Kalismic temperaments: Difference between revisions

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{{Mapping|legend=1| 6 0 0 8 17 | 0 1 0 -2 -4 | 0 0 1 2 3 }}
{{Mapping|legend=1| 6 0 0 8 17 | 0 1 0 -2 -4 | 0 0 1 2 3 }}
: mapping generators: ~55/49, ~3, ~5
: mapping generators: ~55/49, ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~55/49 = 200.0008{{c}}, ~3/2 = 701.9318{{c}}, ~5/4 = 386.3342{{c}}
: [[error map]]: {{val| +0.005 -0.018 +0.031 -0.004 -0.018 }}
* [[CWE]]: ~55/49 = 200.0000{{c}}, ~3/2 = 701.9320{{c}}, ~5/4 = 386.3393{{c}}
: error map: {{val| 0.000 -0.023 +0.026 -0.011 -0.028 }}


{{Optimal ET sequence|legend=1| 12, 42, 48dee, 54c, 60e, 72, 198, 270, 342, 612, 954, 1236, 1506, 1578, 1848, 3426, 4038, 5616, 7464, 17118e, 18966e }}
{{Optimal ET sequence|legend=1| 12, 42, 48dee, 54c, 60e, 72, 198, 270, 342, 612, 954, 1236, 1506, 1578, 1848, 3426, 4038, 5616, 7464, 17118e, 18966e }}


[[Badness]] (Smith): 0.116 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.140


== Lycoris ==
== Lycoris ==
Line 39: Line 45:
[[Comma list]]: 9801/9800, 1771561/1771470
[[Comma list]]: 9801/9800, 1771561/1771470


{{Mapping|legend=1| 2 0 4 -6 1 | 0 1 1 3 2 | 0 0 -6 5 -1 }}
{{Mapping|legend=1| 2 0 -2 1 0 | 0 1 1 3 2 | 0 0 6 -5 1 }}
: mapping generators: ~99/70, ~3, ~81/70
: mapping generators: ~99/70, ~3, ~11/9


[[Optimal tuning]] ([[CTE]]): ~99/70 = 600.0000{{c}}, ~3/2 = 701.9439{{c}}, ~81/70 = 252.6028{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 600.0018{{c}}, ~3/2 = 701.9411{{c}}, ~11/9 = 347.3976{{c}}
: [[error map]]: {{val| +0.004 -0.010 +0.013 -0.018 -0.031 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 701.9414{{c}}, ~11/9 = 347.3969{{c}}
: error map: {{val| 0.000 -0.014 +0.009 +0.014 -0.038 }}


{{Optimal ET sequence|legend=1| 152, 328, 342, 836, 1178, 1354, 1506, 1848, 2684, 4038, 4190, 4532, 11254, 15786e }}
{{Optimal ET sequence|legend=1| 152, 328, 342, 836, 1178, 1354, 1506, 1848, 2684, 4038, 4190, 4532, 11254, 15786e }}


[[Badness]] (Smith): 0.249 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.299


=== Higanbana ===
=== Higanbana ===
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Comma list: 9801/9800, 10648/10647, 1399680/1399489
Comma list: 9801/9800, 10648/10647, 1399680/1399489


Mapping: {{mapping| 2 1 5 -3 3 8 | 0 2 2 6 4 1 | 0 0 -6 5 -1 4 }}
Mapping: {{mapping| 2 1 -1 2 2 4 | 0 2 2 6 4 1 | 0 0 6 -5 1 4 }}
: mapping generators: ~99/70, ~1458/1001, ~81/70
: mapping generators: ~99/70, ~1458/1001, ~81/70


Optimal tuning (CTE): ~99/70 = 600.0000{{c}}, ~1458/1001 = 650.9715{{c}}, ~81/70 = 252.6037{{c}}
Optimal tunings:
* WE: ~99/70 = 599.9996{{c}}, ~1458/1001 = 650.9717{{c}}, ~11/9 = 347.3963{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~1458/1001 = 650.9718{{c}}, ~11/9 = 347.3964{{c}}


{{Optimal ET sequence|legend=0| 166, 190, 304d, 328, 494, 684, 1012, 1178, 1506, 2190, 2684, 4190, 6380, 9064, 15938 }}
{{Optimal ET sequence|legend=0| 166, 190, 304d, 328, 494, 684, 1012, 1178, 1506, 2190, 2684, 4190, 6380, 9064, 15938 }}


Badness (Smith): 0.416 × 10<sup>-3</sup>
Badness (Sintel): 0.389


== Van gogh ==
== Van gogh ==
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{{Mapping|legend=1| 2 0 8 0 11 | 0 1 1 2 1 | 0 0 -9 -1 -10 }}
{{Mapping|legend=1| 2 0 8 0 11 | 0 1 1 2 1 | 0 0 -9 -1 -10 }}
: mapping generators: ~99/70, ~3, ~9/7
: mapping generators: ~99/70, ~3, ~11/10
 
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 600.0022{{c}}, ~3/2 = 701.9464{{c}}, ~11/9 = 164.9319{{c}}
: [[error map]]: {{val| +0.004 -0.004 +0.022 +0.005 -0.046 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 701.9469{{c}}, ~11/9 = 164.9316{{c}}
: error map: {{val| 0.000 -0.008 +0.018 -0.000 -0.055 }}


{{Optimal ET sequence|legend=1| 22, 58, 80, 138cde, 204cde, 226ce, 240d, 262d, 284, 320, 342, 742, 764, 1084, 1106, 1448, 1506, 1848, 4038, 4802, 5144, 6992 }}
{{Optimal ET sequence|legend=1| 22, 58, 80, 138cde, 204cde, 226ce, 240d, 262d, 284, 320, 342, 742, 764, 1084, 1106, 1448, 1506, 1848, 4038, 4802, 5144, 6992 }}


[[Badness]] (Smith): 0.297 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.358


== Sif ==
== Sif ==
Line 82: Line 100:
: mapping generators: ~99/70, ~3
: mapping generators: ~99/70, ~3


[[Optimal tuning]] ([[CTE]]): ~99/70 = 600.0000{{c}}, ~3/2 = 701.9492{{c}}, ~48/35 = 546.6041{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 599.9863{{c}}, ~3/2 = 701.9658{{c}}, ~48/35 = 546.5930{{c}}
: [[error map]]: {{val| -0.027 -0.017 +0.045 +0.052 +0.000 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 701.9755{{c}}, ~48/35 = 546.6052{{c}}
: error map: {{val| 0.000 +0.021 +0.107 +0.124 +0.101 }}


{{Optimal ET sequence|legend=1| 22, 46, 68, 114, 134, 156, 178, 202, 224, 270, 494, 742, 764, 966, 1236, 1506, 2742, 3236, 3506, 4742d, 10990ccdde, 15732ccdddee }}
{{Optimal ET sequence|legend=1| 22, 46, 68, 114, 134, 156, 178, 202, 224, 270, 494, 742, 764, 966, 1236, 1506, 2742, 3236, 3506, 4742d, 10990ccdde, 15732ccdddee }}


[[Badness]] (Smith): 0.423 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.508


=== 13-limit ===
=== 13-limit ===
Line 95: Line 117:
Mapping: {{mapping| 2 0 1 7 11 16 | 0 1 0 1 -1 -3 | 0 0 4 -5 -1 1 }}
Mapping: {{mapping| 2 0 1 7 11 16 | 0 1 0 1 -1 -3 | 0 0 4 -5 -1 1 }}


Optimal tuning (CTE): ~99/70 = 600.0000{{c}}, ~3/2 = 701.9902{{c}}, ~48/35 = 546.6064{{c}}
Optimal tunings:
* WE: ~99/70 = 599.9857{{c}}, ~3/2 = 701.9705{{c}}, ~48/35 = 546.5927{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.9877{{c}}, ~48/35 = 546.6058{{c}}


{{Optimal ET sequence|legend=0| 22, 46, 156f, 178, 202f, 224, 270, 494, 764, 1012, 1236, 1506, 3236, 3506, 4742df, 6248cdf, 8248cdef, 12990ccddeff, 14496ccdddeeff }}
{{Optimal ET sequence|legend=0| 22, 46, 156f, 178, 202f, 224, 270, 494, 764, 1012, 1236, 1506, 3236, 3506, 4742df, 6248cdf, 8248cdef, 12990ccddeff, 14496ccdddeeff }}


Badness (Smith): 0.339 × 10<sup>-3</sup>
Badness (Sintel): 0.317


== Loki ==
== Loki ==
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{{Mapping|legend=1| 2 0 0 -21 -18 | 0 1 0 4 2 | 0 0 1 3 4 }}
{{Mapping|legend=1| 2 0 0 -21 -18 | 0 1 0 4 2 | 0 0 1 3 4 }}
: mapping generators: ~99/70, ~3, ~5
: mapping generators: ~99/70, ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 599.9481{{c}}, ~3/2 = 702.1195{{c}}, ~5/4 = 386.7636{{c}}
: [[error map]]: {{val| -0.104 +0.061 +0.242 -0.005 -0.128 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 702.1647{{c}}, ~5/4 = 386.7768{{c}}
: error map: {{val| 0.000 +0.210 +0.463 +0.163 +0.119 }}


{{Optimal ET sequence|legend=1| 12, 22, 34d, 56d, 74d, 84de, 96d, 118, 130, 152, 248, 270, 670, 822, 940, 1092, 1362c, 2032c, 2302c }}
{{Optimal ET sequence|legend=1| 12, 22, 34d, 56d, 74d, 84de, 96d, 118, 130, 152, 248, 270, 670, 822, 940, 1092, 1362c, 2032c, 2302c }}


[[Badness]] (Smith): 0.493 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.592


== Pessoal ==
== Pessoal ==
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: mapping generators: ~99/70, ~3, ~32/21
: mapping generators: ~99/70, ~3, ~32/21


[[Optimal tuning]] ([[CTE]]): ~99/70 = 600.0000{{c}}, ~3/2 = 702.0759{{c}}, ~32/21 = 728.7910{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 599.9711{{c}}, ~3/2 = 702.0265{{c}}, ~32/21 = 728.7942{{c}}
: [[error map]]: {{val| -0.058 +0.014 +0.040 +0.122 -0.040 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 702.0635{{c}}, ~32/21 = 728.8214{{c}}
: error map: {{val| 0.000 +0.109 +0.150 +0.289 +0.134 }}


{{Optimal ET sequence|legend=1| 36, 46, 84, 94, 130, 224, 270, 494, 764, 1164, 1658, 3586cd, 5244cdde, 6008bcdde }}
{{Optimal ET sequence|legend=1| 36, 46, 84, 94, 130, 224, 270, 494, 764, 1164, 1658, 3586cd, 5244cdde, 6008bcdde }}


[[Badness]] (Smith): 0.499 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.599


=== 13-limit ===
=== 13-limit ===
Line 136: Line 170:
Mapping: {{mapping| 2 0 1 10 14 13 | 0 1 0 -1 -3 -1 | 0 0 3 -1 2 -2 }}
Mapping: {{mapping| 2 0 1 10 14 13 | 0 1 0 -1 -3 -1 | 0 0 3 -1 2 -2 }}


Optimal tuning (CTE): ~99/70 = 600.0000{{c}}, ~3/2 = 702.0637{{c}}, ~32/21 = 728.7786{{c}}
Optimal tunings:
* WE: ~99/70 = 599.9835{{c}}, ~3/2 = 702.0253{{c}}, ~32/21 = 728.7700{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.0477{{c}}, ~32/21 = 728.7882{{c}}


{{Optimal ET sequence|legend=0| 36, 46, 84, 94, 130, 224, 270, 494, 764, 1258, 1882d, 2152d }}
{{Optimal ET sequence|legend=0| 36, 46, 84, 94, 130, 224, 270, 494, 764, 1258, 1882d, 2152d }}


Badness (Smith): 0.391 × 10<sup>-3</sup>
Badness (Sintel): 0.366


== Linus ==
== Linus ==
Line 146: Line 182:
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Linus]].''
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Linus]].''


Linus [[tempering out|tempers out]] the [[linus comma]], {{monzo| 11 -10 -10 10 }} in the 7-limit and can be described as the 80 & 130 & 270 temperament. It tempers out 9801/9800 and 391314/390625 in the 11-limit; 1001/1000, 4225/4224, and 4459/4455 in the 13-limit. The 1/10-octave period interval represents 15/14, three of which represents 16/13, and five of which represents both 99/70 and 140/99.
Linus [[tempering out|tempers out]] the [[linus comma]], {{monzo| 11 -10 -10 10 }} in the 7-limit and can be described as the {{nowrap| 80 & 130 & 270 }} temperament. It tempers out 9801/9800 and 391314/390625 in the 11-limit; 1001/1000, 4225/4224, and 4459/4455 in the 13-limit. The 1/10-octave period interval represents 15/14, three of which represents 16/13, and five of which represents both 99/70 and 140/99.


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 154: Line 190:
{{Mapping|legend=1| 10 0 0 -11 4 | 0 1 0 1 -1 | 0 0 1 1 2 }}
{{Mapping|legend=1| 10 0 0 -11 4 | 0 1 0 1 -1 | 0 0 1 1 2 }}


[[Optimal tuning]] ([[CTE]]): ~15/14 = 120.0000{{c}}, ~3/2 = 702.0010{{c}}, ~5/4 = 386.5927{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~15/14 = 119.9975{{c}}, ~3/2 = 702.0188{{c}}, ~5/4 = 386.6399{{c}}
: [[error map]]: {{val| -0.025 +0.039 +0.276 -0.215 -0.143 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~3/2 = 702.0181{{c}}, ~5/4 = 386.6218{{c}}
: error map: {{val| 0.000 +0.063 +0.308 -0.186 -0.092 }}


{{Optimal ET sequence|legend=1| 50, 60e, 80, 130, 190, 270, 670, 940, 1130, 1400, 1800c, 2070c, 2340c }}
{{Optimal ET sequence|legend=1| 50, 60e, 80, 130, 190, 270, 670, 940, 1130, 1400, 1800c, 2070c, 2340c }}


[[Badness]] (Smith): 1.12 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.34


=== 13-limit ===
=== 13-limit ===
Line 167: Line 207:
Mapping: {{mapping| 10 0 0 -11 4 37 | 0 1 0 1 -1 0 | 0 0 1 1 2 0 }}
Mapping: {{mapping| 10 0 0 -11 4 37 | 0 1 0 1 -1 0 | 0 0 1 1 2 0 }}


Optimal tuning (CTE): ~15/14 = 1200.0000{{c}}, ~3/2 = 702.0010{{c}}, ~5/4 = 386.5928{{c}}
Optimal tunings:
* WE: ~15/14 = 120.0054{{c}}, ~3/2 = 701.9625{{c}}, ~5/4 = 386.4910{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 701.9533{{c}}, ~5/4 = 386.5119{{c}}


{{Optimal ET sequence|legend=0| 50, 60e, 80, 130, 190, 270, 590, 730, 860, 1130, 1590df, 1860def }}
{{Optimal ET sequence|legend=0| 50, 60e, 80, 130, 190, 270, 590, 730, 860, 1130, 1590df, 1860def }}


Badness (Smith): 0.771 × 10<sup>-3</sup>
Badness (Sintel): 0.721


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Kalismic temperaments| ]] <!-- main article -->
[[Category:Kalismic temperaments| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 14:55, 19 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-3 temperaments that temper out the kalisma, 9801/9800. These temperaments always split the octave into halves.

Temperaments discussed elsewhere are:

Considered below are odin, lycoris, van gogh, sif, loki, pessoal, and linus. For the rank-4 temperament, see Rank-4 temperament #Kalismic (9801/9800).

Odin

See also: Landscape family

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 151263/151250

Mapping: [⟨6 0 0 8 17], ⟨0 1 0 -2 -4], ⟨0 0 1 2 3]]

mapping generators: ~55/49, ~3, ~5

Optimal tunings:

  • WE: ~55/49 = 200.0008 ¢, ~3/2 = 701.9318 ¢, ~5/4 = 386.3342 ¢
error map: ⟨+0.005 -0.018 +0.031 -0.004 -0.018]
  • CWE: ~55/49 = 200.0000 ¢, ~3/2 = 701.9320 ¢, ~5/4 = 386.3393 ¢
error map: ⟨0.000 -0.023 +0.026 -0.011 -0.028]

Optimal ET sequence: 12, 42, 48dee, 54c, 60e, 72, 198, 270, 342, 612, 954, 1236, 1506, 1578, 1848, 3426, 4038, 5616, 7464, 17118e, 18966e

Badness (Sintel): 0.140

Lycoris

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 1771561/1771470

Mapping: [⟨2 0 -2 1 0], ⟨0 1 1 3 2], ⟨0 0 6 -5 1]]

mapping generators: ~99/70, ~3, ~11/9

Optimal tunings:

  • WE: ~99/70 = 600.0018 ¢, ~3/2 = 701.9411 ¢, ~11/9 = 347.3976 ¢
error map: ⟨+0.004 -0.010 +0.013 -0.018 -0.031]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.9414 ¢, ~11/9 = 347.3969 ¢
error map: ⟨0.000 -0.014 +0.009 +0.014 -0.038]

Optimal ET sequence: 152, 328, 342, 836, 1178, 1354, 1506, 1848, 2684, 4038, 4190, 4532, 11254, 15786e

Badness (Sintel): 0.299

Higanbana

Subgroup: 2.3.5.7.11.13

Comma list: 9801/9800, 10648/10647, 1399680/1399489

Mapping: [⟨2 1 -1 2 2 4], ⟨0 2 2 6 4 1], ⟨0 0 6 -5 1 4]]

mapping generators: ~99/70, ~1458/1001, ~81/70

Optimal tunings:

  • WE: ~99/70 = 599.9996 ¢, ~1458/1001 = 650.9717 ¢, ~11/9 = 347.3963 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~1458/1001 = 650.9718 ¢, ~11/9 = 347.3964 ¢

Optimal ET sequence: 166, 190, 304d, 328, 494, 684, 1012, 1178, 1506, 2190, 2684, 4190, 6380, 9064, 15938

Badness (Sintel): 0.389

Van gogh

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 199297406/199290375

Mapping: [⟨2 0 8 0 11], ⟨0 1 1 2 1], ⟨0 0 -9 -1 -10]]

mapping generators: ~99/70, ~3, ~11/10

Optimal tunings:

  • WE: ~99/70 = 600.0022 ¢, ~3/2 = 701.9464 ¢, ~11/9 = 164.9319 ¢
error map: ⟨+0.004 -0.004 +0.022 +0.005 -0.046]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.9469 ¢, ~11/9 = 164.9316 ¢
error map: ⟨0.000 -0.008 +0.018 -0.000 -0.055]

Optimal ET sequence: 22, 58, 80, 138cde, 204cde, 226ce, 240d, 262d, 284, 320, 342, 742, 764, 1084, 1106, 1448, 1506, 1848, 4038, 4802, 5144, 6992

Badness (Sintel): 0.358

Sif

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 2097152/2096325

Mapping: [⟨2 0 1 7 11], ⟨0 1 0 1 -1], ⟨0 0 4 -5 -1]]

mapping generators: ~99/70, ~3

Optimal tunings:

  • WE: ~99/70 = 599.9863 ¢, ~3/2 = 701.9658 ¢, ~48/35 = 546.5930 ¢
error map: ⟨-0.027 -0.017 +0.045 +0.052 +0.000]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.9755 ¢, ~48/35 = 546.6052 ¢
error map: ⟨0.000 +0.021 +0.107 +0.124 +0.101]

Optimal ET sequence: 22, 46, 68, 114, 134, 156, 178, 202, 224, 270, 494, 742, 764, 966, 1236, 1506, 2742, 3236, 3506, 4742d, 10990ccdde, 15732ccdddee

Badness (Sintel): 0.508

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4096/4095, 6656/6655, 9801/9800

Mapping: [⟨2 0 1 7 11 16], ⟨0 1 0 1 -1 -3], ⟨0 0 4 -5 -1 1]]

Optimal tunings:

  • WE: ~99/70 = 599.9857 ¢, ~3/2 = 701.9705 ¢, ~48/35 = 546.5927 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.9877 ¢, ~48/35 = 546.6058 ¢

Optimal ET sequence: 22, 46, 156f, 178, 202f, 224, 270, 494, 764, 1012, 1236, 1506, 3236, 3506, 4742df, 6248cdf, 8248cdef, 12990ccddeff, 14496ccdddeeff

Badness (Sintel): 0.317

Loki

Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 9801/9800

Mapping: [⟨2 0 0 -21 -18], ⟨0 1 0 4 2], ⟨0 0 1 3 4]]

mapping generators: ~99/70, ~3, ~5

Optimal tunings:

  • WE: ~99/70 = 599.9481 ¢, ~3/2 = 702.1195 ¢, ~5/4 = 386.7636 ¢
error map: ⟨-0.104 +0.061 +0.242 -0.005 -0.128]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.1647 ¢, ~5/4 = 386.7768 ¢
error map: ⟨0.000 +0.210 +0.463 +0.163 +0.119]

Optimal ET sequence: 12, 22, 34d, 56d, 74d, 84de, 96d, 118, 130, 152, 248, 270, 670, 822, 940, 1092, 1362c, 2032c, 2302c

Badness (Sintel): 0.592

Pessoal

See also: Pessoalisma

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977

Mapping: [⟨2 0 1 10 14], ⟨0 1 0 -1 -3], ⟨0 0 3 -1 2]]

mapping generators: ~99/70, ~3, ~32/21

Optimal tunings:

  • WE: ~99/70 = 599.9711 ¢, ~3/2 = 702.0265 ¢, ~32/21 = 728.7942 ¢
error map: ⟨-0.058 +0.014 +0.040 +0.122 -0.040]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.0635 ¢, ~32/21 = 728.8214 ¢
error map: ⟨0.000 +0.109 +0.150 +0.289 +0.134]

Optimal ET sequence: 36, 46, 84, 94, 130, 224, 270, 494, 764, 1164, 1658, 3586cd, 5244cdde, 6008bcdde

Badness (Sintel): 0.599

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095

Mapping: [⟨2 0 1 10 14 13], ⟨0 1 0 -1 -3 -1], ⟨0 0 3 -1 2 -2]]

Optimal tunings:

  • WE: ~99/70 = 599.9835 ¢, ~3/2 = 702.0253 ¢, ~32/21 = 728.7700 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.0477 ¢, ~32/21 = 728.7882 ¢

Optimal ET sequence: 36, 46, 84, 94, 130, 224, 270, 494, 764, 1258, 1882d, 2152d

Badness (Sintel): 0.366

Linus

Main article: Linus
For the 7-limit version, see Miscellaneous 7-limit temperaments #Linus.

Linus tempers out the linus comma, [11 -10 -10 10⟩ in the 7-limit and can be described as the 80 & 130 & 270 temperament. It tempers out 9801/9800 and 391314/390625 in the 11-limit; 1001/1000, 4225/4224, and 4459/4455 in the 13-limit. The 1/10-octave period interval represents 15/14, three of which represents 16/13, and five of which represents both 99/70 and 140/99.

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 391314/390625

Mapping: [⟨10 0 0 -11 4], ⟨0 1 0 1 -1], ⟨0 0 1 1 2]]

Optimal tunings:

  • WE: ~15/14 = 119.9975 ¢, ~3/2 = 702.0188 ¢, ~5/4 = 386.6399 ¢
error map: ⟨-0.025 +0.039 +0.276 -0.215 -0.143]
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 702.0181 ¢, ~5/4 = 386.6218 ¢
error map: ⟨0.000 +0.063 +0.308 -0.186 -0.092]

Optimal ET sequence: 50, 60e, 80, 130, 190, 270, 670, 940, 1130, 1400, 1800c, 2070c, 2340c

Badness (Sintel): 1.34

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 4225/4224, 4459/4455

Mapping: [⟨10 0 0 -11 4 37], ⟨0 1 0 1 -1 0], ⟨0 0 1 1 2 0]]

Optimal tunings:

  • WE: ~15/14 = 120.0054 ¢, ~3/2 = 701.9625 ¢, ~5/4 = 386.4910 ¢
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 701.9533 ¢, ~5/4 = 386.5119 ¢

Optimal ET sequence: 50, 60e, 80, 130, 190, 270, 590, 730, 860, 1130, 1590df, 1860def

Badness (Sintel): 0.721

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