Schismatic temperament
Encyclopedia
The schismatic temperament is a musical tuning
system that results from tempering the schisma
of 32805:32768 to a unison. It is also called the schismic temperament or Helmholtz 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 (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 temperament
s 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.
and Norwegian composer Eivind Groven
. Helmholtz had a special Physharmonica (a harmonium
by Schiedmayer) 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 much 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.
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 G to B in ascending just fifths and descending just fourths, instead of the prevalent A to C or E to G schemes. This allowed concordant diminished fourths D-G, A-D, E-A, and B-E and augmented seconds G-A, D-E, and A-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-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.
Musical tuning
In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
system that results from tempering the schisma
Schisma
In music, the schisma is the ratio between a Pythagorean comma and a syntonic comma and equals 32805:32768, which is 1.9537 cents...
of 32805:32768 to a unison. It is also called the schismic temperament or Helmholtz temperament.
Comparison with other tunings
The quarter-comma meantone temperamentMeantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...
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 (so that 10/(4/3)8 = 32805/32768 is tempered out). Meantone tunings are often described by what fraction of a syntonic comma
Syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
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 temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
s 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.
History of schismatic temperaments
Historically significant is the 1/8-schisma tuning of Hermann von HelmholtzHermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...
and Norwegian composer Eivind Groven
Eivind Groven
Eivind Groven was a Norwegian microtonal composer and music-theorist. He was from Telemark and had his background in the folk music of the area.- Biography :...
. Helmholtz had a special Physharmonica (a harmonium
Harmonium
A harmonium is a free-standing keyboard instrument similar to a reed organ. Sound is produced by air being blown through sets of free reeds, resulting in a sound similar to that of an accordion...
by Schiedmayer) 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
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
. 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 much 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.
Mark Lindley
Mark Lindley
Mark Lindley is both a noted musicologist and, more recently, an historian of modern India. Born in Washington DC, he studied at Harvard University , Juilliard School of Music and Columbia University...
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
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
, though typically tuned from G to B in ascending just fifths and descending just fourths, instead of the prevalent A to C or E to G schemes. This allowed concordant diminished fourths D-G, A-D, E-A, and B-E and augmented seconds G-A, D-E, and A-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
Wolf interval
In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament...
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
Meantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...
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.
External links
- "Schismic Temperaments", Intonation Information.