Syntonic–kleismic equivalence continuum: Difference between revisions

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@@ Line 17: / Line 17: @@
 |-
 | 0
-| 19 & 19c
+| [[Graywood]]
 | [[19-comma|1162261467/1073741824]]
 | {{Monzo| -30 19 }}
@@ Line 62: / Line 62: @@
 |-
 | 9
-| 19 & 51c
+| [[Xenial]]
 | [[129140163/125000000]]
 | {{Monzo| -6 17 -9 }}
@@ Line 102: / Line 102: @@
 | [[Mowgli]] || 15/2 = 7.5 || {{monzo| 0 22 -15 }}
 |}
+== Graywood ==
+Named by [[CompactStar]] in 2024, graywood tempers out the [[19-comma]], corresponding to {{nowrap| ''n'' {{=}} 0 }}. It takes [[19edo]]'s closed [[circle of fifths]], but adds an independent generator for [[prime interval|prime]] [[5/1|5]]. 19 is the only equal temperament that makes it to the optimal ET sequence as all the small edo tunings, e.g. [[38edo|38c-edo]] or [[57edo|57c-edo]], are not nearly as accurate as 19 itself.
+[[Subgroup]]: 2.3.5
+[[Comma list]]: 1162261467/1073741824
+{{Mapping|legend=1| 19 30 0 | 0 0 1 }}
+: mapping generators: ~2187/2048, ~5
+[[Optimal tuning]]s:
+* [[WE]]: ~2187/2048 = 63.2773{{c}}, ~5/4 = 381.7568{{c}}
+: [[error map]]: {{val| +2.268 -3.637 -0.020 }}
+* [[CWE]]: ~2187/2048 = 63.1579{{c}}, ~5/4 = 382.7889{{c}}
+: error map: {{val| 0.000 -7.218 -3.525 }}
+{{Optimal ET sequence|legend=1| 19 }}
+[[Badness]] (Sintel): 32.4
 == Hogzilla ==
@@ Line 168: / Line 188: @@
 [[Badness]] (Sintel): 2.04
+== Xenial ==
+: ''For extensions, see [[Starling temperaments #Xenial]] and [[Sensamagic clan #Xenia]].''
+Named by [[User:Xenllium|Xenllium]] in 2026, xenial splits the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. It corresponds to {{nowrap| ''n'' {{=}} 9 }}. Its [[ploidacot]] is zeta-enneacot, and from this it derives its name.
+[[Subgroup]]: 2.3.5
+[[Comma list]]: 129140163/125000000
+{{Mapping|legend=1| 1 -6 -12 | 0 9 17 }}
+: mapping generators: ~2, ~9/5
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2802{{c}}, ~9/5 = 1011.2914{{c}}
+: [[error map]]: {{val| +0.280 -2.013 +2.278 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.0762{{c}}
+: error map: {{val| 0.000 -2.269 +1.982 }}
+{{Optimal ET sequence|legend=1| 19, 70, 89, 108, 127 }}
+[[Badness]] (Sintel): 8.84
 == Lalasepyo (8c & 11) ==
@@ Line 206: / Line 248: @@
 [[Badness]] (Sintel): 10.8
-== Parakleismic (5-limit) ==
-{{Main| Parakleismic }}
-: ''For extensions, see [[Ragismic microtemperaments #Parakleismic]].''
-The 5-limit version of parakleismic tempers out the [[parakleisma]]. It corresponds to {{nowrap| ''n'' {{=}} 13/2 }}, and 13 generator steps give the interval class of [[3/1|3]].
-[[Subgroup]]: 2.3.5
-[[Comma list]]: 1224440064/1220703125
-{{Mapping|legend=1| 1 -8 -8 | 0 13 14 }}
-: mapping generators: ~2, ~5/3
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.971{{c}}, ~5/3 = 884.7383{{c}}
-: [[error map]]: {{val| -0.029 -0.127 +0.253 }}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~5/3 = 884.8576{{c}}
-: error map: {{val| 0.000 -0.106 +0.293 }}
-{{Optimal ET sequence|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}
-[[Badness]] (Sintel): 1.02
 == Mowgli ==
@@ Line 251: / Line 270: @@
 [[Badness]] (Sintel): 15.3
+== Enneadecal (5-limit) ==
+: ''For extensions, see [[Ragismic microtemperaments #Enneadecal]].''
+The 5-limit version of enneadecal tempers out the [[enneadeca]], which simply equates a stack of nineteen [[6/5]] minor thirds with five [[2/1|octaves]]. It corresponds to {{nowrap| ''n'' {{=}} 19/3 }}, with a 19th-octave period and a generator of a [[3/2|perfect fifth]].
+[[Subgroup]]: 2.3.5
+[[Comma list]]: 19073486328125/19042491875328
+{{Mapping|legend=1| 19 0 14 | 0 1 1 }}
+: mapping generators: ~648/625, ~3
+[[Optimal tuning]]s:
+* [[WE]]: ~648/625 = 63.1579{{c}}, ~3/2 = 701.9861{{c}}
+: [[error map]]: {{val| +0.013 +0.044 -0.095 }}
+* [[CWE]]: ~648/625 = 63.1579{{c}}, ~3/2 = 701.9900{{c}}
+: error map: {{val| 0.000 -0.035 -0.113 }}
+{{Optimal ET sequence|legend=1| 19, 95, 114, 133, 152, 171, 323, 494, 665, 1159, 1824, 2983, 7125c }}
+[[Badness]] (Sintel): 1.12
 == Countermeantone ==
@@ Line 289: / Line 330: @@
 [[Badness]] (Sintel): 7.45
-== Oviminor ==
-{{See also| Ragismic microtemperaments #Oviminor }}
-Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.
-[[Subgroup]]: 2.3.5
-[[Comma list]]: {{monzo| -134 -185 184 }}
-{{Mapping|legend=1| 1 -134 -134 | 0 184 185 }}
-: mapping generators: ~2, ~5/3
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.0094{{c}}, ~5/3 = 884.2568{{c}}
-: [[error map]]: {{val| +0.009 +0.033 -0.069 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.2499{{c}}
-: error map: {{val| 0.000 +0.026 -0.083 }}
-{{Optimal ET sequence|legend=1| 19, …, 1600, 3219, 4819 }}
-[[Badness]] (Sintel): 751
 [[Category:19edo]]
 [[Category:Equivalence continua]]