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 == | ||
Revision as of 18:01, 5 June 2023
| Interval information |
reduced harmonic
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). Tempering it out gives a 5-limit microtemperament called schismatic, schismic or Helmholtz, which if extended to larger subgroups leads to the schismatic family of temperaments.
Schismic temperaments derivable from its S-expressions
Nestoria
As the schisma is expressible as S19/(S16/S18)2 and (S15/S20)/(S16/S18), we can derive the 12&53 temperament:
Subgroup: 2.3.5.19
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
CTE generator: 701.684 ¢
Garibaldi
As the schisma is also equal to S15/(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 ¢
2.3.5.7.19[53&147] (garibaldi nestoria)
Adding Nestoria to Garibaldi (tempering 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
CTE generator: 702.043 ¢
2.3.5.7.17[12&130&171] (unnamed)
As the schisma also equals S15/S16 * 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
CTE generators: (2/1,) 3/2 = 701.72 ¢, 7/4 = 968.831 ¢
2.3.5.7.17.19[12&130&171] (unnamed Nestoria)
By tempering S16/S18 we equate S15 with 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
CTE generators: (2/1,) 3/2 = 701.705 ¢, 7/4 = 968.928 ¢
Trivia
The schisma explains how the greatly composite numbers 1048576 (220) and 104976 (184) look alike in decimal. The largest common power of two between these numbers is 25, (when 1049760 is written to equalize) and when reduced by that, 1049760/1048576 becomes 32805/32768.