Keemic temperaments: Difference between revisions
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These temper out the keema, {{monzo| -5 -3 3 1 }} = [[875/864]]. Keemic temperaments | {{Technical data page}} | ||
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. | |||
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 == | == Quasitemp == | ||
: ''For the 5-limit version of this temperament, see [[ | : ''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 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 1 5 5 5 | 0 -14 -11 -9 }} | {{Mapping|legend=1| 1 5 5 5 | 0 -14 -11 -9 }} | ||
: | : Mapping generators: ~2, ~25/21 | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 292.710 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 292.710 | ||
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== Chromo == | == Chromo == | ||
: ''For the 5-limit version of this temperament, see [[ | : ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Chromo]].'' | ||
Chromo represents the [[13edf]] chain as a rank-2 temperament, with [[6/5]] and [[5/4]] mapped to 6 and 7 steps, respectively. Since the difference of those two intervals is abbreviated considerably from just, keemic provides the most meaningful 7-limit extension (setting [[7/6]], 6/5, 5/4, [[9/7]] equidistant) so that the temperament then approximates the [[4:5:6:7]] tetrad with 0:7:13:18 generator steps. | |||
Note that if one allows a more complex mapping for prime 7 and wants a larger prime limit, one may prefer [[escapade]]. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 1 1 2 2 | 0 13 7 18 }} | {{Mapping|legend=1| 1 1 2 2 | 0 13 7 18 }} | ||
: | : Mapping generators: ~2, ~25/24 | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/24 = 53.816 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/24 = 53.816 | ||
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[[Badness]]: 0.090769 | [[Badness]]: 0.090769 | ||
== Barbad == | == Barbad == | ||
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{{Mapping|legend=1| 1 9 7 11 | 0 -19 -12 -21 }} | {{Mapping|legend=1| 1 9 7 11 | 0 -19 -12 -21 }} | ||
: | : Mapping generators: ~2, ~98/75 | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98/75 = 468.331 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98/75 = 468.331 | ||
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{{Mapping|legend=1| 1 -3 -2 2 | 0 17 16 3 }} | {{Mapping|legend=1| 1 -3 -2 2 | 0 17 16 3 }} | ||
: | : Mapping generators: ~2, ~6/5 | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 323.780 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 323.780 | ||
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{{Mapping|legend=1| 7 0 -6 53 | 0 1 2 -3 }} | {{Mapping|legend=1| 7 0 -6 53 | 0 1 2 -3 }} | ||
: | : Mapping generators: ~10/9, ~3 | ||
[[Optimal tuning]] ([[POTE]]): ~10/9 = 1\7, ~3/2 = 705.613 | [[Optimal tuning]] ([[POTE]]): ~10/9 = 1\7, ~3/2 = 705.613 | ||
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[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||
[[Category:Pages with mostly numerical content]] | |||
[[Category:Keemic temperaments| ]] <!-- main article --> | [[Category:Keemic temperaments| ]] <!-- main article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||