Misty family: Difference between revisions

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 {{Main| Misty }}
-Misty features a third-octave period, which represents the [[512/405|ptolemaic diminished fourth (512/405)]]. Four [[4/3|perfect fourths]] (i.e. a minor sixth, ~[[128/81]]) minus a period give the interval class of 5. Misty has a [[ploidacot]] of triploid monocot, and equates the [[Pythagorean comma]] with a stack of three [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]] with {{nowrap| ''n'' {{=}} 3 }}.
+Misty features a third-octave period, which represents the [[512/405|ptolemaic diminished fourth (512/405)]]. Four [[4/3|perfect fourths]] (i.e. a minor sixth, [[~]][[128/81]]) minus a period give the interval class of 5. Misty has a [[ploidacot]] of triploid monocot, and equates the [[Pythagorean comma]] with a stack of three [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]] with {{nowrap| ''n'' {{=}} 3 }}.
 [[Subgroup]]: 2.3.5
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 == Septimal misty ==
 {{Main| Misty }}
+Septimal misty tempers out not only [[3136/3125]] and [[5120/5103]], but [[250047/250000]], mapping [[~]][[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 99 }}, and finds [[7/4]] by [[5/4]] plus six extra fourths octave reduced. [[99edo]] makes for an excellent tuning, and [[87edo]] and [[111edo]] are notable alternatives.
+It extends best to the no-11 no-13 [[19-limit]], where the ~[[16/15]] stands in for ~[[17/16]], and the ~[[25/21]], a fifth minus a period, stands in for ~[[19/16]]. The [[S-expression]]-based comma list for the 2.3.5.7.17-subgroup extension is {([[5120/5103|S8/S9]]), [[2025/2023|S15/S17]], [[256/255|S16]], [[1701/1700|S18/S20]], ([[2500/2499|S50]])}, and for the 2.3.5.7.17.19-subgroup extension, {([[5120/5103|S8/S9]]), [[256/255|S16]], [[324/323|S18]], [[400/399|S20]], [[1445/1444|S17/S19]], ([[2500/2499|S50]])}.
 === 7-limit ===
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 ==== 2.3.5.7.17-subgroup ====
-The [[S-expression]]-based comma list for this extension is {([[5120/5103|S8/S9]]), [[2025/2023|S15/S17]], [[256/255|S16]], [[1701/1700|S18/S20]], ([[2500/2499|S50]])}, and for the 2.3.5.7.17.19-subgroup extension, {([[5120/5103|S8/S9]]), [[256/255|S16]], [[324/323|S18]], [[400/399|S20]], [[1445/1444|S17/S19]], ([[2500/2499|S50]])}.
 Subgroup: 2.3.5.7.17
@@ Line 82: / Line 84: @@
 Badness (Sintel): 0.637
-=== Undecimal misty ===
+=== Murky ===
+Murky (87 & 99ef) tempers out [[441/440]] and [[896/891]], which joins misty with [[parapyth]]/[[pele]].
+This extension used to be known as ''undecimal misty'', but was decanonicalized in 2026 in favor of the no-11 no-13 temperament.
 Subgroup: 2.3.5.7.11
@@ Line 142: / Line 148: @@
 Badness (Sintel): 1.21
-==== Mystic ====
+=== Hazy ===
+Subgroup: 2.3.5.7.11
+Comma list: 176/175, 1375/1372, 5120/5103
+Mapping: {{mapping| 3 0 26 56 96 | 0 1 -4 -10 -18 }}
+Optimal tunings:
+* WE: ~63/50 = 399.8097{{c}}, ~3/2 = 702.4039{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 702.7731{{c}}
+{{Optimal ET sequence|legend=0| 12, 87e, 99, 111 }}
+Badness (Sintel): 1.84
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 176/175, 351/350, 847/845, 1375/1372
+Mapping: {{mapping| 3 0 26 56 96 111 | 0 1 -4 -10 -18 -21 }}
+Optimal tunings:
+* WE: ~63/50 = 399.8089{{c}}, ~3/2 = 702.3993{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 702.7661{{c}}
+{{Optimal ET sequence|legend=0| 12f, 87ef, 99, 111 }}
+Badness (Sintel): 1.54
+=== Smoky ===
+Subgroup: 2.3.5.7.11
+Comma list: 385/384, 2200/2187, 3136/3125
+Mapping: {{mapping| 3 0 26 56 -61 | 0 1 -4 -10 15 }}
+Optimal tunings:
+* WE: ~63/50 = 400.0072{{c}}, ~3/2 = 702.3068{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 703.2941{{c}}
+{{Optimal ET sequence|legend=0| 12e, 87, 186, 273 }}
+Badness (Sintel): 2.25
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 325/324, 352/351, 385/384, 3136/3125
+Mapping: {{mapping| 3 0 26 56 -61 -46 | 0 1 -4 -10 15 12 }}
+Optimal tunings:
+* WE: ~63/50 = 400.0184{{c}}, ~3/2 = 703.3420{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 703.3101{{c}}
+{{Optimal ET sequence|legend=0| 12e, 87, 186, 273 }}
+Badness (Sintel): 1.63
+=== Ashy ===
+Subgroup: 2.3.5.7.11
+Comma list: 540/539, 3136/3125, 5120/5103
+Mapping: {{mapping| 3 0 26 56 -80 | 0 1 -4 -10 19 }}
+Optimal tunings:
+* WE: ~63/50 = 399.7776{{c}}, ~3/2 = 702.4764{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 702.8652{{c}}
+{{Optimal ET sequence|legend=0| 12e, 99e, 210e }}
+Badness (Sintel): 1.89
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 441/440, 896/891, 1001/1000
+Comma list: 352/351, 540/539, 729/728, 3136/3125
-Mapping: {{mapping| 3 0 26 56 77 -46 | 0 1 -4 -10 -14 12 }}
+Mapping: {{mapping| 3 0 26 56 -80 -65 | 0 1 -4 -10 19 16 }}
 Optimal tunings:
-* WE: ~44/35 = 399.9543{{c}}, ~3/2 = 702.2310{{c}}
+* WE: ~63/50 = 399.7470{{c}}, ~3/2 = 702.3782{{c}}
-* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 703.3140{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 702.8135{{c}}
-{{Optimal ET sequence|legend=0| 12, 87, 186e, 273e }}
+{{Optimal ET sequence|legend=0| 12e, 99ef, 111, 210ef, 321cdeff, 531ccddeeffff }}
-Badness (Sintel): 1.52
+Badness (Sintel): 1.72
 === Semimist ===
@@ Line 220: / Line 301: @@
 == Fog ==
-The fog temperament ({{nowrap| 12 & 63 }}) tempers out the marvel comma, [[225/224]].
+Named by [[Xenllium]] in 2021, the fog temperament tempers out the marvel comma, [[225/224]], and may be described as {{nowrap| 12 & 63 }}.
 [[Subgroup]]: 2.3.5.7