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all the temps mentioned here except one (which is undocumented AFAIK!) are both non-duplicate (are presented to link and refer to other existing temps and only give an overview) and are here because they are simple and efficient ways of expressing the schisma. i thought it's agreed not to delete info and make the wiki less navigable
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{{Wikipedia| Schisma }}
{{Wikipedia| Schisma }}


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]]).  
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]]).
 
== Other intervals ==
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]], [[Alpharabian schisma]] and [[tridecaschisma]].


== Temperaments ==
== Temperaments ==
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.
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.


== History and etymology ==
=== Nestoria ===
{{See also| No-sevens subgroup temperaments #Nestoria }}
 
Nestoria tempers out [[361/360]] (S19) and [[513/512]] (S15/S20), and can be described as the 12 & 53 temperament in the 2.3.5.19 subgroup. This is derived since the schisma is expressible as [[361/360|S19]]/([[1216/1215|S16/S18]])<sup>2</sup> and ([[513/512|S15/S20]])/([[1216/1215|S16/S18]]).
 
=== Garibaldi ===
{{Main| Garibaldi }}
 
Garibaldi tempers out [[225/224]] (S15) and [[5120/5103]] (S8/S9), and can be described as the 41 & 53 temperament in the 7-limit. This is derived since the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]]).
 
==== 2.3.5.7.19 subgroup ====
{{Main| Garibaldi }}
 
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.
 
=== 2.3.5.7.17 12 & 118 & 171 (unnamed) ===
As the schisma also equals [[57375/57344|S15/S16]] * [[1701/1700|S18/S20]], we can derive the extremely accurate 12 & 118 & 171 temperament:
 
[[Subgroup]]: 2.3.5.7.17
 
[[Comma list]]: 1701/1700, 32805/32768
 
{{mapping|legend=1| 1 0 15 0 -32 | 0 1 -8 0 21 | 0 0 0 1 1 }}
 
: mapping generators: ~2, ~3, ~7
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7197, ~7/4 = 968.8307
 
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 106d, 118, 171, 472, 525, 643, 814, 985, 1799, 2324, 2495, 3138b, 3309bd, 4294bdg }}
 
==== 2.3.5.7.17.19 12 & 118 & 171 (unnamed) ====
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
 
[[Comma list]]: 361/360, 513/512, 1701/1700
 
{{mapping|legend=1| 1 0 15 0 -32 9 | 0 1 -8 0 21 -3 | 0 0 0 1 1 0 }}
 
: mapping generators: ~2, ~3, ~7
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7053, ~7/4 = 968.9281
 
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 106d, 118, 171, 289h, 460hh }}
 
{{Todo| improve readability }}
 
=== 2.3.5.41 53 & 65 (unnamed) ===
The schisma can additionally split into two superparticular commas in the 41-limit: 32805/32768 = ([[1025/1024]])*([[6561/6560]]). Tempering both of these out provides a natural mapping for prime 41, if a little less practical than those for 19 or 7.
 
== 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.
''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.
== Other intervals ==
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]].


== Trivia ==
== Trivia ==
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[[Category:Schismic]]
[[Category:Schismic]]
[[Category:Commas named for their regular temperament properties]]
Retrieved from "https://en.xen.wiki/w/Schisma"