Landscape microtemperaments: Difference between revisions

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== Sextile ==
== Sextile ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Sextile (5-limit)]].''  
: ''For the 5-limit version, see [[Schismic–commatic equivalence continuum #Sextile (5-limit)]].''  


Sextile tempers out the [[garischisma]] with a 1/6-octave period and is the {{nowrap| 12 & 270 }} temperament.  
Sextile tempers out the [[garischisma]] with a 1/6-octave period and is the {{nowrap| 12 & 270 }} temperament.  
Line 34: Line 34:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2347{{c}}
* [[WE]]: ~4096/3645 = 199.9828{{c}}, ~3/2 = 702.1521{{c}}
* [[POTE]]: ~4096/3645 = 200.000{{c}}, ~3/2 = 702.212{{c}}
: [[error map]]: {{val| -0.103 +0.094 +0.173 -0.088 }}
* [[CWE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2187{{c}}
: error map: {{val| 0.000 +0.264 +0.374 +0.112 }}


{{Optimal ET sequence|legend=1| 12, 234d, 246d, 258, 270, 1362c, 1632c, 1902c, 2172c, 2442bc, 2712bc }}
{{Optimal ET sequence|legend=1| 12, …, 258, 270, 1362c, 1632c, …, 2442bc, 2712bc }}


[[Badness]] (Smith): 0.070097
[[Badness]] (Sintel): 1.77


=== 11-limit ===
=== 11-limit ===
Line 49: Line 51:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2225{{c}}
* CTE: ~55/49 = 199.9817{{c}}, ~3/2 = 702.1383{{c}}
* POTE: ~55/49 = 200.000{{c}}, ~3/2 = 702.202{{c}}
* POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2080{{c}}


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


Badness (Smith): 0.029677
Badness (Sintel): 0.981


=== 13-limit ===
=== 13-limit ===
Line 64: Line 66:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2141{{c}}
* WE: ~55/49 = 199.9804{{c}}, ~3/2 = 702.1260{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2001{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2001{{c}}


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


Badness (Smith): 0.0191
Badness (Sintel): 0.788


==== 17-limit ====
==== 17-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2117{{c}}
* WE: ~55/49 = 199.9669{{c}}, ~3/2 = 702.0643{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.1869{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.1869{{c}}


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


Badness (Smith): 0.0209
Badness (Sintel): 1.06


==== 19-limit ====
==== 19-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2118{{c}}
* CTE: ~55/49 = 199.9711{{c}}, ~3/2 = 702.0829{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.1890{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.1890{{c}}


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


Badness (Smith): 0.0156
Badness (Sintel): 0.948


=== Sextilia ===
=== Sextilia ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2296{{c}}
* WE: ~55/49 = 199.9975{{c}}, ~3/2 = 702.2196{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2285{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2285{{c}}


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


Badness (Smith): 0.0391
Badness (Sintel): 1.62


==== 17-limit ====
==== 17-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2264{{c}}
* WE: ~55/49 = 199.9862{{c}}, ~3/2 = 702.1714{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2207{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2207{{c}}


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


Badness (Smith): 0.0384
Badness (Sintel): 1.95


==== 19-limit ====
==== 19-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2266{{c}}
* WE: ~55/49 = 199.9866{{c}}, ~3/2 = 702.1731{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2208{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2208{{c}}


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


Badness (Smith): 0.0252
Badness (Sintel): 1.53


== Septichrome ==
== Septichrome ==
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{{Mapping|legend=1| 3 3 1 0 | 0 5 17 24 }}
{{Mapping|legend=1| 3 3 1 0 | 0 5 17 24 }}
: mapping generators: ~2, ~243/224


[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~243/224 = 140.367{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0100{{c}}, ~243/224 = 140.3702{{c}}
: [[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| 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]] (Smith): 0.016814
[[Badness]] (Sintel): 0.426


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


Optimal tuning (CTE): ~55/49 = 200.0000{{c}}, ~375/308 = 340.3687{{c}} (~1760/1701 = 59.6313{{c}})
Optimal tunings:
* WE: ~55/49 = 200.0058{{c}}, ~375/308 = 340.3742{{c}} (~1760/1701 = 59.6375{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3661{{c}} (~1760/1701 = 59.6339{{c}})


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


Badness (Smith): 0.0194
Badness (Sintel): 0.642


==== 13-limit ====
==== 13-limit ====
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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 = 200.0000{{c}}, ~375/308 = 340.3778{{c}} (~121/117 = 59.6222{{c}})
Optimal tunings:
* WE: ~55/49 = 199.9936{{c}}, ~375/308 = 340.3707{{c}} (~121/117 = 59.6165{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3802{{c}} (~121/117 = 59.6198{{c}})


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


Badness (Smith): 0.0397
Badness (Sintel): 1.64


==== 17-limit ====
==== 17-limit ====
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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 = 200.0000{{c}}, ~375/308 = 340.3763{{c}} (~88/85 = 59.6237{{c}})
Optimal tunings:
* WE: ~55/49 = 199.9865{{c}}, ~375/308 = 340.3619{{c}} (~88/85 = 59.6111{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3821{{c}} (~88/85 = 59.6179{{c}})


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


Badness (Smith): 0.0272
Badness (Sintel): 1.39


==== 19-limit ====
==== 19-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 23409/23408
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 23409/23408
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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 = 200.0000{{c}}, ~162/133 = 340.3767{{c}} (~88/85 = 59.6233{{c}})
Optimal tunings:
* WE: ~55/49 = 199.9837{{c}}, ~162/133 = 340.3589{{c}} (~88/85 = 59.6084{{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 }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness (Smith): 0.0222
Badness (Sintel): 1.35


==== 23-limit ====
==== 23-limit ====
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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 = 200.0000{{c}}, ~162/133 = 340.3757{{c}} (~88/85 = 59.6243{{c}})
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 }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness (Smith): 0.0160
Badness (Sintel): 1.14


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


{{Mapping|legend=1| 3 10 11 6 | 0 -13 -10 6 }}
{{Mapping|legend=1| 3 -3 1 12 | 0 13 10 -6 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~192/175 = 161.399{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0312{{c}}, ~147/128 = 238.6196{{c}} (~192/175 = 161.4116{{c}})
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~147/128 = 238.6038{{c}} (~192/175 = 161.3962{{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]] (Smith): 0.045660
[[Badness]] (Sintel): 1.16


== Domain ==
== Domain ==

Revision as of 05:59, 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, terture, slendscape, akjayland, magnesium, chromium, zinc, and poe.

Sextile

For the 5-limit version, see Schismic–commatic equivalence continuum #Sextile (5-limit).

Sextile tempers out the garischisma with a 1/6-octave period and is the 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:

  • WE: ~4096/3645 = 199.9828 ¢, ~3/2 = 702.1521 ¢
error map: ⟨-0.103 +0.094 +0.173 -0.088]
  • CWE: ~4096/3645 = 200.0000 ¢, ~3/2 = 702.2187 ¢
error map: ⟨0.000 +0.264 +0.374 +0.112]

Optimal ET sequence: 12, …, 258, 270, 1362c, 1632c, …, 2442bc, 2712bc

Badness (Sintel): 1.77

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 = 199.9817 ¢, ~3/2 = 702.1383 ¢
  • POTE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2080 ¢

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

Badness (Sintel): 0.981

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:

  • WE: ~55/49 = 199.9804 ¢, ~3/2 = 702.1260 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2001 ¢

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

Badness (Sintel): 0.788

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:

  • WE: ~55/49 = 199.9669 ¢, ~3/2 = 702.0643 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.1869 ¢

Optimal ET sequence: 12f, 270, 552g

Badness (Sintel): 1.06

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 = 199.9711 ¢, ~3/2 = 702.0829 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.1890 ¢

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

Badness (Sintel): 0.948

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:

  • WE: ~55/49 = 199.9975 ¢, ~3/2 = 702.2196 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2285 ¢

Optimal ET sequence: 12, 258e, 270

Badness (Sintel): 1.62

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:

  • WE: ~55/49 = 199.9862 ¢, ~3/2 = 702.1714 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2207 ¢

Optimal ET sequence: 12, 258e, 270

Badness (Sintel): 1.95

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

Optimal tunings:

  • WE: ~55/49 = 199.9866 ¢, ~3/2 = 702.1731 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 702.2208 ¢

Optimal ET sequence: 12, 258e, 270

Badness (Sintel): 1.53

Septichrome

Subgroup: 2.3.5.7

Comma list: 250047/250000, 2460375/2458624

Mapping: [⟨3 3 1 0], ⟨0 5 17 24]]

mapping generators: ~2, ~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

Pnict

Subgroup: 2.3.5.7

Comma list: 250047/250000, 2100875/2097152

Mapping: [⟨3 -3 1 12], ⟨0 13 10 -6]]

Optimal tunings:

  • WE: ~63/50 = 400.0312 ¢, ~147/128 = 238.6196 ¢ (~192/175 = 161.4116 ¢)
  • CWE: ~63/50 = 400.0000 ¢, ~147/128 = 238.6038 ¢ (~192/175 = 161.3962 ¢)

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

Badness (Sintel): 1.16

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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