Kalismic temperaments: Difference between revisions

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These are rank-3 temperaments tempering out [[9801/9800]]. Temperaments discussed elsewhere are:  
{{Technical data page}}
* ''[[Jubilee]]'' (+50/49 or 99/98) → [[Jubilismic family #Jubilee|Jubilismic family]]
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.
 
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]]
* ''[[Julius]]'' or ''[[varda]]'' (+176/175) → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank three family]]
* ''[[Varda]]'' (+176/175) → [[Diaschismic rank-3 family #Varda|Diaschismic rank-3 family]]
* ''[[Hagrid]]'' (+243/242) → [[Cataharry family #Hagrid|Cataharry family]]
* ''[[Hagrid]]'' (+243/242) → [[Cataharry family #Hagrid|Cataharry family]]
* ''[[Uniwiz]]'' (+385/384) → [[Keenanismic temperaments #Uniwiz|Keenanismic temperaments]]
* ''[[Uniwiz]]'' (+385/384) → [[Keenanismic temperaments #Uniwiz|Keenanismic temperaments]]
Line 12: 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]]
* ''[[Semiporwell]]'' (+6144/6125) → [[Porwell family #Semiporwell|Porwell family]]
* ''[[Semiporwell]]'' (+6144/6125) → [[Porwell family #Semiporwell|Porwell family]]
* ''[[Semicanou]]'' (+14641/14580) → [[Canou family #Semicanou|Canou family]]
* ''[[Semicanou]]'' (+14641/14580) → [[Canou family #Semicanou|Canou family]]
* ''[[Odin]]'' (+151263/151250) → [[Landscape family #Odin|Landscape family]]


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


[[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 }}
[[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 52: Line 49:
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 = 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 68: Line 66:


{{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


== Hnoss ==
== Sif ==
To the wizma {{monzo| -6 -8 2 5 }} = 420175/419904, the kalisma is a natural complement, as their product is the [[tinge]].
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, 41503/41472
{{Mapping|legend=1| 2 0 1 2 6 | 0 1 4 0 2 | 0 0 -5 2 -3 }}
: mapping generators: ~99/70, ~3, ~144/77
{{Optimal ET sequence|legend=1| 22, 50, 72, 166, 176, 198, 248, 270, 342, 612, 954, 1566, 4086dee, 5652cddeee }}
[[Badness]]: 0.368 × 10<sup>-3</sup>
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 17303/17280
Mapping: {{mapping| 2 0 1 2 6 -3 | 0 1 4 0 2 1 | 0 0 -5 2 -3 4 }}
{{Optimal ET sequence|legend=1| 22f, 32cf, 54cff, 72, 166, 198, 270, 634, 904, 1174, 1880ef }}
Badness: 0.867 × 10<sup>-3</sup>
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 715/714, 1089/1088, 1225/1224, 2025/2023
Mapping: {{mapping| 2 0 1 2 6 -3 0 | 0 1 4 0 2 1 6 | 0 0 -5 2 -3 4 -6 }}
{{Optimal ET sequence|legend=1| 22f, 54cffgg, 72, 166g, 198g, 270, 364, 436, 634g, 706f }}
Badness: 0.862 × 10<sup>-3</sup>
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 715/714, 1225/1224, 1540/1539, 2080/2079, 4200/4199
Mapping: {{mapping| 2 0 1 2 6 -3 0 13 | 0 1 4 0 2 1 6 2 | 0 0 -5 2 -3 4 -6 -6 }}
{{Optimal ET sequence|legend=1| 72, 94, 166g, 198g, 270, 436, 634g, 706f }}
Badness: 0.901 × 10<sup>-3</sup>
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 715/714, 1225/1224, 1540/1539, 2080/2079, 2530/2527, 2737/2736
Mapping: {{mapping| 2 0 1 2 6 -3 0 13 19 | 0 1 4 0 2 1 6 2 -2 | 0 0 -5 2 -3 4 -6 -6 -2 }}
{{Optimal ET sequence|legend=1| 72, 94, 166g, 270, 342f, 436, 706fi }}
Badness: 1.14 × 10<sup>-3</sup>
=== Gersemi ===
The extension to 13-limit with [[4225/4224]] is weak but facilitates the use of 18/7 as the equave. [[Fokker block]]s of 128 notes are available for the latter, corresponding to 94edo. 18/7 is split into 4 parts that become ~19/15 in 19-limit. Also, (18/7)<sup>3</sup> ~ 17/1 via the [[5832/5831|chlorisma]]. However, the tones 9/8 and (19/15)/(9/8) = 152/135 have distinct mappings.
Subgroup: 2.3.5.7.11.13
Comma list: 4225/4224, 9801/9800, 41503/41472
Mapping: {{mapping| 2 0 1 2 6 9 | 0 1 9 -2 5 -6 | 0 0 -10 4 -6 7 }}
: mapping generators: ~99/70, ~3, ~154/65
{{Optimal ET sequence|legend=1| 44, 50, 94, 144, 176, 220, 270, 590, 684, 954 }}
Badness: 1.06 × 10<sup>-3</sup>
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 1089/1088, 1225/1224, 2025/2023, 4225/4224
Mapping: {{mapping| 2 0 1 2 6 9 0 | 0 1 9 -2 5 -6 12 | 0 0 -10 4 -6 7 -12 }}
{{Optimal ET sequence|legend=1| 44, 50, 94, 144g, 176g, 220g, 270, 364, 414, 634g, 684 }}
Badness: 1.46 × 10<sup>-3</sup>
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 1089/1088, 1225/1224, 1729/1728, 2926/2925, 3762/3757
Mapping: {{mapping| 2 0 1 2 6 9 0 1 | 0 1 9 -2 5 -6 12 11 | 0 0 -10 4 -6 7 -12 -11 }}
{{Optimal ET sequence|legend=1| 44, 50, 94, 144gh, 176g, 220g, 270, 414h, 590, 634g, 684h }}
Badness: 1.11 × 10<sup>-3</sup>
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 897/896, 1089/1088, 1225/1224, 1729/1728, 2737/2736, 2926/2925
Mapping: {{mapping| 2 0 1 2 6 9 0 1 7 | 0 1 9 -2 5 -6 12 11 3 | 0 0 -10 4 -6 7 -12 -11 -3 }}
{{Optimal ET sequence|legend=1| 44, 50, 94, 144gh, 176g, 220g, 226, 270, 320i, 364i, 414hi }}
Badness: 1.23 × 10<sup>-3</sup>
== Sif ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 188: Line 86:


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


[[Optimal tuning]] ([[CTE]]): ~99/70 = 1\2, ~3/2 = 701.9492, ~48/35 = 546.6041
[[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]]: 0.423 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.508


=== 13-limit ===
=== 13-limit ===
Line 204: Line 105:
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 = 1\2, ~3/2 = 701.9902, ~48/35 = 546.6064
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| 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: 0.339 × 10<sup>-3</sup>
Badness (Sintel): 0.317


== Loki ==
== Loki ==
Line 216: Line 119:


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


== Pessoal ==
== Pessoal ==
{{See also| Pessoalisma }}
{{See also| Pessoalisma }}
Pessoal tempers out the [[olympia]]. It was named by [[Aura]] in 2023, meaning [[Wiktionary: pessoal #Portuguese|"personal"]], for the fact that it is associated with [[abigail]], which is in turn a person's name.


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 231: Line 141:


{{Mapping|legend=1| 2 0 1 10 14 | 0 1 0 -1 -3 | 0 0 3 -1 2 }}
{{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 generators: ~99/70, ~3, ~32/21


[[Optimal tuning]] ([[CTE]]): ~99/70 = 1\2, ~3/2 = 702.0759, ~32/21 = 728.7910
[[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]]: 0.499 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.599


=== 13-limit ===
=== 13-limit ===
Line 247: Line 160:
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 = 1\2, ~3/2 = 702.0637, ~32/21 = 728.7786
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=1| 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: 0.391 × 10<sup>-3</sup>
Badness (Sintel): 0.366


== Rishi ==
== Linus ==
The 7-limit comma {{monzo| 65 -84 10 16 }} ~ 0.13¢ has the ratio of the exponents of 3 and 2 that is close to the one in 81/8. The square root of the latter is close to 35/11. This suggests tempering out (81/8)(35/11)<sup>-2</sup>, which is the kalisma.
{{Main| Linus }}
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Linus]].''


Apart from 35/11, 35/33, and the equivalents of their squares, 81/8 and 9/8, another equave that comes to mind is 3/2, especially after tempering out the [[chalmersia]]. When 3/2 is chosen as the equave, Fokker blocks of 34 notes can be used that are close to [[34edf]] and 58edo.
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


[[Comma list]]: 9801/9800, 572145834917888/571919811374025
[[Comma list]]: 9801/9800, 391314/390625


{{Mapping|legend=1| 2 0 3 -10 -4 | 0 1 2 4 4 | 0 0 8 -5 3 }}
{{Mapping|legend=1| 10 0 0 -11 4 | 0 1 0 1 -1 | 0 0 1 1 2 }}


: mapping generators: ~99/70, ~3, ~17364375/14172488
[[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| 24, 34d, 58, …, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 7414, 9874, 12828e }}
{{Optimal ET sequence|legend=1| 50, 60e, 80, 130, 190, 270, 670, 940, 1130, 1400, 1800c, 2070c, 2340c }}


[[Badness]]: 2.10 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.34


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


Comma list: 9801/9800, 10648/10647, 371293/371250
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: {{mapping| 2 0 3 -10 -4 2 | 0 1 2 4 4 3 | 0 0 8 -5 3 7 }}
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=1| 24, 34d, 58, …, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 5414, 6920, 7414, 9874, 12828e }}
{{Optimal ET sequence|legend=0| 50, 60e, 80, 130, 190, 270, 590, 730, 860, 1130, 1590df, 1860def }}


[[Badness]]: 0.505 × 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:Kalismic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 09:21, 6 July 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, loki, pessoal, 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

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

Pessoal tempers out the olympia. It was named by Aura in 2023, meaning "personal", for the fact that it is associated with abigail, which is in turn a person's name.

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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