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.

Construction

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 ).

:C — G — D — A — E

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]."

:E <math>\approx</math> F

:F — C — G — D — A — E — B — F — C

In his eighth-schisma "Helmholtzian temperament" This was not a temperament but merely 12-tone Pythagorean tuning. Justly tuned fifths and fourths generate a reasonable schismatic tuning and therefore schismatic is in some respects an easier way to introduce approach justly tuned thirds into a Pythagorean harmonic fabric than meantone. However, the result suffers from the same difficulties as just intonation – for example, the wolf B-G here arises all too easily when availing oneself of the concordant schismatic substitutions just outlined – so it is not surprising that meantone temperament became the dominant tuning system by the early Renaissance. Helmholtz's and Groven's systems get around some, but not all, of these difficulties by including multiple tunings for each key on the keyboard, so that a particular note can be tuned as G in some contexts and F in others, for example.

References

  • "Schismic Temperaments", Intonation Information.
  • "Schismic", Tonalsoft(R) - Encyclopedia of microtonal music theory.