Schismatic family: Difference between revisions

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~~__FORCETOC__~~

<span style="display: block; text-align: right;">Other languages: [[:de:Schismatische_Temperaturen|Deutsch]]</span>

=Five limit=

= Five limit =

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 [[Wedgies_and_Multivals|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)^8 * 32805/32768.

The 5-limit version of the temperament is a [[Microtempering|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 ~~Schizma~~, with pure minor thirds and a minutely better 5th, or 2/17 ~~schizma~~, with both thirds flat by 1/17 of a ~~schizma~~, although the differences would be very hard to distinguish unless using a large gamut.

The 5-limit version of the temperament is a [[Microtempering|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.

[[~~POTE_tuning|~~POTE generator]]: ~3/2 = 701.736

[[POTE generator]]: ~3/2 = 701.736

Mapping generator: ~3

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Badness: 0.00426

=Seven limit children=

== Seven limit children ==

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 [[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]].

= Garibaldi =

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[[Category:Schismatic]]

[[Category:Schismatic family| ]]

[[Category:Rank 2]]

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-__FORCETOC__
 <span style="display: block; text-align: right;">Other languages: [[:de:Schismatische_Temperaturen|Deutsch]]</span>
-=Five limit=
+= Five limit =
 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 &lt;&lt;1 -8 -15|| for the [[Wedgies_and_Multivals|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)^8 * 32805/32768.
-The 5-limit version of the temperament is a [[Microtempering|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 Schizma, with pure minor thirds and a minutely better 5th, or 2/17 schizma, with both thirds flat by 1/17 of a schizma, although the differences would be very hard to distinguish unless using a large gamut.
+The 5-limit version of the temperament is a [[Microtempering|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.
-[[POTE_tuning|POTE generator]]: ~3/2 = 701.736
+[[POTE generator]]: ~3/2 = 701.736
 Mapping generator: ~3
@@ Line 18: / Line 16: @@
 Badness: 0.00426
-=Seven limit children=
+== Seven limit children ==
-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 [[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]].
 = Garibaldi =
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 [[Category:Schismatic]]
 [[Category:Schismatic family| ]] <!-- main article -->
+[[Category:Rank 2]]
+{{todo|review}}