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{{Wikipedia| Schisma }}
{{Wikipedia| Schisma }}


The '''schisma''', '''32805/32768''', is the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]). Tempering it out gives a [[5-limit]] microtemperament called [[Schismatic family#Schismatic aka Helmholtz|schismatic, schismic or Helmholtz]], which if extended to larger subgroups leads to the [[schismatic family]] of temperaments.
The '''schisma''', '''32805/32768''', is a small interval about 2 [[cent]]s. It arises as the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]).  


== Schismic temperaments derivable from its S-expressions ==
== History and etymology ==
''Schisma'' is a borrowing of Ancient Greek, meaning "split". The term was first used by [[Boethius]] (6th century), in his ''De institutione musica'', using it to refer to half of the [[Pythagorean comma]]. The modern sense was introduced by [[Helmholtz]]' ''On the Sensations of Tone'', in particular the translation by [[Alexander Ellis]], where it is spelled ''skhisma''. Since it is extremely close to the [[superparticular]] ratio [[887/886]] {{nowrap|(2<sup>-1</sup> 443<sup>-1</sup> 887)}}, it is used interchangably with this interval in some of Helmholtz' writing.


===[[Nestoria]]===
== Temperaments ==
As the schisma is expressible as [[361/360|S19]]/([[1216/1215|S16/S18]])<sup>2</sup> and ([[513/512|S15/S20]])/([[1216/1215|S16/S18]]), we can derive the 12&53 temperament:
{{main|Schismatic family}}
Tempering out this comma gives a [[5-limit]] microtemperament called [[schismic|schismatic, schismic or helmholtz]], which if extended to larger [[subgroup]]s leads to the [[schismatic family]] of temperaments.


[[Subgroup]]: 2.3.5.19
== Other intervals ==


Patent EDO tunings: 12, 17, 24, 29, 36, 41, 53, 65, 77, 82, 89, 94, 101, 106, 118, 130, 135, 142, 147, 154, 159, 171, 183, 195, 207, 219, 248, 260, 272
Commas arising from the difference between a stack of Pythagorean intervals and other primes may also be called schismas. The difference between the [[Pythagorean comma]] and [[septimal comma]] is called the [[septimal schisma]]. Other examples are [[undevicesimal schisma]] and [[Alpharabian schisma]].
 
[[CTE]] generator: 701.684{{cent}}
 
===[[Garibaldi]]===
As the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]]), we can derive the 41&53 temperament:
 
[[Subgroup]]: 2.3.5.7
 
Patent EDO tunings: 12, 29, 41, 53, 82, 94, 106, 135, 147
 
[[CTE]] generator: 702.059{{cent}}
 
==== 2.3.5.7.19[53&147] (garibaldi nestoria) ====
Adding Nestoria to Garibaldi (tempering [[400/399|S20]]) results in an extremely elegant temperament which has all of the same patent tunings that Garibaldi has but which includes a mapping for 19 through Nestoria.
 
[[Subgroup]]: 2.3.5.7.19
 
Patent EDO tunings: 12, 29, 41, 53, 82, 94, 106, 135, 147
 
{{mapping|legend=1| 1 1 7 11 6 | 0 1 -8 -14 -3 }}
 
[[CTE]] generator: 702.043{{cent}}
 
=== 2.3.5.7.17[12&130&171] (unnamed) ===
As the schisma also equals [[57375/57344|S15/S16]] * [[1701/1700|S18/S20]], we can derive the extremely accurate 12&41 temperament:
 
[[Subgroup]]: 2.3.5.7.17
 
Patent EDO tunings < 300 (largest is 2548): 12, 29, 41, 53, 118, 130, 142, 159, 171, 183, 212, 224, 236, 289
 
{{mapping|legend=1| 1 1 7 2 -9 | 0 1 -8 0 21 | 0 0 0 1 1 }}
 
[[CTE]] generators: (2/1,) 3/2 = 701.72{{cent}}, 7/4 = 968.831{{cent}}
 
==== 2.3.5.7.17.19[12&130&171] (unnamed Nestoria) ====
By tempering [[1216/1215|S16/S18]] we equate [[225/224|S15]] with [[400/399|S20]] (tempering the other comma of Nestoria) because of S15~S16~S18~S20, leading to:
 
[[Subgroup]]: 2.3.5.7.17.19
 
Patent EDO tunings: 12, 29, 41, 53, 118, 130, 142, 159, 171, 183
 
{{mapping|legend=1| 1 1 7 2 -9 6 | 0 1 -8 0 21 -3 | 0 0 0 1 1 0 }}
 
[[CTE]] generators: (2/1,) 3/2 = 701.705{{cent}}, 7/4 = 968.928{{cent}}


== Trivia ==
== Trivia ==
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== See also ==
== See also ==
* [[Pythagorean tuning]]
* [[Unnoticeable comma]]
* [[Unnoticeable comma]]


[[Category:Schismatic]]
[[Category:Schismatic]]
[[Category:Commas named for their regular temperament properties]]
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