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Schismatic family: Difference between revisions

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→Term: +intro to note its utility
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== Term ==
== Term ==
Term tempers out the [[landscape comma]], mapping ~63/50 to the 1/3-octave period. It can be described as 12 & 171, and is the unique temperament that equates a syntonic~Pythagorean comma with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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{{Multival|legend=1| 3 -24 -54 -45 -94 -58 }}
{{Multival|legend=1| 3 -24 -54 -45 -94 -58 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~3/2 = 701.742
[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000, ~3/2 = 701.742


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis)]]: 2.5/3
* [[7-odd-limit]] [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis)]]: 2.5/3
* [[9-odd-limit]] eigenmonzo (unchanged-interval) basis: 2.9/7
* [[9-odd-limit]] eigenmonzo (unchanged-interval) basis: 2.9/7


Line 1,488: Line 1,490:


=== Terminal ===
=== Terminal ===
The terminal temperament (12 & 159) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.  
The terminal temperament (12 & 159) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,496: Line 1,498:
Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}
Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 701.824
Optimal tuning (POTE): ~44/35 = 400.000, ~3/2 = 701.824


{{Optimal ET sequence|legend=1| 12, 147de, 159, 330 }}
{{Optimal ET sequence|legend=0| 12, 147de, 159, 330 }}


Badness: 0.059502
Badness: 0.059502
Line 1,509: Line 1,511:
Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}
Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 701.821
Optimal tuning (POTE): ~44/35 = 400.000, ~3/2 = 701.821


{{Optimal ET sequence|legend=1| 12f, 147def, 159, 330 }}
{{Optimal ET sequence|legend=0| 12f, 147def, 159, 330 }}


Badness: 0.037082
Badness: 0.037082
Line 1,522: Line 1,524:
Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}
Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}


Optimal tuning (POTE): ~34/27 = 1\3, ~3/2 = 701.810
Optimal tuning (POTE): ~34/27 = 400.000, ~3/2 = 701.810


{{Optimal ET sequence|legend=1| 12f, 147def, 159, 171, 330 }}
{{Optimal ET sequence|legend=0| 12f, 147def, 159, 171, 330 }}


Badness: 0.027073
Badness: 0.027073
Line 1,535: Line 1,537:
Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}
Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.685
Optimal tuning (POTE): ~63/50 = 400.000, ~3/2 = 701.685


{{Optimal ET sequence|legend=1| 12e, 159e, 171, 183, 354, 537, 891de }}
{{Optimal ET sequence|legend=0| 12e, 159e, 171, 183, 354, 537, 891de }}


Badness: 0.066968
Badness: 0.066968
Line 1,548: Line 1,550:
Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}
Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.689
Optimal tuning (POTE): ~63/50 = 400.000, ~3/2 = 701.689


{{Optimal ET sequence|legend=1| 171, 183, 354, 891de, 1245dee, 1599ddee }}
{{Optimal ET sequence|legend=0| 171, 183, 354, 891de, 1245dee, 1599ddee }}


Badness: 0.035487
Badness: 0.035487
Line 1,561: Line 1,563:
Mapping: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}
Mapping: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.688
Optimal tuning (POTE): ~63/50 = 400.000, ~3/2 = 701.688


{{Optimal ET sequence|legend=1| 171, 183, 354, 891de, 1245dee, 1599ddee }}
{{Optimal ET sequence|legend=0| 171, 183, 354, 891de, 1245dee, 1599ddee }}


Badness: 0.020434
Badness: 0.020434


=== Semiterm ===
=== Semiterm ===
The semiterm temperament (12 & 342) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma).
The semiterm temperament (12 & 342) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma).


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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: mapping generators: ~55/49, ~3
: mapping generators: ~55/49, ~3


Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 701.7460
Optimal tuning (POTE): ~55/49 = 200.000, ~3/2 = 701.7460


{{Optimal ET sequence|legend=1| 12, 330e, 342, 1380, 1722, 2064, 2406c }}
{{Optimal ET sequence|legend=0| 12, 330e, 342, 1380, 1722, 2064, 2406c }}


Badness: 0.029438
Badness: 0.029438
Line 1,591: Line 1,593:
Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}
Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}


Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 701.7256
Optimal tuning (POTE): ~55/49 = 200.000, ~3/2 = 701.7256


{{Optimal ET sequence|legend=1| 12f, 330eff, 342f, 696f }} *
{{Optimal ET sequence|legend=0| 12f, 330eff, 342f, 696f }} *


<nowiki>*</nowiki> optimal patent val: [[354edo|354]]
<nowiki>*</nowiki> optimal patent val: [[354edo|354]]
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: mapping generators: ~63/50, ~693/400
: mapping generators: ~63/50, ~693/400


Optimal tuning (POTE): ~63/50 = 1\3, ~693/400 = 950.872 (~12/11 = 150.872)
Optimal tuning (POTE): ~63/50 = 400.000, ~693/400 = 950.872 (~12/11 = 150.872)


{{Optimal ET sequence|legend=1| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}
{{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }}


Badness: 0.020687
Badness: 0.020687
Line 1,621: Line 1,623:
Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}
Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~26/15 = 950.873 (~12/11 = 150.873)
Optimal tuning (POTE): ~63/50 = 400.000, ~26/15 = 950.873 (~12/11 = 150.873)


{{Optimal ET sequence|legend=1| 24d, 159, 183, 342f }}
{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f }}


Badness: 0.031362
Badness: 0.031362
Line 1,634: Line 1,636:
Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}
Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}


Optimal tuning (POTE): ~34/27 = 1\3, ~26/15 = 950.867 (~12/11 = 150.867)
Optimal tuning (POTE): ~34/27 = 400.000, ~26/15 = 950.867 (~12/11 = 150.867)


{{Optimal ET sequence|legend=1| 24d, 159, 183, 342f, 525f, 867ff }}
{{Optimal ET sequence|legend=0| 24d, 159, 183, 342f, 525f, 867ff }}


Badness: 0.022316
Badness: 0.022316
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