@@ Line 4: / Line 4: @@
 Since {{nowrap|(10/9)<sup>4</sup> {{=}} (4375/4374)⋅(32/21) }}, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have {{nowrap| 7/6 {{=}} (4375/4374)⋅(27/25)<sup>2</sup> }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
-Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, ragitritonic, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
+Temperaments discussed elsewhere are:
 * ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
 * ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
@@ Line 17: / Line 17: @@
 * [[Ennealimmal]] (+2401/2400) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
 * ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]]
+* [[Parakleismic]] (+3136/3125) → [[Parakleismic family #Septimal parakleismic|Parakleismic family]]
 * [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]]
 * [[Pontiac]] (+32805/32768) → [[Schismatic family #Pontiac|Schismatic family]]
@@ Line 30: / Line 31: @@
 * ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]]
 * ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments #Dzelic|37th-octave temperaments]]
+Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, ragitritonic, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, counterkleismic, quincy, sfourth, and trideci.
 == Supermajor ==
@@ Line 1,384: / Line 1,387: @@
 Badness (Sintel): 1.44
-== Parakleismic ==
-{{Main| Parakleismic }}
-: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic (5-limit)]].''
-In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat [[6/5]], 13 of which give 32/3, and 14 give 64/5. While 118 no longer has better than a cent of accuracy in the 7-limit, it is a decent temperament there nonetheless, and this allows an extension adding [[3136/3125]] and 4375/4374, for which [[99edo]], 118edo, and especially [[217edo]] are accurate tunings.
-Parakleismic does not extend easily to the 11- or 13-limit. Possible 11-limit extensions include undecimal parakleismic (99 & 118), paralytic (99e & 118), parkleismic (80 & 99), and paradigmic (80 & 99e).
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 3136/3125, 4375/4374
-{{Mapping|legend=1| 1 -8 -8 -23 | 0 13 14 35 }}
-: mapping generators: ~2, ~5/3
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.7820{{c}}, ~5/3 = 884.6581{{c}}
-: [[error map]]: {{val| -0.218 +0.344 +0.644 -0.779 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8088{{c}}
-: error map: {{val| 0.000 +0.560 +1.010 -0.516 }}
-{{Optimal ET sequence|legend=1| 19, 61d, 80, 99, 217, 316, 415 }}
-[[Badness]] (Sintel): 0.694
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 385/384, 3136/3125, 4375/4374
-Mapping: {{mapping| 1 -8 -8 -23 30 | 0 13 14 35 -36 }}
-Optimal tunings:
-* WE: ~2 = 1200.3296{{c}}, ~5/3 = 884.9921{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7519{{c}}
-{{Optimal ET sequence|legend=0| 19, 99, 118 }}
-Badness (Sintel): 1.64
-=== Paralytic ===
-Paralytic (99e & 118) tempers out [[441/440]], [[5632/5625]], and [[19712/19683]]. In 13-limit, 118 & 217 tempers out 1001/1000, 1575/1573, and 3584/3575.
-Subgroup: 2.3.5.7.11
-Comma list: 441/440, 3136/3125, 4375/4374
-Mapping: {{mapping| 1 -8 -8 -23 -57 | 0 13 14 35 82 }}
-Optimal tunings:
-* WE: ~2 = 1199.9940{{c}}, ~5/3 = 884.7757{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7800{{c}}
-{{Optimal ET sequence|legend=0| 19e, …, 99e, 118, 217, 335 }}
-Badness (Sintel): 1.19
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
-Mapping: {{mapping| 1 -8 -8 -23 -57 59 | 0 13 14 35 82 -75 }}
-Optimal tunings:
-* WE: ~2 = 1199.9218{{c}}, ~5/3 = 884.7285{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7858{{c}}
-{{Optimal ET sequence|legend=0| 99e, 118, 217 }}
-Badness (Sintel): 1.85
-==== Paraklein ====
-Paraklein (19e & 118) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]].
-Subgroup: 2.3.5.7.11.13
-Comma list: 196/195, 352/351, 625/624, 729/728
-Mapping: {{mapping| 1 -8 -8 -23 -57 -28 | 0 13 14 35 82 43 }}
-Optimal tunings:
-* WE: ~2 = 1199.8239{{c}}, ~5/3 = 884.6449{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7709{{c}}
-{{Optimal ET sequence|legend=0| 19e, …, 99ef, 118 }}
-Badness (Sintel): 1.55
-=== Parkleismic ===
-Subgroup: 2.3.5.7.11
-Comma list: 176/175, 1375/1372, 2200/2187
-Mapping: {{mapping| 1 -8 -8 -23 -43 | 0 13 14 35 63 }}
-Optimal tunings:
-* WE: ~2 = 1199.1848{{c}}, ~5/3 = 884.3386{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9158{{c}}
-{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179, 259cd }}
-Badness (Sintel): 1.85
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 176/175, 325/324, 1375/1372
-Mapping: {{mapping| 1 -8 -8 -23 -43 -14 | 0 13 14 35 63 24 }}
-Optimal tunings:
-* WE: ~2 = 1199.5318{{c}}, ~5/3 = 884.5800{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9118{{c}}
-{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179 }}
-Badness (Sintel): 1.51
-=== Paradigmic ===
-Subgroup: 2.3.5.7.11
-Comma list: 540/539, 896/891, 3136/3125
-Mapping: {{mapping| 1 -8 -8 -23 16 | 0 13 14 35 -17 }}
-Optimal tunings:
-* WE: ~2 = 1199.0616{{c}}, ~5/3 = 884.2124{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.8877{{c}}
-{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e, 179e, 457bcddeeee }}
-Badness (Sintel): 1.38
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 325/324, 540/539, 832/825
-Mapping: {{mapping| 1 -8 -8 -23 16 -14 | 0 13 14 35 -17 24 }}
-Optimal tunings:
-* WE: ~2 = 1199.2683{{c}}, ~5/3 = 884.3805{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9061{{c}}
-{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e }}
-Badness (Sintel): 1.48
-=== Semiparakleismic ===
-Subgroup: 2.3.5.7.11
-Comma list: 3025/3024, 3136/3125, 4375/4374
-Mapping: {{mapping| 2 -3 -2 -11 -4 | 0 13 14 35 23 }}
-: mapping generators: ~99/70, ~33/28
-Optimal tunings:
-* WE: ~99/70 = 599.9270{{c}}, ~33/28 = 284.7841{{c}}
-* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8119{{c}}
-{{Optimal ET sequence|legend=0| 80, 118, 198, 316, 514c }}
-Badness (Sintel): 1.13
-==== Semiparamint ====
-This extension was named ''semiparakleismic'' in the earlier materials.
-Subgroup: 2.3.5.7.11.13
-Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
-Mapping: {{mapping| 2 -3 -2 -11 -4 15 | 0 13 14 35 23 -16 }}
-Optimal tunings:
-* WE: ~99/70 = 599.8253{{c}}, ~33/28 = 284.7608{{c}}
-* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8366{{c}}
-{{Optimal ET sequence|legend=0| 80, 118, 198 }}
-Badness (Sintel): 1.40
-==== Semiparawolf ====
-This extension was named ''gentsemiparakleismic'' in the earlier materials.
-Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 325/324, 364/363, 3136/3125
-Mapping: {{mapping| 2 -3 -2 -11 -4 -4 | 0 13 14 35 23 24 }}
-Optimal tunings:
-* WE: ~99/70 = 600.0569{{c}}, ~13/11 = 284.8431{{c}}
-* CWE: ~99/70 = 600.0000{{c}}, ~13/11 = 284.8216{{c}}
-{{Optimal ET sequence|legend=0| 80, 118f, 198f }}
-Badness (Sintel): 1.67
 == Counterkleismic ==
 : ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].''
-In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses]] ((648/625)<sup>6</sup>) fall short of the [[5/4|classic major third (5/4)]]. It can be described as {{nowrap| 19 & 224 }} temperament, tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). It was named by analogy to [[catakleismic]] and parakleismic)
+In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses]] ((648/625)<sup>6</sup>) fall short of the [[5/4|classic major third (5/4)]]. It can be described as {{nowrap| 19 & 224 }} temperament, tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). It was named by analogy to [[catakleismic]] and [[parakleismic]]).
 [[Subgroup]]: 2.3.5.7

Ragismic microtemperaments: Difference between revisions