Schisma: Difference between revisions
m Normalising usage of Infobox Interval |
added temperaments. the information given is NOT duplicate information; there is some commas mentioned not expressed on the pages for the temps and there is CTE generators shown |
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== Trivia == | == 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. | 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. | ||
== Schismic temperaments derivable from its S-expressions == | |||
=== [[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: | |||
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{{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 | |||
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 | |||
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 | |||
CTE generators: (2/1,) 3/2 = 701.705{{cent}}, 7/4 = 968.928{{cent}} | |||
== See also == | == See also == | ||
* [[Unnoticeable comma]] | * [[Unnoticeable comma]] | ||
div_iv( mul_iv(s(15),s(15)), (3125**1,3087**1) ) | |||
[[Category:Schismatic]] | [[Category:Schismatic]] | ||