Schismatic family: Difference between revisions

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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.

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.

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

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

Schismatic family: Difference between revisions

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