View source for Schisma

{{Short description|Musical interval}}
{{Distinguish|Schism (disambiguation){{!}}Schism}}
[[Image:Schisma.png|thumb|Schisma as difference in cents between 8&nbsp;[[perfect fifth]]s plus 1&nbsp;[[just major third]] and 5&nbsp;octaves.]]
[[Image:Schisma on C.png|thumb|right|Schisma on C {{audio|Schisma on C.mid|Play}}. Note that the note depicted lower on the staff (B#++) is higher in pitch (than C{{music|natural}}).]]

In [[music]], the '''schisma''' (also spelled ''skhisma'') is the [[musical interval|interval]] between the [[syntonic comma]] (81:80) and the [[Pythagorean comma]] {{nobr|( 531 441 : 524 288 ),}} which is slightly larger. It equals {{sfrac|&nbsp; 5 × 3{{sup|8}} &nbsp;|&nbsp; 2{{sup|15}} &nbsp;}} or {{nobr| 32 805 : 32 768 }} &approx; 1.00113,<ref>{{cite book |last=Benson |first=Dave |year=2006 |title=Music: A mathematical offering |page=171 |ISBN=0-521-85387-7 }}</ref><ref>{{cite book |editor-link=Willi Apel |editor-last=Apel |editor-first=W. |year=1961 |title=Harvard Dictionary of Music |page=188 |ISBN=0-674-37501-7 }}</ref> which corresponds to 1.9537&nbsp;[[cent (music)|cent]]s ({{Audio|Schisma on C.mid|Play}}). It may also be defined as:
* the difference (in cents) between 8&nbsp;[[just intonation|justly tuned]] [[perfect fifth]]s plus a justly tuned [[major third]] and 5&nbsp;octaves;
* the ratio of [[major limma]] to the [[Pythagorean limma]];
* the ratio of the syntonic comma and the [[diaschisma]].

''Schisma'' is a [[Greek language|Greek]] word meaning a split or crack (see [[wikt:schism|schism]]) whose musical sense was introduced by [[Boethius]] at the beginning of the 6th&nbsp;century in the 3rd&nbsp;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&nbsp;cents (2{{sup|7/12}}).<ref>{{cite web |url=http://www.huygens-fokker.org/docs/measures.html |title=Logarithmic interval measures |website=Huygens-Fokker.org |access-date=2015-06-06 |df=dmy-all}}</ref> This value, 1.955&nbsp;cents, may be well approximated by converting the ratio 886:885 to cents.<ref>{{cite web |last=Monzo |first=Joe |year=2005 |url=http://www.tonalsoft.com/enc/g/grad.aspx |title=Grad |website=TonalSoft.com |access-date=2015-06-06 |df=dmy-all}}</ref> This interval is also sometimes called a ''schisma''.

Curiously, <math>\ 2^{1/12}\ 5^{1/7}\ </math> 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 {{nobr| 16 384 : 10 935 }} exceed seven octaves, and therefore fail to close, by the tiny interval of <math>\ \frac{\ 2^{161}\ }{\ 3^{84}\ 5^{12}\ }\ ,</math> called the ''atom of Kirnberger'' of 0.01536&nbsp;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&nbsp;cents); his schisma divided into a perfect fifth produces 40:27 (680.45&nbsp;cents), and a major sixth times a schisma is 27:16 (905.87&nbsp;cents).<ref>{{cite book |first1=Ruth |last1=Katz |first2=Carl |last2=Dahlhaus |author2-link=Carl Dahlhaus |year=1987 |title=Contemplating Music: Substance |page=523 |ISBN=0-918728-60-6 }}</ref> However, by this definition a "schisma" would be what is better known as the [[syntonic comma]] (81:80).

==See also==
* [[Twelfth root of two]]

==References==
{{reflist|25em}}

==External links==
* {{cite web |first1=Joe |last1=Monzo |first2=Kami |last2=Rousseau |year=2005 |url=http://tonalsoft.com/enc/s/septimal-comma.aspx |title=Septimal-comma |website=TonalSoft.com |series=Encyclopedia of Microtonal Music Theory |access-date=2015-06-06 |df=dmy-all }}
* {{cite web |url=http://www.huygens-fokker.org/docs/intervals.html |title=List of intervals |website=Huygens-Fokker.org |access-date=2015-06-06 |df=dmy-all}}

{{Intervals|state=expanded}}

[[Category:5-limit tuning and intervals]]
[[Category:Commas (music)]]