Keemic temperaments: Difference between revisions

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These temper out the keema, {{monzo| -5 -3 3 1 }} = [[875/864]]~~. Keemic temperaments include~~ [[~~Jubilismic clan #Doublewide|doublewide~~]], [[~~Meantone family #Flattone|flattone~~]], [[~~Porcupine family #Septimal porcupine|porcupine~~]], [[~~Gamelismic clan #Superkleismic|superkleismic~~]], [[~~Magic family #Septimal magic|magic~~]], [[~~Kleismic~~ family #~~Keemun~~|~~keemun~~]], [[~~11th~~-~~octave temperaments #Undeka|undeka~~]], and [[~~Sycamore family #Septimal sycamore|sycamore~~]]~~. Discussed below are quasitemp~~, ~~chromo, barbad, hyperkleismic, and sevond~~.

These temper out the keema, {{monzo| -5 -3 3 1 }} = [[875/864]] = {{S|5/S6}}, whose 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.

~~Tempering out the keema is one of the two main ways septimal harmony is organized in EDOs of medium size, alongside~~ [[~~myna~~]]~~. While myna makes the distance between 5~~/~~4 and 6~~/~~5 twice the distance between 9~~/~~7 and 5~~/~~4, keemic makes the two distances equal, resulting in the distance between the classical major and minor thirds being narrowed, or in other words~~ [[7/6]] - [[6/5]] - [[5/4]] - [[9/7]] ~~being made equidistant. EDOs with this structure include 15~~, 19, and ~~22 (though 15 tunes 9/7 very poorly)~~.

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

* [[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]]

* [[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.

== Quasitemp ==

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

Quasitemp is a full 7-limit strong extension of [[gariberttet]], the 2.5/3.7/3 subgroup temperament 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]] (which is equated to [[13/11]] in the 13-limit extension). 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.

[[Subgroup]]: 2.3.5.7

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: Mapping generators: ~2, ~25/21

~~{{Multival|legend=1| 14 11 9 -15 -25 -10 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 292.710

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: Mapping generators: ~2, ~98/75

~~{{Multival|legend=1| 19 12 21 -25 -20 15 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98/75 = 468.331

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: Mapping generators: ~2, ~6/5

~~{{Multival|legend=1| 17 16 3 -14 -43 -38 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 323.780

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[[Category:Temperament collections]]

[[Category:Pages with mostly numerical content]]

[[Category:Keemic temperaments| ]]

[[Category:Rank 2]]

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 {{Technical data page}}
-These temper out the keema, {{monzo| -5 -3 3 1 }} = [[875/864]]. Keemic temperaments include [[Jubilismic clan #Doublewide|doublewide]], [[Meantone family #Flattone|flattone]], [[Porcupine family #Septimal porcupine|porcupine]], [[Gamelismic clan #Superkleismic|superkleismic]], [[Magic family #Septimal magic|magic]], [[Kleismic family #Keemun|keemun]], [[11th-octave temperaments #Undeka|undeka]], and [[Sycamore family #Septimal sycamore|sycamore]]. Discussed below are quasitemp, chromo, barbad, hyperkleismic, and sevond.
+These temper out the keema, {{monzo| -5 -3 3 1 }} = [[875/864]] = {{S|5/S6}}, whose 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.
-Tempering out the keema is one of the two main ways septimal harmony is organized in EDOs of medium size, alongside [[myna]]. While myna makes the distance between 5/4 and 6/5 twice the distance between 9/7 and 5/4, keemic makes the two distances equal, resulting in the distance between the classical major and minor thirds being narrowed, or in other words [[7/6]] - [[6/5]] - [[5/4]] - [[9/7]] being made equidistant. EDOs with this structure include 15, 19, and 22 (though 15 tunes 9/7 very poorly).
+Full [[7-limit]] keemic temperaments discussed elsewhere are:
+* [[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]]
+* [[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.
 == Quasitemp ==
 : ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Quasitemp]].''
+Quasitemp is a full 7-limit strong extension of [[gariberttet]], the 2.5/3.7/3 subgroup temperament 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]] (which is equated to [[13/11]] in the 13-limit extension). 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.
 [[Subgroup]]: 2.3.5.7
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 : Mapping generators: ~2, ~25/21
-{{Multival|legend=1| 14 11 9 -15 -25 -10 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 292.710
@@ Line 103: / Line 113: @@
 : Mapping generators: ~2, ~98/75
-{{Multival|legend=1| 19 12 21 -25 -20 15 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98/75 = 468.331
@@ Line 146: / Line 154: @@
 : Mapping generators: ~2, ~6/5
-{{Multival|legend=1| 17 16 3 -14 -43 -38 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 323.780
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 [[Category:Temperament collections]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Keemic temperaments| ]] <!-- main article -->
 [[Category:Rank 2]]