Slendric: Difference between revisions

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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]].

A single slendric generator, therefore, is approximable by acoustic phi divided by [[sqrt(2)]]. This can also be explained by 18 being the 6th Lucas number, and therefore a close approximation to φ6; approximating 181/6 by φ gives us φ/√2 as an approximation of (3/2)1/3. This interval's precise value is about 233.0903{{c}}, and using it 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 (233.0935{{c}}) tuning in [[139edo]].

== Chords ==

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 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]].
+A single slendric generator, therefore, is approximable by acoustic phi divided by [[sqrt(2)]]. This can also be explained by 18 being the 6th Lucas number, and therefore a close approximation to φ<sup>6</sup>; approximating 18<sup>1/6</sup> by φ gives us φ/√2 as an approximation of (3/2)<sup>1/3</sup>. This interval's precise value is about 233.0903{{c}}, and using it 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 (233.0935{{c}}) tuning in [[139edo]].
 == Chords ==