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Schismatic temperament
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Schismatic temperament
A schismatic temperament is a musical tuning system that results from tempering the schisma of 32805:32768 (1.9537 cents) to a unison. It is also called the schismic temperament, Helmholtz temperament, or quasi-Pythagorean temperament.
In Pythagorean tuning all notes are tuned as a number of perfect fifths (701.96 cents ⓘ). The major third above C, E, is considered four fifths above C. This causes the Pythagorean major third, E+ (407.82 cents ⓘ), to differ from the just major third, E♮ (386.31 cents ⓘ): the Pythagorean third is sharper than the just third by 21.51 cents (a syntonic comma ⓘ).
Ellis's "skhismic temperament" instead uses the note eight fifths below C, F♭-- (384.36 cents ⓘ), the Pythagorean diminished fourth or schismatic major third. Though spelled "incorrectly" for a major third, this note is only 1.95 cents (a schisma) flat of E♮, and thus more in tune than the Pythagorean major third. As Ellis puts it, "the Fifths should be perfect and the Skhisma should be disregarded [accepted/ignored]."
In his eighth-schisma "Helmholtzian temperament" the note eight fifths below C is also used as the major third above C. However, in the "skhismic temperament" pure perfect fifths are used to construct an approximate major third, while in the "Helmholtzian temperament" approximate perfect fifths are used to construct a pure major third. To raise the Pythagorean diminished fourth 1.95 cents to a just major third each fifth must be narrowed, or tempered, by 1.95/8 = 0.24 cents. Thus, the fifth becomes 701.71 cents instead of 701.96 cents. As Ellis puts it, "the major Thirds are taken perfect, and the Skhisma is disregarded [tempered out]."
Whereas schismatic temperaments achieve a ratio with a number of fifths, each tempered by a fraction of the schisma; meantone temperaments achieve a ratio with fifths, each tempered by a fraction of the syntonic comma (81:80, 21.51 cents). As meantone temperaments are often described by what fraction of the syntonic comma is used to alter the perfect fifths, schismatic temperaments are often described by what fraction of the schisma is used to alter the perfect fifths (thus quarter-comma meantone temperament, eighth-schisma temperament, etc.).
In both eighth-schisma tuning and quarter-comma meantone the octave and major third are just, but eighth-schisma has much much more accurate perfect fifths and minor thirds (less than a quarter of a cent off from just intonation). However, quarter-comma meantone has a large advantage in that the major third and minor third are spelled as such, whereas in schismatic tunings, they're represented by the diminished fourth and augmented second (if spelled according to their construction in the tuning). This places them well outside the span of a single diatonic scale and requires both a larger number of pitches and more microtonal pitch-shifting when attempting common-practice Western music.
Various equal temperaments lead to schismatic tunings which can be described in the same terms. Dividing the octave by 53 provides an approximately 1/29-schisma temperament; by 65 a 1/5-schisma temperament, by 118 a 2/15-schisma temperament, and by 171 a 1/10-schisma temperament. The last named, 171, produces very accurate septimal intervals, but they are hard to reach, as to get to a 7/4 requires 39 fifths. The −1/11-schisma temperament of 94, with sharp rather than flat fifths, gets to a less accurate but more available 7:4 by means of 14 fourths. Eduardo Sabat-Garibaldi also had an approximation of 7:4 by means of 14 fourths in mind when he derived his 1/9-schisma tuning.
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Schismatic temperament AI simulator
(@Schismatic temperament_simulator)
Schismatic temperament
A schismatic temperament is a musical tuning system that results from tempering the schisma of 32805:32768 (1.9537 cents) to a unison. It is also called the schismic temperament, Helmholtz temperament, or quasi-Pythagorean temperament.
In Pythagorean tuning all notes are tuned as a number of perfect fifths (701.96 cents ⓘ). The major third above C, E, is considered four fifths above C. This causes the Pythagorean major third, E+ (407.82 cents ⓘ), to differ from the just major third, E♮ (386.31 cents ⓘ): the Pythagorean third is sharper than the just third by 21.51 cents (a syntonic comma ⓘ).
Ellis's "skhismic temperament" instead uses the note eight fifths below C, F♭-- (384.36 cents ⓘ), the Pythagorean diminished fourth or schismatic major third. Though spelled "incorrectly" for a major third, this note is only 1.95 cents (a schisma) flat of E♮, and thus more in tune than the Pythagorean major third. As Ellis puts it, "the Fifths should be perfect and the Skhisma should be disregarded [accepted/ignored]."
In his eighth-schisma "Helmholtzian temperament" the note eight fifths below C is also used as the major third above C. However, in the "skhismic temperament" pure perfect fifths are used to construct an approximate major third, while in the "Helmholtzian temperament" approximate perfect fifths are used to construct a pure major third. To raise the Pythagorean diminished fourth 1.95 cents to a just major third each fifth must be narrowed, or tempered, by 1.95/8 = 0.24 cents. Thus, the fifth becomes 701.71 cents instead of 701.96 cents. As Ellis puts it, "the major Thirds are taken perfect, and the Skhisma is disregarded [tempered out]."
Whereas schismatic temperaments achieve a ratio with a number of fifths, each tempered by a fraction of the schisma; meantone temperaments achieve a ratio with fifths, each tempered by a fraction of the syntonic comma (81:80, 21.51 cents). As meantone temperaments are often described by what fraction of the syntonic comma is used to alter the perfect fifths, schismatic temperaments are often described by what fraction of the schisma is used to alter the perfect fifths (thus quarter-comma meantone temperament, eighth-schisma temperament, etc.).
In both eighth-schisma tuning and quarter-comma meantone the octave and major third are just, but eighth-schisma has much much more accurate perfect fifths and minor thirds (less than a quarter of a cent off from just intonation). However, quarter-comma meantone has a large advantage in that the major third and minor third are spelled as such, whereas in schismatic tunings, they're represented by the diminished fourth and augmented second (if spelled according to their construction in the tuning). This places them well outside the span of a single diatonic scale and requires both a larger number of pitches and more microtonal pitch-shifting when attempting common-practice Western music.
Various equal temperaments lead to schismatic tunings which can be described in the same terms. Dividing the octave by 53 provides an approximately 1/29-schisma temperament; by 65 a 1/5-schisma temperament, by 118 a 2/15-schisma temperament, and by 171 a 1/10-schisma temperament. The last named, 171, produces very accurate septimal intervals, but they are hard to reach, as to get to a 7/4 requires 39 fifths. The −1/11-schisma temperament of 94, with sharp rather than flat fifths, gets to a less accurate but more available 7:4 by means of 14 fourths. Eduardo Sabat-Garibaldi also had an approximation of 7:4 by means of 14 fourths in mind when he derived his 1/9-schisma tuning.