatic family]]
* ''[[Caleb]]'' (+33075/32768) → [[Mirwomo temperaments #Caleb|Mirwomo temperaments]]
* [[Mutt temperament|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 }}) → [[Tricot family #Tritricot|Tricot 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, terture, akjayland and terture.
Considered below are septichrome, akjayland, pnict, atomic, slendscape, avicenna, zinc, magnesium, poe, and chromium in the order of increasing [[badness]].
 
== Sextile ==
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
 
{{Multival|legend=1| 6 -36 -84 -71 -150 -94 }}


[[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


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: 2.3.5.7.11.13.17.19.23
 
Comma list: 936/935, 1701/1700, 1716/1715, 1863/1862, 2024/2023, 2025/2023, 2376/2375
 
Mapping: {{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{{c}}, ~162/133 = 340.3576{{c}} (~88/85 = 59.6081{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~162/133 = 340.3844{{c}} (~88/85 = 59.6156{{c}})
 
{{Optimal ET sequence|legend=0| 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, {{monzo| 43 -1 -13 -4 }}
 
{{Mapping|legend=1| 21 1 38 102 | 0 3 1 -4 }}
: mapping generators: ~1323/1280, ~131072/91875
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
 
[[Badness]] (Sintel): 0.838
 
=== 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.
 
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 102487/102400, {{monzo| 39 -4 -11 -5 2 }}
 
Mapping: {{mapping| 21 4 39 98 58 | 0 6 2 -8 3 3 }}
: mapping generators: ~1323/1280, ~6615/5632
 
Optimal tunings:
* WE: ~1323/1280 = 57.1436{{c}}, ~6615/5632 = 278.9017{{c}}
* CWE: ~1323/1280 = 57.1429{{c}}, ~6615/5632 = 278.8985{{c}}
 
{{Optimal ET sequence|legend=0| 168, 357, 525, 882, 1407, 2289e }}
 
Badness (Sintel): 3.14
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2296
* WE: ~336/325 = 57.1426{{c}}, ~168/143 = 278.9047{{c}}
* CWE: ~55/49 = 1\6, ~3/2 = 702.2285
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9060{{c}}


{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}


Badness: 0.0391
Badness (Sintel): 2.28


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


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


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2264
* WE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9037{{c}}
* CWE: ~55/49 = 1\6, ~3/2 = 702.2207
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9036{{c}}


{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}


Badness: 0.0384
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: 715/714, 1001/1000, 1216/1215, 1701/1700, 1729/1728, 2912/2907
Comma list: 2376/2375, 2601/2600, 2926/2925, 3025/3024, 3213/3211, 4096/4095


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2266
* WE: ~336/325 = 57.1425{{c}}, ~168/143 = 278.8960{{c}}
* CWE: ~55/49 = 1\6, ~3/2 = 702.2208
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.8976{{c}}


{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}


Badness: 0.0252
Badness (Sintel): 1.64


== Septichrome ==
==== 23-limit ====
[[Subgroup]]: 2.3.5.7
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


[[Comma list]]: 250047/250000, 2460375/2458624
Mapping: {{mapping| 21 4 39 98 58 107 120 16 95 | 0 6 2 -8 3 -6 -7 15 0 }}


{{Mapping|legend=1| 3 3 1 0 | 0 5 17 24 }}
Optimal tunings:
* WE: ~336/325 = 57.1422{{c}}, ~168/143 = 278.8949{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.8980{{c}}


{{Multival|legend=1| 15 51 72 46 72 24 }}
{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~243/224 = 140.367
Badness (Sintel): 1.43


{{Optimal ET sequence|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}
== Pnict ==
[[Subgroup]]: 2.3.5.7


[[Badness]]: 0.016814
[[Comma list]]: 250047/250000, 2100875/2097152


=== Semiseptichrome ===
{{Mapping|legend=1| 3 -3 1 12 | 0 13 10 -6 }}
Subgroup: 2.3.5.7.11
: mapping generators: ~63/50, ~147/128


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


Mapping: {{mapping| 6 1 -15 -24 -32 | 0 5 17 24 31 }}
{{Optimal ET sequence|legend=1| 15, 141, 156, 171, 2409cd, 2580cd, …, 4461bccddd, 4632bccddd }}


: mapping generators: ~55/49, ~375/308
[[Badness]] (Sintel): 1.16


Optimal tuning (CTE): ~55/49 = 1\6, ~375/308 = 340.3687 (~1760/1701 = 59.6313)
== Atomic ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Atomic]].''


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


Badness: 0.0194
[[Subgroup]]: 2.3.5.7


==== 13-limit ====
[[Comma list]]: 250047/250000, {{monzo| -55 30 2 1 }}
Subgroup: 2.3.5.7.11.13


Comma list: 1716/1715, 2080/2079, 34398/34375, 85293/85184
{{Mapping|legend=1| 12 0 161 338 | 0 1 -7 -16 }}


Mapping: {{mapping| 6 1 -15 -24 -32 -68 | 0 5 17 24 31 53 }}
[[Optimal tuning]]s:  
* [[WE]]: ~30375/28672 = 99.999866{{c}}, ~3/2 = 701.948670{{c}} (~32805/32768 = 1.949605{{c}})
: [[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 tuning (CTE): ~55/49 = 1\6, ~375/308 = 340.3778 (~121/117 = 59.6222)
{{Optimal ET sequence|legend=1| 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}


Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}
[[Badness]] (Sintel): 1.16


Badness: 0.0397
=== 11-limit ===
Subgroup: 2.3.5.7.11


==== 17-limit ====
Comma list: 9801/9800, 151263/151250, 184549376/184528125
Subgroup: 2.3.5.7.11.13


Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 61965/61952
Mapping: {{mapping| 12 0 161 338 517 | 0 1 -7 -16 -25 }}


Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 | 0 5 17 24 31 53 15 }}
Optimal tunings:  
* WE: ~30375/28672 = 99.999760{{c}}, ~3/2 = 701.946301{{c}} (~32805/32768 = 1.947983{{c}})
* CWE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.948121{{c}} (~32805/32768 = 1.948121{{c}})


Optimal tuning (CTE): ~55/49 = 1\6, ~375/308 = 340.3763 (~88/85 = 59.6237)
{{Optimal ET sequence|legend=0| 12, …, 600e, 612, 1236, 1848 }}


Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}
Badness (Sintel): 0.530


Badness: 0.0272
==== Minutes ====
Minutes (600e & 2460) splits the 1/12-octave period into five 1/60-octave parts.  


==== 19-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 23409/23408
Comma list: 9801/9800, 151263/151250, 371293/371250, 184549376/184528125
 
Mapping: {{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{{c}}, ~3/2 = 701.946866{{c}}
* CWE: ~2704/2673 = 20.000000{{c}}, ~3/2 = 701.948111{{c}}


Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 | 0 5 17 24 31 53 15 -5 }}
{{Optimal ET sequence|legend=0| 600e, …, 1860, 2460, 6780, 9240 }}


Optimal tuning (CTE): ~55/49 = 1\6, ~162/133 = 340.3767 (~88/85 = 59.6233)
Badness (Sintel): 2.82


Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0222
Comma list: 9801/9800, 12376/12375, 28561/28560, 151263/151250, 11275335/11275264


==== 23-limit ====
Mapping: {{mapping| 60 0 805 1690 2585 1173 1957 | 0 1 7 -16 -25 -10 -18 }}
Subgroup: 2.3.5.7.11.13.17.19.23
 
Optimal tunings:
* WE: ~2704/2673 = 19.999974{{c}}, ~3/2 = 701.946956{{c}}
* CWE: ~2704/2673 = 20.000000{{c}}, ~3/2 = 701.947926{{c}}


Comma list: 936/935, 1701/1700, 1716/1715, 1863/1862, 2024/2023, 2025/2023, 2376/2375
{{Optimal ET sequence|legend=0| 600e, …, 1860, 2460, 6780, 9240 }}


Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 -12 | 0 5 17 24 31 53 15 -5 23 }}
Badness (Sintel): 1.67


Optimal tuning (CTE): ~55/49 = 1\6, ~162/133 = 340.3757 (~88/85 = 59.6243)
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}
Comma list: 9801/9800, 12376/12375, 12636/12635, 23409/23408, 28561/28560, 151263/151250


Badness: 0.0160
Mapping: {{mapping| 60 60 385 730 1085 573 877 -61 | 0 1 7 -16 -25 -10 -18 9 }}


== Pnict ==
Optimal tunings:
[[Subgroup]]: 2.3.5.7
* WE: ~2704/2673 = 19.999962{{c}}, ~3/2 = 701.946354{{c}}
* CWE: ~2704/2673 = 20.000000{{c}}, ~3/2 = 701.947751{{c}}


[[Comma list]]: 250047/250000, 2100875/2097152
{{Optimal ET sequence|legend=0| 600e, 1860, 2460, 4320, 6780, 9240 }}


{{Mapping|legend=1| 3 10 11 6 | 0 -13 -10 6 }}
Badness (Sintel): 1.50


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


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~192/175 = 161.399
Subgroup: 2.3.5.7.11.13


{{Optimal ET sequence|legend=1| 15, 141, 156, 171, 2409cd, 2580cd, …, 4461bccddd, 4632bccddd }}
Comma list: 9801/9800, 151263/151250, 184549376/184528125, 308915776/308828625


[[Badness]]: 0.045660
Mapping: {{mapping| 72 0 966 2028 3102 2777 | 0 1 -7 -16 -25 -22 }}
: mapping generators: ~105/104, ~3


== Domain ==
Optimal tunings:
{{See also| Minortonic family }}
* WE: ~105/104 = 16.666629{{c}}, ~3/2 = 701.945668{{c}}
* CWE: ~105/104 = 16.666667{{c}}, ~3/2 = 701.947388{{c}}


[[Subgroup]]: 2.3.5.7
{{Optimal ET sequence|legend=0| 1224, 3096e, 4320, 5544, 9864c }}


[[Comma list]]: 250047/250000, 645700815/645657712
Badness (Sintel): 4.76


{{Mapping|legend=1| 3 -3 -9 -8 | 0 17 35 36 }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


{{Multival|legend=1| 51 105 108 48 28 -44 }}
Comma list: 9801/9800, 12376/12375, 151263/151250, 1713660/1713481, 97144749/97140736


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467
Mapping: {{mapping| 72 0 966 2028 3102 2777 979 | 0 1 -7 -16 -25 -22 -6 }}
: mapping generators: ~105/104, ~3


{{Optimal ET sequence|legend=1| 171, 1164, 1335, 1506, 1677, 1848, 2019, 2190, 11943, 13962, 15981, 18000, 20019, 22038 }}
Optimal tunings:
* WE: ~105/104 = 16.666615{{c}}, ~3/2 = 701.944968{{c}}
* CWE: ~105/104 = 16.666667{{c}}, ~3/2 = 701.947293{{c}}


[[Badness]]: 0.013979
{{Optimal ET sequence|legend=0| 1224, 3096e, 4320, 5544, 9864c }}


== Avicenna ==
Badness (Sintel): 2.73
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 29360128/29296875
===== 2.3.5.7.11.13.17.31 subgroup =====
Subgroup: 2.3.5.7.11.13.17.31


{{Mapping|legend=1| 3 2 8 16 | 0 8 -3 -22 }}
Comma list: 9801/9800, 10881/10880, 12376/12375, 57629/57624, 179712/179707, 61456384/61448625


{{Multival|legend=1| 24 -9 -66 -70 -172 -128 }}
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, ~1024/945 = 137.768
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| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}
{{Optimal ET sequence|legend=0| 1224, 3096e, 4320, 5544 }}


[[Badness]]: 0.062187
Badness (Sintel): 2.36


=== 11-limit ===
===== 2.3.5.7.11.13.17.31.37 subgroup =====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17.31.37


Comma list: 3025/3024, 5632/5625, 102487/102400
Comma list: 9801/9800, 10881/10880, 12376/12375, 16576/16575, 57629/57624, 93093/93092, 179712/179707


Mapping: {{mapping| 3 2 8 16 9 | 0 8 -3 -22 4 }}
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, ~693/640 = 137.771
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| 87, 183, 270, 1263, 1533, 1803c, 2073c }}
{{Optimal ET sequence|legend=0| 1224, 3096el, 4320, 5544 }}


Badness: 0.023085
Badness (Sintel): 1.98


=== 13-limit ===
== Slendscape ==
Subgroup: 2.3.5.7.11.13
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.


Comma list: 676/675, 1001/1000, 3025/3024, 4096/4095
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 3 2 8 16 9 8 | 0 8 -3 -22 4 9 }}
[[Comma list]]: 250047/250000, 12884901888/12867859375


Optimal tuning (POTE): ~63/50 = 1\3, ~13/12 = 137.777
{{Mapping|legend=1| 15 0 17 54 | 0 4 3 -2 }}
: mapping generators: ~8575/8192, ~1152/875


{{Optimal ET sequence|legend=1| 87, 183, 270 }}
[[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 }}


Badness: 0.015557
{{Optimal ET sequence|legend=1| 15, 240, 255, 270, 795, 1065, 1335, 2400 }}


=== 17-limit ===
[[Badness]] (Sintel): 1.47
Subgroup: 2.3.5.7.11.13.17


Comma list: 676/675, 715/714, 1001/1000, 3025/3024, 4096/4095
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: {{mapping| 3 2 8 16 9 8 4 | 0 8 -3 -22 4 9 24 }}
Comma list: 3025/3024, 102487/102400, 180224/180075


Optimal tuning (POTE): ~63/50 = 1\3, ~13/12 = 137.777
Mapping: {{mapping| 15 0 17 54 40 | 0 4 3 -2 2 }}


{{Optimal ET sequence|legend=1| 87, 183, 270 }}
Optimal tunings:
* WE: ~22/21 = 79.9991{{c}}, ~968/735 = 475.4915{{c}}
* CWE: ~22/21 = 80.0000{{c}}, ~968/735 = 475.4955{{c}}


Badness: 0.015557
{{Optimal ET sequence|legend=0| 15, 240, 255, 270, 795, 1065, 2400e }}


=== 19-limit ===
Badness (Sintel): 0.868
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 3025/3024, 4096/4095
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: {{mapping| 3 2 8 16 9 8 4 0 | 0 8 -3 -22 4 9 24 37 }}
Comma list: 1716/1715, 3025/3024, 4096/4095, 14641/14625


Optimal tuning (POTE): ~63/50 = 1\3, ~13/12 = 137.777
Mapping: {{mapping| 15 0 17 54 40 109 | 0 4 3 -2 2 -9 }}


{{Optimal ET sequence|legend=1| 87, 183, 270 }}
Optimal tunings:
* WE: ~22/21 = 79.9993{{c}}, ~154/117 = 475.4902{{c}}
* CWE: ~22/21 = 80.0000{{c}}, ~154/117 = 475.4943{{c}}


Badness: 0.015557
{{Optimal ET sequence|legend=0| 255, 270, 795, 1065 }}


== Terture ==
Badness (Sintel): 0.877
{{See also| Vulture family }}


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


[[Comma list]]: 250047/250000, 359661568/358722675
[[Comma list]]: 250047/250000, 29360128/29296875


{{Mapping|legend=1| 3 4 3 2 | 0 4 21 34 }}
{{Mapping|legend=1| 3 2 8 16 | 0 8 -3 -22 }}
: mapping generators: ~63/50, ~1024/945


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~392/375 = 75.555
[[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| 111, 159, 270 }}
{{Optimal ET sequence|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}


[[Badness]]: 0.087156
[[Badness]] (Sintel): 1.57


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


Comma list: 3025/3024, 19712/19683, 102487/102400
Comma list: 3025/3024, 5632/5625, 102487/102400


Mapping: {{mapping| 3 4 3 2 10 | 0 4 21 34 2 }}
Mapping: {{mapping| 3 2 8 16 9 | 0 8 -3 -22 4 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~392/375 = 75.550
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=1| 111, 159, 270, 1239, 1509, 1779, 2049, 2319 }}
{{Optimal ET sequence|legend=0| 87, 183, 270, 1263, 1533, 1803c }}


Badness: 0.029326
Badness (Sintel): 0.763


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


Mapping: {{mapping| 3 4 3 2 10 6 | 0 4 21 34 2 27 }}
Mapping: {{mapping| 3 2 8 16 9 8 | 0 8 -3 -22 4 9 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~117/112 = 75.553
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=1| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 87, 183, 270 }}


Badness: 0.018647
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: 676/675, 715/714, 936/935, 1001/1000, 4928/4913
Comma list: 676/675, 715/714, 936/935, 1001/1000, 4096/4095


Mapping: {{mapping| 3 4 3 2 10 6 10 | 0 4 21 34 2 27 12 }}
Mapping: {{mapping| 3 2 8 16 9 8 4 | 0 8 -3 -22 4 9 24 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~117/112 = 75.560
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=1| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 87, 183, 270, 453 }}


Badness: 0.018705
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: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1617/1615
Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1729/1728


Mapping: {{mapping| 3 4 3 2 10 6 10 5 | 0 4 21 34 2 27 12 41 }}
Mapping: {{mapping| 3 2 8 16 9 8 4 0 | 0 8 -3 -22 4 9 24 37 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~95/91 = 75.560
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=1| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 87, 183, 270 }}


Badness: 0.013902
Badness (Sintel): 0.928


=== 23-limit ===
== Zinc ==
Subgroup: 2.3.5.7.11.13.17.19.23
{{See also| 10th-octave temperaments #Neon }}


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


Mapping: {{mapping| 3 4 3 2 10 6 10 5 13 | 0 4 21 34 2 27 12 41 3 }}
[[Subgroup]]: 2.3.5.7


Optimal tuning (POTE): ~63/50 = 1\3, ~24/23 = 75.548
[[Comma list]]: 250047/250000, {{monzo| -53 -12 2 24 }}


{{Optimal ET sequence|legend=1| 111, 159, 270 }}
{{Mapping|legend=1| 30 2 15 66 | 0 5 6 2 }}
: mapping generators: ~53747712/52521875, ~216/175


Badness: 0.014915
[[Optimal tuning]]s:  
* [[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 }}


== Akjayland ==
{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650 }}
{{See also| 21st-octave temperaments }}


Akjayland tempers out the [[akjaysma]] in addition to landscape comma, and thereby features a period of 1\21.  
[[Badness]] (Sintel): 1.88


[[Subgroup]]: 2.3.5.7
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Comma list]]: 250047/250000, {{monzo| 43 -1 -13 -4 }}
Comma list: 9801/9800, 151263/151250, {{monzo| -27 -6 4 6 3 }}


{{Mapping|legend=1| 21 1 38 102 | 0 3 1 -4 }}
Mapping: {{mapping| 30 2 15 66 122 | 0 5 6 2 -2 }}


: mapping generators: ~1323/1280, ~131072/91875
Optimal tunings:  
* 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}})


[[Optimal tuning]] ([[CTE]]): ~1323/1280 = 1\21, ~131072/91875 = 614.9354
{{Optimal ET sequence|legend=0| 270, 1110, 1380, 1650, 1920, 2190 }}


{{Optimal ET sequence|legend=1| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
Badness (Sintel): 0.727


[[Badness]]: 0.0309
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== Vasca ===
Comma list: 9801/9800, 10648/10647, 105644/105625, 196625/196608
Vasca can be described as the 357 & 525 temperament, extended as high as the 23-limit. It tempers out the {{monzo| 95 0 0 0 0 0 0 0 -21 }}, and sets a stack of twenty-one [[23/16]]'s equal with eleven octaves. The name derives from elements vanadium (23) and scandium (21), since this uses the 23rd harmonic, which itself is extremely well represented in 21edo.


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 30 2 15 66 122 193 | 0 5 6 2 -2 -9 }}


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


Mapping: {{mapping| 21 4 39 98 58 | 0 6 2 -8 3 3 }}
{{Optimal ET sequence|legend=0| 270, 1380, 1650, 1920, 2190, 4650, 6840e, 11490de }}


: mapping generators: ~1323/1280, ~6615/5632
Badness (Sintel): 0.640


Optimal tuning (CTE): ~1323/1280 = 1\21, ~6615/5632 = 278.8998
==== 2.3.5.7.11.13.19 subgroup (neozinc) ====
Subgroup: 2.3.5.7.11.13.19


{{Optimal ET sequence|legend=1| 168, 357, 525, 882, 1407, 2289e }}
Comma list: 5929/5928, 6860/6859, 9801/9800, 10241/10240, 89376/89375


Badness: 0.0949
Mapping: {{mapping| 30 2 15 66 122 193 91 | 0 5 6 2 -2 -9 4 }}


==== 13-limit ====
Optimal tunings:
Subgroup: 2.3.5.7.11.13
* 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}})
Comma list: 3025/3024, 4096/4095, 14641/14625, 85750/85683
 
Mapping: {{mapping| 21 4 39 98 58 107 | 0 6 2 -8 3 -6 }}


Optimal tuning (CTE): ~336/325 = 1\21, ~168/143 = 278.9058
{{Optimal ET sequence|legend=0| 270, 1380, 1650, 1920, 2190, 4650, 6840e }}


{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
Badness (Sintel): 0.477


Badness: 0.0551
=== Neodymium ===
Neodymium (540 & 1920) splits the period into 1/60-octave halves for prime 17.  


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


Mapping: {{mapping| 21 4 39 98 58 107 120 | 0 6 2 -8 3 -6 -7 }}
Mapping: {{mapping| 60 4 30 132 244 386 391 | 0 5 6 2 -2 -9 -8 }}
: mapping generators: ~612/605, ~216/175


Optimal tuning (CTE): ~336/325 = 1\21, ~168/143 = 278.9036
Optimal tunings:
* WE: ~612/605 = 20.0002{{c}}, ~216/175 = 364.3894{{c}}
* CWE: ~612/605 = 20.0000{{c}}, ~216/175 = 364.3864{{c}}


{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=0| 540, 1380, 1920, 2460, 4380, 6840e }}


Badness: 0.0319
Badness (Sintel): 1.23


==== 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: 5929/5928, 6860/6859, 9801/9800, 10241/10240, 23409/23408, 89376/89375
 
Mapping: {{mapping| 60 4 30 132 244 386 391 182 | 0 5 6 2 -2 -9 -8 4 }}


Mapping: {{mapping| 21 4 39 98 58 107 120 16 | 0 6 2 -8 3 -6 -7 15 }}
Optimal tunings:  
* WE: ~612/605 = 20.0001{{c}}, ~216/175 = 364.3883{{c}}
* CWE: ~612/605 = 20.0000{{c}}, ~216/175 = 364.3860{{c}}


Optimal tuning (CTE): ~336/325 = 1\21, ~168/143 = 278.9036
{{Optimal ET sequence|legend=0| 540, 1380, 1920, 2460, 4380, 6840e }}


{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
Badness (Sintel): 1.02


Badness: 0.0270
== Magnesium ==
: ''For the 5-limit version, see [[12th-octave temperaments #Magnesium (5-limit)]].


==== 23-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: 2.3.5.7.11.13.17.19.23


Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 21 4 39 98 58 107 120 16 95 | 0 -6 -2 8 -3 6 7 -15 0 }}
[[Comma list]]: 250047/250000, {{monzo| -59 2 18 5 }}


Optimal tuning (CTE): ~336/325 = 1\21, ~168/143 = 278.8971
{{Mapping|legend=1| 12 2 23 58 | 0 7 2 -10 }}
: mapping generators: ~138915/131072, ~3145728/2734375


{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
[[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 }}


Badness: 0.0199
{{Optimal ET sequence|legend=1| 84, 360d, 444, 528, 612, 1920, 2532, 10740cd, 13272bcdd, 15804bcdd, 18336bcddd }}


== Magnesium ==
Badness (Sintel): 2.44
: ''For the 5-limit version, see [[12th-octave temperaments#Magnesium (5-limit)]].


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]].
== Poe ==
Named by [[Tristan Bay]] in 2025, poe may be described as the {{nowrap| 60 & 270 }} temperament.  


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


[[Comma list]]: 250047/250000, {{monzo| -59 2 18 5 }}
[[Comma list]]: 250047/250000, {{monzo| 15 -16 -4 7 }}
 
{{Mapping|legend=1| 30 0 -73 -106 | 0 1 3 4 }}
: mapping generators: ~2240/2187, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2240/2187 = 39.9982{{c}}, ~3/2 = 702.1533{{c}}
: [[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| 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: {{mapping| 30 0 -73 -106 -134 | 0 1 3 4 5 }}
 
Optimal tunings:
* WE: ~45/44 = 39.9976{{c}}, ~3/2 = 702.1955{{c}}
* CWE: ~45/44 = 40.0000{{c}}, ~3/2 = 702.2129{{c}}
 
{{Optimal ET sequence|legend=0| 60e, …, 210e, 270 }}
 
Badness (Sintel): 1.31
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Mapping|legend=1| 12 2 23 58 | 0 7 2 -10 }}
Comma list: 1001/1000, 4225/4224, 4459/4455, 19712/19683


: mapping generators: ~138915/131072, 3145728/2734375
Mapping: {{mapping| 30 0 -73 -106 -134 111 | 0 1 3 4 5 0 }}


Optimal tuning (CTE): ~138915/131072 = 1\12, ~3145728/2734375 = 243.130
Optimal tunings:
* WE: ~45/44 = 40.0008{{c}}, ~3/2 = 702.1778{{c}}
* CWE: ~45/44 = 40.0000{{c}}, ~3/2 = 702.1699{{c}}


{{Optimal ET sequence|legend=1| 84, 360d, 444, 528, 612, 1920, 2532, 10740cd, 13272bcdd, 15804bcdd, 18336bcddd }}
{{Optimal ET sequence|legend=0| 60e, 210e, 270, 1410ef, 1680ef }}


Badness: 0.0964
Badness (Sintel): 1.19


== Chromium ==
== Chromium ==
: ''For the 5-limit version, see [[24th-octave temperaments#Chromium (5-limit)]].
: ''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. Named after the 24th element for being period 24.
[[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
[[Subgroup]]: 2.3.5.7
Line 524: Line 694:


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


[[Optimal tuning]] ([[CTE]]): ~250/243 = 1\24, ~10/7 = 617.2710
[[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 }}
{{Optimal ET sequence|legend=1| 72, …, 480, 552, 624, 1320, 1944d, 3264d }}


[[Badness]]: 0.139
[[Badness]] (Sintel): 3.52


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


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


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


Optimal tuning (CTE): ~250/243 = 1\24, ~10/7 = 617.2597
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=1| 72, …, 480, 552, 624 }}
{{Optimal ET sequence|legend=0| 72, …, 480, 552, 624 }}


Badness: 0.0398
Badness (Sintel): 1.32


=== 13-limit ===
=== 13-limit ===
Line 553: Line 728:
Mapping: {{mapping| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 }}
Mapping: {{mapping| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 }}


Optimal tuning (CTE): ~250/243 = 1\24, ~10/7 = 617.2869
Optimal 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=1| 72, …, 480f, 552, 624, 1176de, 1800cdee }}
{{Optimal ET sequence|legend=0| 72, …, 480f, 552, 624, 1176de, 1800cdee }}


Badness: 0.0293
Badness (Sintel): 1.21


=== 17-limit ===
=== 17-limit ===
Line 566: Line 743:
Mapping: {{mapping| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 9 }}
Mapping: {{mapping| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 9 }}


Optimal tuning (CTE): ~35/34 = 1\24, ~10/7 = 617.2732
Optimal tunings:
 
* WE: ~35/34 = 49.9959{{c}}, ~10/7 = 617.2685{{c}}
{{Optimal ET sequence|legend=1| 72, …, 480fgg, 552g, 624 }}
* CWE: ~35/34 = 50.0000{{c}}, ~10/7 = 617.3015{{c}}
 
Badness: 0.0209
 
== Zinc ==
{{See also|30th-octave temperaments#Zinc}}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 250047/250000, {{monzo| -45 -24 14 18 }}
 
{{Mapping|legend=1| 30 2 15 66 | 0 5 6 2 }}
 
[[Optimal tuning]] ([[CTE]]): ~53747712/52521875 = 1\30, ~216/175 = 364.389
 
{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650 }}
 
[[Badness]]: 0.0742
=== 11-limit===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 250047/250000, 97869261875/97844723712
 
Mapping: {{mapping| 30 2 15 66 122 | 0 5 6 2 -2 }}
 
Optimal tuning (CTE): ~18865/18432 = 1\30, ~216/175 = 364.389
 
{{Optimal ET sequence|legend=1| 270, 1110, 1380, 1650, 1920, 2190 }}
 
Badness: 0.0220
===13-limit===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 123201/123200, 250047/250000, 1990656/1990625
 
Mapping: {{mapping| 30 2 15 66 122 193 | 0 5 6 2 -2 -9 }}


Optimal tuning (CTE): ~351/343 = 1\30, ~216/175 = 364.389
{{Optimal ET sequence|legend=0| 72, …, 480fgg, 552g, 624 }}


{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650, 6840e, 11490de }}
Badness (Sintel): 1.06


Badness: 0.0150
[[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 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

Retrieved from "https://en.xen.wiki/index.php?title=Landscape_microtemperaments&oldid=230430"