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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 [[Jubilismic clan #Doublewide|doublewide]], [[Meantone family #Flattone|flattone]], [[Porcupine family #Porcupine|porcupine]], [[Gamelismic clan #Superkleismic|superkleismic]], [[Magic family #Magic|magic]], [[Kleismic family #Keemun|keemun]], 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:
== 5-limit ==
* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
Comma list: 6103515625/5804752896
* ''[[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.


Mapping: [{{val| 1 5 5 }}, {{val| 0 -14 -11 }}]
== Quasitemp ==
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Quasitemp]].''


{{Val list|legend=1| 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
Comma list: 875/864, 2401/2400


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


Mapping: [{{val| 1 5 5 5 }}, {{val| 0 -14 -11 -9 }}]
: Mapping generators: ~2, ~25/21


{{Multival|legend=1| 14 11 9 -15 -25 -10 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 292.710


{{Val list|legend=1| 4, …, 37, 41 }}
{{Optimal ET sequence|legend=1| 4, 37, 41 }}


Badness: 0.0603
[[Badness]]: 0.060269
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 100/99, 385/384, 1375/1372
Comma list: 100/99, 385/384, 1375/1372


POTE generator: ~25/21 = 292.547
Mapping: {{mapping| 1 5 5 5 2 | 0 -14 -11 -9 6 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.547


Mapping: [{{val| 1 5 5 5 2 }}, {{val| 0 -14 -11 -9 6 }}]
{{Optimal ET sequence|legend=1| 4, 37, 41, 119 }}


{{Val list|legend=1| 37, 41, 119 }}
Badness: 0.043209


Badness: 0.0432
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 13-limit ===
Comma list: 100/99, 196/195, 275/273, 385/384
Comma list: 100/99, 196/195, 275/273, 385/384


POTE generator: ~13/11 = 292.457
Mapping: {{mapping| 1 5 5 5 2 2 | 0 -14 -11 -9 6 7 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 292.457


Mapping: [{{val| 1 5 5 5 2 2 }}, {{val| 0 -14 -11 -9 6 7 }}]
{{Optimal ET sequence|legend=1| 4, 37, 41, 78, 119f }}


{{Val list|legend=1| 37, 41, 78, 119f }}
Badness: 0.032913


Badness: 0.0329
=== Quato ===
Subgroup: 2.3.5.7.11


== Quato ==
Comma list: 243/242, 441/440, 625/616
Comma list: 243/242, 441/440, 625/616


POTE generator: ~25/21 = 292.851
Mapping: {{mapping| 1 5 5 5 12 | 0 -14 -11 -9 -35 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.851


Mapping: [{{val| 1 5 5 5 12 }}, {{val| 0 -14 -11 -9 -35 }}]
{{Optimal ET sequence|legend=1| 41, 127cd, 168cd }}


{{Val list|legend=1| 41, 127cd, 168cd }}
Badness: 0.041170


Badness: 0.0412
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 13-limit ===
Comma list: 105/104, 243/242, 275/273, 325/324
Comma list: 105/104, 243/242, 275/273, 325/324


POTE generator: ~13/11 = 292.928
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 }}


Mapping: [{{val| 1 5 5 5 12 12 }}, {{val| 0 -14 -11 -9 -35 -34 }}]
[[Badness]]: 0.090769


{{Val list|legend=1| 41, 86ce, 127cd }}
== Barbad ==
[[Subgroup]]: 2.3.5.7


Badness: 0.0301
[[Comma list]]: 875/864, 16875/16807


= Barbad =
{{Mapping|legend=1| 1 9 7 11 | 0 -19 -12 -21 }}
Comma list: 875/864, 16875/16807


POTE generator: ~75/49 = 468.331
: Mapping generators: ~2, ~98/75


Mapping: [{{val| 1 9 7 11 }}, {{val| 0 -19 -12 -21 }}]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98/75 = 468.331


{{Multival|legend=1| 19 12 21 -25 -20 15 }}
{{Optimal ET sequence|legend=1| 18, 23d, 41 }}


{{Val list|legend=1| 18, 23d, 41 }}
[[Badness]]: 0.110448


Badness: 0.1104
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 245/242, 540/539, 625/616
Comma list: 245/242, 540/539, 625/616


POTE generator: ~98/75 = 468.367
Mapping: {{mapping| 1 9 7 11 14 | 0 -19 -12 -21 -27 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~98/75 = 468.367


Mapping: [{{val| 1 9 7 11 14 }}, {{val| 0 -19 -12 -21 -27 }}]
{{Optimal ET sequence|legend=1| 18e, 23de, 41, 228ccdd }}


{{Val list|legend=1| 18e, 23de, 41, 228ccdd }}
Badness: 0.050105


Badness: 0.0501
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 13-limit ==
Comma list: 144/143, 196/195, 245/242, 275/273
Comma list: 144/143, 196/195, 245/242, 275/273


POTE generator: ~13/10 = 468.270
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
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 323.780
 
{{Optimal ET sequence|legend=1| 26, 37, 63 }}
 
[[Badness]]: 0.157830
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 385/384, 2420/2401
 
Mapping: {{mapping| 1 -3 -2 2 4 | 0 17 16 3 -2}}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.796
 
{{Optimal ET sequence|legend=1| 26, 37, 63 }}
 
Badness: 0.065356
 
=== 13-limit ===
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 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.790
 
{{Optimal ET sequence|legend=1| 26, 37, 63 }}
 
Badness: 0.035724
 
== 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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 875/864, 327680/321489
 
{{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 ET sequence|legend=1| 7, 56, 63, 119 }}
 
[[Badness]]: 0.206592
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 385/384, 6655/6561
 
Mapping: {{mapping| 7 0 -6 53 2 | 0 1 2 -3 2 }}
 
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
 
Comma list: 100/99, 169/168, 352/351, 385/384
 
Mapping: {{mapping| 7 0 -6 53 2 37 | 0 1 2 -3 2 -1 }}


Mapping: [{{val| 1 9 7 11 14 8 }}, {{val| 0 -19 -12 -21 -27 -11 }}]
Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.344


{{Val list|legend=1| 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]]
Retrieved from "https://en.xen.wiki/w/Keemic_temperaments"