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{{Technical data page}}
{{Technical data page}}
'''The Quartercache''' is a collection of temperaments of different ranks, including subgroup temperaments, that all temper out the [[quartisma]]- the unnoticeable comma with the ratio 117440512/117406179, and a monzo of {{monzo|24 -6 0 1 -5}}. Among the members of this family are quartic, Altierran, meanquarter, coin, escapismic, dietismic, kleirtismic, doublefour and quarterframe.
'''The Quartercache''' is a collection of temperaments of different ranks, including [[subgroup temperaments]], that all temper out the [[quartisma]] – the unnoticeable comma with the [[ratio]] 117440512/117406179, and a [[monzo]] of {{monzo| 24 -6 0 1 -5 }}. Among the members of this family are quartic, altierran, meanquarter, coin, escapismic, dietismic, kleirtismic, doublefour and quarterframe.


== Quartismic ==
== Quartismic ==
: ''For extensions, see [[Quartismic family]].
See [[Catalog of rank-4 temperaments #Quartismic (117440512/117406179)]].  
 
The 11-limit parent comma for the [[quartismic family]] and for the Quartercache is the quartisma with a ratio of 117440512/117406179 and a monzo of {{monzo| 24 -6 0 1 -5 }}. As the quartisma is an unnoticeable comma, this rank-4 temperament is a [[microtemperament]].
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 117440512/117406179
 
[[Mapping]]: [{{val| 1 0 0 1 5 }}, {{val| 0 1 0 1 -1 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 5 1 }}]
 
Mapping generators: ~2, ~3, ~5, ~33/32
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9742, ~5/4 = 386.3137, ~33/32 = 53.3683
 
{{Optimal ET sequence|legend=1| 21, 22, 43, 46, 65d, 68, 89, 111, 159, 202, 224, 270, 494, 742, 764, 966, 1236, 1506, 2159, 2653, 3125, 3395, 7060, 7554, 10949e, 14614e, 15850ee, 22168bdee, 23404bcdee, 26799bcdeee, 34353bcdeeee }}
 
[[Badness]]: 0.274 × 10<sup>-6</sup>


== Quartic ==
== Quartic ==
There are some temperaments that temper out the quartisma despite having limited accuracy in their approximations of five-limit intervals. This particular temperament is the parent temperament of all such no-fives children, and is referred to as '''Saquinlu-azo temperament''' in color notation.
There are some temperaments that temper out the quartisma despite having limited accuracy in their approximations of five-limit intervals. This particular temperament is the parent temperament of all such no-fives children, and is referred to as ''Saquinlu-azo'' in color notation.


Subgroup: 2.3.7.11
[[Subgroup]]: 2.3.7.11


[[Comma list]]: 117440512/117406179
[[Comma list]]: 117440512/117406179


[[Sval]] [[mapping]]: [{{val| 1 0 1 5 }}, {{val| 0 1 1 -1 }}, {{val| 0 0 5 1 }}]
{{Mapping|legend=1| 1 0 1 5 | 0 1 1 -1 | 0 0 5 1 }}
: mapping generators: ~2, ~3, ~33/32


[[POTE generator]]s: ~3/2 = 701.9826, ~33/32 = 53.3748
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9875{{c}}, ~3/2 = 701.9753{{c}}, ~33/32 = 53.3743{{c}}
: [[error map]]: {{val| -0.012 +0.008 -0.004 +0.031 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9802{{c}}, ~33/32 = 53.3730{{c}}
: error map: {{val| 0.000 +0.025 +0.019 +0.075 }}


{{Optimal ET sequence|legend=1| 21, 22, 24, 43, 46, 89, 135, 359, 494, 629, 742, 877, 1012, 1506, 2248, 2383, 2518, 7419, 8431e, 10949e, 13467e }}
{{Optimal ET sequence|legend=1| 21, 22, 24, 43, 46, 89, 135, 359, 494, 629, 742, 877, 1012, 1506, 2248, 2383, 2518, 7419, 8431e, 10949e, 13467e }}


The following unnamed rank-2 quartismic temperament MOS scales have been found
[[Badness]] (Sintel): 0.416
 
The following rank-2 [[mos scale]]s of quartismic have been found:
* [https://sevish.com/scaleworkshop/?name=Rank%202%20scale%20(106.71461627796054%2C%201200.0)%2C%205%7C5&data=106.714616%0A213.429233%0A320.143849%0A426.858465%0A533.573081%0A666.426919%0A773.141535%0A879.856151%0A986.570767%0A1093.285384%0A1200.000000&freq=440&midi=69&vert=9&horiz=1&colors=&waveform=triangle&ampenv=organ Rank 2 scale (106.71461627796054, 1200.0), 5|5]   
* [https://sevish.com/scaleworkshop/?name=Rank%202%20scale%20(106.71461627796054%2C%201200.0)%2C%205%7C5&data=106.714616%0A213.429233%0A320.143849%0A426.858465%0A533.573081%0A666.426919%0A773.141535%0A879.856151%0A986.570767%0A1093.285384%0A1200.000000&freq=440&midi=69&vert=9&horiz=1&colors=&waveform=triangle&ampenv=organ Rank 2 scale (106.71461627796054, 1200.0), 5|5]   
* The following scale tree has been found: [http://www.microtonalsoftware.com/scale-tree.html?left=12&right=11&rr=1200&ioi=106.71461627796054 1200-106.71461627796054-12-11 Scale Tree]
* The following scale tree has been found: [http://www.microtonalsoftware.com/scale-tree.html?left=12&right=11&rr=1200&ioi=106.71461627796054 1200-106.71461627796054-12-11 Scale Tree]


== Direct quartismic ==
== Direct quartismic ==
Instead of 11-limit, defined directly in the 2.33/2.7/6 subgroup, as the quartisma itself was discovered via a representation where the generator is 33/32 and five of them stack to 7/6. [[45edo]] is an excellent tuning.
Instead of 11-limit, direct quartismic is defined directly in the 2.7/3.33 subgroup, as the quartisma itself was discovered via a representation where the generator is 33/32 and five of them stack to 7/6. [[45edo]] is an excellent tuning. It was named by [[Eliora]] in 2023.  


[[Subgroup]]: 2.33/32.7/6
[[Subgroup]]: 2.7/3.33


[[Comma list]]: 117440512/117406179
[[Comma list]]: 117440512/117406179


[[Sval]] [[mapping]]: [{{val|1 0 0}}, {{val|0 1 5}}]
{{Mapping|legend=2| 1 1 5 | 0 5 1 }}
: mapping generators: ~2, ~33/32
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9875{{c}}, ~33/32 = 53.3743{{c}}
: [[error map]]: {{val| -0.012 -0.012 +0.039 }}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~33/32 = 53.3730{{c}}
: error map: {{val| 0.000 -0.006 +0.100 }}
 
{{Optimal ET sequence|legend=1| 21, 22, 45, 337, 382, 427, 472, 517, 562, 607, 652, 1911, 2563, 3215, 5778, 8993* }}


[[Sval]] [[mapping]] [[generators]]: ~2 = 1\1, ~33/32 = 53.368
<nowiki/>* wart for 33


== Altierran ==
== Altierran ==
In altierran, both the schisma and the quartisma are tempered out.
In altierran, both the schisma and the quartisma are tempered out.


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 32805/32768, 161280/161051
[[Comma list]]: 32805/32768, 161280/161051


[[Mapping]]: [{{val| 1 0 15 1 5 }}, {{val| 0 1 -8 1 -1 }}, {{val| 0 0 0 5 1 }}]
{{Mapping|legend=1| 1 0 15 1 5 | 0 1 -8 1 -1 | 0 0 0 5 1 }}


[[POTE generator]]s: ~3/2 = 701.7299, ~33/32 = 53.3889  
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.7299{{c}}, ~33/32 = 53.3889{{c}}


{{Optimal ET sequence|legend=1| 24, 46c, 65d, 89, 135, 159, 224, 383, 472, 696, 1168, 1327, 1551, 2023e }}
{{Optimal ET sequence|legend=1| 24, 46c, 65d, 89, 135, 159, 224, 383, 472, 696, 1168, 1327, 1551, 2023e }}


[[Badness]]: 4.563 × 10<sup>-3</sup>
[[Badness]] (Smith): 4.563 × 10<sup>-3</sup>


=== Tenierian ===
=== Tenierian ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 10985/10976, 32805/32768, 161280/161051
Comma list: 10985/10976, 32805/32768, 161280/161051


[[Mapping]]: [{{val| 1 2 -1 3 3 5 }}, {{val| 0 -3 24 -3 3 -11 }}, {{val| 0 0 0 5 1 5 }}]
Mapping: {{mapping| 1 2 -1 3 3 5 | 0 -3 24 -3 3 -11 | 0 0 0 5 1 5 }}
: mapping generators: ~2, ~11/10, ~33/32


[[POTE generator]]s: ~11/10 = 166.0628, ~33/32 = 53.4151  
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~11/10 = 166.0628{{c}}, ~33/32 = 53.4151{{c}}


[[Badness]]: 16.903 × 10<sup>-3</sup>
Badness (Smith): 16.903 × 10<sup>-3</sup>


== Meanquarter ==
== Meanquarter ==
In meanquarter, both the meantone comma and the quartisma are tempered out.  
In meanquarter, both the meantone comma and the quartisma are tempered out.  


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 81/80, 4128768/4026275
[[Comma list]]: 81/80, 4128768/4026275


[[Mapping]]: [{{val| 1 0 -4 1 5 }}, {{val| 0 1 4 1 -1 }}, {{val| 0 0 5 1 }}]
{{Mapping|legend=1| 1 0 -4 1 5 | 0 1 4 1 -1 | 0 0 5 1 }}
: mapping generators: ~2, ~3, ~33/32


[[POTE generator]]s: ~3/2 = 697.3325, ~33/32 = 54.1064
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 697.3325{{c}}, ~33/32 = 54.1064{{c}}


{{Optimal ET sequence|legend=1| 24, 43, 67, 110c }}
{{Optimal ET sequence|legend=1| 24, 43, 67, 110c }}


[[Badness]]: 15.125 × 10<sup>-3</sup>
[[Badness]] (Smith): 15.125 × 10<sup>-3</sup>


== Coin ==
== Coin ==
In coin, both the magic comma and the quartisma are tempered out.
In coin, both the magic comma and the quartisma are tempered out.


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 3125/3072, 117440512/117406179
[[Comma list]]: 3125/3072, 117440512/117406179


[[Mapping]]: [{{val| 1 0 2 1 5 }}, {{val| 0 5 1 0 -6 }}, {{val| 0 0 0 5 1 }}]
{{Mapping|legend=1| 1 0 2 1 5 | 0 5 1 0 -6 | 0 0 0 5 1 }}
: mapping generators: ~2, ~5/4, ~9/7


[[POTE generator]]s: ~5/4 = 380.3623, ~9/7 = 433.3120
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~5/4 = 380.3623{{c}}, ~9/7 = 433.3120{{c}}


{{Optimal ET sequence|legend=1| 19d, 22 }}
{{Optimal ET sequence|legend=1| 19d, 22 }}


[[Badness]]: 70.470 × 10<sup>-3</sup>
[[Badness]] (Smith): 70.470 × 10<sup>-3</sup>


== Escapismic ==
== Escapismic ==
In escapisimic, both the escapade comma and the quartisma are tempered out, thus, it is essentially an [[Escapade family|escapade expansion]].
In escapisimic, both the escapade comma and the quartisma are tempered out, thus, it is essentially an [[escapade]] [[expansion]].


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 117440512/117406179, 4294967296/4271484375
[[Comma list]]: 117440512/117406179, 4294967296/4271484375


[[Mapping]]: [{{val| 1 2 2 3 3 }}, {{val| 0 -9 7 -4 10 }}, {{val| 0 0 0 5 1 }}]
{{Mapping|legend=1| 1 2 2 3 3 | 0 -9 7 -4 10 | 0 0 0 5 1 }}
: mapping generators: ~2, ~?, ~33/32


[[POTE generator]]s: ~33/32 = 55.3538
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~33/32 = 55.3538{{c}}


{{Optimal ET sequence|legend=1| 21, 22, 43, 65d, 521d, 543, 564, 586, 629c, 651 }}
{{Optimal ET sequence|legend=1| 21, 22, 43, 65d, 521d, 543, 564, 586, 629c, 651 }}


[[Badness]]: 64.233 × 10<sup>-3</sup>
[[Badness]] (Smith): 64.233 × 10<sup>-3</sup>


== Dietismic ==
== Dietismic ==
In dietismic, both the diaschisma and the quartisma are tempered out. Dietismic can easily be further tempered to [[Diaschismic family #Shrutar|shrutar]], and in fact, it is rather unusual to find a different tempering option.
In dietismic, both the diaschisma and the quartisma are tempered out. Dietismic can easily be further tempered to [[Diaschismic family #Shrutar|shrutar]], and in fact, it is rather unusual to find a different tempering option.


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 2048/2025, 117440512/117406179
[[Comma list]]: 2048/2025, 117440512/117406179


[[POTE generator]]s: ~3/2 = 704.5238, ~33/32 = 53.4408
{{Mapping|legend=1| 2 0 11 2 10 | 0 1 -2 1 -1 | 0 0 0 5 1 }}
: mapping generators: ~45/32, ~3, ~33/32


[[Mapping]]: [{{val| 2 0 11 2 10 }}, {{val| 0 1 -2 1 -1 }}, {{val| 0 0 0 5 1 }}]
[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.0000{{c}}, ~3/2 = 704.5238{{c}}, ~33/32 = 53.4408{{c}}


{{Optimal ET sequence|legend=1| 22, 46, 68, 114 }}
{{Optimal ET sequence|legend=1| 22, 46, 68, 114 }}


[[Badness]]: 23.250 × 10<sup>-3</sup>
[[Badness]] (Smith): 23.250 × 10<sup>-3</sup>


Scales:  
Scales:  
Line 143: Line 147:


== Kleirtismic ==
== Kleirtismic ==
In kleirtismic, both the kleisma and the quartisma are tempered out. The "kleir-" in "kleirtismic" is pronounced the same as "Clair".  
In kleirtismic, both the kleisma and the quartisma are tempered out. The ''kleir-'' in ''kleirtismic'' is meant to be pronounced the same as ''Clair''.  


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 15625/15552, 117440512/117406179
[[Comma list]]: 15625/15552, 117440512/117406179


[[Mapping]]: [{{val| 1 0 1 1 5 }}, {{val| 0 6 5 1 -7 }}, {{val| 0 0 0 5 1 }}]
{{Mapping|legend=1| 1 0 1 1 5 | 0 6 5 1 -7 | 0 0 0 5 1 }}
: mapping generators: ~2, ~6/5, ~68/55


[[POTE generator]]s: ~6/5 = 317.0291, ~68/55 = 370.2940
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~6/5 = 317.0291{{c}}, ~68/55 = 370.2940{{c}}


{{Optimal ET sequence|legend=1| 68, 91, 159, 246, 337, 405 }}
{{Optimal ET sequence|legend=1| 68, 91, 159, 246, 337, 405 }}


[[Badness]]: 26.882 × 10<sup>-3</sup>
[[Badness]] (Smith): 26.882 × 10<sup>-3</sup>


== Doublefour ==
== Doublefour ==
In doublefour, both the tetracot comma and the quartisma are tempered out.
In doublefour, both the tetracot comma and the quartisma are tempered out.


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 20000/19683, 100656875/99090432
[[Comma list]]: 20000/19683, 100656875/99090432


[[Mapping]]: [{{val| 1 1 1 2 4 }}, {{val| 0 4 9 4 -4 }}, {{val| 0 0 0 5 1 }}]
{{Mapping|legend=1| 1 1 1 2 4 | 0 4 9 4 -4 | 0 0 0 5 1 }}
: mapping generators: ~2, ~425/384, ~33/32


[[POTE generator]]s: ~425/384 = 175.9566, ~33/32 = 52.9708
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~425/384 = 175.9566{{c}}, ~33/32 = 52.9708{{c}}


{{Optimal ET sequence|legend=1| 48d, 68, 116d, 157c, 225 }}
{{Optimal ET sequence|legend=1| 48d, 68, 116d, 157c, 225 }}


[[Badness]]: 81.083 × 10<sup>-3</sup>
[[Badness]] (Smith): 81.083 × 10<sup>-3</sup>


== Quarterframe ==
== Quarterframe ==
This is actually a microtemperament involving the [[3025/3024|lehmerisma]] and the [[frameshift comma]]. It is also a weak extension of the [[Very high accuracy temperaments|monzismic temperament]].
This is actually a microtemperament involving the [[lehmerisma]] and the [[frameshift comma]]. It is also a weak extension of the [[monzismic]] temperament.


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 3025/3024, 26214400/26198073, 29296875/29218112
[[Comma list]]: 3025/3024, 26214400/26198073, 29296875/29218112


[[Mapping]]: [{{val| 1 4 47 130 26 }}, {{val| 0 -6 -111 -316 -56 }}]
{{Mapping|legend=1| 1 4 47 130 26 | 0 -6 -111 -316 -56 }}
: mapping generators: ~2, ~160/121


[[POTE generator]]: ~160/121 = 483.0098
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~160/121 = 483.0098{{c}}


{{Optimal ET sequence|legend=1| 159, 559d, 718, 877 }}
{{Optimal ET sequence|legend=1| 159, 559d, 718, 877 }}


[[Badness]]: 0.154578
[[Badness]] (Smith): 0.154578


== Ravine ==
== Ravine ==
Initially defined upon the 33/32 generator in 1619edo, producing a 832 & 1619 temperament.
This temperament was initially defined upon the 33/32 generator in 1619edo, producing a 832 & 1619 temperament.


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 194: Line 201:
[[Comma list]]: 514714375/514434888, 117440512/117406179, 1220703125/1219784832
[[Comma list]]: 514714375/514434888, 117440512/117406179, 1220703125/1219784832


[[Mapping]]: [{{val| 1 26 21 27 -21 }}, {{val| 0 -549 -420 -544 550 }}]
{{Mapping|legend=1| 1 26 21 27 -21 | 0 -549 -420 -544 550 }}
: mapping generators: ~2, ~33/32


[[Optimal tuning]] ([[CTE]]): ~33/32 = 53.366
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~33/32 = 53.366{{c}}


{{Optimal ET sequence|legend=1| 832, 1619 }}
{{Optimal ET sequence|legend=1| 832, 1619 }}
Line 205: Line 213:
Comma list: 196625/196608, 200000/199927, 2912000/2910897, 3764768/3764475
Comma list: 196625/196608, 200000/199927, 2912000/2910897, 3764768/3764475


Mapping: [{{val| 1 26 21 27 -21 21 }}, {{val| 0 -549 -420 -544 550 -389 }}]
Mapping: {{mapping| 1 26 21 27 -21 21 | 0 -549 -420 -544 550 -389 }}


Optimal tuning (CTE): ~33/32 = 53.366
Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~33/32 = 53.366


{{Optimal ET sequence|legend=1| 832, 1619 }}
{{Optimal ET sequence|legend=0| 832, 1619 }}


== Prequartismic ==
== Prequartismic ==
Named because 3125edo was the only one confirmed for tempering out the quartisma before it's discovery as significance of the difference between 5 33/32s and 7/6. Defined upon the 33/32 (or 64/33) generator in 3125edo, in terms of patent vals it can be described as 3125 & 4991 or 3125 & 1866.
This temperament was named because 3125edo was the only one confirmed for tempering out the quartisma before its discovery as significance of the difference between five 33/32's and 7/6. Defined upon the 33/32 (or 64/33) generator in 3125edo, in terms of patent vals it can be described as 3125 & 4991 or 3125 & 1866.


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


[[Comma list]]: {{monzo|24 -6 0 1 -5}}, {{monzo|-1 4 11 -11 0}}, {{monzo|-19 -25 14  13 -3}}
[[Comma list]]: {{monzo| 24 -6 0 1 -5 }}, {{monzo| -1 4 11 -11 0 }}, {{monzo| -19 -25 14  13 -3 }}


[[Mapping]]: [{{val|1 1389 890 1395 -1383}}, {{val|0 -1452 -929 -1457 1451}}]
{{Mapping|legend=1| 1 1389 890 1395 -1383 | 0 -1452 -929 -1457 1451 }}
: mapping generators: ~2, ~64/33


[[Optimal tuning]] ([[CTE]]): ~64/33 = 1146.624
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~64/33 = 1146.624{{c}


{{Optimal ET sequence|legend=1|1259e, 1866, 3125, 4384e, 4991, 6250e, 8116d, 7509ee, 9375e, 11241de}}
{{Optimal ET sequence|legend=1| 1259e, 1866, 3125, 4384e, 4991, 6250e, 8116d, 7509ee, 9375e, 11241de }}


=== 13-limit ===
=== 13-limit ===
[[Subgroup]]: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 1990656/1990625, 14236560/14235529, 117440512/117406179, 1181640625/1181599328
Comma list: 1990656/1990625, 14236560/14235529, 117440512/117406179, 1181640625/1181599328


[[Mapping]]: [{{val|1 1389 890 1395 -1383 -282}}, {{val|0 -1452 -929 -1457 1451 299 }}]
Mapping: {{mapping| 1 1389 890 1395 -1383 -282 | 0 -1452 -929 -1457 1451 299 }}


[[Optimal tuning]] (CTE): ~64/33 = 1146.624
Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~64/33 = 1146.624{{c}}


[[Category:Regular temperament theory]]
[[Category:Commatic realms]]
[[Category:Commatic realms]]
[[Category:The Quartercache| ]] <!-- main article -->
[[Category:The Quartercache| ]] <!-- main article -->
[[Category:Microtemperaments]]
[[Category:Quartismic]]
[[Category:Quartismic]]

Latest revision as of 15:31, 17 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.

The Quartercache is a collection of temperaments of different ranks, including subgroup temperaments, that all temper out the quartisma – the unnoticeable comma with the ratio 117440512/117406179, and a monzo of [24 -6 0 1 -5⟩. Among the members of this family are quartic, altierran, meanquarter, coin, escapismic, dietismic, kleirtismic, doublefour and quarterframe.

Quartismic

See Catalog of rank-4 temperaments #Quartismic (117440512/117406179).

Quartic

There are some temperaments that temper out the quartisma despite having limited accuracy in their approximations of five-limit intervals. This particular temperament is the parent temperament of all such no-fives children, and is referred to as Saquinlu-azo in color notation.

Subgroup: 2.3.7.11

Comma list: 117440512/117406179

Mapping: [⟨1 0 1 5], ⟨0 1 1 -1], ⟨0 0 5 1]]

mapping generators: ~2, ~3, ~33/32

Optimal tunings:

  • WE: ~2 = 1199.9875 ¢, ~3/2 = 701.9753 ¢, ~33/32 = 53.3743 ¢
error map: ⟨-0.012 +0.008 -0.004 +0.031]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9802 ¢, ~33/32 = 53.3730 ¢
error map: ⟨0.000 +0.025 +0.019 +0.075]

Optimal ET sequence: 21, 22, 24, 43, 46, 89, 135, 359, 494, 629, 742, 877, 1012, 1506, 2248, 2383, 2518, 7419, 8431e, 10949e, 13467e

Badness (Sintel): 0.416

The following rank-2 mos scales of quartismic have been found:

Direct quartismic

Instead of 11-limit, direct quartismic is defined directly in the 2.7/3.33 subgroup, as the quartisma itself was discovered via a representation where the generator is 33/32 and five of them stack to 7/6. 45edo is an excellent tuning. It was named by Eliora in 2023.

Subgroup: 2.7/3.33

Comma list: 117440512/117406179

Subgroup-val mapping: [⟨1 1 5], ⟨0 5 1]]

mapping generators: ~2, ~33/32

Optimal tunings:

  • Subgroup WE: ~2 = 1199.9875 ¢, ~33/32 = 53.3743 ¢
error map: ⟨-0.012 -0.012 +0.039]
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~33/32 = 53.3730 ¢
error map: ⟨0.000 -0.006 +0.100]

Optimal ET sequence: 21, 22, 45, 337, 382, 427, 472, 517, 562, 607, 652, 1911, 2563, 3215, 5778, 8993*

* wart for 33

Altierran

In altierran, both the schisma and the quartisma are tempered out.

Subgroup: 2.3.5.7.11

Comma list: 32805/32768, 161280/161051

Mapping: [⟨1 0 15 1 5], ⟨0 1 -8 1 -1], ⟨0 0 0 5 1]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 701.7299 ¢, ~33/32 = 53.3889 ¢

Optimal ET sequence: 24, 46c, 65d, 89, 135, 159, 224, 383, 472, 696, 1168, 1327, 1551, 2023e

Badness (Smith): 4.563 × 10-3

Tenierian

Subgroup: 2.3.5.7.11.13

Comma list: 10985/10976, 32805/32768, 161280/161051

Mapping: [⟨1 2 -1 3 3 5], ⟨0 -3 24 -3 3 -11], ⟨0 0 0 5 1 5]]

mapping generators: ~2, ~11/10, ~33/32

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~11/10 = 166.0628 ¢, ~33/32 = 53.4151 ¢

Badness (Smith): 16.903 × 10-3

Meanquarter

In meanquarter, both the meantone comma and the quartisma are tempered out.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 4128768/4026275

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

mapping generators: ~2, ~3, ~33/32

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 697.3325 ¢, ~33/32 = 54.1064 ¢

Optimal ET sequence: 24, 43, 67, 110c

Badness (Smith): 15.125 × 10-3

Coin

In coin, both the magic comma and the quartisma are tempered out.

Subgroup: 2.3.5.7.11

Comma list: 3125/3072, 117440512/117406179

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

mapping generators: ~2, ~5/4, ~9/7

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~5/4 = 380.3623 ¢, ~9/7 = 433.3120 ¢

Optimal ET sequence: 19d, 22

Badness (Smith): 70.470 × 10-3

Escapismic

In escapisimic, both the escapade comma and the quartisma are tempered out, thus, it is essentially an escapade expansion.

Subgroup: 2.3.5.7.11

Comma list: 117440512/117406179, 4294967296/4271484375

Mapping: [⟨1 2 2 3 3], ⟨0 -9 7 -4 10], ⟨0 0 0 5 1]]

mapping generators: ~2, ~?, ~33/32

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~33/32 = 55.3538 ¢

Optimal ET sequence: 21, 22, 43, 65d, 521d, 543, 564, 586, 629c, 651

Badness (Smith): 64.233 × 10-3

Dietismic

In dietismic, both the diaschisma and the quartisma are tempered out. Dietismic can easily be further tempered to shrutar, and in fact, it is rather unusual to find a different tempering option.

Subgroup: 2.3.5.7.11

Comma list: 2048/2025, 117440512/117406179

Mapping: [⟨2 0 11 2 10], ⟨0 1 -2 1 -1], ⟨0 0 0 5 1]]

mapping generators: ~45/32, ~3, ~33/32

Optimal tuning (POTE): ~45/32 = 600.0000 ¢, ~3/2 = 704.5238 ¢, ~33/32 = 53.4408 ¢

Optimal ET sequence: 22, 46, 68, 114

Badness (Smith): 23.250 × 10-3

Scales:

Rank 2 scale (52.6800, 2/1), 13|9

Rank 2 scale (53.3742, 2/1), 13|9

Kleirtismic

In kleirtismic, both the kleisma and the quartisma are tempered out. The kleir- in kleirtismic is meant to be pronounced the same as Clair.

Subgroup: 2.3.5.7.11

Comma list: 15625/15552, 117440512/117406179

Mapping: [⟨1 0 1 1 5], ⟨0 6 5 1 -7], ⟨0 0 0 5 1]]

mapping generators: ~2, ~6/5, ~68/55

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~6/5 = 317.0291 ¢, ~68/55 = 370.2940 ¢

Optimal ET sequence: 68, 91, 159, 246, 337, 405

Badness (Smith): 26.882 × 10-3

Doublefour

In doublefour, both the tetracot comma and the quartisma are tempered out.

Subgroup: 2.3.5.7.11

Comma list: 20000/19683, 100656875/99090432

Mapping: [⟨1 1 1 2 4], ⟨0 4 9 4 -4], ⟨0 0 0 5 1]]

mapping generators: ~2, ~425/384, ~33/32

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~425/384 = 175.9566 ¢, ~33/32 = 52.9708 ¢

Optimal ET sequence: 48d, 68, 116d, 157c, 225

Badness (Smith): 81.083 × 10-3

Quarterframe

This is actually a microtemperament involving the lehmerisma and the frameshift comma. It is also a weak extension of the monzismic temperament.

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 26214400/26198073, 29296875/29218112

Mapping: [⟨1 4 47 130 26], ⟨0 -6 -111 -316 -56]]

mapping generators: ~2, ~160/121

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~160/121 = 483.0098 ¢

Optimal ET sequence: 159, 559d, 718, 877

Badness (Smith): 0.154578

Ravine

This temperament was initially defined upon the 33/32 generator in 1619edo, producing a 832 & 1619 temperament.

Subgroup: 2.3.5.7.11

Comma list: 514714375/514434888, 117440512/117406179, 1220703125/1219784832

Mapping: [⟨1 26 21 27 -21], ⟨0 -549 -420 -544 550]]

mapping generators: ~2, ~33/32

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~33/32 = 53.366 ¢

Optimal ET sequence: 832, 1619

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196625/196608, 200000/199927, 2912000/2910897, 3764768/3764475

Mapping: [⟨1 26 21 27 -21 21], ⟨0 -549 -420 -544 550 -389]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~33/32 = 53.366

Optimal ET sequence: 832, 1619

Prequartismic

This temperament was named because 3125edo was the only one confirmed for tempering out the quartisma before its discovery as significance of the difference between five 33/32's and 7/6. Defined upon the 33/32 (or 64/33) generator in 3125edo, in terms of patent vals it can be described as 3125 & 4991 or 3125 & 1866.

Subgroup: 2.3.5.7.11

Comma list: [24 -6 0 1 -5⟩, [-1 4 11 -11 0⟩, [-19 -25 14  13 -3⟩

Mapping: [⟨1 1389 890 1395 -1383], ⟨0 -1452 -929 -1457 1451]]

mapping generators: ~2, ~64/33

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~64/33 = 1146.624{{c}

Optimal ET sequence: 1259e, 1866, 3125, 4384e, 4991, 6250e, 8116d, 7509ee, 9375e, 11241de

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1990656/1990625, 14236560/14235529, 117440512/117406179, 1181640625/1181599328

Mapping: [⟨1 1389 890 1395 -1383 -282], ⟨0 -1452 -929 -1457 1451 299]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~64/33 = 1146.624 ¢

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