Schismatic Temperament - Comparison With Other Tunings

Comparison With Other Tunings

The quarter-comma meantone temperament obtains the ratio 5:1 with four tempered fifths (so that the difference (3/2)4/5 = 81/80 is tempered out). The schismatic temperament is analogous; it obtains the ratio 10:1 with eight tempered fourths Play (so that 10/(4/3)8 = 32805/32768 is tempered out). Meantone tunings are often described by what fraction of a syntonic comma the fifth has been flattened. In the same way, schismatic tunings can be described by what fraction of a schisma the fifth is flattened – or even sharpened.

An advantage of meantone over schismatic tunings is that in meantone, the interval ratios of 5:4 and 6:5 are represented by the major third and minor third, respectively. 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.