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Complete data for quartic and direct quartismic
- altierrran (addressed in schismic rank-3 family); sort the rest
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'''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 ==
== Rank-4 temperaments ==
=== Quartismic ===
See [[Catalog of rank-4 temperaments #Quartismic (117440512/117406179)]].  
See [[Catalog of rank-4 temperaments #Quartismic (117440512/117406179)]].  


== Quartic ==
== Rank-3 temperaments ==
=== 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.
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.


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* 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 ==
=== Meanquarter ===
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
 
{{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* }}
 
<nowiki/>* 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|legend=1| 1 0 15 1 5 | 0 1 -8 1 -1 | 0 0 0 5 1 }}
 
[[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 }}
 
[[Badness]] (Smith): 4.563 × 10<sup>-3</sup>
 
=== Tenierian ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 10985/10976, 32805/32768, 161280/161051
 
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
 
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~11/10 = 166.0628{{c}}, ~33/32 = 53.4151{{c}}
 
Badness (Smith): 16.903 × 10<sup>-3</sup>
 
== 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.  


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[[Badness]] (Smith): 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.


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[[Badness]] (Smith): 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]] [[expansion]].
In escapisimic, both the escapade comma and the quartisma are tempered out, thus, it is essentially an [[escapade]] [[expansion]].


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[[Badness]] (Smith): 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.


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[https://sevish.com/scaleworkshop/?name=Rank%202%20scale%20(53.37418112074753%2C%202%2F1)%2C%2013%7C9&data=53.374181%0A106.748362%0A160.122543%0A213.496724%0A266.870906%0A320.245087%0A373.619268%0A426.993449%0A480.367630%0A533.741811%0A587.115992%0A640.490173%0A693.864355%0A719.632370%0A773.006551%0A826.380732%0A879.754913%0A933.129094%0A986.503276%0A1039.877457%0A1093.251638%0A1146.625819%0A1200.000000&freq=440&midi=69&vert=9&horiz=1&colors=white%20black%20white%20white%20black%20white%20black%20white%20white%20black%20white%20black&waveform=triangle&ampenv=organ Rank 2 scale (53.3742, 2/1), 13|9]
[https://sevish.com/scaleworkshop/?name=Rank%202%20scale%20(53.37418112074753%2C%202%2F1)%2C%2013%7C9&data=53.374181%0A106.748362%0A160.122543%0A213.496724%0A266.870906%0A320.245087%0A373.619268%0A426.993449%0A480.367630%0A533.741811%0A587.115992%0A640.490173%0A693.864355%0A719.632370%0A773.006551%0A826.380732%0A879.754913%0A933.129094%0A986.503276%0A1039.877457%0A1093.251638%0A1146.625819%0A1200.000000&freq=440&midi=69&vert=9&horiz=1&colors=white%20black%20white%20white%20black%20white%20black%20white%20white%20black%20white%20black&waveform=triangle&ampenv=organ Rank 2 scale (53.3742, 2/1), 13|9]


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


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[[Badness]] (Smith): 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.


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


== Quarterframe ==
=== 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]]: {{monzo| 24 -6 0 1 -5 }}, {{monzo| -1 4 11 -11 0 }}, {{monzo| -19 -25 14  13 -3 }}
 
{{Mapping|legend=1| 1 1389 890 1395 -1383 | 0 -1452 -929 -1457 1451 }}
: mapping generators: ~2, ~64/33
 
[[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 }}
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1990656/1990625, 14236560/14235529, 117440512/117406179, 1181640625/1181599328
 
Mapping: {{mapping| 1 1389 890 1395 -1383 -282 | 0 -1452 -929 -1457 1451 299 }}
 
Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~64/33 = 1146.624{{c}}
 
== Rank-2 temperaments ==
=== 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
 
{{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* }}
 
<nowiki/>* wart for 33
 
=== Quarterframe ===
This is actually a microtemperament involving the [[lehmerisma]] and the [[frameshift comma]]. It is also a weak extension of the [[monzismic]] temperament.
This is actually a microtemperament involving the [[lehmerisma]] and the [[frameshift comma]]. It is also a weak extension of the [[monzismic]] temperament.


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


== Ravine ==
=== Ravine ===
This temperament was 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.


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


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


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{{Optimal ET sequence|legend=0| 832, 1619 }}
{{Optimal ET sequence|legend=0| 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]]: {{monzo| 24 -6 0 1 -5 }}, {{monzo| -1 4 11 -11 0 }}, {{monzo| -19 -25 14  13 -3 }}
{{Mapping|legend=1| 1 1389 890 1395 -1383 | 0 -1452 -929 -1457 1451 }}
: mapping generators: ~2, ~64/33
[[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 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1990656/1990625, 14236560/14235529, 117440512/117406179, 1181640625/1181599328
Mapping: {{mapping| 1 1389 890 1395 -1383 -282 | 0 -1452 -929 -1457 1451 299 }}
Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~64/33 = 1146.624{{c}}


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

Revision as of 07:13, 17 September 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.

Rank-4 temperaments

Quartismic

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

Rank-3 temperaments

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:

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

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 ¢

Rank-2 temperaments

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

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

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