Xenharmonic Wiki

Keemic temperaments: Difference between revisions

← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled
4,516 edits
mNo edit summary
 
(28 intermediate revisions by 9 users not shown)
Line 1: Line 1:
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.
{{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.


== Quasitemp ==
Full [[7-limit]] keemic temperaments discussed elsewhere are:
Subgroup: 2.3.5
* [[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]]


[[Comma list]]: 6103515625/5804752896
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]].''
[[POTE generator]]: ~3125/2592 = 292.702
 
{{Val list|legend=1| 4, 37, 41 }}


[[Badness]]: 0.767824
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.


=== 7-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 875/864, 2401/2400
[[Comma list]]: 875/864, 2401/2400


[[Mapping]]: [{{val| 1 5 5 5 }}, {{val| 0 -14 -11 -9 }}]
{{Mapping|legend=1| 1 5 5 5 | 0 -14 -11 -9 }}


{{Multival|legend=1| 14 11 9 -15 -25 -10 }}
: Mapping generators: ~2, ~25/21


[[POTE generator]]: ~25/21 = 292.710
[[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]]: 0.060269
[[Badness]]: 0.060269
Line 34: Line 38:
Comma list: 100/99, 385/384, 1375/1372
Comma list: 100/99, 385/384, 1375/1372


Mapping: [{{val| 1 5 5 5 2 }}, {{val| 0 -14 -11 -9 6 }}]
Mapping: {{mapping| 1 5 5 5 2 | 0 -14 -11 -9 6 }}


POTE generator: ~25/21 = 292.547
Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.547


Vals: {{Val list| 4, 37, 41, 119 }}
{{Optimal ET sequence|legend=1| 4, 37, 41, 119 }}


Badness: 0.043209
Badness: 0.043209
Line 47: Line 51:
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 }}
 
Mapping: [{{val| 1 5 5 5 2 2 }}, {{val| 0 -14 -11 -9 6 7 }}]


POTE generator: ~13/11 = 292.457
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 292.457


Vals: {{Val list| 4, 37, 41, 78, 119f }}
{{Optimal ET sequence|legend=1| 4, 37, 41, 78, 119f }}


Badness: 0.032913
Badness: 0.032913
Line 62: Line 64:
Comma list: 243/242, 441/440, 625/616
Comma list: 243/242, 441/440, 625/616


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


POTE generator: ~25/21 = 292.851
Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.851


Vals: {{Val list| 41, 127cd, 168cd }}
{{Optimal ET sequence|legend=1| 41, 127cd, 168cd }}


Badness: 0.041170
Badness: 0.041170
Line 75: Line 77:
Comma list: 105/104, 243/242, 275/273, 325/324
Comma list: 105/104, 243/242, 275/273, 325/324


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


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


Vals: {{Val list| 41, 86ce, 127cd }}
{{Optimal ET sequence|legend=1| 41, 86ce, 127cd }}


Badness: 0.030081
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 ==
== Barbad ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 9 7 11 }}, {{val| 0 -19 -12 -21 }}]
{{Mapping|legend=1| 1 9 7 11 | 0 -19 -12 -21 }}


{{Multival|legend=1| 19 12 21 -25 -20 15 }}
: Mapping generators: ~2, ~98/75


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


{{Val list|legend=1| 18, 23d, 41 }}
{{Optimal ET sequence|legend=1| 18, 23d, 41 }}


[[Badness]]: 0.110448
[[Badness]]: 0.110448
Line 103: Line 125:
Comma list: 245/242, 540/539, 625/616
Comma list: 245/242, 540/539, 625/616


Mapping: [{{val| 1 9 7 11 14 }}, {{val| 0 -19 -12 -21 -27 }}]
Mapping: {{mapping| 1 9 7 11 14 | 0 -19 -12 -21 -27 }}


POTE generator: ~98/75 = 468.367
Optimal tuning (POTE): ~2 = 1\1, ~98/75 = 468.367


Vals: {{Val list| 18e, 23de, 41, 228ccdd }}
{{Optimal ET sequence|legend=1| 18e, 23de, 41, 228ccdd }}


Badness: 0.050105
Badness: 0.050105
Line 116: Line 138:
Comma list: 144/143, 196/195, 245/242, 275/273
Comma list: 144/143, 196/195, 245/242, 275/273


Mapping: [{{val| 1 9 7 11 14 8 }}, {{val| 0 -19 -12 -21 -27 -11 }}]
Mapping: {{mapping| 1 9 7 11 14 8 | 0 -19 -12 -21 -27 -11 }}


POTE generator: ~13/10 = 468.270
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 468.270


Vals: {{Val list| 18e, 23de, 41 }}
{{Optimal ET sequence|legend=1| 18e, 23de, 41 }}


Badness: 0.039183
Badness: 0.039183


[[Category:Regular temperament theory]]
== Hyperkleismic ==
[[Category:Temperament collection]]
[[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 }}
 
Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.344
 
{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
 
Badness: 0.041238
 
[[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]]
Retrieved from "https://en.xen.wiki/w/Keemic_temperaments"