Landscape microtemperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (2/). - domain (covered in minortonic family)
Switch to Sintel's badness, WE & CWE tunings (3/)
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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
: mapping generators: ~63/50, ~243/224


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


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~63/50 = 400.000{{c}}, ~1024/945 = 137.772{{c}}
* [[WE]]: ~63/50 = 399.9681{{c}}, ~1024/945 = 137.7570{{c}}
* [[POTE]]: ~63/50 = 400.000{{c}}, ~1024/945 = 137.768{{c}}
: [[error map]]: {{val| -0.096 +0.037 +0.160 +0.010 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~1024/945 = 137.7689{{c}}
: error map: {{val| 0.000 +0.196 +0.380 +0.259 }}


{{Optimal ET sequence|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}
{{Optimal ET sequence|legend=1| 87, 183, 270, 723, 993, 1263, 2796cd, 4059bccd }}


[[Badness]] (Smith): 0.062187
[[Badness]] (Sintel): 1.57


=== 11-limit ===
=== 11-limit ===
Line 283: Line 287:


Optimal tunings:  
Optimal tunings:  
* CTE: ~63/50 = 400.000{{c}}, ~693/640 = 137.773{{c}}
* WE: ~63/50 = 399.9798{{c}}, ~693/640 = 137.7643{{c}}
* POTE: ~63/50 = 400.000{{c}}, ~693/640 = 137.771{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~693/640 = 137.7716{{c}}


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


Badness (Smith): 0.023085
Badness (Sintel): 0.763


=== 13-limit ===
=== 13-limit ===
Line 298: Line 302:


Optimal tunings:  
Optimal tunings:  
* CTE: ~63/50 = 400.000{{c}}, ~13/12 = 137.777{{c}}
* WE: ~63/50 = 399.9921{{c}}, ~13/12 = 137.7743{{c}}
* POTE: ~63/50 = 400.000{{c}}, ~13/12 = 137.777{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~13/12 = 137.7770{{c}}


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


Badness (Smith): 0.015557
Badness (Sintel): 0.643


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


Comma list: 676/675, 715/714, 1001/1000, 3025/3024, 4096/4095
Comma list: 676/675, 715/714, 936/935, 1001/1000, 4096/4095


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 tunings:  
Optimal tunings:  
* CTE: ~63/50 = 400.000{{c}}, ~13/12 = 137.758{{c}}
* WE: ~34/27 = 399.9776{{c}}, ~13/12 = 137.7535{{c}}
* POTE: ~63/50 = 400.000{{c}}, ~13/12 = 137.761{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~13/12 = 137.7608{{c}}


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


Badness (Smith): 0.0171
Badness (Sintel): 0.869


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


Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 1729/1728, 3025/3024
Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1729/1728


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 tunings:  
Optimal tunings:  
* CTE: ~63/50 = 400.000{{c}}, ~13/12 = 137.763{{c}}
* WE: ~34/27 = 399.9804{{c}}, ~13/12 = 137.7602{{c}}
* POTE: ~63/50 = 400.000{{c}}, ~13/12 = 137.767{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~13/12 = 137.7664{{c}}


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


Badness (Smith): 0.0153
Badness (Sintel): 0.928


== Terture ==
== Terture ==
Line 342: Line 346:
[[Comma list]]: 250047/250000, 359661568/358722675
[[Comma list]]: 250047/250000, 359661568/358722675


{{Mapping|legend=1| 3 4 3 2 | 0 4 21 34 }}
{{Mapping|legend=1| 3 0 -18 -32 | 0 4 21 34 }}
: mapping generators: ~63/50, ~320/243


[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~392/375 = 75.555{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 399.9723{{c}}, ~320/243 = 475.5221{{c}} (~392/375 = 75.5499{{c}})
: [[error map]]: {{val| -0.083 +0.134 +0.151 -0.185 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~320/243 = 475.5519{{c}} (~392/375 = 75.5519{{c}})
: error map: {{val| 0.000 +0.253 +0.276 -0.061 }}


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


[[Badness]] (Smith): 0.087156
[[Badness]] (Sintel): 2.21


=== 11-limit ===
=== 11-limit ===
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Comma list: 3025/3024, 19712/19683, 102487/102400
Comma list: 3025/3024, 19712/19683, 102487/102400


Mapping: {{mapping| 3 4 3 2 10 | 0 4 21 34 2 }}
Mapping: {{mapping| 3 0 -18 -32 8 | 0 4 21 34 2 }}


Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~392/375 = 75.550{{c}}
Optimal tunings:
* WE: ~63/50 = 399.9902{{c}}, ~320/243 = 475.5383{{c}} (~392/375 = 75.5481{{c}})
* CWE: ~63/50 = 400.0000{{c}}, ~320/243 = 475.5490{{c}} (~392/375 = 75.5490{{c}})


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


Badness (Smith): 0.029326
Badness (Sintel): 0.969


=== 13-limit ===
=== 13-limit ===
Line 368: Line 379:
Comma list: 676/675, 1001/1000, 3025/3024, 10985/10976
Comma list: 676/675, 1001/1000, 3025/3024, 10985/10976


Mapping: {{mapping| 3 4 3 2 10 6 | 0 4 21 34 2 27 }}
Mapping: {{mapping| 3 0 -18 -32 8 -21 | 0 4 21 34 2 27 }}


Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~117/112 = 75.553{{c}}
Optimal tunings:
* WE: ~63/50 = 399.9958{{c}}, ~154/117 = 475.5485{{c}} (~117/112 = 75.5527{{c}})
* CWE: ~63/50 = 400.0000{{c}}, ~154/117 = 475.5531{{c}} (~117/112 = 75.5531{{c}})


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


Badness (Smith): 0.018647
Badness (Sintel): 0.771


=== 17-limit ===
=== 17-limit ===
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Comma list: 676/675, 715/714, 936/935, 1001/1000, 4928/4913
Comma list: 676/675, 715/714, 936/935, 1001/1000, 4928/4913


Mapping: {{mapping| 3 4 3 2 10 6 10 | 0 4 21 34 2 27 12 }}
Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 | 0 4 21 34 2 27 12 }}


Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~117/112 = 75.560{{c}}
Optimal tunings:
* WE: ~34/27 = 399.9664{{c}}, ~112/85 = 475.5198{{c}} (~117/112 = 75.5534{{c}})
* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5568{{c}} (~117/112 = 75.5568{{c}})


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


Badness (Smith): 0.018705
Badness (Sintel): 0.953


=== 19-limit ===
=== 19-limit ===
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Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1617/1615
Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1617/1615


Mapping: {{mapping| 3 4 3 2 10 6 10 5 | 0 4 21 34 2 27 12 41 }}
Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 -36 | 0 4 21 34 2 27 12 41 }}


Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~95/91 = 75.560{{c}}
Optimal tunings:
* WE: ~34/27 = 399.9665{{c}}, ~112/85 = 475.5198{{c}} (~95/91 = 75.5533{{c}})
* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5568{{c}} (~95/91 = 75.5568{{c}})


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


Badness (Smith): 0.013902
Badness (Sintel): 0.846


=== 23-limit ===
=== 23-limit ===
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Comma list: 460/459, 529/528, 676/675, 715/714, 936/935, 1001/1000, 1216/1215
Comma list: 460/459, 529/528, 676/675, 715/714, 936/935, 1001/1000, 1216/1215


Mapping: {{mapping| 3 4 3 2 10 6 10 5 13 | 0 4 21 34 2 27 12 41 3 }}
Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 -36 10 | 0 4 21 34 2 27 12 41 3 }}


Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~24/23 = 75.548{{c}}
Optimal tunings:
* WE: ~34/27 = 400.0026{{c}}, ~112/85 = 475.5510{{c}} (~24/23 = 75.5485{{c}})
* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5482{{c}} (~24/23 = 75.5482{{c}})


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


Badness (Smith): 0.014915
Badness (Sintel): 1.07


== Slendscape ==
== Slendscape ==

Revision as of 06:34, 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, 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: ~63/50, ~243/224

Optimal tunings:

  • WE: ~63/50 = 400.0100 ¢, ~243/224 = 140.3702 ¢
error map: ⟨+0.030 -0.074 -0.010 +0.059]
  • CWE: ~63/50 = 400.0000 ¢, ~243/224 = 140.3685 ¢
error map: ⟨0.000 -0.113 -0.050 +0.017]

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

Badness (Sintel): 0.426

Semiseptichrome

Subgroup: 2.3.5.7.11

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

Mapping: [⟨6 1 -15 -24 -32], ⟨0 5 17 24 31]]

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

Optimal tunings:

  • WE: ~55/49 = 200.0058 ¢, ~375/308 = 340.3742 ¢ (~1760/1701 = 59.6375 ¢)
  • CWE: ~55/49 = 200.0000 ¢, ~375/308 = 340.3661 ¢ (~1760/1701 = 59.6339 ¢)

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

Badness (Sintel): 0.642

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 34398/34375, 85293/85184

Mapping: [⟨6 1 -15 -24 -32 -68], ⟨0 5 17 24 31 53]]

Optimal tunings:

  • WE: ~55/49 = 199.9936 ¢, ~375/308 = 340.3707 ¢ (~121/117 = 59.6165 ¢)
  • CWE: ~55/49 = 200.0000 ¢, ~375/308 = 340.3802 ¢ (~121/117 = 59.6198 ¢)

Optimal ET sequence: 282, 342f, 624

Badness (Sintel): 1.64

17-limit

Subgroup: 2.3.5.7.11.13

Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 61965/61952

Mapping: [⟨6 1 -15 -24 -32 -68 -1], ⟨0 5 17 24 31 53 15]]

Optimal tunings:

  • WE: ~55/49 = 199.9865 ¢, ~375/308 = 340.3619 ¢ (~88/85 = 59.6111 ¢)
  • CWE: ~55/49 = 200.0000 ¢, ~375/308 = 340.3821 ¢ (~88/85 = 59.6179 ¢)

Optimal ET sequence: 282, 342f, 624

Badness (Sintel): 1.39

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 23409/23408

Mapping: [⟨6 1 -15 -24 -32 -68 -1 34], ⟨0 5 17 24 31 53 15 -5]]

Optimal tunings:

  • WE: ~55/49 = 199.9837 ¢, ~162/133 = 340.3589 ¢ (~88/85 = 59.6084 ¢)
  • CWE: ~55/49 = 200.0000 ¢, ~162/133 = 340.3844 ¢ (~88/85 = 59.6156 ¢)

Optimal ET sequence: 282, 342f, 624

Badness (Sintel): 1.35

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 936/935, 1701/1700, 1716/1715, 1863/1862, 2024/2023, 2025/2023, 2376/2375

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

Optimal tunings:

  • WE: ~55/49 = 199.9829 ¢, ~162/133 = 340.3576 ¢ (~88/85 = 59.6081 ¢)
  • CWE: ~55/49 = 200.0000 ¢, ~162/133 = 340.3844 ¢ (~88/85 = 59.6156 ¢)

Optimal ET sequence: 282, 342f, 624

Badness (Sintel): 1.14

Pnict

Subgroup: 2.3.5.7

Comma list: 250047/250000, 2100875/2097152

Mapping: [⟨3 -3 1 12], ⟨0 13 10 -6]]

mapping generators: ~63/50, ~147/128

Optimal tunings:

  • WE: ~63/50 = 400.0312 ¢, ~147/128 = 238.6196 ¢ (~192/175 = 161.4116 ¢)
error map: ⟨+0.094 +0.006 -0.087 -0.169]
  • CWE: ~63/50 = 400.0000 ¢, ~147/128 = 238.6038 ¢ (~192/175 = 161.3962 ¢)
error map: ⟨0.000 -0.106 -0.276 -0.449]

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

Badness (Sintel): 1.16

Avicenna

Subgroup: 2.3.5.7

Comma list: 250047/250000, 29360128/29296875

Mapping: [⟨3 2 8 16], ⟨0 8 -3 -22]]

mapping generators: ~63/50, ~1024/945

Optimal tunings:

  • WE: ~63/50 = 399.9681 ¢, ~1024/945 = 137.7570 ¢
error map: ⟨-0.096 +0.037 +0.160 +0.010]
  • CWE: ~63/50 = 400.0000 ¢, ~1024/945 = 137.7689 ¢
error map: ⟨0.000 +0.196 +0.380 +0.259]

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

Badness (Sintel): 1.57

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~63/50 = 399.9798 ¢, ~693/640 = 137.7643 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~693/640 = 137.7716 ¢

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

Badness (Sintel): 0.763

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

  • WE: ~63/50 = 399.9921 ¢, ~13/12 = 137.7743 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~13/12 = 137.7770 ¢

Optimal ET sequence: 87, 183, 270

Badness (Sintel): 0.643

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 936/935, 1001/1000, 4096/4095

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

Optimal tunings:

  • WE: ~34/27 = 399.9776 ¢, ~13/12 = 137.7535 ¢
  • CWE: ~34/27 = 400.0000 ¢, ~13/12 = 137.7608 ¢

Optimal ET sequence: 87, 183, 270, 453

Badness (Sintel): 0.869

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

Optimal tunings:

  • WE: ~34/27 = 399.9804 ¢, ~13/12 = 137.7602 ¢
  • CWE: ~34/27 = 400.0000 ¢, ~13/12 = 137.7664 ¢

Optimal ET sequence: 87, 183, 270

Badness (Sintel): 0.928

Terture

See also: Vulture family

Subgroup: 2.3.5.7

Comma list: 250047/250000, 359661568/358722675

Mapping: [⟨3 0 -18 -32], ⟨0 4 21 34]]

mapping generators: ~63/50, ~320/243

Optimal tunings:

  • WE: ~63/50 = 399.9723 ¢, ~320/243 = 475.5221 ¢ (~392/375 = 75.5499 ¢)
error map: ⟨-0.083 +0.134 +0.151 -0.185]
  • CWE: ~63/50 = 400.0000 ¢, ~320/243 = 475.5519 ¢ (~392/375 = 75.5519 ¢)
error map: ⟨0.000 +0.253 +0.276 -0.061]

Optimal ET sequence: 111, 159, 270

Badness (Sintel): 2.21

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 0 -18 -32 8], ⟨0 4 21 34 2]]

Optimal tunings:

  • WE: ~63/50 = 399.9902 ¢, ~320/243 = 475.5383 ¢ (~392/375 = 75.5481 ¢)
  • CWE: ~63/50 = 400.0000 ¢, ~320/243 = 475.5490 ¢ (~392/375 = 75.5490 ¢)

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

Badness (Sintel): 0.969

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨3 0 -18 -32 8 -21], ⟨0 4 21 34 2 27]]

Optimal tunings:

  • WE: ~63/50 = 399.9958 ¢, ~154/117 = 475.5485 ¢ (~117/112 = 75.5527 ¢)
  • CWE: ~63/50 = 400.0000 ¢, ~154/117 = 475.5531 ¢ (~117/112 = 75.5531 ¢)

Optimal ET sequence: 111, 159, 270

Badness (Sintel): 0.771

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨3 0 -18 -32 8 -21 -2], ⟨0 4 21 34 2 27 12]]

Optimal tunings:

  • WE: ~34/27 = 399.9664 ¢, ~112/85 = 475.5198 ¢ (~117/112 = 75.5534 ¢)
  • CWE: ~34/27 = 400.0000 ¢, ~112/85 = 475.5568 ¢ (~117/112 = 75.5568 ¢)

Optimal ET sequence: 111, 159, 270

Badness (Sintel): 0.953

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 0 -18 -32 8 -21 -2 -36], ⟨0 4 21 34 2 27 12 41]]

Optimal tunings:

  • WE: ~34/27 = 399.9665 ¢, ~112/85 = 475.5198 ¢ (~95/91 = 75.5533 ¢)
  • CWE: ~34/27 = 400.0000 ¢, ~112/85 = 475.5568 ¢ (~95/91 = 75.5568 ¢)

Optimal ET sequence: 111, 159, 270

Badness (Sintel): 0.846

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 0 -18 -32 8 -21 -2 -36 10], ⟨0 4 21 34 2 27 12 41 3]]

Optimal tunings:

  • WE: ~34/27 = 400.0026 ¢, ~112/85 = 475.5510 ¢ (~24/23 = 75.5485 ¢)
  • CWE: ~34/27 = 400.0000 ¢, ~112/85 = 475.5482 ¢ (~24/23 = 75.5482 ¢)

Optimal ET sequence: 111, 159, 270

Badness (Sintel): 1.07

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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