Schismic–Pythagorean equivalence continuum: Difference between revisions

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 == Python ==
-Python is generated by a fifth, which is typically flatter than 7\12. The ~5/4 is reached by sixteen fifths octave-reduced, which is a double augmented second (C–Dx). It can be described as {{nowrap| 12 & 91 }}, and 103edo is a good tuning.
+Python is generated by a fifth, which is typically flatter than 7\12. The ~5/4 is reached by +16 fifths octave reduced, which is a double-augmented second (C–Dx). It can be described as {{nowrap| 12 & 91 }}, and [[103edo]] is a good tuning. It corresponds to {{nowrap| ''m'' {{=}} -1 }} and {{nowrap| ''n'' {{=}} 1/2 }}.
 [[Subgroup]]: 2.3.5
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 [[Badness]] (Sintel): 6.92
-== Sextile ==
+== Gracecordial (5-limit) ==
+: ''For extensions, see [[Marvel temperaments #Gracecordial]].''
+The 5-limit version of gracecordial is generated by a fifth, which is typically sharp of 7\12 but flat of just. The ~5/4 is reached by -20 fifths octave reduced, which is a triple-diminished fifth (C–Gbbb). It can be described as {{nowrap| 12 & 125 }}, and [[137edo]] is a good tuning. It corresponds to {{nowrap| ''n'' {{=}} -1 }} and {{nowrap| ''m'' {{=}} 1/2 }}.
+[[Subgroup]]: 2.3.5
+[[Comma list]]: 17433922005/17179869184
+{{Mapping|legend=1| 1 0 34 | 0 1 -20 }}
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.691{{c}}
+* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.734{{c}}
+{{Optimal ET sequence|legend=1| 12, 113, 125, 137, 1221bbcc }}
+[[Badness]] (Sintel): 7.20
+== Sextile (5-limit) ==
 {{See also| Landscape microtemperaments #Sextile }}
-The 5-limit version of sextile reaches the interval class of 5 by −6 perfect fifths minus a period of 1/6-octave.
+The 5-limit version of sextile reaches the interval class of 5 by −6 perfect fifths (i.e. a diminished fifth) minus a period of 1/6-octave. It corresponds to {{nowrap| ''n'' {{=}} 6 }}, meaning the Pythagorean comma is equated with a stack of six schismas.
 [[Subgroup]]: 2.3.5
@@ Line 254: / Line 273: @@
 [[Badness]] (Sintel): 13.0
+== Wronecki ==
+Wronecki equates a stack of six ~10/9's with the octave. It reaches the interval class of 5 by +2 perfect fifths (i.e. a major second) plus a period of 1/6-octave. It corresponds to {{nowrap| ''m'' {{=}} 6 }}, meaning the Pythagorean comma is equated with a stack of six syntonic commas.
+[[Subgroup]]: 2.3.5
+[[Comma list]]: 531441/500000
+{{Mapping|legend=1| 6 0 -5 | 0 1 2 }}
+: mapping generators: ~10/9, ~3
+[[Optimal tuning]]s:
+* [[CTE]]: ~10/9 = 200.000{{c}}, ~3/2 = 696.229{{c}}
+* [[POTE]]: ~10/9 = 200.000{{c}}, ~3/2 = 695.040{{c}}
+{{Optimal ET sequence|legend=1| 12, 66b, 78b, 90b, 102b }}
+[[Badness]] (Sintel): 8.02
 == Heptacot ==
-Heptacot tempers out the [[heptacot comma]] and divides the fifth into seven equal parts, the most notable example being [[12edo]] (7\12).
+Heptacot tempers out the [[heptacot comma]] and divides the fifth into seven equal parts, the most notable example being [[12edo]] (7\12). It corresponds to {{nowrap| ''n'' {{=}} 7 }}, meaning the Pythagorean comma is equated with a stack of seven schismas.
 [[Subgroup]]: 2.3.5