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Keemic temperaments: Difference between revisions

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{{Technical data page}}
These temper out the keema, {{monzo| -5 -3 3 1 }} = [[875/864]]. Keemic temperaments include [[Magic family #Magic|magic]], [[Kleismic family #Keemun|keemun]], [[Meantone family #Flattone|flattone]], [[Porcupine family #Porcupine|porcupine]], [[Jubilismic clan #Doublewide|doublewide]], [[Gamelismic clan #Superkleismic|superkleismic]], and [[Sycamore family #Sycamore|sycamore]]. Discussed below are quasitemp and barbad.
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.


= Quasitemp =
Full [[7-limit]] keemic temperaments discussed elsewhere are:
Comma: 6103515625/5804752896
* [[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]]


POTE generator: ~3125/2592 = 292.702
Discussed below are quasitemp, chromo, barbad, hyperkleismic, and sevond.


Map: [&lt;1 5 5|, &lt;0 -14 -11|]
== Quasitemp ==
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Quasitemp]].''


EDOs: {{EDOs| 4, 37, 41}}
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.


Badness: 0.7678
[[Subgroup]]: 2.3.5.7


== 7-limit ==
[[Comma list]]: 875/864, 2401/2400
Commas: 875/864, 2401/2400


POTE generator ~25/21 = 292.710
{{Mapping|legend=1| 1 5 5 5 | 0 -14 -11 -9 }}


Map: [&lt;1 5 5 5|, &lt;0 -14 -11 -9|]
: Mapping generators: ~2, ~25/21


Wedgie: &lt;&lt;14 11 9 -15 -25 -10||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 292.710


EDOs: {{EDOs| 4, 37, 41 }}
{{Optimal ET sequence|legend=1| 4, 37, 41 }}


Badness: 0.0603
[[Badness]]: 0.060269


== 11-limit ==
=== 11-limit ===
Commas: 100/99, 385/384, 1375/1372
Subgroup: 2.3.5.7.11


POTE generator: ~25/21 = 292.547
Comma list: 100/99, 385/384, 1375/1372


Map: [&lt;1 5 5 5 2|, &lt;0 -14 -11 -9 6|]
Mapping: {{mapping| 1 5 5 5 2 | 0 -14 -11 -9 6 }}


EDOs: {{EDOs| 4, 37, 41, 119 }}
Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.547


Badness: 0.0432
{{Optimal ET sequence|legend=1| 4, 37, 41, 119 }}
 
Badness: 0.043209
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 196/195, 275/273, 385/384
 
Mapping: {{mapping| 1 5 5 5 2 2 | 0 -14 -11 -9 6 7 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 292.457
 
{{Optimal ET sequence|legend=1| 4, 37, 41, 78, 119f }}
 
Badness: 0.032913
 
=== Quato ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 625/616
 
Mapping: {{mapping| 1 5 5 5 12 | 0 -14 -11 -9 -35 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.851
 
{{Optimal ET sequence|legend=1| 41, 127cd, 168cd }}
 
Badness: 0.041170
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 243/242, 275/273, 325/324
 
Mapping: {{mapping| 1 5 5 5 12 12 | 0 -14 -11 -9 -35 -34 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 292.928
 
{{Optimal ET sequence|legend=1| 41, 86ce, 127cd }}
 
Badness: 0.030081
 
== Chromo ==
: ''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
 
[[Comma list]]: 875/864, 2430/2401
 
{{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 ET sequence|legend=1| 22, 45, 67c }}
 
[[Badness]]: 0.090769
 
== Barbad ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 875/864, 16875/16807
 
{{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 ET sequence|legend=1| 18, 23d, 41 }}
 
[[Badness]]: 0.110448
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/242, 540/539, 625/616
 
Mapping: {{mapping| 1 9 7 11 14 | 0 -19 -12 -21 -27 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~98/75 = 468.367
 
{{Optimal ET sequence|legend=1| 18e, 23de, 41, 228ccdd }}
 
Badness: 0.050105


=== 13-limit ===
=== 13-limit ===
Commas: 100/99, 196/195, 275/273, 385/384
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 196/195, 245/242, 275/273
 
Mapping: {{mapping| 1 9 7 11 14 8 | 0 -19 -12 -21 -27 -11 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 468.270
 
{{Optimal ET sequence|legend=1| 18e, 23de, 41 }}
 
Badness: 0.039183
 
== Hyperkleismic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 875/864, 51200/50421
 
{{Mapping|legend=1| 1 -3 -2 2 | 0 17 16 3 }}
 
: Mapping generators: ~2, ~6/5


POTE generator: ~13/11 = 292.457
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 323.780


Map: [&lt;1 5 5 5 2 2|, &lt;0 -14 -11 -9 6 7|]
{{Optimal ET sequence|legend=1| 26, 37, 63 }}


EDOs: {{EDOs| 4, 37, 41, 78, 119f }}
[[Badness]]: 0.157830


Badness: 0.0329
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Quato ==
Comma list: 100/99, 385/384, 2420/2401
Commas: 243/242, 441/440, 625/616


POTE generator: ~25/21 = 292.851
Mapping: {{mapping| 1 -3 -2 2 4 | 0 17 16 3 -2}}


Map: [&lt;1 5 5 5 12|, &lt;0 -14 -11 -9 -35|]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.796


EDOs: {{EDOs| 41, 127cd, 168cd }}
{{Optimal ET sequence|legend=1| 26, 37, 63 }}


Badness: 0.0412
Badness: 0.065356


=== 13-limit ===
=== 13-limit ===
Commas: 105/104, 243/242, 275/273, 325/324
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 169/168, 275/273, 385/384
 
Mapping: {{mapping| 1 -3 -2 2 4 1 | 0 17 16 3 -2 10 }}


POTE generator: ~13/11 = 292.928
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.790


Map: [&lt;1 5 5 5 12 12|, &lt;0 -14 -11 -9 -35 -34|]
{{Optimal ET sequence|legend=1| 26, 37, 63 }}


EDOs: {{EDOs| 41, 86ce, 127cd }}
Badness: 0.035724


Badness: 0.0301
== Sevond ==
10/9 is tempered to be exactly 1\7 of an octave. Therefore 3/2 is 1 generator sharp of a 7edo step and 5/4 is 2 generators sharp.


= Barbad =
[[Subgroup]]: 2.3.5.7
Commas: 875/864, 16875/16807


POTE generator: ~75/49 = 468.331
[[Comma list]]: 875/864, 327680/321489


Map: [&lt;1 9 7 11|, &lt;0 -19 -12 -21|]
{{Mapping|legend=1| 7 0 -6 53 | 0 1 2 -3 }}


Wedgie: &lt;&lt;19 12 21 -25 -20 15||
: Mapping generators: ~10/9, ~3


EDOs: {{EDOs| 18, 23d, 41 }}
[[Optimal tuning]] ([[POTE]]): ~10/9 = 1\7, ~3/2 = 705.613


Badness: 0.1104
{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}


== 11-limit ==
[[Badness]]: 0.206592
Commas: 245/242, 540/539, 625/616


POTE generator: ~98/75 = 468.367
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 9 7 11 14|, &lt;0 -19 -12 -21 -27|]
Comma list: 100/99, 385/384, 6655/6561


EDOs: {{EDOs| 18e, 23de, 41, 228ccdd }}
Mapping: {{mapping| 7 0 -6 53 2 | 0 1 2 -3 2 }}


Badness: 0.0501
Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.518
 
{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
 
Badness: 0.070437
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 13-limit ==
Comma list: 100/99, 169/168, 352/351, 385/384
Commas: 144/143, 196/195, 245/242, 275/273


POTE generator: ~13/10 = 468.270
Mapping: {{mapping| 7 0 -6 53 2 37 | 0 1 2 -3 2 -1 }}


Map: [&lt;1 9 7 11 14 8|, &lt;0 -19 -12 -21 -27 -11|]
Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.344


EDOs: {{EDOs| 18e, 23de, 41 }}
{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}


Badness: 0.0392
Badness: 0.041238


[[Category:Theory]]
[[Category:Temperament collections]]
[[Category:Temperament]]
[[Category:Pages with mostly numerical content]]
[[Category:Keemic]]
[[Category:Keemic temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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