Schismic temperament
Schismic temperament is the temperament which results from tempering the schisma of 32805/32768 to a unison. Analogously to the way in which meantone temperament approximates an interval of 5 by four fifths, so that (3/2)4/5 = 81/80 is tempered to a unison, schismic temperament approximates an interval of 10 by eight fourths, so that 10/(4/3)8 = 32805/32786 is tempered to a unison. Meantone tunings are often described in terms of what fraction of a syntonic comma the fifth has been flattened by; in the same way schismic tunings can be described in terms of what fraction of a schisma the fifth is flattened--or even sharpened--by.Historically significant is the 1/8-schisma tuning of Hermann von Helmhotz and Norwegian composer Eivind Groven. For this Helmholtz built a reed organ he called a Harmonium, with 24 tones to the octave, and Groven an organ with 36 tones to the octave which had the ability to further adjust its tuning during performances. It may be considered the analog of 1/4-comma meantone, as it also has pure major thirds; though of course with vastly more accurate fifths and minor thirds, less than a quarter of a cent off from just intonation. A 1/9-schismic tuning has also been proposed by Eduard Sabat-Garibaldi, who together with his students uses a 53-tone to the octave guitar with this tuning.
Various equal temperaments lead to schismic tunings which can be described in the same terms. Equal division by 12 leads to the 1-schisma tuning, but this fails to take advantage of the nearly pure harmonies of schismic temperament. Dividing the octave by 53 is approximately a 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.
Mark Lindley and Ronald Turner-Smith argue that schismic temperament was briefly in use during the late medieval period. Justly tuned fifths are a reasonble schismic tuning and therefore schismic is in some respects an easier way to introduce justly tuned thirds into a Pythagorean harmonic fabric than meantone.