Kalismic temperaments: Difference between revisions

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__FORCETOC__
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{{Technical data page}}
{{Technical data page}}
These are [[rank-3 temperament]]s tempering out [[9801/9800]]. Temperaments discussed elsewhere are:  
This is a collection of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] that [[tempering out|temper out]] the kalisma, [[9801/9800]]. These temperaments always split the octave into halves.
* ''[[Jubilee]]'' (+50/49 or 99/98) → [[Jubilismic family #Jubilee|Jubilismic family]]
 
Temperaments discussed elsewhere are:  
* ''[[Jubilismic]]'' (+50/49) → [[Jubilismic family #Undecimal jubilismic|Jubilismic family]]
* ''[[Fantastic]]'' (+225/224) → [[Marvel family #Fantastic|Marvel family]]
* ''[[Fantastic]]'' (+225/224) → [[Marvel family #Fantastic|Marvel family]]
* ''[[Bisector]]'' (+121/120 or 245/243) → [[Sensamagic family #Bisector|Sensamagic family]]
* ''[[Bisector]]'' (+121/120 or 245/243) → [[Sensamagic family #Bisector|Sensamagic family]]
Line 13: Line 15:
* ''[[Baldur]]'' (+2401/2400) → [[Breed family #Baldur|Breed family]]
* ''[[Baldur]]'' (+2401/2400) → [[Breed family #Baldur|Breed family]]
* ''[[Thor]]'' (+3025/3024 or 4375/4374) → [[Ragismic family #Thor|Ragismic family]]
* ''[[Thor]]'' (+3025/3024 or 4375/4374) → [[Ragismic family #Thor|Ragismic family]]
* ''[[Semihemimean]]'' (+3136/3125) → [[Hemimean family #Semihemimean|Hemimean family]]
* ''[[Loki]]'' (+5632/5625) → [[Vishdelismic clan #Loki|Vishdelismic clan]]
* ''[[Semiporwell]]'' (+6144/6125) → [[Porwell family #Semiporwell|Porwell family]]
* ''[[Semiporwell]]'' (+6144/6125) → [[Porwell family #Semiporwell|Porwell family]]
* ''[[Semicanou]]'' (+14641/14580) → [[Canou family #Semicanou|Canou family]]
* ''[[Semicathart]]'' (+14641/14580) → [[Cathartic family #Semicathart|Cathartic family]]
* ''[[Pessoal]]'' (+131072/130977) → [[Olympic clan #Pessoal|Olympic clan]]
* ''[[Odin]]'' (+151263/151250) → [[Landscape family #Odin|Landscape family]]


Considered below are odin, lycoris, van gogh, sif, loki, pessoal, and linus. For the rank-4 temperament, see [[Rank-4 temperament #Kalismic (9801/9800)]].
Considered below are lycoris, van gogh, sif, diaspheric, and linus, in the order of increasing [[badness]]. For the rank-4 temperament, see [[Rank-4 temperament #Kalismic (9801/9800)]].


== Odin ==
== Lycoris ==
{{See also| Landscape family }}
Lycoris tempers out the [[parimo]] in addition to the kalisma, and splits the [[syntonic comma]] into three equal parts, one for [[121/120]], and two for [[243/242]]. It is therefore [[support]]ed by third-comma equal temperaments. [[342edo]] shows an excellent example of this, but it can be tuned much more accurate.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 9801/9800, 151263/151250
 
{{Mapping|legend=1| 6 0 0 8 17 | 0 1 0 -2 -4 | 0 0 1 2 3 }}
 
: mapping generators: ~55/49, ~3, ~5


{{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 }}
It was named by [[Flora Canou]] in 2023 after the flower associated with afterlife in Japanese culture, under the impression that a temperament with such intricacy will never be fully explored in a lifetime.


[[Badness]]: 0.116 × 10<sup>-3</sup>
== Lycoris ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[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, ~11/9
: mapping generators: ~99/70, ~3, ~81/70


[[Optimal tuning]] ([[CTE]]): ~99/70 = 1\2, ~3/2 = 701.9439, ~81/70 = 252.6028
[[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]]: 0.249 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.299


=== Higanbana ===
=== Higanbana ===
Line 53: Line 51:
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|legend=0| 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 = 1\2, ~1458/1001 = 650.9715, ~81/70 = 252.6037
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=1| 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: 0.416 × 10<sup>-3</sup>
Badness (Sintel): 0.389


== Van gogh ==
== Van gogh ==
Line 69: Line 68:


{{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, ~11/10


: mapping generators: ~99/70, ~3, ~9/7
[[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]]: 0.297 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.358


== Sif ==
== Sif ==
Sif tempers out 2097152/2096325, and extends to a strong [[13-limit]] temperament by virtue of the identity 2097152/2096325 = ([[4096/4095]])⋅([[6656/6655]]). It was named by [[Flora Canou]] in 2023 as a sharp-tending counterpart of [[thor]].
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 82: Line 88:


{{Mapping|legend=1| 2 0 1 7 11 | 0 1 0 1 -1 | 0 0 4 -5 -1 }}
{{Mapping|legend=1| 2 0 1 7 11 | 0 1 0 1 -1 | 0 0 4 -5 -1 }}
: mapping generators: ~99/70, ~3, ~48/35


: mapping generators: ~99/70, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~99/70 = 599.9863{{c}}, ~3/2 = 701.9658{{c}}, ~48/35 = 546.5930{{c}}
[[Optimal tuning]] ([[CTE]]): ~99/70 = 1\2, ~3/2 = 701.9492, ~48/35 = 546.6041
: [[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]]: 0.423 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.508


=== 13-limit ===
=== 13-limit ===
Line 96: Line 105:
Comma list: 4096/4095, 6656/6655, 9801/9800
Comma list: 4096/4095, 6656/6655, 9801/9800


Mapping: {{mapping| 2 0 1 7 11 16 | 0 1 0 1 -1 -3 | 0 0 4 -5 -1 1 }}
{{Mapping|legend=0| 2 0 1 7 11 16 | 0 1 0 1 -1 -3 | 0 0 4 -5 -1 1 }}
 
Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 701.9902, ~48/35 = 546.6064
 
{{Optimal ET sequence|legend=1| 22, 46, 156f, 178, 202f, 224, 270, 494, 764, 1012, 1236, 1506, 3236, 3506, 4742df, 6248cdf, 8248cdef, 12990ccddeff, 14496ccdddeeff }}
 
Badness: 0.339 × 10<sup>-3</sup>
 
== Loki ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 5632/5625, 9801/9800
 
{{Mapping|legend=1| 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 = 599.9857{{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=1| 12, 22, 34d, 56d, 74d, 84de, 96d, 118, 130, 152, 248, 270, 670, 822, 940, 1092, 1362c, 2032c, 2302c }}
{{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]]: 0.493 × 10<sup>-3</sup>
Badness (Sintel): 0.317


== Pessoal ==
== Diaspheric ==
{{See also| Pessoalisma }}
Diaspheric is notable for detempering [[diaschismic family #Septimal diaschismic|diaschismic]]. It was named by [[Eufalesio]] in 2026 as a variant of ''diaschismic''.


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 9801/9800, 131072/130977
[[Comma list]]: 9801/9800, 2252800/2250423
 
{{Mapping|legend=1| 2 0 1 10 14 | 0 1 0 -1 -3 | 0 0 3 -1 2 }}


: mapping generators: ~99/70, ~3, ~32/21
{{Mapping|legend=1| 2 0 1 16 20 | 0 1 0 -5 -7 | 0 0 2 3 5 }}
: mapping generators: ~99/70, ~3, ~1323/704


[[Optimal tuning]] ([[CTE]]): ~99/70 = 1\2, ~3/2 = 702.0759, ~32/21 = 728.7910
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 599.9742{{c}}, ~3/2 = 702.0681{{c}}, ~1323/704 = 1093.1024{{c}}
: [[error map]]: {{val| -0.052 +0.061 -0.015 +0.166 -0.138 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 702.0803{{c}}, ~1323/704 = 1093.1807{{c}}
: error map: {{val| 0.000 +0.133 +0.048 +0.275 -0.033 }}


{{Optimal ET sequence|legend=1| 36, 46, 84, 94, 130, 224, 270, 494, 764, 1164, 1658, 3586cd, 5244cdde, 6008bcdde }}
{{Optimal ET sequence|legend=1| 46, 58, 104c, 154ce, 166, 212, 224, 270, 494, 764 }}


[[Badness]]: 0.499 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.950


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1716/1715, 2080/2079, 4096/4095
Comma list: 1716/1715, 2080/2079, 10648/10647


Mapping: {{mapping| 2 0 1 10 14 13 | 0 1 0 -1 -3 -1 | 0 0 3 -1 2 -2 }}
{{Mapping|legend=0| 2 0 1 16 20 25 | 0 1 0 -5 -7 -9 | 0 0 2 3 5 6 }}


Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 702.0637, ~32/21 = 728.7786
Optimal tunings:
* WE: ~99/70 = 599.9720{{c}}, ~3/2 = 702.0544{{c}}, ~1014/539 = 1093.1699{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.0759{{c}}, ~1014/539 = 1093.1900{{c}}


{{Optimal ET sequence|legend=1| 36, 46, 84, 94, 130, 224, 270, 494, 764, 1258, 1882d, 2152d }}
{{Optimal ET sequence|legend=0| 46, 58, 104c, 154cef, 166, 212, 224, 270, 494, 764, 1258, 1810d, 2304d, 2574d }}


Badness: 0.391 × 10<sup>-3</sup>
Badness (Sintel): 0.454


== Linus ==
== Linus ==
{{Main| Linus }}
{{Main| 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.
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 578509309952/576650390625
{{Mapping|legend=1| 10 0 0 -11 | 0 1 0 1 | 0 0 1 1 }}
: mapping generators: ~15/14, ~3, ~5
[[Optimal tuning]] ([[CTE]]): ~15/14 = 1\10, ~3/2 = 702.0437, ~5/4 = 386.5040
{{Optimal ET sequence|legend=1| 50, 60, 80, 130, 270, 1270, 1540, 1810, 1940, 2080, 2210c, 2480c }}
[[Badness]]: 1.12 × 10<sup>-3</sup>
=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 174: Line 162:
{{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 = 1\10, ~3/2 = 702.0010, ~5/4 = 386.5927
[[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]]: 1.12 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.34


=== 13-limit ===
=== 13-limit ===
Line 185: Line 177:
Comma list: 1001/1000, 4225/4224, 4459/4455
Comma list: 1001/1000, 4225/4224, 4459/4455


Mapping: {{mapping| 10 0 0 -11 4 37 | 0 1 0 1 -1 0 | 0 0 1 1 2 0 }}
{{Mapping|legend=0| 10 0 0 -11 4 37 | 0 1 0 1 -1 0 | 0 0 1 1 2 0 }}


Optimal tuning (CTE): ~15/14 = 1\10, ~3/2 = 702.0010, ~5/4 = 386.5928
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: {{Optimal ET sequence| 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: 0.771 × 10<sup>-3</sup>
Badness (Sintel): 0.721


[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Kalismic temperaments| ]] <!-- main article -->
[[Category:Kalismic temperaments| ]] <!-- main article -->
[[Category:Kalismic| ]] <!-- key article -->
[[Category:Kalismic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Temperament collections]]
[[Category:Catalogs of rank-3 temperaments]]

Latest revision as of 13:35, 24 October 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 lycoris, van gogh, sif, diaspheric, and linus, in the order of increasing badness. For the rank-4 temperament, see Rank-4 temperament #Kalismic (9801/9800).

Lycoris

Lycoris tempers out the parimo in addition to the kalisma, and splits the syntonic comma into three equal parts, one for 121/120, and two for 243/242. It is therefore supported by third-comma equal temperaments. 342edo shows an excellent example of this, but it can be tuned much more accurate.

It was named by Flora Canou in 2023 after the flower associated with afterlife in Japanese culture, under the impression that a temperament with such intricacy will never be fully explored in a lifetime.

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

Sif tempers out 2097152/2096325, and extends to a strong 13-limit temperament by virtue of the identity 2097152/2096325 = (4096/4095)⋅(6656/6655). It was named by Flora Canou in 2023 as a sharp-tending counterpart of thor.

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, ~48/35

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

Diaspheric

Diaspheric is notable for detempering diaschismic. It was named by Eufalesio in 2026 as a variant of diaschismic.

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 2252800/2250423

Mapping: [⟨2 0 1 16 20], ⟨0 1 0 -5 -7], ⟨0 0 2 3 5]]

mapping generators: ~99/70, ~3, ~1323/704

Optimal tunings:

  • WE: ~99/70 = 599.9742 ¢, ~3/2 = 702.0681 ¢, ~1323/704 = 1093.1024 ¢
error map: ⟨-0.052 +0.061 -0.015 +0.166 -0.138]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.0803 ¢, ~1323/704 = 1093.1807 ¢
error map: ⟨0.000 +0.133 +0.048 +0.275 -0.033]

Optimal ET sequence: 46, 58, 104c, 154ce, 166, 212, 224, 270, 494, 764

Badness (Sintel): 0.950

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 0 1 16 20 25], ⟨0 1 0 -5 -7 -9], ⟨0 0 2 3 5 6]]

Optimal tunings:

  • WE: ~99/70 = 599.9720 ¢, ~3/2 = 702.0544 ¢, ~1014/539 = 1093.1699 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.0759 ¢, ~1014/539 = 1093.1900 ¢

Optimal ET sequence: 46, 58, 104c, 154cef, 166, 212, 224, 270, 494, 764, 1258, 1810d, 2304d, 2574d

Badness (Sintel): 0.454

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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