Schisma (liverwort)
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In
music Music is the arrangement of sound to create some combination of Musical form, form, harmony, melody, rhythm, or otherwise Musical expression, expressive content. Music is generally agreed to be a cultural universal that is present in all hum ...
, the schisma (also spelled ''skhisma'') is the interval between the
syntonic comma In music theory Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first i ...
(81:80) and the
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 ...
which is slightly larger. It equals or ≈ 1.00113, which corresponds to 1.9537  cents (). It may also be defined as: * the difference (in cents) 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 f ...
s plus a justly tuned
major third In music theory, a third is a Interval (music), musical interval encompassing three staff positions (see Interval (music)#Number, Interval number for more details), and the major third () is a third spanning four Semitone, half steps or two ...
and 5 octaves; * the ratio of
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 ...
to the
Pythagorean limma A semitone, also called a minor second, 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 t ...
; * the ratio of the syntonic comma and the
diaschisma The diaschisma (or diacisma) is a small interval (music), musical interval defined as the difference between three octaves and four perfect fifths plus two just major third, major thirds (in just intonation). It can be represented by the ratio 2 ...
. ''Schisma'' is a
Greek Greek may refer to: 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 of all kno ...
word meaning a split or crack (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 simply as Boethius (; Latin: ''Boetius''; 480–524 AD), was a Roman Roman Senate, senator, Roman consul, consul, ''magister officiorum'', polymath, historian, and philosopher of the Early Middl ...
at the beginning of the 6th century in the 3rd book of his ''De institutione musica''. Boethius was also the first to define the
diaschisma The diaschisma (or diacisma) is a small interval (music), musical interval defined as the difference between three octaves and four perfect fifths plus two just major third, major thirds (in just intonation). It can be represented by the ratio 2 ...
.
Andreas Werckmeister Andreas Werckmeister (November 30, 1645 – October 26, 1706) was a German organist, music theorist, and composer of the Baroque era. He was responsible for a temperament that resulted in all tonalities sounding acceptable on the keyboard. This im ...
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 well approximated by converting the ratio 886:885 to cents. This interval is also sometimes called a ''schisma''. Curiously, \ 2^\ 5^\ is very close to 4:3, the just
perfect fourth A fourth is a interval (music), 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 int ...
. 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 a tuning system in which the space between notes' frequencies (called intervals) is a whole number ratio. Intervals spaced in this way are said to be pure, and are called just intervals. Just inter ...
version of
equal temperament An equal temperament is a musical temperament or Musical tuning#Tuning systems, tuning system that approximates Just intonation, just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequency, frequencie ...
may be obtained 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 studied the organ with Johann Peter Kellner and Heinrich Nicolaus Gerber, and st ...
, a pupil of
Bach Johann Sebastian Bach (German: joːhan zeˈbasti̯an baχ ( – 28 July 1750) was a German composer and musician of the late Baroque period. He is known for his prolific output across a variety of instruments and forms, including the or ...
. Twelve of these Kirnberger fifths of exceed seven octaves, and therefore fail to close, by the tiny interval of \ \frac\ , called the ''atom of Kirnberger'' of 0.01536 cents. Tempering out the ''schisma'' leads to a
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. Construct ...
. Descartes used the word ''schisma'' to mean that which multiplied by a perfect fourth produces 27:20 (519.55 cents); his schisma divided into a perfect fifth produces 40:27 (680.45 cents), and a major sixth times a schisma is 27:16 (905.87 cents). However, by this definition a "schisma" would be what is better known as the
syntonic comma In music theory Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first i ...
(81:80).


See also

*
Twelfth root of two The twelfth root of two or \sqrt 2/math> (or equivalently 2^) is an algebraic irrational number, approximately equal to 1.0594631. It is most important in Western music theory, where it represents the frequency ratio ( musical interval) of a se ...


References


External links

* * {{Intervals, state=expanded 5-limit tuning and intervals Commas (music)