Keemic temperaments: Difference between revisions

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This is a collection of [[rank-2 temperament|linear]] [[regular temperament|temperaments]] that [[tempering out|temper out]] the [[keema]] ({{monzo|legend=1| -5 -3 3 1 }}, [[ratio]]: 875/864), with [[S-expression]] S5/S6. Its fundamental equivalence entails that [[6/5]] is sharpened so that it stacks three times to reach [[7/4]], and the interval between 6/5 and [[5/4]] is compressed so that [[7/6]]–6/5–5/4–[[9/7]] are set equidistant from each other. As the canonical extension of rank-3 [[keemic]] to the [[11-limit]] tempers out the commas [[100/99]] and [[385/384]] (whereby ([[6/5]])<sup>2</sup> is identified with [[16/11]]), this provides a clean way to extend the various keemic temperaments to the 11-limit as well.

Full [[7-limit]] keemic temperaments discussed elsewhere are:

* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]

* ''[[Mujannabic]]'' (+25/24) → [[Dicot family #Dicot|Dicot family]]

* [[Porcupine]] (+64/63) → [[Porcupine family #Septimal porcupine|Porcupine family]]

* [[Monkey]] (+5120/5103) → [[Tetracot family #Monkey|Tetracot family]]

* [[Magic]] (+225/224) → [[Magic family #Septimal magic|Magic family]]

* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]

* ''[[Doublewide]]'' (+50/49) → [[Jubilismic clan #Doublewide|Jubilismic clan]]

* [[~~Porcupine~~]] (+64/63) → [[~~Porcupine family~~ #~~Septimal porcupine~~|~~Porcupine family]]~~

* [[Superkleismic]] (+1029/1024) → [[Gamelismic clan #Superkleismic|Gamelismic clan]]

* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]

* [[Magic]] (+225/224) → [[Magic family #Septimal magic|Magic family]]

* ''[[Sycamore]]'' (+686/675) → [[Sycamore family #Septimal sycamore|Sycamore family]]

* [[Superkleismic]] (+1029/1024) → [[Gamelismic clan #Superkleismic|Gamelismic clan]]

* ''[[Undeka]]'' (+3200/3087) → [[11th-octave temperaments #Undeka|11th-octave temperaments]]

Discussed below are quasitemp, chromo, barbad, hyperkleismic, and sevond.

Discussed below are quasitemp, chromo, barbad, pentadecal, hyperkleismic, and sevond, in the order of increasing [[TE logflat badness]].

== Quasitemp ==

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasitemp]].''

Quasitemp tempers out [[2401/2400]] in addition to 875/864 and may be described as the {{nowrap| 37 & 41 }} temperament. It can be thought of as a [[strong extension]] of the 2.5/3.7/3 subgroup temperament called [[gariberttet]], which is defined by tempering out [[3125/3087]]. In gariberttet, three generators reach [[5/3]] and five reach [[7/3]], so that the generator itself has the interpretation of [[25/21]]. This implies that 3:5:7 and 5:6:7 chords are reached rather quickly. Quasitemp tempering out 875/864 entails that [[8/7]] is found after 9 generators, from which the mappings of 3 and 5 follow.

Quasitemp tempers out [[2401/2400]] in addition to 875/864 and may be described as the {{nowrap| 37 & 41 }} temperament. It is characterized by equating the interval between the pental and septimal thirds ([[36/35]]) with the classical chromatic semitone ([[25/24]]), and by tempering together the septimal dieses of [[49/48]] and [[50/49]]. In that sense, it is opposed to [[orwellismic temperaments]], in particular [[myna]], where the distance between the pental and septimal thirds is the same as the septimal dieses and different from the classical chromatic semitone.

Quasitemp can also be thought of as a [[strong extension]] of the 2.5/3.7/3-subgroup temperament called [[gariberttet]], which is defined by tempering out [[3125/3087]]. In gariberttet, three generators reach [[5/3]] and five reach [[7/3]], so that the generator itself has the interpretation of [[25/21]]. This implies that 3:5:7 and 5:6:7 chords are reached rather quickly. Quasitemp tempering out 875/864 entails that [[8/7]] is found after 9 generators, from which the mappings of 3 and 5 follow.

Note that the generator is given as 25/21's octave complement, 42/25, in the data that follow, since a stack of 14 such generators octave-reduced is the perfect fifth, whence the temperament's [[ploidacot]] is iota-14-cot. This generator is equated to [[22/13]] for the 13-limit extension, tempering out [[275/273]].

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Badness (Sintel): 1.62

== Pentadecal ==

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Pentadecal]].''

Named by [[Xenllium]] in 2021, pentadecal tempers out the 15-5/3-comma ({{monzo| -11 -15 15 }}) in the 5-limit. This temperament can be described as {{nowrap| 15 & 60 }} temperament, tempering out the [[cloudy comma]], 16807/16384 and the [[keema]], 875/864 in the 7-limit.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 875/864, 16807/16384

: mapping generators: ~21/20, ~3

[[Optimal tuning]]s:

* [[WE]]: ~21/20 = 80.1141{{c}}, ~3/2 = 700.2213{{c}} (~126/125 = 19.8053{{c}})

: [[error map]]: {{val| +1.711 +0.977 -2.127 -4.035 }}

* [[CWE]]: ~21/20 = 80.0000{{c}}, ~3/2 = 701.2357{{c}} (~126/125 = 19.7643{{c}})

: error map: {{val| 0.000 -0.719 -5.078 -8.826 }}

[[Badness]] (Sintel): 2.91

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 1372/1331

Mapping: {{mapping| 15 0 11 42 52 | 0 1 1 0 0 }}

Optimal tunings:

* WE: ~21/20 = 80.0213{{c}}, ~3/2 = 702.9194{{c}} (~56/55 = 17.2721{{c}})

* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 702.8751{{c}} (~56/55 = 17.1249{{c}})

Badness (Sintel): 2.56

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 1372/1331

Mapping: {{mapping| 15 0 11 42 52 8 | 0 1 1 0 0 2 }}

Optimal tunings:

* WE: ~21/20 = 80.0207{{c}}, ~3/2 = 701.8966{{c}} (~91/90 = 18.2900{{c}})

* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 701.8545{{c}} (~91/90 = 18.1455{{c}})

Badness (Sintel): 2.14

=== Quindecal ===

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 875/864

Mapping: {{mapping| 15 0 11 42 28 | 0 1 1 0 1 }}

Optimal tunings:

* WE: ~21/20 = 80.1322{{c}}, ~3/2 = 701.4751{{c}} (~126/125 = 19.7148{{c}})

* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 701.5453{{c}} (~126/125 = 18.4547{{c}})

Badness (Sintel): 1.47

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 196/195, 352/351, 875/864

Mapping: {{mapping| 15 0 11 42 28 103 | 0 1 1 0 1 -2 }}

Optimal tunings:

* WE: ~21/20 = 80.1015{{c}}, ~3/2 = 702.5374{{c}} (~126/125 = 18.3763{{c}})

* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 701.8504{{c}} (~126/125 = 18.1496{{c}})

Badness (Sintel): 2.29

== Hyperkleismic ==

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 This is a collection of [[rank-2 temperament|linear]] [[regular temperament|temperaments]] that [[tempering out|temper out]] the [[keema]] ({{monzo|legend=1| -5 -3 3 1 }}, [[ratio]]: 875/864), with [[S-expression]] S5/S6. Its fundamental equivalence entails that [[6/5]] is sharpened so that it stacks three times to reach [[7/4]], and the interval between 6/5 and [[5/4]] is compressed so that [[7/6]]–6/5–5/4–[[9/7]] are set equidistant from each other. As the canonical extension of rank-3 [[keemic]] to the [[11-limit]] tempers out the commas [[100/99]] and [[385/384]] (whereby ([[6/5]])<sup>2</sup> is identified with [[16/11]]), this provides a clean way to extend the various keemic temperaments to the 11-limit as well.
 Full [[7-limit]] keemic temperaments discussed elsewhere are:
+* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]
+* ''[[Mujannabic]]'' (+25/24) → [[Dicot family #Dicot|Dicot family]]
+* [[Porcupine]] (+64/63) → [[Porcupine family #Septimal porcupine|Porcupine family]]
+* [[Monkey]] (+5120/5103) → [[Tetracot family #Monkey|Tetracot family]]
+* [[Magic]] (+225/224) → [[Magic family #Septimal magic|Magic family]]
 * [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
 * ''[[Doublewide]]'' (+50/49) → [[Jubilismic clan #Doublewide|Jubilismic clan]]
-* [[Porcupine]] (+64/63) → [[Porcupine family #Septimal porcupine|Porcupine family]]
+* [[Superkleismic]] (+1029/1024) → [[Gamelismic clan #Superkleismic|Gamelismic clan]]
-* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]
-* [[Magic]] (+225/224) → [[Magic family #Septimal magic|Magic family]]
 * ''[[Sycamore]]'' (+686/675) → [[Sycamore family #Septimal sycamore|Sycamore family]]
-* [[Superkleismic]] (+1029/1024) → [[Gamelismic clan #Superkleismic|Gamelismic clan]]
 * ''[[Undeka]]'' (+3200/3087) → [[11th-octave temperaments #Undeka|11th-octave temperaments]]
-Discussed below are quasitemp, chromo, barbad, hyperkleismic, and sevond.
+Discussed below are quasitemp, chromo, barbad, pentadecal, hyperkleismic, and sevond, in the order of increasing [[TE logflat badness]].
 == Quasitemp ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasitemp]].''
-Quasitemp tempers out [[2401/2400]] in addition to 875/864 and may be described as the {{nowrap| 37 & 41 }} temperament. It can be thought of as a [[strong extension]] of the 2.5/3.7/3 subgroup temperament called [[gariberttet]], which is defined by tempering out [[3125/3087]]. In gariberttet, three generators reach [[5/3]] and five reach [[7/3]], so that the generator itself has the interpretation of [[25/21]]. This implies that 3:5:7 and 5:6:7 chords are reached rather quickly. Quasitemp tempering out 875/864 entails that [[8/7]] is found after 9 generators, from which the mappings of 3 and 5 follow.
+Quasitemp tempers out [[2401/2400]] in addition to 875/864 and may be described as the {{nowrap| 37 & 41 }} temperament. It is characterized by equating the interval between the pental and septimal thirds ([[36/35]]) with the classical chromatic semitone ([[25/24]]), and by tempering together the septimal dieses of [[49/48]] and [[50/49]]. In that sense, it is opposed to [[orwellismic temperaments]], in particular [[myna]], where the distance between the pental and septimal thirds is the same as the septimal dieses and different from the classical chromatic semitone.
+Quasitemp can also be thought of as a [[strong extension]] of the 2.5/3.7/3-subgroup temperament called [[gariberttet]], which is defined by tempering out [[3125/3087]]. In gariberttet, three generators reach [[5/3]] and five reach [[7/3]], so that the generator itself has the interpretation of [[25/21]]. This implies that 3:5:7 and 5:6:7 chords are reached rather quickly. Quasitemp tempering out 875/864 entails that [[8/7]] is found after 9 generators, from which the mappings of 3 and 5 follow.
 Note that the generator is given as 25/21's octave complement, 42/25, in the data that follow, since a stack of 14 such generators octave-reduced is the perfect fifth, whence the temperament's [[ploidacot]] is iota-14-cot. This generator is equated to [[22/13]] for the 13-limit extension, tempering out [[275/273]].
@@ Line 169: / Line 173: @@
 Badness (Sintel): 1.62
+== Pentadecal ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Pentadecal]].''
+Named by [[Xenllium]] in 2021, pentadecal tempers out the 15-5/3-comma ({{monzo| -11 -15 15 }}) in the 5-limit. This temperament can be described as {{nowrap| 15 & 60 }} temperament, tempering out the [[cloudy comma]], 16807/16384 and the [[keema]], 875/864 in the 7-limit.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 875/864, 16807/16384
+{{Mapping|legend=1| 15 0 11 42 | 0 1 1 0 }}
+: mapping generators: ~21/20, ~3
+[[Optimal tuning]]s:
+* [[WE]]: ~21/20 = 80.1141{{c}}, ~3/2 = 700.2213{{c}} (~126/125 = 19.8053{{c}})
+: [[error map]]: {{val| +1.711 +0.977 -2.127 -4.035 }}
+* [[CWE]]: ~21/20 = 80.0000{{c}}, ~3/2 = 701.2357{{c}} (~126/125 = 19.7643{{c}})
+: error map: {{val| 0.000 -0.719 -5.078 -8.826 }}
+{{Optimal ET sequence|legend=1| 15, 45, 60 }}
+[[Badness]] (Sintel): 2.91
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 100/99, 385/384, 1372/1331
+Mapping: {{mapping| 15 0 11 42 52 | 0 1 1 0 0 }}
+Optimal tunings:
+* WE: ~21/20 = 80.0213{{c}}, ~3/2 = 702.9194{{c}} (~56/55 = 17.2721{{c}})
+* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 702.8751{{c}} (~56/55 = 17.1249{{c}})
+{{Optimal ET sequence|legend=0| 15, 45, 60, 75de, 135de }}
+Badness (Sintel): 2.56
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 100/99, 105/104, 144/143, 1372/1331
+Mapping: {{mapping| 15 0 11 42 52 8 | 0 1 1 0 0 2 }}
+Optimal tunings:
+* WE: ~21/20 = 80.0207{{c}}, ~3/2 = 701.8966{{c}} (~91/90 = 18.2900{{c}})
+* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 701.8545{{c}} (~91/90 = 18.1455{{c}})
+{{Optimal ET sequence|legend=0| 15, 45f, 60, 135de, 195cddee }}
+Badness (Sintel): 2.14
+=== Quindecal ===
+Subgroup: 2.3.5.7.11
+Comma list: 121/120, 441/440, 875/864
+Mapping: {{mapping| 15 0 11 42 28 | 0 1 1 0 1 }}
+Optimal tunings:
+* WE: ~21/20 = 80.1322{{c}}, ~3/2 = 701.4751{{c}} (~126/125 = 19.7148{{c}})
+* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 701.5453{{c}} (~126/125 = 18.4547{{c}})
+{{Optimal ET sequence|legend=0| 15, 45e, 60e }}
+Badness (Sintel): 1.47
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 121/120, 196/195, 352/351, 875/864
+Mapping: {{mapping| 15 0 11 42 28 103 | 0 1 1 0 1 -2 }}
+Optimal tunings:
+* WE: ~21/20 = 80.1015{{c}}, ~3/2 = 702.5374{{c}} (~126/125 = 18.3763{{c}})
+* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 701.8504{{c}} (~126/125 = 18.1496{{c}})
+{{Optimal ET sequence|legend=0| 15f, 60e, 135deef }}
+Badness (Sintel): 2.29
 == Hyperkleismic ==