Very high accuracy temperaments: Difference between revisions

Line 352:

[[Comma list]]: {{monzo| 71 -99 37 }}

: mapping generators: ~2, ~6561~~/4000~~

: mapping generators: ~2, ~8000/6561

[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1200.000000{{c}}, ~8000/6561 = 343.296094{{c}}

* [[WE]]: ~2 = 1199.999880537{{c}}, ~8000/6561 = 343.296063356{{c}}

* [[POTE]]: ~2 = 1200.000000{{c}}, ~8000/6561 = 343.296098{{c}}

: [[error map]]: {{val| -0.000119 +0.000418 -0.000336 }}

* [[CWE]]: ~2 = 1200.000000000{{c}}, ~8000/6561 = 343.296095985{{c}}

: error map: {{val| 0.000000 +0.000551 -0.000211 }}

<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~8000/6561 = 343.296094{{c}}

* [[POTE]]: ~2 = 1200.000000{{c}}, ~8000/6561 = 343.296098{{c}} -->

{{Optimal ET sequence|legend=1| 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860 }}

{{Optimal ET sequence|legend=1| 7, …, 381, 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860 }}

[[Badness]] (Sintel): TBD

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[[Comma list]]: {{monzo| -90 -15 49 }}

: ~~Mapping~~ generators: ~2, ~{{monzo| -~~23 -4~~ 13 }}

: mapping generators: ~2, ~{{monzo| 24 4 -13 }}

[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 24 4 -13 }} = 185.754186{{c}}

* [[WE]]: ~2 = 1200.000195604{{c}}, ~{{monzo| 24 4 -13 }} = 185.754209314{{c}}

* [[POTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 24 4 -13 }} = 185.754179{{c}}

: [[error map]]: {{val| -0.000196 +0.000418 -0.000336 }}

* [[CWE]]: ~2 = 1200.000000000{{c}}, ~{{monzo| 24 4 -13 }} = 185.754209314{{c}}

: error map: {{val| 0.000000 +0.000082 -0.000574 }}

<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 24 4 -13 }} = 185.754186{{c}}

* [[POTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 24 4 -13 }} = 185.754179{{c}} -->

{{Optimal ET sequence|legend=1| 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 … }}

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Atomic observes the [[schisma]], and the 3/2 is one schisma sharp of its [[12edo]] value. In atomic, since twelve fifths are sharp of seven octaves by twelve schismas, the [[Pythagorean comma]] is twelve schismas, and hence [[81/80]], the Didymus comma, is eleven schismas. In fact eleven schismas is sharp of 81/80, and twelve schismas of the Pythaorean comma, by the microscopic interval of the atom, which atomic tempers out. Extremely accurate.

Atomic extensions discussed elsewhere include [[minutes]] and [[72nd-octave temperaments#Hafnium|hafnium]].

Atomic extensions discussed elsewhere include [[minutes]] and [[72nd-octave temperaments #Hafnium|hafnium]].

[[Subgroup]]: 2.3.5

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[[Optimal tuning]]s:

* [[CTE]]: ~{{monzo| -67 35 5 }} = 100.000000{{c}}, ~3/2 = 701.955176{{c}} (~32805/32768 = 1.955176{{c}})

* [[WE]]: ~{{monzo| -67 35 5 }} = 99.999995361{{c}}, ~3/2 = 701.955129505{{c}} (~32805/32768 = 1.955161980{{c}})

* [[POTE]]: ~{{monzo| -67 35 5 }} = 100.000000{{c}}, ~3/2 = 701.955162{{c}} (~32805/32768 = 1.955162{{c}})

: [[error map]]: {{val| -0.000056 +0.000072 +0.000022 }}

* [[CWE]]: ~{{monzo| -67 35 5 }} = 100.000000000{{c}}, ~3/2 = 701.955166184{{c}} (~32805/32768 = 1.955166184{{c}})

: error map: {{val| 0.000000 +0.000165 +0.000123 }}

<!-- * [[CTE]]: ~{{monzo| -67 35 5 }} = 100.000000{{c}}, ~3/2 = 701.955176{{c}} (~32805/32768 = 1.955176{{c}})

* [[POTE]]: ~{{monzo| -67 35 5 }} = 100.000000{{c}}, ~3/2 = 701.955162{{c}} (~32805/32768 = 1.955162{{c}}) -->

{{Optimal ET sequence|legend=1| 12, …, 576, 588, 600, 612, 2460, 3072, 3684, 4296, 12276, 16572, 20868, 25164, 46032 }}

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Optimal tunings:

* CTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.949917{{c}} (or ~32805/32768 = 1.949917{{c}})

* WE: ~30375/28672 = 99.999866{{c}}, ~3/2 = 701.948670{{c}} (~32805/32768 = 1.949605{{c}})

* POTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.949608{{c}} (or ~32805/32768 = 1.949608{{c}})

* CWE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.949698{{c}} (~32805/32768 = 1.949698{{c}})

<!-- * CTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.949917{{c}} (~32805/32768 = 1.949917{{c}})

* POTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.949608{{c}} (~32805/32768 = 1.949608{{c}}) -->

{{Optimal ET sequence|legend=0| 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}

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Optimal tunings:

* CTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.948456{{c}} (or ~32805/32768 = 1.948456{{c}})

* WE: ~30375/28672 = 99.999760{{c}}, ~3/2 = 701.946301{{c}} (~32805/32768 = 1.947983{{c}})

* POTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.947988{{c}} (or ~32805/32768 = 1.947988{{c}})

* CWE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.948121{{c}} (~32805/32768 = 1.948121{{c}})

<!-- * CTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.948456{{c}} (or ~32805/32768 = 1.948456{{c}})

* POTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.947988{{c}} (or ~32805/32768 = 1.947988{{c}}) -->

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Subgroup: 2.3.7

Comma list: ~~11399736556781568/11398895185373143~~

Comma list: {{monzo| 47 4 -19 }}

: mapping generators: ~2, ~~~24576~~/~~16807~~

: mapping generators: ~2, ~16807/12288

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = 1200.000000{{c}}, ~~~24576~~/~~16807~~ = ~~657~~.~~791865~~{{c}}

* [[WE]]: ~2 = 1199.998821{{c}}, ~16807/12288 = 542.207710{{c}}

* [[~~CWE~~]]: ~2 = 1200.000000{{c}}, ~~~24576~~/~~16807~~ = ~~657~~.~~791796~~{{c}}

: [[error map]]: {{val| -0.0012 -0.0003 +0.0038 }}

* [[CWE]]: ~2 = 1200.000000{{c}}, ~16807/12288 = 542.208204{{c}}

: error map: {{val| 0.0000 +0.0009 +0.0069 }}

{{Optimal ET sequence|legend=1| 31, 73, 104, 135, 436, 571, 706, 1277, 8233, 9510, 10787, 12064, 13341, 14618, 15895, 33067, 48962, 113819d, 162781d, … }}

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Comma list: 645700815/645657712, {{monzo| 51 -13 -1 -10 }}

Mapping: {{mapping| 1 12 -~~155 5~~ | 0 -19 287 -4 }}

Mapping: {{mapping| 1 -7 132 1 | 0 19 -287 4 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000000~~{{c}}, ~~~24576~~/~~16807~~ = ~~657~~.~~792032~~{{c}}

* WE: ~2 = 1200.000110{{c}}, ~16807/12288 = 542.208018{{c}}

* CWE: ~2 = 1200.000000{{c}}, ~~~24576~~/16807 = ~~657~~.~~792032~~{{c}}

* CWE: ~2 = 1200.000000{{c}}, ~16807/12288 = 542.207968{{c}}

{{Optimal ET sequence|legend=0| 135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794, 121605d }}

@@ Line 352: / Line 352: @@
 [[Comma list]]: {{monzo| 71 -99 37 }}
-{{Mapping|legend=1| 1 28 73 | 0 -37 -99 }}
+{{Mapping|legend=1| 1 -9 -26 | 0 37 99 }}
-: mapping generators: ~2, ~6561/4000
+: mapping generators: ~2, ~8000/6561
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000000{{c}}, ~8000/6561 = 343.296094{{c}}
+* [[WE]]: ~2 = 1199.999880537{{c}}, ~8000/6561 = 343.296063356{{c}}
-* [[POTE]]: ~2 = 1200.000000{{c}}, ~8000/6561 = 343.296098{{c}}
+: [[error map]]: {{val| -0.000119 +0.000418 -0.000336 }}
+* [[CWE]]: ~2 = 1200.000000000{{c}}, ~8000/6561 = 343.296095985{{c}}
+: error map: {{val| 0.000000 +0.000551 -0.000211 }}
+<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~8000/6561 = 343.296094{{c}}
+* [[POTE]]: ~2 = 1200.000000{{c}}, ~8000/6561 = 343.296098{{c}} -->
-{{Optimal ET sequence|legend=1| 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860 }}
+{{Optimal ET sequence|legend=1| 7, …, 381, 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860 }}
 [[Badness]] (Sintel): TBD
@@ Line 370: / Line 374: @@
 [[Comma list]]: {{monzo| -90 -15 49 }}
-{{Mapping|legend=1| 1 43 15 | 0 -49 -15 }}
+{{Mapping|legend=1| 1 -6 0 | 0 49 15 }}
-: Mapping generators: ~2, ~{{monzo| -23 -4 13 }}
+: mapping generators: ~2, ~{{monzo| 24 4 -13 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 24 4 -13 }} = 185.754186{{c}}
+* [[WE]]: ~2 = 1200.000195604{{c}}, ~{{monzo| 24 4 -13 }} = 185.754209314{{c}}
-* [[POTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 24 4 -13 }} = 185.754179{{c}}
+: [[error map]]: {{val| -0.000196 +0.000418 -0.000336 }}
+* [[CWE]]: ~2 = 1200.000000000{{c}}, ~{{monzo| 24 4 -13 }} = 185.754209314{{c}}
+: error map: {{val| 0.000000 +0.000082 -0.000574 }}
+<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 24 4 -13 }} = 185.754186{{c}}
+* [[POTE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 24 4 -13 }} = 185.754179{{c}} -->
 {{Optimal ET sequence|legend=1| 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 … }}
@@ Line 386: / Line 394: @@
 Atomic observes the [[schisma]], and the 3/2 is one schisma sharp of its [[12edo]] value. In atomic, since twelve fifths are sharp of seven octaves by twelve schismas, the [[Pythagorean comma]] is twelve schismas, and hence [[81/80]], the Didymus comma, is eleven schismas. In fact eleven schismas is sharp of 81/80, and twelve schismas of the Pythaorean comma, by the microscopic interval of the atom, which atomic tempers out. Extremely accurate.
-Atomic extensions discussed elsewhere include [[minutes]] and [[72nd-octave temperaments#Hafnium|hafnium]].
+Atomic extensions discussed elsewhere include [[minutes]] and [[72nd-octave temperaments #Hafnium|hafnium]].
 [[Subgroup]]: 2.3.5
@@ Line 396: / Line 404: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~{{monzo| -67 35 5 }} = 100.000000{{c}}, ~3/2 = 701.955176{{c}} (~32805/32768 = 1.955176{{c}})
+* [[WE]]: ~{{monzo| -67 35 5 }} = 99.999995361{{c}}, ~3/2 = 701.955129505{{c}} (~32805/32768 = 1.955161980{{c}})
-* [[POTE]]: ~{{monzo| -67 35 5 }} = 100.000000{{c}}, ~3/2 = 701.955162{{c}} (~32805/32768 = 1.955162{{c}})
+: [[error map]]: {{val| -0.000056 +0.000072 +0.000022 }}
+* [[CWE]]: ~{{monzo| -67 35 5 }} = 100.000000000{{c}}, ~3/2 = 701.955166184{{c}} (~32805/32768 = 1.955166184{{c}})
+: error map: {{val| 0.000000 +0.000165 +0.000123 }}
+<!-- * [[CTE]]: ~{{monzo| -67 35 5 }} = 100.000000{{c}}, ~3/2 = 701.955176{{c}} (~32805/32768 = 1.955176{{c}})
+* [[POTE]]: ~{{monzo| -67 35 5 }} = 100.000000{{c}}, ~3/2 = 701.955162{{c}} (~32805/32768 = 1.955162{{c}}) -->
 {{Optimal ET sequence|legend=1| 12, …, 576, 588, 600, 612, 2460, 3072, 3684, 4296, 12276, 16572, 20868, 25164, 46032 }}
@@ Line 411: / Line 423: @@
 Optimal tunings:
-* CTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.949917{{c}} (or ~32805/32768 = 1.949917{{c}})
+* WE: ~30375/28672 = 99.999866{{c}}, ~3/2 = 701.948670{{c}} (~32805/32768 = 1.949605{{c}})
-* POTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.949608{{c}} (or ~32805/32768 = 1.949608{{c}})
+* CWE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.949698{{c}} (~32805/32768 = 1.949698{{c}})
+<!-- * CTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.949917{{c}} (~32805/32768 = 1.949917{{c}})
+* POTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.949608{{c}} (~32805/32768 = 1.949608{{c}}) -->
 {{Optimal ET sequence|legend=0| 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}
@@ Line 426: / Line 440: @@
 Optimal tunings:
-* CTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.948456{{c}} (or ~32805/32768 = 1.948456{{c}})
+* WE: ~30375/28672 = 99.999760{{c}}, ~3/2 = 701.946301{{c}} (~32805/32768 = 1.947983{{c}})
-* POTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.947988{{c}} (or ~32805/32768 = 1.947988{{c}})
+* CWE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.948121{{c}} (~32805/32768 = 1.948121{{c}})
+<!-- * CTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.948456{{c}} (or ~32805/32768 = 1.948456{{c}})
+* POTE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.947988{{c}} (or ~32805/32768 = 1.947988{{c}}) -->
 {{Optimal ET sequence|legend=0| 12, …, 600e, 612, 1236, 1848 }}
@@ Line 438: / Line 454: @@
 Subgroup: 2.3.7
-Comma list: 11399736556781568/11398895185373143
+Comma list: {{monzo| 47 4 -19 }}
-{{Mapping|legend=2| 1 12 5 | 0 -19 -4 }}
+{{Mapping|legend=2| 1 -7 1 | 0 19 4 }}
-: mapping generators: ~2, ~24576/16807
+: mapping generators: ~2, ~16807/12288
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000000{{c}}, ~24576/16807 = 657.791865{{c}}
+* [[WE]]: ~2 = 1199.998821{{c}}, ~16807/12288 = 542.207710{{c}}
-* [[CWE]]: ~2 = 1200.000000{{c}}, ~24576/16807 = 657.791796{{c}}
+: [[error map]]: {{val| -0.0012 -0.0003 +0.0038 }}
+* [[CWE]]: ~2 = 1200.000000{{c}}, ~16807/12288 = 542.208204{{c}}
+: error map: {{val| 0.0000 +0.0009 +0.0069 }}
+<!-- * [[CTE]]: ~2 = 1200.000000{{c}}, ~16807/12288 = 542.208135{{c}} -->
 {{Optimal ET sequence|legend=1| 31, 73, 104, 135, 436, 571, 706, 1277, 8233, 9510, 10787, 12064, 13341, 14618, 15895, 33067, 48962, 113819d, 162781d, … }}
@@ Line 458: / Line 477: @@
 Comma list: 645700815/645657712, {{monzo| 51 -13 -1 -10 }}
-Mapping: {{mapping| 1 12 -155 5 | 0 -19 287 -4 }}
+Mapping: {{mapping| 1 -7 132 1 | 0 19 -287 4 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000000{{c}}, ~24576/16807 = 657.792032{{c}}
+* WE: ~2 = 1200.000110{{c}}, ~16807/12288 = 542.208018{{c}}
-* CWE: ~2 = 1200.000000{{c}}, ~24576/16807 = 657.792032{{c}}
+* CWE: ~2 = 1200.000000{{c}}, ~16807/12288 = 542.207968{{c}}
+<!-- * CTE: ~2 = 1200.000000{{c}}, ~16807/12288 = 542.207968{{c}} -->
 {{Optimal ET sequence|legend=0| 135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794, 121605d }}