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Very high accuracy temperaments: Difference between revisions

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== Septarium ==
== Septarium ==
Septarium ({{nowrap| 441 & 2513 }}) tempers out the [[septarium comma]], {{monzo| 73 77 -84 }}.
Septarium splits the octave in seven, and tempers out the [[septarium comma]], {{monzo| 73 77 -84 }}. It can be described as {{nowrap| 441 & 2513 }} temperament, which tempers out 78125000/78121827 and {{monzo| 47 -7 -7 -7 }} ([[akjaysma]]) in the 7-limit.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 267: Line 267:
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Comma list: 78125000/78121827, {{monzo| 47 -7 -7 -7 }}
Comma list: 78125000/78121827, 140737488355328/140710042265625


Mapping: {{Mapping| 7 1 7 39 | 0 12 11 -23 }}
Mapping: {{Mapping| 7 1 7 39 | 0 12 11 -23 }}
Line 318: Line 318:


[[Badness]] (Sintel): 0.133
[[Badness]] (Sintel): 0.133
== Counterquectismic ==
{{See also| Counterquectisma }}
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| -500 314 1 }}
{{Mapping|legend=1| 1 0 500 | 0 1 -314 }}
: mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.000184840{{c}}, ~3/2 = 701.954524421{{c}}
: [[error map]]: {{val| +0.000185 -0.000292 -0.000002 }}
* [[CWE]]: ~2 = 1200.000000000{{c}}, ~3/2 = 701.954415845{{c}}
: error map: {{val| 0.000000 -0.000585 -0.000289 }}
{{Optimal ET sequence|legend=1| 306c, 359, 665, 2354, 3019, 3684, 11717, 15401, 19085, 34486 }}
[[Badness]] (Sintel): 11.971
== Quectismic ==
{{See also| Quectisma }}
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| 554 -351 1 }}
{{Mapping|legend=1| 1 0 -554 | 0 1 351 }}
: mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.999902567{{c}}, ~3/2 = 701.955253373{{c}}
: [[error map]]: {{val| -0.000097 +0.000155 -0.000001 }}
* [[CWE]]: ~2 = 1200.000000000{{c}}, ~3/2 = 701.955310152{{c}}
: error map: {{val| 0.000000 +0.000309 +0.000149 }}
{{Optimal ET sequence|legend=1| 306, 359c, 665, 2301, 2966, 3631, 4296, 46591, 50887, 55183, 59479, 63775, 68071, 72367, 76663, 80959, 85255, 89551 }}
[[Badness]] (Sintel): 8.736
=== Nanoquectismic ===
Subgroup: 2.3.5.7
Comma list: 43046721/43025920, {{monzo| 82 -19 -3 -16 }}
Mapping: {{mapping| 1 0 -554 109 | 0 1 351 -67 }}
Optimal tunings:
* WE: ~2 = 1199.979407{{c}}, ~3/2 = 701.943437{{c}}
* CWE: ~2 = 1200.000000{{c}}, ~3/2 = 701.955457{{c}}
{{Optimal ET sequence|legend=1| 306, 665, 1636, 2301, 2966 }}
Badness (Sintel): 15.285
=== Conanoquectismic ===
Subgroup: 2.3.5.7
Comma list: {{monzo| 33 -26 12 -7 }}, {{monzo| -32 13 17 -10 }}
Mapping: {{mapping| 1 0 -554 -945 | 0 1 351 598 }}
Optimal tunings:
* WE: ~2 = 1200.010750{{c}}, ~3/2 = 701.961041{{c}}
* CWE: ~2 = 1200.000000{{c}}, ~3/2 = 701.954771{{c}}
{{Optimal ET sequence|legend=1| 306d, 359cd, 665 }}
Badness (Sintel): 54.974


== Atomic ==
== Atomic ==
{{See also | Kirnberger's atom }}
: ''For extensions, see [[Landscape microtemperaments #Atomic]].''
 
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 tempers out [[Kirnberger's atom]], the microscopic interval separating eleven [[schisma]]s and a [[syntonic comma]], or twelfth schismas and a [[Pythagorean comma]]. It is a member of the [[schismic–commatic equivalence continuum]] with {{nowrap| ''n'' {{=}} 12 }}. In this extremely accurate temperament, the [[3/2|perfect fifth]] is one [[schisma]] sharp of its [[12edo]] value.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 412: Line 340:


[[Badness]] (Sintel): 0.0892
[[Badness]] (Sintel): 0.0892
=== 7-limit ===
Subgroup: 2.3.5.7
Comma list: 250047/250000, {{monzo| -55 30 2 1 }}
Mapping: {{mapping| 12 0 161 338 | 0 1 -7 -16 }}
Optimal tunings:
* WE: ~30375/28672 = 99.999866{{c}}, ~3/2 = 701.948670{{c}} (~32805/32768 = 1.949605{{c}})
* CWE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.949698{{c}} (~32805/32768 = 1.949698{{c}})
{{Optimal ET sequence|legend=0| 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}
Badness (Sintel): 1.160
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 151263/151250, 184549376/184528125
Mapping: {{mapping| 12 0 161 338 517 | 0 1 -7 -16 -25 }}
Optimal tunings:
* WE: ~30375/28672 = 99.999760{{c}}, ~3/2 = 701.946301{{c}} (~32805/32768 = 1.947983{{c}})
* CWE: ~30375/28672 = 100.000000{{c}}, ~3/2 = 701.948121{{c}} (~32805/32768 = 1.948121{{c}})
{{Optimal ET sequence|legend=0| 12, …, 600e, 612, 1236, 1848 }}
Badness (Sintel): 0.530


== Revopentic ==
== Revopentic ==
Line 521: Line 419:


= Rank-3 temperaments =
= Rank-3 temperaments =
== Akjaysmic ==
{{See also| Akjaysma }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: {{monzo| 47 -7 -7 -7 }}
{{Mapping|legend=1| 7 0 0 47 | 0 1 0 -1 | 0 0 1 -1 }}
: mapping generators: ~1157625/1048576, ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~1157625/1048576 = 171.427811{{c}}, ~3/2 = 701.962313{{c}}, ~5/4 = 386.328628{{c}}
: [[error map]]: {{val| -0.0053 +0.0020 +0.0043 +0.0062 }}
* [[CWE]]: ~1157625/1048576 = 171.428571{{c}}, ~3/2 = 701.964859{{c}}, ~5/4 = 386.330310{{c}}
: error map: {{val| 0.0000 +0.0099 +0.0166 +0.0218 }}
{{Optimal ET sequence|legend=1| 140, 217, 224, 301, 441, 966, 1106, 1407, 1547, 1848, 2513, 2954, 6349, 9303, 11151, 14105, 17500, 20454 }}
[[Badness]] (Sintel): 2.22
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 184549376/184528125, 199297406/199290375
Mapping: {{mapping| 7 0 0 47 -168 | 0 1 0 -1 10 | 0 0 1 -1 5 }}
: mapping generators: ~29160/26411, ~3, ~5
Optimal tunings:
* WE: ~29160/26411 = 171.427802{{c}}, ~3/2 = 701.964561{{c}}, ~5/4 = 386.329837{{c}}
* CWE: ~29160/26411 = 171.428571{{c}}, ~3/2 = 701.967291{{c}}, ~5/4 = 386.331624{{c}}
{{Optimal ET sequence|legend=0| 301, 441, 665e, 742, 1106, 1547, 1848, 3395, 4501, 5243, 6349, 17941, 24290, 30639, 45185cde, 63126bcde, 69475bccdde, 75824bccddee }}
Badness (Sintel): 1.32
== Sasaquinbizo-atriyo (171 & 1106 & 3566) ==
== Sasaquinbizo-atriyo (171 & 1106 & 3566) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Badness]] (Sintel): 1.24
[[Badness]] (Sintel): 1.24


== Saquadtrizo-asepgu (171 & 2019 & 3566) ==
== Izarismic ==
This temperament is identical to the [[izar]] in the no-twos subgroup, but has an independent generator for prime 2. It can be described as {{nowrap| 41 & 49 & 130 }} temperament, and tempers out the [[izar comma]], {{monzo| 0 -11 -7 12 }}.
 
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: {{monzo| 0 -11 -7 12 }} (13841287201/13839609375)
[[Comma list]]: 13841287201/13839609375


{{Mapping|legend=1| 1 0 0 0 | 0 1 7 5 | 0 0 -12 -7 }}
{{Mapping|legend=1| 1 0 0 0 | 0 1 7 5 | 0 0 -12 -7 }}
: mapping generators: ~2, ~3, ~16807/10125


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9584{{c}}, ~16807/10125 = 877.2825{{c}}
[[Optimal tuning]]s:
* [[TE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9584{{c}}, ~16807/10125 = 877.2825{{c}}
* [[WE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9584{{c}}, ~16807/10125 = 877.2825{{c}}


{{Optimal ET sequence|legend=1| 171, 814, 985, 1034, 1205, 1376, 1547, 1718, 1848, 2019, 3224, 3395, 3566, 8980, 9151, 12717, 47131, 50697, 59848d, 63414d, 76131d }}
{{Optimal ET sequence|legend=1| 171, 814, 985, 1034, 1205, 1376, 1547, 1718, 1848, 2019, 3224, 3395, 3566, 8980, 9151, 12717, 47131, 50697, 59848d, 63414d, 76131d }}
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