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Schismatic family: Difference between revisions

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This defies the tradition of tertian harmony, as the just major triad on C is {{nowrap|{{dash|C, F♭, G|hair|med}}}}, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as {{nowrap|{{dash|C, vE, G|hair|med}}}}.  
This defies the tradition of tertian harmony, as the just major triad on C is {{nowrap|{{dash|C, F♭, G|hair|med}}}}, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as {{nowrap|{{dash|C, vE, G|hair|med}}}}.  


== Schismatic a.k.a. helmholtz ==
== Schismic ==
The 5-limit version of the temperament is a [[microtemperament]], sometimes called '''helmholtz''', '''schismic''' or '''schismatic''', which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244{{cent}}, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut.
The 5-limit version of the temperament is a [[microtemperament]], sometimes called '''schismic, helmholtz''', or historically '''schismatic''', which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244{{cent}}, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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: Mapping generators: ~2, ~3
: Mapping generators: ~2, ~3
{{Multival|legend=1| 1 -8 -14 -15 -25 -10 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.085
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.085
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* [[CTE]]: ~2 = 1\1, ~3/2 = 702.2696
* [[CTE]]: ~2 = 1\1, ~3/2 = 702.2696
* [[POTE]]: ~2 = 1\1, ~3/2 = 701.556
* [[POTE]]: ~2 = 1\1, ~3/2 = 701.556
{{Multival|legend=1| 1 -8 -2 -15 -6 18 }}


{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
{{Optimal ET sequence|legend=1| 5c, 7c, 12 }}
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{{Mapping|legend=1| 1 0 15 -59 | 0 1 -8 39 }}
{{Mapping|legend=1| 1 0 15 -59 | 0 1 -8 39 }}
{{Multival|legend=1| 1 -8 39 -15 59 113 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.757
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.757
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: Mapping generators: ~2, ~3
: Mapping generators: ~2, ~3
{{Multival|legend=1| 1 -8 -26 -15 -44 -38 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.239
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.239
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: Mapping generators: ~567/400, ~3
: Mapping generators: ~567/400, ~3
{{Multival|legend=1| 2 -16 -40 -30 -69 -48 }}


[[Optimal tuning]] ([[CTE]]): ~567/400 = 1\2, ~3/2 = 701.5899
[[Optimal tuning]] ([[CTE]]): ~567/400 = 1\2, ~3/2 = 701.5899
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: Mapping generators: ~1225/864, ~35/24
: Mapping generators: ~1225/864, ~35/24
{{Multival|legend=1| 4 -32 38 -60 49 178 }}


[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~35/24 = 650.920 (~36/35 = 50.920)
[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~35/24 = 650.920 (~36/35 = 50.920)
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: Mapping generators: ~2, ~140/81
: Mapping generators: ~2, ~140/81
{{Multival|legend=1| 2 -16 25 -30 34 103 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~140/81 = 950.797
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~140/81 = 950.797
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{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }}
{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }}
{{Multival|legend=1| 3 -24 -13 -45 -29 37 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~160/147 = 166.140
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~160/147 = 166.140
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{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}
{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}
{{Multival|legend=1| 3 -24 52 -45 74 188 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~192/175 = 166.019
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~192/175 = 166.019
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: Mapping generators: ~343/243, ~9/7
: Mapping generators: ~343/243, ~9/7
{{Multival|legend=1| 6 -48 10 -90 -1 158 }}


[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 433.901
[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 433.901
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: Mapping generators: ~63/50, ~3
: Mapping generators: ~63/50, ~3
{{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 = 1＼3, ~3/2 = 701.742
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: Mapping generators: ~2, ~448/405
: Mapping generators: ~2, ~448/405
{{Multival|legend=1| 4 -32 -15 -60 -35 55 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 175.434
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 175.434
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{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}
{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}
{{Multival|legend=1| 5 -40 -82 -75 -144 -78 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~625/588 = 99.625
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~625/588 = 99.625
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{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}
{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}
{{Multival|legend=1| 5 -40 -94 -75 -163 -106 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~200/189 = 99.664
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~200/189 = 99.664
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{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}
{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}


: Mapping generators: ~2, ~21/20
: Mapping generators: ~2, ~21/2
 
{{Multival|legend=1| 6 -48 -55 -90 -104 7 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 83.053
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 83.053
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{{Mapping|legend=1| 1 3 -9 2 | 0 -7 -56 4 }}
{{Mapping|legend=1| 1 3 -9 2 | 0 -7 -56 4 }}
{{Multival|legend=1| 7 56 -4 231 -26 -76 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~147/128 = 242.614
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~147/128 = 242.614
Line 2,179: Line 2,147:


: Mapping generators: ~2, ~243/224
: Mapping generators: ~2, ~243/224
{{Multival|legend=1| 5 -40 24 -75 24 168 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/224 = 140.350
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/224 = 140.350
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: Mapping generators: ~2, ~56/45
: Mapping generators: ~2, ~56/45
{{Multival|legend=1| 5 -40 -29 -75 -60 45 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~56/45 = 380.355
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~56/45 = 380.355
Line 2,267: Line 2,231:


: Mapping generators: ~25/21, ~3
: Mapping generators: ~25/21, ~3
{{Multival|legend=1| 4 -32 -68 -60 -119 -68 }}


[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~3/2 = 701.8234
[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~3/2 = 701.8234
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: Mapping generators: ~8575/7776, ~3
: Mapping generators: ~8575/7776, ~3
{{Multival|legend=1| 7 -56 49 -105 58 271 }}


[[Optimal tuning]] ([[POTE]]): ~8575/7776 = 1\7, ~3/2 = 701.702
[[Optimal tuning]] ([[POTE]]): ~8575/7776 = 1\7, ~3/2 = 701.702
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: Mapping generators: ~42875/39366, ~3
: Mapping generators: ~42875/39366, ~3
{{Multival|legend=1| 8 -64 88 -120 117 384 }}


[[Optimal tuning]] ([[POTE]]): ~42875/39366 = 1\8, ~3/2 = 701.713
[[Optimal tuning]] ([[POTE]]): ~42875/39366 = 1\8, ~3/2 = 701.713
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