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, 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.
* ''[[Fantastic]]'', {225/224, 4375/4356} → [[Marvel family #Fantastic|Marvel family]]
* ''[[Bisector]]'', {121/120, 245/243} → [[Sensamagic family #Bisector|Sensamagic family]]
* ''[[Julius]]'' or ''[[varda]]'', {176/175, 896/891} → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank three family]]
* ''[[Hagrid]]'', {243/242, 9801/9800} → [[Cataharry family #Hagrid|Cataharry family]]
* ''[[Uniwiz]]'', {385/384, 9801/9800} → [[Keenanismic temperaments #Uniwiz|Keenanismic temperaments]]
* ''[[Varuna]]'', {441/440, 8019/8000} → [[Werckismic temperaments #Varuna|Werckismic temperaments]]
* ''[[Hades]]'', {540/539, 4000/3993} → [[Swetismic temperaments #Hades|Swetismic temperaments]]
* ''[[Dimcomp]]'', {1375/1372, 6250/6237} → [[Dimcomp family #Undecimal dimcomp|Dimcomp family]]
* ''[[Baldur]]'', {2401/2400, 9801/9800} → [[Breed family #Baldur|Breed family]]
* ''[[Thor]]'', {3025/3024, 4375/4374} → [[Ragismic family #Thor|Ragismic family]]
* ''[[Semiporwell]]'', {6144/6125, 9801/9800} → [[Porwell family #Semiporwell|Porwell family]]
* ''[[Semicanou]]'', {9801/9800, 14641/14580} → [[Canou family #Semicanou|Canou family]]


Considered below are odin, loki, van gogh, rishi, hnoss, and gersemi, but we can begin by looking at the rank-4 temperament.
Temperaments discussed elsewhere are:
* ''[[Jubilismic]]'' (+50/49) → [[Jubilismic family #Undecimal jubilismic|Jubilismic family]]
* ''[[Fantastic]]'' (+225/224) → [[Marvel family #Fantastic|Marvel family]]
* ''[[Bisector]]'' (+121/120 or 245/243) → [[Sensamagic family #Bisector|Sensamagic family]]
* ''[[Varda]]'' (+176/175) → [[Diaschismic rank-3 family #Varda|Diaschismic rank-3 family]]
* ''[[Hagrid]]'' (+243/242) → [[Cataharry family #Hagrid|Cataharry family]]
* ''[[Uniwiz]]'' (+385/384) → [[Keenanismic temperaments #Uniwiz|Keenanismic temperaments]]
* ''[[Varuna]]'' (+441/440) → [[Werckismic temperaments #Varuna|Werckismic temperaments]]
* ''[[Hades]]'' (+540/539) → [[Swetismic temperaments #Hades|Swetismic temperaments]]
* ''[[Dimcomp]]'' (+1375/1372) → [[Dimcomp family #Undecimal dimcomp|Dimcomp family]]
* ''[[Baldur]]'' (+2401/2400) → [[Breed family #Baldur|Breed 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]]
* ''[[Semicanou]]'' (+14641/14580) → [[Canou family #Semicanou|Canou family]]
* ''[[Odin]]'' (+151263/151250) → [[Landscape family #Odin|Landscape family]]


== Kalismic ==
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)]].
Subgroup: 2.3.5.7.11


[[Comma list]]: 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 [[support]]ed by third-comma equal temperaments. [[342edo]] shows an excellent example of this, but it can be tuned much more accurate.


[[Mapping]]: [{{val| 2 0 0 0 3 }}, {{val| 0 1 0 0 -2 }}, {{val| 0 0 1 0 1 }}, {{val| 0 0 0 1 1 }}]
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.


Mapping generators: ~99/70, ~3, ~5, ~7
[[Subgroup]]: 2.3.5.7.11


{{Val list|legend=1| 8d, 10, 12, 22, 34d, 46, 58, 72, 118, 130, 152, 224, 270, 342, 612, 836, 1084, 1106, 1236, 1506, 1578, 1848, 2684, 4038, 4190, 4532, 11254, 15786e, 21896e }}
[[Comma list]]: 9801/9800, 1771561/1771470


== Odin ==
{{Mapping|legend=1| 2 0 -2 1 0 | 0 1 1 3 2 | 0 0 6 -5 1 }}
{{see also| Landscape family }}
: mapping generators: ~99/70, ~3, ~11/9


Subgroup: 2.3.5.7.11
[[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 }}


[[Comma list]]: 9801/9800, 151263/151250
{{Optimal ET sequence|legend=1| 152, 328, 342, 836, 1178, 1354, 1506, 1848, 2684, 4038, 4190, 4532, 11254, 15786e }}


[[Mapping]]: [{{val| 6 0 0 8 17 }}, {{val| 0 1 0 -2 -4 }}, {{val| 0 0 1 2 3 }}]
[[Badness]] (Sintel): 0.299


Mapping generators: ~55/49, ~3, ~5
=== Higanbana ===
 
Subgroup: 2.3.5.7.11.13
{{Val list|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>
Comma list: 9801/9800, 10648/10647, 1399680/1399489


== Loki ==
Mapping: {{mapping| 2 1 -1 2 2 4 | 0 2 2 6 4 1 | 0 0 6 -5 1 4 }}
Subgroup: 2.3.5.7.11
: mapping generators: ~99/70, ~1458/1001, ~81/70


[[Comma list]]: 5632/5625, 9801/9800
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}}


[[Mapping]]: [{{val| 2 0 0 -21 -18 }}, {{val| 0 1 0 4 2 }}, {{val| 0 0 1 3 4 }}]
{{Optimal ET sequence|legend=0| 166, 190, 304d, 328, 494, 684, 1012, 1178, 1506, 2190, 2684, 4190, 6380, 9064, 15938 }}


Mapping generators: ~99/70, ~3, ~5
Badness (Sintel): 0.389


{{Val list|legend=1| 12, 22, 34d, 56d, 74d, 84de, 96d, 118, 130, 152, 248, 270, 670, 822, 940, 1092, 1362c, 2032c, 2302c }}
== Van gogh ==
 
[[Subgroup]]: 2.3.5.7.11
[[Badness]]: 0.493 × 10<sup>-3</sup>
 
== Van Gogh ==
Subgroup: 2.3.5.7.11


[[Comma list]]: 9801/9800, 199297406/199290375
[[Comma list]]: 9801/9800, 199297406/199290375


[[Mapping]]: [{{val| 2 0 8 0 11 }}, {{val| 0 1 1 2 1 }}, {{val| 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
 
{{Val list|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>
[[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 }}


== Rishi ==
{{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 }}


The 7-limit comma [65 -84 10 16⟩ ~ 0.13c 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.
[[Badness]] (Sintel): 0.358


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.
== 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


[[Comma list]]: 9801/9800, 572145834917888/571919811374025
[[Comma list]]: 9801/9800, 2097152/2096325


[[Mapping]]: [{{val| 2 0 3 -10 -4 }}, {{val| 0 1 2 4 4 }}, {{val| 0 0 8 -5 3 }}]
{{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, ~17364375/14172488
[[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 }}


{{Val list|legend=1| 24, 34d, 58, 150cdee, 208ccddee, 252ccddeee, 262ccdee, 286ccdee, 310cddee, 320ccee, 344cdee, 378ce, 402de, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 7414, 9874, 12828e }}
{{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]]: 2.10 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.508


=== 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: 4096/4095, 6656/6655, 9801/9800


[[Mapping]]: [{{val| 2 0 3 -10 -4 2 }}, {{val| 0 1 2 4 4 3 }}, {{val| 0 0 8 -5 3 7 }}]
Mapping: {{mapping| 2 0 1 7 11 16 | 0 1 0 1 -1 -3 | 0 0 4 -5 -1 1 }}


Mapping generators: ~99/70, ~3, ~364/297
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}}


{{Val list|legend=1| 24, 34d, 58, 150cdeef, 208ccddeeff, 252ccddeeefff, 262ccdeefff, 286ccdeeff, 310cddeeff, 320cceeff, 344cdeef, 378cef, 402def, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 5414, 6920, 7414, 9874, 12828e }}
{{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]]: 0.505 × 10<sup>-3</sup>
Badness (Sintel): 0.317


== Hnoss ==
== Loki ==
[[Subgroup]]: 2.3.5.7.11


To the wizma [-6 -8 2 5⟩ = 420175/419904, the kalisma is a natural complement, as their product is the [[tinge]].
[[Comma list]]: 5632/5625, 9801/9800


Subgroup: 2.3.5.7.11
{{Mapping|legend=1| 2 0 0 -21 -18 | 0 1 0 4 2 | 0 0 1 3 4 }}
: mapping generators: ~99/70, ~3, ~5


[[Comma list]]: 9801/9800, 41503/41472
[[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 }}


[[Mapping]]: [{{val| 2 0 1 2 6 }}, {{val| 0 1 4 0 2 }}, {{val| 0 0 -5 2 -3 }}]
{{Optimal ET sequence|legend=1| 12, 22, 34d, 56d, 74d, 84de, 96d, 118, 130, 152, 248, 270, 670, 822, 940, 1092, 1362c, 2032c, 2302c }}


Mapping generators: ~99/70, ~3, ~144/77
[[Badness]] (Sintel): 0.592


{{Val list|legend=1| 10c, 18bcd, 22, 50, 72, 166, 176, 198, 248, 270, 342, 612, 954, 1566, 4086dee, 5652cddeee }}
== Pessoal ==
{{See also| Pessoalisma }}


[[Badness]]: 0.368 × 10<sup>-3</sup>
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.  


=== 13-limit ===
[[Subgroup]]: 2.3.5.7.11


Subgroup: 2.3.5.7.11.13
[[Comma list]]: 9801/9800, 131072/130977


[[Comma list]]: 1716/1715, 2080/2079, 17303/17280
{{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]]: [{{val| 2 0 1 2 6 -3 }}, {{val| 0 1 4 0 2 1 }}, {{val| 0 0 -5 2 -3 4 }}]
[[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 }}


Mapping generators: ~99/70, ~3, ~144/77
{{Optimal ET sequence|legend=1| 36, 46, 84, 94, 130, 224, 270, 494, 764, 1164, 1658, 3586cd, 5244cdde, 6008bcdde }}


{{Val list|legend=1| 10c, 22f, 32cf, 54cff, 72, 166, 198, 270, 634, 904, 1174, 1880ef }}
[[Badness]] (Sintel): 0.599
 
[[Badness]]: 0.867 × 10<sup>-3</sup>
 
=== Gersemi ===
 
18/7 is a possible equave. Fokker blocks of 128 notes are available for it, 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.
 
==== 13-limit ====


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


[[Comma list]]: 4225/4224, 9801/9800, 41503/41472
Comma list: 1716/1715, 2080/2079, 4096/4095


[[Mapping]]: [{{val| 2 0 1 2 6 9 }}, {{val| 0 1 9 -2 5 -6 }}, {{val| 0 0 -10 4 -6 7 }}]
Mapping: {{mapping| 2 0 1 10 14 13 | 0 1 0 -1 -3 -1 | 0 0 3 -1 2 -2 }}


Mapping generators: ~99/70, ~3, ~154/65
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}}


{{Val list|legend=1| 44, 50, 94, 144, 176, 220, 270, 590, 684, 954 }}
{{Optimal ET sequence|legend=0| 36, 46, 84, 94, 130, 224, 270, 494, 764, 1258, 1882d, 2152d }}


[[Badness]]: 1.06 × 10<sup>-3</sup>
Badness (Sintel): 0.366


==== 17-limit ====
== Linus ==
{{Main| Linus }}
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Linus]].''


Subgroup: 2.3.5.7.11.13.17
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.


[[Comma list]]: 1089/1088, 1225/1224, 2025/2023, 4225/4224
[[Subgroup]]: 2.3.5.7.11


[[Mapping]]: [{{val| 2 0 1 2 6 9 0 }}, {{val| 0 1 9 -2 5 -6 12 }}, {{val| 0 0 -10 4 -6 7 -12 }}]
[[Comma list]]: 9801/9800, 391314/390625


Mapping generators: ~99/70, ~3, ~154/65
{{Mapping|legend=1| 10 0 0 -11 4 | 0 1 0 1 -1 | 0 0 1 1 2 }}


{{Val list|legend=1| 44, 50, 94, 144g, 176g, 220g, 270, 364, 414, 634g, 684 }}
[[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 }}


[[Badness]]: 1.46 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=1| 50, 60e, 80, 130, 190, 270, 670, 940, 1130, 1400, 1800c, 2070c, 2340c }}


==== 19-limit ====
[[Badness]] (Sintel): 1.34


Subgroup: 2.3.5.7.11.13.19
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
[[Comma list]]: 1089/1088, 1225/1224, 1729/1728, 2926/2925, 3762/3757
 
[[Mapping]]: [{{val| 2 0 1 2 6 9 0 1 }}, {{val| 0 1 9 -2 5 -6 12 11 }}, {{val| 0 0 -10 4 -6 7 -12 -11 }}]
 
Mapping generators: ~99/70, ~3, ~45/19
 
{{Val list|legend=1| 44, 50, 94, 144gh, 176g, 220g, 270, 414h, 590, 634g, 684h }}
 
[[Badness]]: 1.11 × 10<sup>-3</sup>
 
==== 23-limit ====
 
Subgroup: 2.3.5.7.11.13.19.23


[[Comma list]]: 897/896, 1089/1088, 1225/1224, 1729/1728, 2737/2736, 2926/2925
Comma list: 1001/1000, 4225/4224, 4459/4455


[[Mapping]]: [{{val| 2 0 1 2 6 9 0 1 7 }}, {{val| 0 1 9 -2 5 -6 12 11 3 }}, {{val| 0 0 -10 4 -6 7 -12 -11 -3 }}]
Mapping: {{mapping| 10 0 0 -11 4 37 | 0 1 0 1 -1 0 | 0 0 1 1 2 0 }}


Mapping generators: ~99/70, ~3, ~45/19
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}}


{{Val list|legend=1| 44, 50, 94, 144gh, 176g, 220g, 226, 270, 320i, 364i, 414hi }}
{{Optimal ET sequence|legend=0| 50, 60e, 80, 130, 190, 270, 590, 730, 860, 1130, 1590df, 1860def }}


[[Badness]]: 1.23 × 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]]

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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