Schismatic family: Difference between revisions
m +interwiki |
Re-title 5-limit as schismatic aka helmholtz; edo list reviewed |
||
| Line 7: | Line 7: | ||
The 5-limit parent comma for the '''schismatic family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[Didymus comma]] (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its [[monzo]] is {{monzo| -15 8 1 }}, and flipping that yields <<1 -8 -15|| for the [[wedgie]]. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)<sup>8</sup> × 32805/32768. | The 5-limit parent comma for the '''schismatic family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[Didymus comma]] (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its [[monzo]] is {{monzo| -15 8 1 }}, and flipping that yields <<1 -8 -15|| for the [[wedgie]]. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)<sup>8</sup> × 32805/32768. | ||
= | = Schismatic aka Helmholtz = | ||
The 5-limit version of the temperament is a [[microtemperament]], sometimes called '''Helmholtz''' 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 cents, 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 '''Helmholtz''' 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 cents, 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. | ||
| Line 14: | Line 14: | ||
Mapping generator: ~3 | Mapping generator: ~3 | ||
Map: [ | Map: [{{val| 1 0 15 }}, {{val| 0 1 -8 }}] | ||
{{EDOs|legend=1| 12, 29, 41, 53 | {{EDOs|legend=1| 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bc }} | ||
Badness: 0.00426 | Badness: 0.00426 | ||
== Seven limit | == Seven-limit extensions == | ||
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding [[garischisma|{{monzo|25 -14 0 -1}}]] gives garibaldi, {{monzo|-44 26 0 1}} grackle, [[64/63|{{monzo|6 -2 0 -1}}]] schism and {{monzo|-59 39 0 -1}} pontiac; these all have a fifth as generator. Bischismic adds {{monzo|-69 40 0 2}} and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = {{monzo|-10 1 0 3}}, with an 8/7 generator, three of which give the fifth, and term adds {{monzo|-94 54 0 3}} with a 1/3 octave period. Sesquiquartififths adds {{monzo|-35 15 0 4}} and slices the fifth in four. | The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding [[garischisma|{{monzo|25 -14 0 -1}}]] gives garibaldi, {{monzo|-44 26 0 1}} grackle, [[64/63|{{monzo|6 -2 0 -1}}]] schism and {{monzo|-59 39 0 -1}} pontiac; these all have a fifth as generator. Bischismic adds {{monzo|-69 40 0 2}} and has a fifth generator with a half-octave period. Guiron adds [[1029/1024]] = {{monzo|-10 1 0 3}}, with an 8/7 generator, three of which give the fifth, and term adds {{monzo|-94 54 0 3}} with a 1/3 octave period. Sesquiquartififths adds {{monzo|-35 15 0 4}} and slices the fifth in four. | ||
Temperaments not discussed here include [[Sensamagic clan #Salsa|salsa]], [[Porwell temperaments #Hemischis|hemischis]] and [[Turkish maqam music temperaments #Karadeniz temperament|karadeniz]]. | Temperaments not discussed here include [[Sensamagic clan #Salsa|salsa]], [[Porwell temperaments #Hemischis|hemischis]] and [[Turkish maqam music temperaments #Karadeniz temperament|karadeniz]]. | ||