Keemic temperaments: Difference between revisions

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This is a collection of [[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 ~~[[Keemic family #Undecimal supermagic|~~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.

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, 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 ~~has~~ a [[strong ~~restriction~~]] to the 2.5/3.7/3 subgroup, 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. ~~In 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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* [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 907.3471{{c}}

: error map: {{val| 0.000 +0.905 -5.495 -2.702 }}

~~~~

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* WE: ~2 = 1199.9585{{c}}, ~42/25 = 907.4221{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.4521{{c}}

~~~~

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* WE: ~2 = 1199.4376{{c}}, ~22/13 = 907.1175{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 907.5314{{c}}

~~~~

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* WE: ~2 = 1201.2729{{c}}, ~42/25 = 908.1116{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.2109{{c}}

~~~~

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* WE: ~2 = 1201.4078{{c}}, ~42/25 = 908.1362{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.1370{{c}}

~~~~

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 53.9055{{c}}

: error map: {{val| 0.000 -1.183 -8.975 +1.474 }}

~~~~

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/49 = 731.7183{{c}}

: error map: {{val| 0.000 +0.692 -5.694 -2.742 }}

~~~~

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* WE: ~2 = 1200.8513{{c}}, ~75/49 = 732.1519{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~75/49 = 731.6740{{c}}

~~~~

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* WE: ~2 = 1199.7960{{c}}, ~20/13 = 731.6053{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 731.7208{{c}}

~~~~

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 323.7816{{c}}

: error map: {{val| 0.000 +2.332 -5.808 +2.519 }}

~~~~

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* WE: ~2 = 1199.9010{{c}}, ~6/5 = 323.7691{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 323.7931{{c}}

~~~~

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* WE: ~2 = 1200.0524{{c}}, ~6/5 = 323.8039{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 323.7912{{c}}

~~~~

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* [[CWE]]: ~10/9 = 171.4286{{c}}, ~3/2 = 705.6057{{c}}

: error map: {{val| 0.000 +3.651 -3.674 +0.071 }}

~~~~

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* WE: ~11/10 = 171.3859{{c}}, ~3/2 = 705.3421{{c}}

* CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.4973{{c}}

~~~~

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* WE: ~11/10 = 171.4163{{c}}, ~3/2 = 705.2930{{c}}

* CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.3402{{c}}

~~~~

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[[Category:Temperament collections]]

~~[[Category:Pages with mostly numerical content]]~~

[[Category:Keemic temperaments| ]]

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-This is a collection of [[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 [[Keemic family #Undecimal supermagic|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.
+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, 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 has a [[strong restriction]] to the 2.5/3.7/3 subgroup, 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. In 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 33: / Line 37: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 907.3471{{c}}
 : error map: {{val| 0.000 +0.905 -5.495 -2.702 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~42/25 = 907.290{{c}} -->
 {{Optimal ET sequence|legend=1| 4, …, 37, 41 }}
@@ Line 49: / Line 52: @@
 * WE: ~2 = 1199.9585{{c}}, ~42/25 = 907.4221{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.4521{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~42/25 = 907.453{{c}} -->
 {{Optimal ET sequence|legend=0| 4, 37, 41, 119 }}
@@ Line 65: / Line 67: @@
 * WE: ~2 = 1199.4376{{c}}, ~22/13 = 907.1175{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~22/13 = 907.5314{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~22/13 = 907.543{{c}} -->
 {{Optimal ET sequence|legend=0| 4, 37, 41, 78, 119f }}
@@ Line 81: / Line 82: @@
 * WE: ~2 = 1201.2729{{c}}, ~42/25 = 908.1116{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.2109{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~42/25 = 907.149{{c}} -->
 {{Optimal ET sequence|legend=0| 41, 127cd, 168cd }}
@@ Line 97: / Line 97: @@
 * WE: ~2 = 1201.4078{{c}}, ~42/25 = 908.1362{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.1370{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~22/13 = 907.072{{c}} -->
 {{Optimal ET sequence|legend=0| 41, 86ce }}
@@ Line 122: / Line 121: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 53.9055{{c}}
 : error map: {{val| 0.000 -1.183 -8.975 +1.474 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~36/35 = 53.816{{c}} -->
 {{Optimal ET sequence|legend=1| 22, 45, 67c }}
@@ Line 141: / Line 139: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~75/49 = 731.7183{{c}}
 : error map: {{val| 0.000 +0.692 -5.694 -2.742 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~75/49 = 731.669{{c}} -->
 {{Optimal ET sequence|legend=0| 18, 23d, 41 }}
@@ Line 157: / Line 154: @@
 * WE: ~2 = 1200.8513{{c}}, ~75/49 = 732.1519{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~75/49 = 731.6740{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~75/49 = 731.633{{c}} -->
 {{Optimal ET sequence|legend=0| 18e, 23de, 41 }}
@@ Line 173: / Line 169: @@
 * WE: ~2 = 1199.7960{{c}}, ~20/13 = 731.6053{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~20/13 = 731.7208{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~20/13 = 731.730{{c}} -->
 {{Optimal ET sequence|legend=0| 18e, 23de, 41 }}
@@ Line 192: / Line 187: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 323.7816{{c}}
 : error map: {{val| 0.000 +2.332 -5.808 +2.519 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 323.780{{c}} -->
 {{Optimal ET sequence|legend=1| 26, 37, 63 }}
@@ Line 208: / Line 202: @@
 * WE: ~2 = 1199.9010{{c}}, ~6/5 = 323.7691{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 323.7931{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 323.796{{c}} -->
 {{Optimal ET sequence|legend=0| 26, 37, 63 }}
@@ Line 224: / Line 217: @@
 * WE: ~2 = 1200.0524{{c}}, ~6/5 = 323.8039{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 323.7912{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 323.790{{c}} -->
 {{Optimal ET sequence|legend=0| 26, 37, 63 }}
@@ Line 247: / Line 239: @@
 * [[CWE]]: ~10/9 = 171.4286{{c}}, ~3/2 = 705.6057{{c}}
 : error map: {{val| 0.000 +3.651 -3.674 +0.071 }}
-<!-- * [[POTE]]: ~10/9 = 171.429{{c}}, ~3/2 = 705.613{{c}} -->
 {{Optimal ET sequence|legend=1| 7, …, 56, 63, 119 }}
@@ Line 263: / Line 254: @@
 * WE: ~11/10 = 171.3859{{c}}, ~3/2 = 705.3421{{c}}
 * CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.4973{{c}}
-<!-- * POTE: ~11/10 = 171.429{{c}}, ~3/2 = 705.518{{c}} -->
 {{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}
@@ Line 279: / Line 269: @@
 * WE: ~11/10 = 171.4163{{c}}, ~3/2 = 705.2930{{c}}
 * CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.3402{{c}}
-<!-- * POTE: ~10/9 = 171.429{{c}}, ~3/2 = 705.344{{c}} -->
 {{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}
@@ Line 286: / Line 275: @@
 [[Category:Temperament collections]]
-[[Category:Pages with mostly numerical content]]
 [[Category:Keemic temperaments| ]] <!-- main article -->
 [[Category:Rank 2]]