Landscape microtemperaments: Difference between revisions

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'''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 #Augene|Augmented family]]
* ''[[Compton]]'' (+225/224) → [[Compton family #Compton|Compton family]]
* ''[[Compton]]'' (+225/224) → [[Compton family #Compton|Compton family]]
* ''[[Tritikleismic]]'' (+1029/1024) → [[Kleismic family #Tritikleismic|Kleismic family]]
* ''[[Tritikleismic]]'' (+1029/1024) → [[Kleismic family #Tritikleismic|Kleismic family]]
* ''[[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]]
* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Misty]]'' (+3136/3125 or 5120/5103) → [[Misty family]]
* ''[[Misty]]'' (+3136/3125 or 5120/5103) → [[Misty family]]
* ''[[Nessafof]]'' (+6144/6125) → [[Porwell temperaments #Nessafof|Porwell temperaments]]
* ''[[Nessafof]]'' (+6144/6125) → [[Porwell temperaments #Nessafof|Porwell temperaments]]
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* ''[[Caleb]]'' (+33075/32768) → [[Mirwomo temperaments #Caleb|Mirwomo temperaments]]
* ''[[Caleb]]'' (+33075/32768) → [[Mirwomo temperaments #Caleb|Mirwomo temperaments]]
* [[Mutt temperament|Mutt]] (+65625/65536) → [[Horwell temperaments #Mutt|Horwell temperaments]]
* [[Mutt temperament|Mutt]] (+65625/65536) → [[Horwell temperaments #Mutt|Horwell temperaments]]
* ''[[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]]
* ''[[Tritricot]]'' (+{{monzo| 35 -23 -3 3 }}) → [[Alphatricot family #Tritricot|Alphatricot family]]
* ''[[Zinc]]'' (+{{monzo| -45 -24 14 18 }}) → [[30th-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 sextile, septichrome, pnict, domain, avicenna, terture, akjayland and terture.


== Sextile ==
== Sextile ==
Sextile tempers out the [[garischisma]] with a 1/6-octave period and is the 12 & 270 temperament.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 28: Line 32:
: mapping generators: ~4096/3645, ~3
: mapping generators: ~4096/3645, ~3


{{Multival|legend=1| 6 -36 -84 -71 -150 -94 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~4096/3645 = 1\6, ~3/2 = 702.2347
[[Optimal tuning]] ([[CTE]]): ~4096/3645 = 1\6, ~3/2 = 702.2347
* [[POTE]]: ~4096/3645 = 1\6, ~3/2 = 702.212


{{Optimal ET sequence|legend=1| 12, 234d, 246d, 258, 270, 1362c, 1632c, 1902c, 2172c, 2442bc, 2712bc }}
{{Optimal ET sequence|legend=1| 12, 234d, 246d, 258, 270, 1362c, 1632c, 1902c, 2172c, 2442bc, 2712bc }}
Line 43: Line 47:
Mapping: {{mapping| 6 0 71 150 230 | 0 1 -6 -14 -22 }}
Mapping: {{mapping| 6 0 71 150 230 | 0 1 -6 -14 -22 }}


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 702.2225
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 702.2225
* POTE: ~55/49 = 1\6, ~3/2 = 702.202


{{Optimal ET sequence|legend=1| 12, 246dee, 258e, 270, 822, 1092, 1362c }}
{{Optimal ET sequence|legend=1| 12, 246dee, 258e, 270, 822, 1092, 1362c }}
Line 56: Line 62:
Mapping: {{mapping| 6 0 71 150 230 279 | 0 1 -6 -14 -22 -27 }}
Mapping: {{mapping| 6 0 71 150 230 279 | 0 1 -6 -14 -22 -27 }}


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 702.2141
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 702.2141
* CWE: ~55/49 = 1\6, ~3/2 = 702.2001


{{Optimal ET sequence|legend=1| 12f, 258ef, 270, 552, 822, 1092, 1914cde }}
{{Optimal ET sequence|legend=1| 12f, 258ef, 270, 552, 822, 1092, 1914cde }}
Line 69: Line 77:
Mapping: {{mapping| 6 0 71 150 230 279 -4 | 0 1 -6 -14 -22 -27 3 }}
Mapping: {{mapping| 6 0 71 150 230 279 -4 | 0 1 -6 -14 -22 -27 3 }}


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 702.2117
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 702.2117
* CWE: ~55/49 = 1\6, ~3/2 = 702.1869


{{Optimal ET sequence|legend=1| 12f, 270, 552g }}
{{Optimal ET sequence|legend=1| 12f, 270, 552g }}
Line 82: Line 92:
Mapping: {{mapping| 6 0 71 150 230 279 -4 35 | 0 1 -6 -14 -22 -27 3 -1 }}
Mapping: {{mapping| 6 0 71 150 230 279 -4 35 | 0 1 -6 -14 -22 -27 3 -1 }}


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 702.2118
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 702.2118
* CWE: ~55/49 = 1\6, ~3/2 = 702.1890


{{Optimal ET sequence|legend=1| 12f, 270, 552g, 822gg }}
{{Optimal ET sequence|legend=1| 12f, 270, 552g, 822gg }}
Line 95: Line 107:
Mapping: {{mapping| 6 0 71 150 230 -149 | 0 1 -6 -14 -22 18 }}
Mapping: {{mapping| 6 0 71 150 230 -149 | 0 1 -6 -14 -22 18 }}


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 702.2296
Optimal tunings:
* CTE: ~55/49 = 1\6, ~3/2 = 702.2296
* CWE: ~55/49 = 1\6, ~3/2 = 702.2285


{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
Line 108: Line 122:
Mapping: {{mapping| 6 0 71 150 230 -149 -4 | 0 1 -6 -14 -22 18 3 }}
Mapping: {{mapping| 6 0 71 150 230 -149 -4 | 0 1 -6 -14 -22 18 3 }}


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 702.2264
Optimal tunings:
* 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 }}
{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
Line 121: Line 137:
Mapping: {{mapping| 6 0 71 150 230 -149 -4 35 | 0 1 -6 -14 -22 18 3 -1 }}
Mapping: {{mapping| 6 0 71 150 230 -149 -4 35 | 0 1 -6 -14 -22 18 3 -1 }}


Optimal tuning (CTE): ~55/49 = 1\6, ~3/2 = 702.2266
Optimal 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 }}
{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
Line 133: Line 151:


{{Mapping|legend=1| 3 3 1 0 | 0 5 17 24 }}
{{Mapping|legend=1| 3 3 1 0 | 0 5 17 24 }}
{{Multival|legend=1| 15 51 72 46 72 24 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~243/224 = 140.367
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~243/224 = 140.367
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[[Badness]]: 0.016814
[[Badness]]: 0.016814
=== Semiseptichrome ===
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 151263/151250, 234375/234256
Mapping: {{mapping| 6 1 -15 -24 -32 | 0 5 17 24 31 }}
: mapping generators: ~55/49, ~375/308
Optimal tuning (CTE): ~55/49 = 1\6, ~375/308 = 340.3687 (~1760/1701 = 59.6313)
Optimal ET sequence: {{Optimal ET sequence| 60e, 222cdee, 282, 342, 966, 1308, 1650, 4608b, 6258bc }}
Badness: 0.0194
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 34398/34375, 85293/85184
Mapping: {{mapping| 6 1 -15 -24 -32 -68 | 0 5 17 24 31 53 }}
Optimal tuning (CTE): ~55/49 = 1\6, ~375/308 = 340.3778 (~121/117 = 59.6222)
Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}
Badness: 0.0397
==== 17-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 61965/61952
Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 | 0 5 17 24 31 53 15 }}
Optimal tuning (CTE): ~55/49 = 1\6, ~375/308 = 340.3763 (~88/85 = 59.6237)
Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}
Badness: 0.0272
==== 19-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 23409/23408
Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 | 0 5 17 24 31 53 15 -5 }}
Optimal tuning (CTE): ~55/49 = 1\6, ~162/133 = 340.3767 (~88/85 = 59.6233)
Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}
Badness: 0.0222
==== 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 tuning (CTE): ~55/49 = 1\6, ~162/133 = 340.3757 (~88/85 = 59.6243)
Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}
Badness: 0.0160


== Pnict ==
== Pnict ==
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{{Mapping|legend=1| 3 10 11 6 | 0 -13 -10 6 }}
{{Mapping|legend=1| 3 10 11 6 | 0 -13 -10 6 }}
{{Multival|legend=1| 39 30 -18 -43 -138 -126 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~192/175 = 161.399
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~192/175 = 161.399
Line 165: Line 246:


{{Mapping|legend=1| 3 -3 -9 -8 | 0 17 35 36 }}
{{Mapping|legend=1| 3 -3 -9 -8 | 0 17 35 36 }}
{{Multival|legend=1| 51 105 108 48 28 -44 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467
Line 181: Line 260:
{{Mapping|legend=1| 3 2 8 16 | 0 8 -3 -22 }}
{{Mapping|legend=1| 3 2 8 16 | 0 8 -3 -22 }}


{{Multival|legend=1| 24 -9 -66 -70 -172 -128 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~63/50 = 1\3, ~1024/945 = 137.772
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~1024/945 = 137.768
* [[POTE]]: ~63/50 = 1\3, ~1024/945 = 137.768


{{Optimal ET sequence|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}
{{Optimal ET sequence|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}
Line 196: Line 275:
Mapping: {{mapping| 3 2 8 16 9 | 0 8 -3 -22 4 }}
Mapping: {{mapping| 3 2 8 16 9 | 0 8 -3 -22 4 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~693/640 = 137.771
Optimal tunings:
* CTE: ~63/50 = 1\3, ~693/640 = 137.773
* POTE: ~63/50 = 1\3, ~693/640 = 137.771


{{Optimal ET sequence|legend=1| 87, 183, 270, 1263, 1533, 1803c, 2073c }}
Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270, 1263, 1533, 1803c }}


Badness: 0.023085
Badness: 0.023085
Line 209: Line 290:
Mapping: {{mapping| 3 2 8 16 9 8 | 0 8 -3 -22 4 9 }}
Mapping: {{mapping| 3 2 8 16 9 8 | 0 8 -3 -22 4 9 }}


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


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


Badness: 0.015557
Badness: 0.015557
Line 222: Line 305:
Mapping: {{mapping| 3 2 8 16 9 8 4 | 0 8 -3 -22 4 9 24 }}
Mapping: {{mapping| 3 2 8 16 9 8 4 | 0 8 -3 -22 4 9 24 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~13/12 = 137.777
Optimal tunings:
* CTE: ~63/50 = 1\3, ~13/12 = 137.758
* POTE: ~63/50 = 1\3, ~13/12 = 137.761


{{Optimal ET sequence|legend=1| 87, 183, 270 }}
Optimal ET sequence: {{Optimal ET sequence| 87, 183, 270, 453 }}


Badness: 0.015557
Badness: 0.0171


=== 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, 3025/3024, 4096/4095
Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 1729/1728, 3025/3024


Mapping: {{mapping| 3 2 8 16 9 8 4 0 | 0 8 -3 -22 4 9 24 37 }}
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, ~13/12 = 137.777
Optimal tunings:
* CTE: ~63/50 = 1\3, ~13/12 = 137.763
* POTE: ~63/50 = 1\3, ~13/12 = 137.767


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


Badness: 0.015557
Badness: 0.0153


== Terture ==
== Terture ==
Line 320: Line 407:


Badness: 0.014915
Badness: 0.014915
== Slendscape ==
Slendscape tempers out the [[slendroschisma]] (68719476736/68641485507) in addition to landscape comma, and thereby features a period of 1\15.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 250047/250000, 12884901888/12867859375
{{Mapping|legend=1| 15 0 17 54 | 0 4 3 -2 }}
: Mapping generators: ~8575/8192, ~1152/875
[[Optimal tuning]] ([[CTE]]): ~8575/8192 = 80.0000, ~1152/875 = 475.4847 (~6/5 = 315.4847 or ~21/20 = 84.5153)
{{Optimal ET sequence|legend=1| 255, 270, 525, 795, 1065 }}
[[Badness]]: 0.058002
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 102487/102400, 180224/180075
Mapping: {{Mapping| 15 0 17 54 40 | 0 4 3 -2 2 }}
Optimal tuning (CTE): ~22/21 = 80.0000, ~968/735 = 475.4912 (~6/5 = 315.4912 or ~21/20 = 84.5088)
Optimal ET sequence: {{EDOs| 255, 270, 525, 795, 1065 }}
Badness: 0.026246
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 3025/3024, 4096/4095, 14641/14625
Mapping: {{Mapping| 15 0 17 54 40 109 | 0 4 3 -2 2 -9 }}
Optimal tuning (CTE): ~22/21 = 80.0000, ~154/117 = 475.4935 (~6/5 = 315.4935 or ~21/20 = 84.5065)
Optimal ET sequence: {{EDOs| 255, 270, 795, 1065 }}
Badness: 0.021230


== Akjayland ==
== Akjayland ==
{{See also| 21st-octave temperaments }}
{{See also| 21st-octave temperaments }}


Akjayland tempers out the [[akjaysma]] in addition to landscape comma, and thereby features a period of 1\21.  
Akjayland tempers out the [[akjaysma]] in addition to landscape comma, and thereby features a period of 1\21.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 334: Line 464:
: mapping generators: ~1323/1280, ~131072/91875
: mapping generators: ~1323/1280, ~131072/91875


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


{{Optimal ET sequence|legend=1| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
{{Optimal ET sequence|legend=1| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
Line 341: Line 471:


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


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 351: Line 481:
: mapping generators: ~1323/1280, ~6615/5632
: mapping generators: ~1323/1280, ~6615/5632


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


{{Optimal ET sequence|legend=1| 168, 357, 525, 882, 1407, 2289e }}
{{Optimal ET sequence|legend=1| 168, 357, 525, 882, 1407, 2289e }}
Line 364: Line 494:
Mapping: {{mapping| 21 4 39 98 58 107 | 0 6 2 -8 3 -6 }}
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 tuning (CTE): ~336/325 = 1\21, ~168/143 = 278.9058


{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
Line 377: Line 507:
Mapping: {{mapping| 21 4 39 98 58 107 120 | 0 6 2 -8 3 -6 -7 }}
Mapping: {{mapping| 21 4 39 98 58 107 120 | 0 6 2 -8 3 -6 -7 }}


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


{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
Line 390: Line 520:
Mapping: {{mapping| 21 4 39 98 58 107 120 16 | 0 6 2 -8 3 -6 -7 15 }}
Mapping: {{mapping| 21 4 39 98 58 107 120 16 | 0 6 2 -8 3 -6 -7 15 }}


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


{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
Line 403: Line 533:
Mapping: {{mapping| 21 4 39 98 58 107 120 16 95 | 0 -6 -2 8 -3 6 7 -15 0 }}
Mapping: {{mapping| 21 4 39 98 58 107 120 16 95 | 0 -6 -2 8 -3 6 7 -15 0 }}


Optimal tuning (CTE): ~336/325 = 1\21, ~168/143 = 278.8971
Optimal tuning (CTE): ~336/325 = 1\21, ~168/143 = 278.8971


{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
{{Optimal ET sequence|legend=1| 168h, 357, 525, 882, 1407 }}
Line 429: Line 559:


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 484: Line 616:
Badness: 0.0209
Badness: 0.0209


== Zinc ==
{{See also| 10th-octave temperaments #Neon }}
{{See also| 60th-octave temperaments #Neodymium }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 250047/250000, {{monzo| -53 -12 2 24 }}
{{Mapping|legend=1| 30 2 15 66 | 0 5 6 2 }}
[[Optimal tuning]]s:
* [[CTE]]: ~53747712/52521875 = 1\30, ~216/175 = 364.3896 (~{{monzo| 21 3 1 -10 }} = 4.3896)
* [[CWE]]: ~53747712/52521875 = 1\30, ~216/175 = 364.3879 (~{{monzo| 21 3 1 -10 }} = 4.3879)
{{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, 151263/151250, {{monzo| -27 -6 4 6 3 }}
Mapping: {{mapping| 30 2 15 66 122 | 0 5 6 2 -2 }}
Optimal tunings:
* CTE: ~18865/18432 = 1\30, ~216/175 = 364.3886 (~385/384 = 4.3886)
* CWE: ~18865/18432 = 1\30, ~216/175 = 364.3849 (~385/384 = 4.3849)
Optimal ET sequence: {{Optimal ET sequence| 270, 1110, 1380, 1650, 1920, 2190 }}
Badness: 0.0220
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 9801/9800, 10648/10647, 105644/105625, 196625/196608
Mapping: {{mapping| 30 2 15 66 122 193 | 0 5 6 2 -2 -9 }}
Optimal tunings:
* CTE: ~351/343 = 1\30, ~216/175 = 364.3879 (~385/384 = 4.3879)
* CWE: ~351/343 = 1\30, ~216/175 = 364.3867 (~385/384 = 4.3867)
Optimal ET sequence: {{Optimal ET sequence| 270, 1380, 1650, 1920, 2190, 4650, 6840e, 11490de }}
Badness: 0.0155
=== 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: {{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
[[Comma list]]: 250047/250000, {{monzo| 15 -16 -4 7 }}
{{Mapping|legend=1| 30 0 -73 -106 | 0 1 3 4 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2240/2187 = 1\30, ~3/2 = 702.141 (~81/80 = 22.141)
* [[CWE]]: ~2240/2187 = 1\30, ~3/2 = 702.166 (~81/80 = 22.166)
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 19712/19683, 250047/250000
Mapping: {{mapping| 30 0 -73 -106 -134 | 0 1 3 4 5 }}
Optimal tunings:
* CTE: ~45/44 = 1\30, ~3/2 = 702.182 (~81/80 = 22.182)
* CWE: ~45/44 = 1\30, ~3/2 = 702.213 (~81/80 = 22.213)
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 19712/19683, 1001/1000, 4459/4455, 4225/4224
Mapping: {{mapping| 30 0 -73 -106 -134 111 | 0 1 3 4 5 0 }}


Optimal tunings:
* CTE: ~45/44 = 1\30, ~3/2 = 702.182 (~81/80 = 22.182)
* CWE: ~45/44 = 1\30, ~3/2 = 702.170 (~81/80 = 22.170)


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