Slendric: Difference between revisions

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For technical data, see [[Gamelismic clan #Slendric]].

== ~~Intervals~~ ==

== Theory ==

=== Interval categories ===

It is possible to define the intervals of slendric in terms of diatonic categories, for at three steps is the perfect fifth, and at every three steps further are all of the standard fifth-generated intervals. For the remaining steps, a single pair of inflections suffices: "super"/"sub", which can be abbreviated with the prefixes S and s, respectively. We define the slendric generator to be the supermajor second, and therefore the 2-generator interval is a subfourth (as a major second and a perfect fourth together reach a perfect fifth) as well as a supersupermajor third. Between a major third and perfect fourth is a minor second, which is therefore equivalent to three repetitions of "super" (implying that "super" is rigorously an inflection by the "quark" of 49/48~64/63); because of this equivalence, it is never necessary to attach more than one "super"/"sub" to a diatonic interval.

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| 27/14

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=== Relationship with acoustic phi ===

One may note that the two-generator interval is tempered somewhat flat of 21/16 in most tunings of slendric. This allows it to be interpretable as [[17/13]], tempering out [[273/272]] and [[833/832]], into which 1029/1024 factors. Combining that with the mapping of 55/32 at four generators, we obtain the representations 21/16, 17/13, [[55/42]], and [[72/55]] for this interval. If we look one octave higher, a pattern becomes clear: [[21/8]], [[34/13]], [[55/21]], and [[144/55]] are all ratios of two-apart Fibonacci numbers, which therefore closely approximate the square of [[acoustic phi]], entailing that acoustic phi squared over 2 is close to two slendric generators.

A single slendric generator, therefore, is approximable by acoustic phi divided by [[sqrt(2)]], whose precise value is about 233.0903 cents. Using this interval as a generator produces a form of slendric too sharp to be mothra but flat of 36edo, with a fifth about 2.7 cents flat, quite close to the 27\139 tuning in [[139edo]].

== Chords ==

@@ Line 21: / Line 21: @@
 For technical data, see [[Gamelismic clan #Slendric]].
-== Intervals ==
+== Theory ==
 === Interval categories ===
 It is possible to define the intervals of slendric in terms of diatonic categories, for at three steps is the perfect fifth, and at every three steps further are all of the standard fifth-generated intervals. For the remaining steps, a single pair of inflections suffices: "super"/"sub", which can be abbreviated with the prefixes S and s, respectively. We define the slendric generator to be the supermajor second, and therefore the 2-generator interval is a subfourth (as a major second and a perfect fourth together reach a perfect fifth) as well as a supersupermajor third. Between a major third and perfect fourth is a minor second, which is therefore equivalent to three repetitions of "super" (implying that "super" is rigorously an inflection by the "quark" of 49/48~64/63); because of this equivalence, it is never necessary to attach more than one "super"/"sub" to a diatonic interval.
@@ Line 246: / Line 246: @@
 | 27/14
 |}
+=== Relationship with acoustic phi ===
+One may note that the two-generator interval is tempered somewhat flat of 21/16 in most tunings of slendric. This allows it to be interpretable as [[17/13]], tempering out [[273/272]] and [[833/832]], into which 1029/1024 factors. Combining that with the mapping of 55/32 at four generators, we obtain the representations 21/16, 17/13, [[55/42]], and [[72/55]] for this interval. If we look one octave higher, a pattern becomes clear: [[21/8]], [[34/13]], [[55/21]], and [[144/55]] are all ratios of two-apart Fibonacci numbers, which therefore closely approximate the square of [[acoustic phi]], entailing that acoustic phi squared over 2 is close to two slendric generators.
+A single slendric generator, therefore, is approximable by acoustic phi divided by [[sqrt(2)]], whose precise value is about 233.0903 cents. Using this interval as a generator produces a form of slendric too sharp to be mothra but flat of 36edo, with a fifth about 2.7 cents flat, quite close to the 27\139 tuning in [[139edo]].
 == Chords ==