Schisma: Difference between revisions

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The '''schisma''', '''32805/32768''', is the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])4/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])3/([[64/45]]).

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]])4/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])3/([[64/45]]).

== 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|(2-1 443-1 887)}}, it is used interchangably with this interval in some of Helmholtz' writing.

== Temperaments ==

Tempering out this comma 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.~~

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.

~~=== Nestoria ===~~

~~{{See also| No-sevens~~ subgroup ~~temperaments #Nestoria }}~~

~~As the schisma is expressible as [[361/360|S19~~]]~~/([[1216/1215|S16/S18]])2 and ([[513/512|S15/S20]])/([[1216/1215|S16/S18]]), we can derive the 12 & 53 temperament:~~

~~=== Garibaldi ===~~

~~{{Main| Garibaldi }}~~

~~As the schisma is also equal~~ to ~~[[225/224|S15]]/([[5120/5103|S8/S9]]), we can derive~~ the ~~41&53 temperament:~~

~~==== 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&41 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~~

== Other intervals ==

~~{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 106d, 118, 171, 289h, 460hh }}~~

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 ==

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== See also ==

* [[Pythagorean tuning]]

* [[Unnoticeable comma]]

[[Category:Schismatic]]

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

@@ Line 13: / Line 13: @@
 {{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]]).
+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]]).
+== 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.
 == Temperaments ==
-Tempering out this comma 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.
+{{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.
-=== Nestoria ===
-{{See also| No-sevens subgroup temperaments #Nestoria }}
-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:
-=== Garibaldi ===
-{{Main| Garibaldi }}
-As the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]]), we can derive the 41&53 temperament:
-==== 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&41 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
+== Other intervals ==
-{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 106d, 118, 171, 289h, 460hh }}
+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 ==
@@ Line 67: / Line 30: @@
 == See also ==
+* [[Pythagorean tuning]]
 * [[Unnoticeable comma]]
 [[Category:Schismatic]]
+[[Category:Commas named for their regular temperament properties]]