In music, the **schisma** (also spelled *skhisma*) is the interval between the syntonic comma (81:80) and the Pythagorean comma ( 531 441 : 524 288 ), which is slightly larger. It equals 5 × 3^{8} / 2^{15} or 32 805 : 32 768 ≈ 1.00113,^{[1]}^{[2]} which corresponds to 1.9537 cents ( ). It may also be defined as:

- the difference (in cents) between 8 justly tuned perfect fifths plus a justly tuned major third and 5 octaves;
- the ratio of major limma to the Pythagorean limma;
- the ratio of the syntonic comma and the diaschisma.

*Schisma* is a Greek word meaning a split or crack (see schism) whose musical sense was introduced by Boethius 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.

Andreas Werckmeister 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^{7/12}).^{[3]} This value, 1.955 cents, may be well approximated by converting the ratio 886:885 to cents.^{[4]} This interval is also sometimes called a *schisma*.

Curiously, is very close to 4:3, the just perfect fourth. This is because the difference between a grad and a schisma is so small. So, a rational intonation version of equal temperament may be obtained by flattening the fifth by a *schisma* rather than a *grad*, a fact first noted by Johann Kirnberger, a pupil of Bach. Twelve of these Kirnberger fifths of 16 384 : 10 935 exceed seven octaves, and therefore fail to close, by the tiny interval of called the *atom of Kirnberger* of 0.01536 cents.

Tempering out the *schisma* leads to a schismatic temperament.

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).^{[5]} However, by this definition a "schisma" would be what is better known as the syntonic comma (81:80).

**^**Benson, Dave (2006).*Music: A mathematical offering*. p. 171. ISBN 0-521-85387-7.**^**Apel, W., ed. (1961).*Harvard Dictionary of Music*. p. 188. ISBN 0-674-37501-7.**^**"Logarithmic interval measures".*Huygens-Fokker.org*. Retrieved 6 June 2015.**^**Monzo, Joe (2005). "Grad".*TonalSoft.com*. Retrieved 6 June 2015.**^**Katz, Ruth; Dahlhaus, Carl (1987).*Contemplating Music: Substance*. p. 523. ISBN 0-918728-60-6.

- Monzo, Joe; Rousseau, Kami (2005). "Septimal-comma".
*TonalSoft.com*. Encyclopedia of Microtonal Music Theory. Retrieved 6 June 2015. - "List of intervals".
*Huygens-Fokker.org*. Retrieved 6 June 2015.

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Other intervals |
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