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]])<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 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. | ||
== Schismic temperaments derivable from its S-expressions == | == Schismic temperaments derivable from its S-expressions == | ||
=== [[Nestoria]] === | ===[[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: | 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: | ||
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CTE generator: 701.684{{cent}} | CTE generator: 701.684{{cent}} | ||
=== [[Garibaldi]] === | ===[[Garibaldi]]=== | ||
As the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]]), we can derive the 41&53 temperament: | As the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]]), we can derive the 41&53 temperament: | ||
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CTE generators: (2/1,) 3/2 = 701.705{{cent}}, 7/4 = 968.928{{cent}} | CTE generators: (2/1,) 3/2 = 701.705{{cent}}, 7/4 = 968.928{{cent}} | ||
== Trivia == | |||
The schisma explains how the greatly composite numbers 1048576 (2<sup>20</sup>) and 104976 (18<sup>4</sup>) look alike in decimal. The largest common power of two between these numbers is 2<sup>5</sup>, (when 1049760 is written to equalize) and when reduced by that, 1049760/1048576 becomes 32805/32768. | |||
== See also == | == See also == | ||