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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
[[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 }}


{{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 }}
{{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


== Rishi ==
== 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]].


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.
[[Subgroup]]: 2.3.5.7.11


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.
[[Comma list]]: 9801/9800, 2097152/2096325


Subgroup: 2.3.5.7.11
{{Mapping|legend=1| 2 0 1 7 11 | 0 1 0 1 -1 | 0 0 4 -5 -1 }}
: mapping generators: ~99/70, ~3


[[Comma list]]: 9801/9800, 572145834917888/571919811374025
[[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 }}


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


Mapping generators: ~99/70, ~3, ~17364375/14172488
[[Badness]] (Sintel): 0.508
 
{{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 }}
 
[[Badness]]: 2.10 × 10<sup>-3</sup>


=== 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
 
[[Comma list]]: 9801/9800, 41503/41472
 
[[Mapping]]: [{{val| 2 0 1 2 6 }}, {{val| 0 1 4 0 2 }}, {{val| 0 0 -5 2 -3 }}]


Mapping generators: ~99/70, ~3, ~144/77
{{Mapping|legend=1| 2 0 0 -21 -18 | 0 1 0 4 2 | 0 0 1 3 4 }}
: mapping generators: ~99/70, ~3, ~5


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


[[Badness]]: 0.368 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=1| 12, 22, 34d, 56d, 74d, 84de, 96d, 118, 130, 152, 248, 270, 670, 822, 940, 1092, 1362c, 2032c, 2302c }}


=== 13-limit ===
[[Badness]] (Sintel): 0.592


Subgroup: 2.3.5.7.11.13
== Pessoal ==
{{See also| Pessoalisma }}


[[Comma list]]: 1716/1715, 2080/2079, 17303/17280
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.


[[Mapping]]: [{{val| 2 0 1 2 6 -3 }}, {{val| 0 1 4 0 2 1 }}, {{val| 0 0 -5 2 -3 4 }}]
[[Subgroup]]: 2.3.5.7.11


Mapping generators: ~99/70, ~3, ~144/77
[[Comma list]]: 9801/9800, 131072/130977


{{Val list|legend=1| 10c, 22f, 32cf, 54cff, 72, 166, 198, 270, 634, 904, 1174, 1880ef }}
{{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


[[Badness]]: 0.867 × 10<sup>-3</sup>
[[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 }}


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


The extension to 13-limit with [[4225/4224]] is weaker but facilitates the use of 18/7 as the equave. Fokker blocks 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.
[[Badness]] (Sintel): 0.599


=== 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]]
Retrieved from "https://en.xen.wiki/w/Kalismic_temperaments"