Landscape microtemperaments: Difference between revisions

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{{Mapping|legend=1| 6 0 71 150 | 0 1 -6 -14 }}
{{Mapping|legend=1| 6 0 71 150 | 0 1 -6 -14 }}
: mapping generators: ~4096/3645, ~3
: mapping generators: ~4096/3645, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~4096/3645 = 1\6, ~3/2 = 702.2347
* [[CTE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2347{{c}}
* [[POTE]]: ~4096/3645 = 1\6, ~3/2 = 702.212
* [[POTE]]: ~4096/3645 = 200.000{{c}}, ~3/2 = 702.212{{c}}


{{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 }}


[[Badness]]: 0.070097
[[Badness]] (Smith): 0.070097


=== 11-limit ===
=== 11-limit ===
Line 50: Line 49:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2225
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2225{{c}}
* POTE: ~55/49 = 1\6, ~3/2 = 702.202
* POTE: ~55/49 = 200.000{{c}}, ~3/2 = 702.202{{c}}


{{Optimal ET sequence|legend=1| 12, 246dee, 258e, 270, 822, 1092, 1362c }}
{{Optimal ET sequence|legend=0| 12, 246dee, 258e, 270, 822, 1092, 1362c }}


Badness: 0.029677
Badness (Smith): 0.029677


=== 13-limit ===
=== 13-limit ===
Line 65: Line 64:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2141
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2141{{c}}
* CWE: ~55/49 = 1\6, ~3/2 = 702.2001
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2001{{c}}


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


Badness: 0.0191
Badness (Smith): 0.0191


==== 17-limit ====
==== 17-limit ====
Line 80: Line 79:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2117
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2117{{c}}
* CWE: ~55/49 = 1\6, ~3/2 = 702.1869
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.1869{{c}}


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


Badness: 0.0209
Badness (Smith): 0.0209


==== 19-limit ====
==== 19-limit ====
Line 95: Line 94:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2118
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2118{{c}}
* CWE: ~55/49 = 1\6, ~3/2 = 702.1890
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.1890{{c}}


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


Badness: 0.0156
Badness (Smith): 0.0156


=== Sextilia ===
=== Sextilia ===
Line 110: Line 109:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2296
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2296{{c}}
* CWE: ~55/49 = 1\6, ~3/2 = 702.2285
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2285{{c}}


{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
{{Optimal ET sequence|legend=0| 12, 258e, 270 }}


Badness: 0.0391
Badness (Smith): 0.0391


==== 17-limit ====
==== 17-limit ====
Line 125: Line 124:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2264
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2264{{c}}
* CWE: ~55/49 = 1\6, ~3/2 = 702.2207
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2207{{c}}


{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
{{Optimal ET sequence|legend=0| 12, 258e, 270 }}


Badness: 0.0384
Badness (Smith): 0.0384


==== 19-limit ====
==== 19-limit ====
Line 140: Line 139:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 1\6, ~3/2 = 702.2266
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2266{{c}}
* CWE: ~55/49 = 1\6, ~3/2 = 702.2208
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2208{{c}}


{{Optimal ET sequence|legend=1| 12, 258e, 270 }}
{{Optimal ET sequence|legend=0| 12, 258e, 270 }}


Badness: 0.0252
Badness (Smith): 0.0252


== Septichrome ==
== Septichrome ==
Line 154: Line 153:
{{Mapping|legend=1| 3 3 1 0 | 0 5 17 24 }}
{{Mapping|legend=1| 3 3 1 0 | 0 5 17 24 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~243/224 = 140.367
[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~243/224 = 140.367{{c}}


{{Optimal ET sequence|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}
{{Optimal ET sequence|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}


[[Badness]]: 0.016814
[[Badness]] (Smith): 0.016814


=== Semiseptichrome ===
=== Semiseptichrome ===
Line 166: Line 165:


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


Optimal tuning (CTE): ~55/49 = 1\6, ~375/308 = 340.3687 (~1760/1701 = 59.6313)
Optimal tuning (CTE): ~55/49 = 200.0000{{c}}, ~375/308 = 340.3687{{c}} (~1760/1701 = 59.6313{{c}})


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


Badness: 0.0194
Badness (Smith): 0.0194


==== 13-limit ====
==== 13-limit ====
Line 182: Line 180:
Mapping: {{mapping| 6 1 -15 -24 -32 -68 | 0 5 17 24 31 53 }}
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 tuning (CTE): ~55/49 = 200.0000{{c}}, ~375/308 = 340.3778{{c}} (~121/117 = 59.6222{{c}})


Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness: 0.0397
Badness (Smith): 0.0397


==== 17-limit ====
==== 17-limit ====
Line 195: Line 193:
Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 | 0 5 17 24 31 53 15 }}
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 tuning (CTE): ~55/49 = 200.0000{{c}}, ~375/308 = 340.3763{{c}} (~88/85 = 59.6237{{c}})


Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness: 0.0272
Badness (Smith): 0.0272


==== 19-limit ====
==== 19-limit ====
Line 208: Line 206:
Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 | 0 5 17 24 31 53 15 -5 }}
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 tuning (CTE): ~55/49 = 200.0000{{c}}, ~162/133 = 340.3767{{c}} (~88/85 = 59.6233{{c}})


Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness: 0.0222
Badness (Smith): 0.0222


==== 23-limit ====
==== 23-limit ====
Line 221: Line 219:
Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 -12 | 0 5 17 24 31 53 15 -5 23 }}
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 tuning (CTE): ~55/49 = 200.0000{{c}}, ~162/133 = 340.3757{{c}} (~88/85 = 59.6243{{c}})


Optimal ET sequence: {{Optimal ET sequence| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness: 0.0160
Badness (Smith): 0.0160


== Pnict ==
== Pnict ==
Line 234: Line 232:
{{Mapping|legend=1| 3 10 11 6 | 0 -13 -10 6 }}
{{Mapping|legend=1| 3 10 11 6 | 0 -13 -10 6 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~192/175 = 161.399
[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~192/175 = 161.399{{c}}


{{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]] (Smith): 0.045660


== Domain ==
== Domain ==
Line 249: Line 247:
{{Mapping|legend=1| 3 -3 -9 -8 | 0 17 35 36 }}
{{Mapping|legend=1| 3 -3 -9 -8 | 0 17 35 36 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467
[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~10/9 = 182.467{{c}}


{{Optimal ET sequence|legend=1| 171, 1164, 1335, 1506, 1677, 1848, 2019, 2190, 11943, 13962, 15981, 18000, 20019, 22038 }}
{{Optimal ET sequence|legend=1| 171, 1164, 1335, 1506, 1677, 1848, 2019, 2190, 11943, 13962, 15981, 18000, 20019, 22038 }}


[[Badness]]: 0.013979
[[Badness]] (Smith): 0.013979


== Avicenna ==
== Avicenna ==
Line 263: Line 261:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~63/50 = 1\3, ~1024/945 = 137.772
* [[CTE]]: ~63/50 = 400.000{{c}}, ~1024/945 = 137.772{{c}}
* [[POTE]]: ~63/50 = 1\3, ~1024/945 = 137.768
* [[POTE]]: ~63/50 = 400.000{{c}}, ~1024/945 = 137.768{{c}}


{{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 }}


[[Badness]]: 0.062187
[[Badness]] (Smith): 0.062187


=== 11-limit ===
=== 11-limit ===
Line 278: Line 276:


Optimal tunings:  
Optimal tunings:  
* CTE: ~63/50 = 1\3, ~693/640 = 137.773
* CTE: ~63/50 = 400.000{{c}}, ~693/640 = 137.773{{c}}
* POTE: ~63/50 = 1\3, ~693/640 = 137.771
* POTE: ~63/50 = 400.000{{c}}, ~693/640 = 137.771{{c}}


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


Badness: 0.023085
Badness (Smith): 0.023085


=== 13-limit ===
=== 13-limit ===
Line 293: Line 291:


Optimal tunings:  
Optimal tunings:  
* CTE: ~63/50 = 1\3, ~13/12 = 137.777
* CTE: ~63/50 = 400.000{{c}}, ~13/12 = 137.777{{c}}
* POTE: ~63/50 = 1\3, ~13/12 = 137.777
* POTE: ~63/50 = 400.000{{c}}, ~13/12 = 137.777{{c}}


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


Badness: 0.015557
Badness (Smith): 0.015557


=== 17-limit ===
=== 17-limit ===
Line 308: Line 306:


Optimal tunings:  
Optimal tunings:  
* CTE: ~63/50 = 1\3, ~13/12 = 137.758
* CTE: ~63/50 = 400.000{{c}}, ~13/12 = 137.758{{c}}
* POTE: ~63/50 = 1\3, ~13/12 = 137.761
* POTE: ~63/50 = 400.000{{c}}, ~13/12 = 137.761{{c}}


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


Badness: 0.0171
Badness (Smith): 0.0171


=== 19-limit ===
=== 19-limit ===
Line 323: Line 321:


Optimal tunings:  
Optimal tunings:  
* CTE: ~63/50 = 1\3, ~13/12 = 137.763
* CTE: ~63/50 = 400.000{{c}}, ~13/12 = 137.763{{c}}
* POTE: ~63/50 = 1\3, ~13/12 = 137.767
* POTE: ~63/50 = 400.000{{c}}, ~13/12 = 137.767{{c}}


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


Badness: 0.0153
Badness (Smith): 0.0153


== Terture ==
== Terture ==
Line 339: Line 337:
{{Mapping|legend=1| 3 4 3 2 | 0 4 21 34 }}
{{Mapping|legend=1| 3 4 3 2 | 0 4 21 34 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~392/375 = 75.555
[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~392/375 = 75.555{{c}}


{{Optimal ET sequence|legend=1| 111, 159, 270 }}
{{Optimal ET sequence|legend=1| 111, 159, 270 }}


[[Badness]]: 0.087156
[[Badness]] (Smith): 0.087156


=== 11-limit ===
=== 11-limit ===
Line 352: Line 350:
Mapping: {{mapping| 3 4 3 2 10 | 0 4 21 34 2 }}
Mapping: {{mapping| 3 4 3 2 10 | 0 4 21 34 2 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~392/375 = 75.550
Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~392/375 = 75.550{{c}}


{{Optimal ET sequence|legend=1| 111, 159, 270, 1239, 1509, 1779, 2049, 2319 }}
{{Optimal ET sequence|legend=0| 111, 159, 270, 1239, 1509, 1779, 2049, 2319 }}


Badness: 0.029326
Badness (Smith): 0.029326


=== 13-limit ===
=== 13-limit ===
Line 365: Line 363:
Mapping: {{mapping| 3 4 3 2 10 6 | 0 4 21 34 2 27 }}
Mapping: {{mapping| 3 4 3 2 10 6 | 0 4 21 34 2 27 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~117/112 = 75.553
Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~117/112 = 75.553{{c}}


{{Optimal ET sequence|legend=1| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 111, 159, 270 }}


Badness: 0.018647
Badness (Smith): 0.018647


=== 17-limit ===
=== 17-limit ===
Line 378: Line 376:
Mapping: {{mapping| 3 4 3 2 10 6 10 | 0 4 21 34 2 27 12 }}
Mapping: {{mapping| 3 4 3 2 10 6 10 | 0 4 21 34 2 27 12 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~117/112 = 75.560
Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~117/112 = 75.560{{c}}


{{Optimal ET sequence|legend=1| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 111, 159, 270 }}


Badness: 0.018705
Badness (Smith): 0.018705


=== 19-limit ===
=== 19-limit ===
Line 391: Line 389:
Mapping: {{mapping| 3 4 3 2 10 6 10 5 | 0 4 21 34 2 27 12 41 }}
Mapping: {{mapping| 3 4 3 2 10 6 10 5 | 0 4 21 34 2 27 12 41 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~95/91 = 75.560
Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~95/91 = 75.560{{c}}


{{Optimal ET sequence|legend=1| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 111, 159, 270 }}


Badness: 0.013902
Badness (Smith): 0.013902


=== 23-limit ===
=== 23-limit ===
Line 404: Line 402:
Mapping: {{mapping| 3 4 3 2 10 6 10 5 13 | 0 4 21 34 2 27 12 41 3 }}
Mapping: {{mapping| 3 4 3 2 10 6 10 5 13 | 0 4 21 34 2 27 12 41 3 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~24/23 = 75.548
Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~24/23 = 75.548{{c}}


{{Optimal ET sequence|legend=1| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 111, 159, 270 }}


Badness: 0.014915
Badness (Smith): 0.014915


== Slendscape ==
== Slendscape ==
Line 418: Line 416:


{{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]] ([[CTE]]): ~8575/8192 = 80.0000{{c}}, ~1152/875 = 475.4847{{c}}
 
[[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 }}
{{Optimal ET sequence|legend=1| 255, 270, 525, 795, 1065 }}


[[Badness]]: 0.058002
[[Badness]] (Smith): 0.058002


=== 11-limit ===
=== 11-limit ===
Line 434: Line 431:
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 tuning (CTE): ~22/21 = 80.0000{{c}}, ~968/735 = 475.4912{{c}}


Optimal ET sequence: {{EDOs| 255, 270, 525, 795, 1065 }}
{{Optimal ET sequence|legend=0| 255, 270, 525, 795, 1065 }}


Badness: 0.026246
Badness (Smith): 0.026246


=== 13-limit ===
=== 13-limit ===
Line 447: Line 444:
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 tuning (CTE): ~22/21 = 80.0000{{c}}, ~154/117 = 475.4935{{c}}


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


Badness: 0.021230
Badness (Smith): 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 463: Line 460:


{{Mapping|legend=1| 21 1 38 102 | 0 3 1 -4 }}
{{Mapping|legend=1| 21 1 38 102 | 0 3 1 -4 }}
: 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 = 57.1429{{c}}, ~131072/91875 = 614.9354{{c}}


{{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 }}


[[Badness]]: 0.0309
[[Badness]] (Smith): 0.0309


=== 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 {{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
Subgroup: 2.3.5.7.11
Line 480: Line 476:


Mapping: {{mapping| 21 4 39 98 58 | 0 6 2 -8 3 3 }}
Mapping: {{mapping| 21 4 39 98 58 | 0 6 2 -8 3 3 }}
: 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 = 57.1429{{c}}, ~6615/5632 = 278.8998{{c}}


{{Optimal ET sequence|legend=1| 168, 357, 525, 882, 1407, 2289e }}
{{Optimal ET sequence|legend=0| 168, 357, 525, 882, 1407, 2289e }}


Badness: 0.0949
Badness (Smith): 0.0949


==== 13-limit ====
==== 13-limit ====
Line 496: Line 491:
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 = 57.1429{{c}}, ~168/143 = 278.9058{{c}}


{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}


Badness: 0.0551
Badness (Smith): 0.0551


==== 17-limit ====
==== 17-limit ====
Line 509: Line 504:
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 = 57.1429{{c}}, ~168/143 = 278.9036{{c}}


{{Optimal ET sequence|legend=1| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}


Badness: 0.0319
Badness (Smith): 0.0319


==== 19-limit ====
==== 19-limit ====
Line 522: Line 517:
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 = 57.1429{{c}}, ~168/143 = 278.9036{{c}}


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


Badness: 0.0270
Badness (Smith): 0.0270


==== 23-limit ====
==== 23-limit ====
Line 535: Line 530:
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 = 57.1429{{c}}, ~168/143 = 278.8971{{c}}


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


Badness: 0.0199
Badness (Smith): 0.0199


== Magnesium ==
== Magnesium ==
: ''For the 5-limit version, see [[12th-octave temperaments#Magnesium (5-limit)]].
: ''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]].
Magnesium is named after element 12 for being period 12; however, it is 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]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 551: Line 546:


{{Mapping|legend=1| 12 2 23 58 | 0 7 2 -10 }}
{{Mapping|legend=1| 12 2 23 58 | 0 7 2 -10 }}
: mapping generators: ~138915/131072, 3145728/2734375
: mapping generators: ~138915/131072, 3145728/2734375


Optimal tuning (CTE): ~138915/131072 = 1\12, ~3145728/2734375 = 243.130
Optimal tuning (CTE): ~138915/131072 = 100.000{{c}}, ~3145728/2734375 = 243.130{{c}}


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


Badness: 0.0964
Badness (Smith): 0.0964


== 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. Named after the 24th element for being period 24.
Line 570: Line 564:


{{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]] ([[CTE]]): ~250/243 = 50.0000{{c}}, ~10/7 = 617.2710{{c}}


{{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]] (Smith): 0.139


=== 11-limit ===
=== 11-limit ===
Line 586: Line 579:
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 tuning (CTE): ~250/243 = 50.0000{{c}}, ~10/7 = 617.2597{{c}}


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


Badness: 0.0398
Badness (Smith): 0.0398


=== 13-limit ===
=== 13-limit ===
Line 599: Line 592:
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 tuning (CTE): ~250/243 = 50.0000{{c}}, ~10/7 = 617.2869{{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 (Smith): 0.0293


=== 17-limit ===
=== 17-limit ===
Line 612: Line 605:
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 tuning (CTE): ~35/34 = 50.0000{{c}}, ~10/7 = 617.2732{{c}}


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


Badness: 0.0209
Badness (Smith): 0.0209


== Zinc ==
== Zinc ==
Line 629: Line 622:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~53747712/52521875 = 1\30, ~216/175 = 364.3896 (~{{monzo| 21 3 1 -10 }} = 4.3896)
* [[CTE]]: ~53747712/52521875 = 40.0000{{c}}, ~216/175 = 364.3896{{c}} (~{{monzo| 21 3 1 -10 }} = 4.3896{{c}})
* [[CWE]]: ~53747712/52521875 = 1\30, ~216/175 = 364.3879 (~{{monzo| 21 3 1 -10 }} = 4.3879)
* [[CWE]]: ~53747712/52521875 = 40.0000{{c}}, ~216/175 = 364.3879{{c}} (~{{monzo| 21 3 1 -10 }} = 4.3879{{c}})


{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650 }}
{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650 }}


[[Badness]]: 0.0742
[[Badness]] (Smith): 0.0742


=== 11-limit===
=== 11-limit===
Line 644: Line 637:


Optimal tunings:  
Optimal tunings:  
* CTE: ~18865/18432 = 1\30, ~216/175 = 364.3886 (~385/384 = 4.3886)
* CTE: ~18865/18432 = 40.0000{{c}}, ~216/175 = 364.3886{{c}} (~385/384 = 4.3886{{c}})
* CWE: ~18865/18432 = 1\30, ~216/175 = 364.3849 (~385/384 = 4.3849)
* CWE: ~18865/18432 = 40.0000{{c}}, ~216/175 = 364.3849{{c}} (~385/384 = 4.3849{{c}})


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


Badness: 0.0220
Badness (Smith): 0.0220


=== 13-limit ===
=== 13-limit ===
Line 659: Line 652:


Optimal tunings:  
Optimal tunings:  
* CTE: ~351/343 = 1\30, ~216/175 = 364.3879 (~385/384 = 4.3879)
* CTE: ~351/343 = 40.0000{{c}}, ~216/175 = 364.3879{{c}} (~385/384 = 4.3879{{c}})
* CWE: ~351/343 = 1\30, ~216/175 = 364.3867 (~385/384 = 4.3867)
* CWE: ~351/343 = 40.0000{{c}}, ~216/175 = 364.3867{{c}} (~385/384 = 4.3867{{c}})


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


Badness: 0.0155
Badness (Smith): 0.0155


=== 2.3.5.7.11.13.19 subgroup (neozinc) ===
=== 2.3.5.7.11.13.19 subgroup (neozinc) ===
Line 674: Line 667:


Optimal tunings:  
Optimal tunings:  
* CTE: ~175/171 = 1\30, ~216/175 = 364.3876 (~400/399 = 4.3876)
* CTE: ~175/171 = 40.0000{{c}}, ~216/175 = 364.3876{{c}} (~400/399 = 4.3876{{c}})
* CWE: ~175/171 = 1\30, ~216/175 = 364.3864 (~400/399 = 4.3864)
* CWE: ~175/171 = 40.0000{{c}}, ~216/175 = 364.3864{{c}} (~400/399 = 4.3864{{c}})


Optimal ET sequence: {{Optimal ET sequence| 270, 1380, 1650, 1920, 2190, 4650, 6840e }}
{{Optimal ET sequence|legend=0| 270, 1380, 1650, 1920, 2190, 4650, 6840e }}


Badness: 0.00921
Badness (Smith): 0.00921


== Poe ==
== Poe ==
Line 689: Line 682:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2240/2187 = 1\30, ~3/2 = 702.141 (~81/80 = 22.141)
* [[CTE]]: ~2240/2187 = 40.000{{c}}, ~3/2 = 702.141{{c}}
* [[CWE]]: ~2240/2187 = 1\30, ~3/2 = 702.166 (~81/80 = 22.166)
* [[CWE]]: ~2240/2187 = 40.000{{c}}, ~3/2 = 702.166{{c}}


{{Optimal ET sequence|legend=1| 60, 210, 270 }}
{{Optimal ET sequence|legend=1| 60, 210, 270 }}


[[Badness]] (Sintel): 2.900
[[Badness]] (Sintel): 2.90


=== 11-limit ===
=== 11-limit ===
Line 704: Line 697:


Optimal tunings:  
Optimal tunings:  
* CTE: ~45/44 = 1\30, ~3/2 = 702.182 (~81/80 = 22.182)
* CTE: ~45/44 = 40.000, ~3/2 = 702.182{{c}}
* CWE: ~45/44 = 1\30, ~3/2 = 702.213 (~81/80 = 22.213)
* CWE: ~45/44 = 40.000{{c}}, ~3/2 = 702.213{{c}}


{{Optimal ET sequence|legend=1| 60e, 210e, 270 }}
{{Optimal ET sequence|legend=0| 60e, 210e, 270 }}


Badness (Sintel): 1.307
Badness (Sintel): 1.31


=== 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: 1001/1000, 4225/4224, 4459/4455, 19712/19683


Mapping: {{mapping| 30 0 -73 -106 -134 111 | 0 1 3 4 5 0 }}
Mapping: {{mapping| 30 0 -73 -106 -134 111 | 0 1 3 4 5 0 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~45/44 = 1\30, ~3/2 = 702.182 (~81/80 = 22.182)
* CTE: ~45/44 = 40.000{{c}}, ~3/2 = 702.182{{c}}
* CWE: ~45/44 = 1\30, ~3/2 = 702.170 (~81/80 = 22.170)
* CWE: ~45/44 = 40.000{{c}}, ~3/2 = 702.170{{c}}


{{Optimal ET sequence|legend=1| 60e, 210e, 270 }}
{{Optimal ET sequence|legend=0| 60e, 210e, 270 }}


Badness (Sintel): 1.194
Badness (Sintel): 1.19


[[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]]

Revision as of 04:06, 23 April 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 sextile, septichrome, pnict, domain, avicenna, akjayland and terture.

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.

Subgroup: 2.3.5.7

Comma list: 250047/250000, 33554432/33480783

Mapping: [⟨6 0 71 150], ⟨0 1 -6 -14]]

mapping generators: ~4096/3645, ~3

Optimal tunings:

  • CTE: ~4096/3645 = 200.0000 ¢, ~3/2 = 702.2347 ¢
  • POTE: ~4096/3645 = 200.000 ¢, ~3/2 = 702.212 ¢

Optimal ET sequence: 12, 234d, 246d, 258, 270, 1362c, 1632c, 1902c, 2172c, 2442bc, 2712bc

Badness (Smith): 0.070097

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨6 0 71 150 230], ⟨0 1 -6 -14 -22]]

Optimal tunings:

  • CTE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2225 ¢
  • POTE: ~55/49 = 200.000 ¢, ~3/2 = 702.202 ¢

Optimal ET sequence: 12, 246dee, 258e, 270, 822, 1092, 1362c

Badness (Smith): 0.029677

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 5632/5625, 10648/10647

Mapping: [⟨6 0 71 150 230 279], ⟨0 1 -6 -14 -22 -27]]

Optimal tunings:

  • CTE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2141 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2001 ¢

Optimal ET sequence: 12f, 258ef, 270, 552, 822, 1092, 1914cde

Badness (Smith): 0.0191

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 936/935, 1701/1700, 1716/1715, 5632/5625, 7744/7735

Mapping: [⟨6 0 71 150 230 279 -4], ⟨0 1 -6 -14 -22 -27 3]]

Optimal tunings:

  • CTE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2117 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.1869 ¢

Optimal ET sequence: 12f, 270, 552g

Badness (Smith): 0.0209

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 936/935, 1216/1215, 1701/1700, 1716/1715, 2376/2375, 4200/4199

Mapping: [⟨6 0 71 150 230 279 -4 35], ⟨0 1 -6 -14 -22 -27 3 -1]]

Optimal tunings:

  • CTE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2118 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.1890 ¢

Optimal ET sequence: 12f, 270, 552g, 822gg

Badness (Smith): 0.0156

Sextilia

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 4096/4095, 4459/4455, 20449/20412

Mapping: [⟨6 0 71 150 230 -149], ⟨0 1 -6 -14 -22 18]]

Optimal tunings:

  • CTE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2296 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2285 ¢

Optimal ET sequence: 12, 258e, 270

Badness (Smith): 0.0391

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 1001/1000, 1701/1700, 4096/4095, 4459/4455

Mapping: [⟨6 0 71 150 230 -149 -4], ⟨0 1 -6 -14 -22 18 3]]

Optimal tunings:

  • CTE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2264 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2207 ¢

Optimal ET sequence: 12, 258e, 270

Badness (Smith): 0.0384

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 715/714, 1001/1000, 1216/1215, 1701/1700, 1729/1728, 2912/2907

Mapping: [⟨6 0 71 150 230 -149 -4 35], ⟨0 1 -6 -14 -22 18 3 -1]]

Optimal tunings:

  • CTE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2266 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2208 ¢

Optimal ET sequence: 12, 258e, 270

Badness (Smith): 0.0252

Septichrome

Subgroup: 2.3.5.7

Comma list: 250047/250000, 2460375/2458624

Mapping: [⟨3 3 1 0], ⟨0 5 17 24]]

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~243/224 = 140.367 ¢

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

Badness (Smith): 0.016814

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 tuning (CTE): ~55/49 = 200.0000 ¢, ~375/308 = 340.3687 ¢ (~1760/1701 = 59.6313 ¢)

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

Badness (Smith): 0.0194

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 tuning (CTE): ~55/49 = 200.0000 ¢, ~375/308 = 340.3778 ¢ (~121/117 = 59.6222 ¢)

Optimal ET sequence: 282, 342f, 624

Badness (Smith): 0.0397

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 tuning (CTE): ~55/49 = 200.0000 ¢, ~375/308 = 340.3763 ¢ (~88/85 = 59.6237 ¢)

Optimal ET sequence: 282, 342f, 624

Badness (Smith): 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: [⟨6 1 -15 -24 -32 -68 -1 34], ⟨0 5 17 24 31 53 15 -5]]

Optimal tuning (CTE): ~55/49 = 200.0000 ¢, ~162/133 = 340.3767 ¢ (~88/85 = 59.6233 ¢)

Optimal ET sequence: 282, 342f, 624

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

Optimal tuning (CTE): ~55/49 = 200.0000 ¢, ~162/133 = 340.3757 ¢ (~88/85 = 59.6243 ¢)

Optimal ET sequence: 282, 342f, 624

Badness (Smith): 0.0160

Pnict

Subgroup: 2.3.5.7

Comma list: 250047/250000, 2100875/2097152

Mapping: [⟨3 10 11 6], ⟨0 -13 -10 6]]

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~192/175 = 161.399 ¢

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

Badness (Smith): 0.045660

Domain

See also: Minortonic family

Subgroup: 2.3.5.7

Comma list: 250047/250000, 645700815/645657712

Mapping: [⟨3 -3 -9 -8], ⟨0 17 35 36]]

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~10/9 = 182.467 ¢

Optimal ET sequence: 171, 1164, 1335, 1506, 1677, 1848, 2019, 2190, 11943, 13962, 15981, 18000, 20019, 22038

Badness (Smith): 0.013979

Avicenna

Subgroup: 2.3.5.7

Comma list: 250047/250000, 29360128/29296875

Mapping: [⟨3 2 8 16], ⟨0 8 -3 -22]]

Optimal tunings:

  • CTE: ~63/50 = 400.000 ¢, ~1024/945 = 137.772 ¢
  • POTE: ~63/50 = 400.000 ¢, ~1024/945 = 137.768 ¢

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

Badness (Smith): 0.062187

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:

  • CTE: ~63/50 = 400.000 ¢, ~693/640 = 137.773 ¢
  • POTE: ~63/50 = 400.000 ¢, ~693/640 = 137.771 ¢

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

Badness (Smith): 0.023085

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:

  • CTE: ~63/50 = 400.000 ¢, ~13/12 = 137.777 ¢
  • POTE: ~63/50 = 400.000 ¢, ~13/12 = 137.777 ¢

Optimal ET sequence: 87, 183, 270

Badness (Smith): 0.015557

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 1001/1000, 3025/3024, 4096/4095

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

Optimal tunings:

  • CTE: ~63/50 = 400.000 ¢, ~13/12 = 137.758 ¢
  • POTE: ~63/50 = 400.000 ¢, ~13/12 = 137.761 ¢

Optimal ET sequence: 87, 183, 270, 453

Badness (Smith): 0.0171

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

Optimal tunings:

  • CTE: ~63/50 = 400.000 ¢, ~13/12 = 137.763 ¢
  • POTE: ~63/50 = 400.000 ¢, ~13/12 = 137.767 ¢

Optimal ET sequence: 87, 183, 270

Badness (Smith): 0.0153

Terture

See also: Vulture family

Subgroup: 2.3.5.7

Comma list: 250047/250000, 359661568/358722675

Mapping: [⟨3 4 3 2], ⟨0 4 21 34]]

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~392/375 = 75.555 ¢

Optimal ET sequence: 111, 159, 270

Badness (Smith): 0.087156

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 4 3 2 10], ⟨0 4 21 34 2]]

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~392/375 = 75.550 ¢

Optimal ET sequence: 111, 159, 270, 1239, 1509, 1779, 2049, 2319

Badness (Smith): 0.029326

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3025/3024, 10985/10976

Mapping: [⟨3 4 3 2 10 6], ⟨0 4 21 34 2 27]]

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~117/112 = 75.553 ¢

Optimal ET sequence: 111, 159, 270

Badness (Smith): 0.018647

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 936/935, 1001/1000, 4928/4913

Mapping: [⟨3 4 3 2 10 6 10], ⟨0 4 21 34 2 27 12]]

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~117/112 = 75.560 ¢

Optimal ET sequence: 111, 159, 270

Badness (Smith): 0.018705

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨3 4 3 2 10 6 10 5], ⟨0 4 21 34 2 27 12 41]]

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~95/91 = 75.560 ¢

Optimal ET sequence: 111, 159, 270

Badness (Smith): 0.013902

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

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

Mapping: [⟨3 4 3 2 10 6 10 5 13], ⟨0 4 21 34 2 27 12 41 3]]

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~24/23 = 75.548 ¢

Optimal ET sequence: 111, 159, 270

Badness (Smith): 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: [⟨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 ¢

Optimal ET sequence: 255, 270, 525, 795, 1065

Badness (Smith): 0.058002

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 tuning (CTE): ~22/21 = 80.0000 ¢, ~968/735 = 475.4912 ¢

Optimal ET sequence: 255, 270, 525, 795, 1065

Badness (Smith): 0.026246

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 tuning (CTE): ~22/21 = 80.0000 ¢, ~154/117 = 475.4935 ¢

Optimal ET sequence: 255, 270, 795, 1065

Badness (Smith): 0.021230

Akjayland

See also: 21st-octave temperaments

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 tuning (CTE): ~1323/1280 = 57.1429 ¢, ~131072/91875 = 614.9354 ¢

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

Badness (Smith): 0.0309

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 tuning (CTE): ~1323/1280 = 57.1429 ¢, ~6615/5632 = 278.8998 ¢

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

Badness (Smith): 0.0949

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 tuning (CTE): ~336/325 = 57.1429 ¢, ~168/143 = 278.9058 ¢

Optimal ET sequence: 168, 357, 525, 882

Badness (Smith): 0.0551

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 tuning (CTE): ~336/325 = 57.1429 ¢, ~168/143 = 278.9036 ¢

Optimal ET sequence: 168, 357, 525, 882

Badness (Smith): 0.0319

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 tuning (CTE): ~336/325 = 57.1429 ¢, ~168/143 = 278.9036 ¢

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

Badness (Smith): 0.0270

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 tuning (CTE): ~336/325 = 57.1429 ¢, ~168/143 = 278.8971 ¢

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

Badness (Smith): 0.0199

Magnesium

For the 5-limit version, see 12th-octave temperaments#Magnesium (5-limit).

Magnesium is named after element 12 for being period 12; however, it is not an extension of the atomic – the associated comma is [-157 -24 84⟩ in the 5-limit and the 7 generators together with 12edo major second reach the just perfect fifth, 3/2.

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 tuning (CTE): ~138915/131072 = 100.000 ¢, ~3145728/2734375 = 243.130 ¢

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

Badness (Smith): 0.0964

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. Named after the 24th element for being period 24.

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 tuning (CTE): ~250/243 = 50.0000 ¢, ~10/7 = 617.2710 ¢

Optimal ET sequence: 72, …, 480, 552, 624, 1320, 1944d, 3264d

Badness (Smith): 0.139

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨24 1 -6 18 46], ⟨0 3 5 4 3]]

Optimal tuning (CTE): ~250/243 = 50.0000 ¢, ~10/7 = 617.2597 ¢

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

Badness (Smith): 0.0398

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 tuning (CTE): ~250/243 = 50.0000 ¢, ~10/7 = 617.2869 ¢

Optimal ET sequence: 72, …, 480f, 552, 624, 1176de, 1800cdee

Badness (Smith): 0.0293

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 tuning (CTE): ~35/34 = 50.0000 ¢, ~10/7 = 617.2732 ¢

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

Badness (Smith): 0.0209

Zinc

See also: 10th-octave temperaments § Neon
See also: 60th-octave temperaments § Neodymium

Subgroup: 2.3.5.7

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

Mapping: [⟨30 2 15 66], ⟨0 5 6 2]]

Optimal tunings:

  • CTE: ~53747712/52521875 = 40.0000 ¢, ~216/175 = 364.3896 ¢ (~[21 3 1 -10⟩ = 4.3896 ¢)
  • CWE: ~53747712/52521875 = 40.0000 ¢, ~216/175 = 364.3879 ¢ (~[21 3 1 -10⟩ = 4.3879 ¢)

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

Badness (Smith): 0.0742

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:

  • CTE: ~18865/18432 = 40.0000 ¢, ~216/175 = 364.3886 ¢ (~385/384 = 4.3886 ¢)
  • CWE: ~18865/18432 = 40.0000 ¢, ~216/175 = 364.3849 ¢ (~385/384 = 4.3849 ¢)

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

Badness (Smith): 0.0220

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:

  • CTE: ~351/343 = 40.0000 ¢, ~216/175 = 364.3879 ¢ (~385/384 = 4.3879 ¢)
  • 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 (Smith): 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: [⟨30 2 15 66 122 193 91], ⟨0 5 6 2 -2 -9 4]]

Optimal tunings:

  • CTE: ~175/171 = 40.0000 ¢, ~216/175 = 364.3876 ¢ (~400/399 = 4.3876 ¢)
  • CWE: ~175/171 = 40.0000 ¢, ~216/175 = 364.3864 ¢ (~400/399 = 4.3864 ¢)

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

Badness (Smith): 0.00921

Poe

Subgroup: 2.3.5.7

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

Mapping: [⟨30 0 -73 -106], ⟨0 1 3 4]]

Optimal tunings:

  • CTE: ~2240/2187 = 40.000 ¢, ~3/2 = 702.141 ¢
  • CWE: ~2240/2187 = 40.000 ¢, ~3/2 = 702.166 ¢

Optimal ET sequence: 60, 210, 270

Badness (Sintel): 2.90

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 19712/19683, 250047/250000

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

Optimal tunings:

  • CTE: ~45/44 = 40.000, ~3/2 = 702.182 ¢
  • CWE: ~45/44 = 40.000 ¢, ~3/2 = 702.213 ¢

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:

  • CTE: ~45/44 = 40.000 ¢, ~3/2 = 702.182 ¢
  • CWE: ~45/44 = 40.000 ¢, ~3/2 = 702.170 ¢

Optimal ET sequence: 60e, 210e, 270

Badness (Sintel): 1.19

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