music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspect ...

, the schisma (also spelled ''skhisma'') is the interval between a

Pythagorean comma In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C ...

(531441:524288) and a

syntonic comma In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ( ...

(81:80) and equals or 32805:32768 = 1.00113, which is 1.9537

cent Cent may refer to: Currency * Cent (currency), a one-hundredth subdivision of several units of currency * Penny (Canadian coin), a Canadian coin removed from circulation in 2013 * 1 cent (Dutch coin), a Dutch coin minted between 1941 and 1944 * ...

s (). It may also be defined as: * the difference between 8 justly tuned

perfect fifth In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval fro ...

s plus a justly tuned

major third In classical music, a third is a musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...

and 5 octaves; * the difference between

major limma In music, a major limma is an interval with the ratio of 135:128, which is the difference between two major tones (a ditone) and a minor third. It is equal to about 92.18 cents. Composer Ben Johnston uses a "−" with a "" as an accidental to ...

and

Pythagorean limma A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...

; * the difference between the syntonic comma and the

diaschisma The diaschisma (or diacisma) is a small musical interval defined as the difference between three octaves and four perfect fifths plus two major thirds (in just intonation). It can be represented by the ratio 2048:2025 and is about 19.5 cents. ...

. ''Schisma'' is a

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

word meaning a split (see

schism A schism ( , , or, less commonly, ) is a division between people, usually belonging to an organization, movement, or religious denomination. The word is most frequently applied to a split in what had previously been a single religious body, suc ...

) whose musical sense was introduced by

Boethius Anicius Manlius Severinus Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, ''magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the tr ...

at the beginning of the 6th century in the 3rd book of his 'De institutione musica'. Boethius was also the first to define

Andreas Werckmeister Andreas Werckmeister (November 30, 1645 – October 26, 1706) was a German organist, music theorist, and composer of the Baroque era. He was amongst the earliest advocates of equal temperament, and through this advocacy was highly influential to t ...

defined the ''grad'' as the twelfth root of the Pythagorean comma, or equivalently the difference between the justly tuned fifth (3/2) and the equally tempered fifth of 700 cents (2). This value, 1.955 cents, may be approximated by the ratio 886:885. This interval is also sometimes called a schisma. Curiously,

2^\times5^

appears very close to 4:3, the just

perfect fourth A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to ...

. This is because the difference between a grad and a schisma is so small. So, a

rational intonation In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and ch ...

version of

equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...

may be realized by flattening the fifth by a schisma rather than a grad, a fact first noted by

Johann Kirnberger Johann Philipp Kirnberger (also ''Kernberg''; 24 April 1721, Saalfeld – 27 July 1783, Berlin) was a musician, composer (primarily of fugues), and music theorist. He was a student of Johann Sebastian Bach. According to Ingeborg Allihn, Kirnberg ...

, a pupil of

Bach Johann Sebastian Bach (28 July 1750) was a German composer and musician of the late Baroque period. He is known for his orchestral music such as the '' Brandenburg Concertos''; instrumental compositions such as the Cello Suites; keyboard w ...

. Twelve of these Kirnberger fifths of 16384:10935 exceed seven octaves, and therefore fail to close, by the tiny interval of

2^\times3^\times5^

, the ''atom of Kirnberger'' of 0.01536 cents. Tempering out the schisma leads to

schismatic temperament 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. Construc ...

. As used by Descartes, a ''schisma'' added to a perfect fourth = 27:20 (519.55 cents), a schisma subtracted from a perfect fifth = 40:27 (680.45 cents), and a major sixth plus a schisma = 27:16 (= 81:48 = 905.87 cents).Ruth Katz,

Carl Dahlhaus Carl Dahlhaus (10 June 1928 – 13 March 1989) was a German musicologist who was among the leading postwar musicologists of the mid to late 20th-century. A prolific scholar, he had broad interests though his research focused on 19th- and 20th- ...

(1987). ''Contemplating Music: Substance'', p.523. . By this definition is a "schisma" is what is known as the

(81:80).

References

External links

*Joe Monzo, Kami Rousseau (2005).
Septimal-Comma
, ''Tonalsoft: Encyclopedia of Microtonal Music Theory''. Accessed 2015-06-06.

, ''Huygens-Fokker.org''. Accessed 2015-06-06. {{Intervals, state=expanded 5-limit tuning and intervals Commas (music)

See also

References

External links