The Quartercache: Difference between revisions

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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.

== Quartismic ==

== Rank-4 temperaments ==

=== Quartismic ===

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.

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[[Comma list]]: 117440512/117406179

~~[[Sval]] [[mapping]]: [~~{{~~val~~| 1 0 1 5 ~~}}, {{val~~| 0 1 1 -1 ~~}}, {{val~~| 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 }}

[[Badness]] (Sintel): 0.416

The following rank-2 [[mos scale]]s of quartismic have been found:

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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]

== ~~Direct quartismic~~ ==

=== Meanquarter ===

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.

~~[[Subgroup]]: 2.33/32.7/6~~

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

~~{{Mapping|legend=2| 1 0 0 | 0 1 5 }}~~

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

~~== 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 × 10-3~~

~~=== 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 × 10-3~~

~~== Meanquarter~~ ==

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

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[[Badness]] (Smith): 15.125 × 10-3

== Coin ==

=== Coin ===

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

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[[Badness]] (Smith): 70.470 × 10-3

== Escapismic ==

=== Escapismic ===

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 × 10-3

== 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.

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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]

== 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''.

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[[Badness]] (Smith): 26.882 × 10-3

== Doublefour ==

=== Doublefour ===

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

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[[Badness]] (Smith): 81.083 × 10-3

== 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|legend=0| 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 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.

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[[Badness]] (Smith): 0.154578

== Ravine ==

=== Ravine ===

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

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=== 13-limit ===

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

~~Mapping:~~ {{~~mapping~~| 1 26 21 27 -21 21 | 0 -549 -420 -544 550 -389 }}

{{Mapping|legend=0| 1 26 21 27 -21 21 | 0 -549 -420 -544 550 -389 }}

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

~~== 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:The Quartercache| ]]

[[Category:Quartismic]]

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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.
-== Quartismic ==
+== Rank-4 temperaments ==
+=== Quartismic ===
 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.
@@ Line 12: / Line 14: @@
 [[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 }}
+[[Badness]] (Sintel): 0.416
 The following rank-2 [[mos scale]]s of quartismic have been found:
@@ Line 22: / Line 31: @@
 * 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, 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.
-[[Subgroup]]: 2.33/32.7/6
-[[Comma list]]: 117440512/117406179
-{{Mapping|legend=2| 1 0 0 | 0 1 5 }}
-: mapping generators: ~2, ~33/32
-== 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.
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 [[Badness]] (Smith): 15.125 × 10<sup>-3</sup>
-== Coin ==
+=== Coin ===
 In coin, both the magic comma and the quartisma are tempered out.
@@ Line 91: / Line 63: @@
 [[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]].
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 [[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.
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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]
-== 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''.
@@ Line 145: / Line 117: @@
 [[Badness]] (Smith): 26.882 × 10<sup>-3</sup>
-== Doublefour ==
+=== Doublefour ===
 In doublefour, both the tetracot comma and the quartisma are tempered out.
@@ Line 161: / Line 133: @@
 [[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|legend=0| 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=3| 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.
@@ Line 177: / Line 193: @@
 [[Badness]] (Smith): 0.154578
-== Ravine ==
+=== Ravine ===
 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 }}
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
 Comma list: 196625/196608, 200000/199927, 2912000/2910897, 3764768/3764475
-Mapping: {{mapping| 1 26 21 27 -21 21 | 0 -549 -420 -544 550 -389 }}
+{{Mapping|legend=0| 1 26 21 27 -21 21 | 0 -549 -420 -544 550 -389 }}
 Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~33/32 = 53.366
 {{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:The Quartercache| ]] <!-- main article -->
 [[Category:Quartismic]]