Schismatic temperament is the temperament which results from tempering the schisma of 32805:32768 to a unison. ("Schismic" is an idiosyncratic spelling or abbreviation, promulgated in the late 1990s by Graham Breed, and having no precedent in the literature.)

Analogously to the way in which meantone temperament approximates the ratio 5:1 by four fifths, so that (3/2)⁴/5 = 81/80 is tempered to a unison, schismatic temperament approximates the ratio 10:1 by eight fourths, so that 10/(4/3)⁸ = 32805/32768 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 schismatic 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. Groven built an organ internally equipped with 36 tones to the octave which had the ability to adjust its tuning automatically during performances; the performer plays a familiar 12-key (per octave) keyboard and in most cases the mechanism will choose from among the three tunings for each key so that the chords played sound virtually in just intonation. Abstractly, 1/8-schisma tuning may be considered the analog, among schismatic tunings, of 1/4-comma meantone among meantone tunings, as it also has pure interval ratios of 2:1 and 5:4, though with vastly more accurate interval ratios of 3:2 and 6:5 (less than a quarter of a cent off from just intonation) than its meantone counterpart. A 1/9-schisma tuning has also been proposed by Eduardo Sabat-Garibaldi, who together with his students uses a 53-tone to the octave guitar with this tuning.

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, thus requiring both a larger number of pitches and more microtonal pitch-shifting when attempting to realize common-practice Western music in a schismatic tuning.

Various equal temperaments lead to schismatic 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 schismatic 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. 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.

Mark Lindley and Ronald Turner-Smith argue that schismatic tuning was briefly in use during the late medieval period. This was not temperament but merely 12-tone Pythagorean tuning, though typically tuned from Gb to B in ascending just fifths and descending just fourths, instead of the prevalent Ab to C# or Eb to G# schemes. This allowed concordant diminished fourths D-Gb, A-Db, E-Ab, and B-Eb and augmented seconds Gb-A, Db-E, and Ab-B to be used in place of the discordant Pythagorean major thirds D-F#, A-C#, E-G#, and B-D# and minor thirds F#-A, C#-E, and G#-B, respectively. Justly tuned fifths and fourths generate a reasonable schismatic tuning and therefore schismatic is in some respects an easier way to introduce 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-Gb here arises all too easily when availing oneself of the concordant schismatic substitutions just outlined -- so it is not suprising 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 Gb in some contexts and F# in others, for example.

External links

http://www.microtonal.co.uk/schismic.htm

http://www.utoronto.ca/conf2000/abs4-6.html

http://tonalsoft.com/monzo/groven/groven.htm

References

Lindley, Mark and Turner-Smith, Ronald, Mathematical Models of Musical Scales: A New Approach Orpheus-Schriftenreihe zu Grundfragen der Musik vol. 66, Verlag für systematische Musikwissenschaft, Bonn-Bad Godesberg, 1993