@@ 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.
+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 [[High badness temperaments #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
+[[Subgroup]]: 2.3.5.7
 [[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
 {{Optimal ET sequence|legend=1| 4, 37, 41 }}
@@ Line 23: / Line 38: @@
 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
 {{Optimal ET sequence|legend=1| 4, 37, 41, 119 }}
@@ Line 36: / Line 51: @@
 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
 {{Optimal ET sequence|legend=1| 4, 37, 41, 78, 119f }}
@@ Line 51: / Line 64: @@
 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
 {{Optimal ET sequence|legend=1| 41, 127cd, 168cd }}
@@ Line 64: / Line 77: @@
 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
 {{Optimal ET sequence|legend=1| 41, 86ce, 127cd }}
@@ Line 73: / Line 86: @@
 == Chromo ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #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
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 875/864, 2430/2401
-[[Mapping]]: [{{Val|1 1 2 2}}, {{Val|0 13 7 18}}]
+{{Mapping|legend=1| 1 1 2 2 | 0 13 7 18 }}
+: Mapping generators: ~2, ~25/24
-[[POTE generator]]: ~25/24 = 53.816
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/24 = 53.816
 {{Optimal ET sequence|legend=1| 22, 45, 67c }}
 [[Badness]]: 0.090769
-== Undeka ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Undeka]].''
-Subgroup: 2.3.5.7
-[[Comma list]]: 875/864, 3200/3087
-[[Mapping]]: [{{Val|11 0 8 31}}, {{Val|0 1 1 0}}]
-{{Multival|legend=1|11 11 0 -8 -31 -31}}
-[[POTE generator]]: ~3/2 = 708.792
-{{Optimal ET sequence|legend=1| 11c, 22 }}
-[[Badness]]: 0.141782
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 100/99, 352/343, 385/384
-Mapping: [{{Val|11 0 8 31 38}}, {{Val|0 1 1 0 0}}]
-POTE generator: ~3/2 = 706.768
-{{Optimal ET sequence|legend=1| 11c, 22 }}
-Badness: 0.068672
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 65/63, 100/99, 169/165, 352/343
-Mapping: [{{Val|11 0 8 31 38 23}}, {{Val|0 1 1 0 0 1}}]
-POTE generator: ~3/2 = 707.764
-{{Optimal ET sequence|legend=1| 11cf, 22 }}
-Badness: 0.056528
 == Barbad ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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
 {{Optimal ET sequence|legend=1| 18, 23d, 41 }}
@@ Line 150: / Line 125: @@
 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
 {{Optimal ET sequence|legend=1| 18e, 23de, 41, 228ccdd }}
@@ Line 163: / Line 138: @@
 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
 {{Optimal ET sequence|legend=1| 18e, 23de, 41 }}
@@ Line 172: / Line 147: @@
 == Hyperkleismic ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 875/864, 51200/50421
-[[Mapping]]: [{{val|1 -3 -2 2}}, {{val|0 17 16 3}}]
+{{Mapping|legend=1| 1 -3 -2 2 | 0 17 16 3 }}
-{{Multival|legend=1|17 16 3 -14 -43 -38}}
+: Mapping generators: ~2, ~6/5
-[[POTE generator]]: ~6/5 = 323.780
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 323.780
 {{Optimal ET sequence|legend=1| 26, 37, 63 }}
@@ Line 189: / Line 164: @@
 Subgroup: 2.3.5.7.11
-Comma list: 100/99, 385/384, 2420/2401<br>
+Comma list: 100/99, 385/384, 2420/2401
-Mapping: [{{val|1 -3 -2 2 4}}, {{val|0 17 16 3 -2}}]<br>
+Mapping: {{mapping| 1 -3 -2 2 4 | 0 17 16 3 -2}}
-POTE generator: ~6/5 = 323.796<br>
+Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.796
 {{Optimal ET sequence|legend=1| 26, 37, 63 }}
@@ Line 204: / Line 179: @@
 Comma list: 100/99, 169/168, 275/273, 385/384
-Mapping: [{{val|1 -3 -2 2 4 1}}, {{val|0 17 16 3 -2 10}}]
+Mapping: {{mapping| 1 -3 -2 2 4 1 | 0 17 16 3 -2 10 }}
-POTE generator: ~6/5 = 323.790
+Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.790
 {{Optimal ET sequence|legend=1| 26, 37, 63 }}
@@ Line 213: / Line 188: @@
 == Sevond ==
-Subgroup: 2.3.5.7
+/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]]: [{{val| 7 0 -6 53 }}, {{val| 0 1 2 -3 }}]
+{{Mapping|legend=1| 7 0 -6 53 | 0 1 2 -3 }}
+: Mapping generators: ~10/9, ~3
-[[POTE generator]]: ~3/2 = 705.613
+[[Optimal tuning]] ([[POTE]]): ~10/9 = 1\7, ~3/2 = 705.613
 {{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
@@ Line 230: / Line 209: @@
 Comma list: 100/99, 385/384, 6655/6561
-Mapping: [{{val| 7 0 -6 53 2 }}, {{val| 0 1 2 -3 2 }}]
+Mapping: {{mapping| 7 0 -6 53 2 | 0 1 2 -3 2 }}
-POTE generator: ~3/2 = 705.518
+Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.518
 {{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
@@ Line 243: / Line 222: @@
 Comma list: 100/99, 169/168, 352/351, 385/384
-Mapping: [{{val| 7 0 -6 53 2 37 }}, {{val| 0 1 2 -3 2 -1 }}]
+Mapping: {{mapping| 7 0 -6 53 2 37 | 0 1 2 -3 2 -1 }}
-POTE generator: ~3/2 = 705.344
+Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.344
 {{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
@@ Line 252: / Line 231: @@
 [[Category:Temperament collections]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Keemic temperaments| ]] <!-- main article -->
 [[Category:Rank 2]]

Keemic temperaments: Difference between revisions