Perfect balance: Difference between revisions
Created page with "A non-empty set of real numbers ''S'' in the range <math>[0, 1)</math> is called '''perfectly balanced''' if a wheel with an equal weight placed at angle <math>2\pi x</math> f..." |
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It is easy to construct perfectly balanced scales that are not a subset of any EDO, by superimposing two scales within EDOs transposed by an irrational amount. There also exists a continuum of perfectly balanced scales that have no such decomposition. | It is easy to construct perfectly balanced scales that are not a subset of any EDO, by superimposing two scales within EDOs transposed by an irrational amount. There also exists a continuum of perfectly balanced scales that have no such decomposition. | ||
Milne at al. showed that an efficient convex optimization procedure exists that, given an arbitrary scale, computes the closest perfectly balanced scale according to a simple squared-difference metric. | Milne at al. showed that an efficient convex optimization procedure exists that, given an arbitrary scale, computes the closest perfectly balanced scale according to a simple squared-difference metric. For example, a perfectly balanced approximation to Ptolemy's diatonic scale [1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8] displaces by the following cent values: [0, +8.61, +20.00, +23.60, +20.03, +9.97, +0.72]. The resulting scale is given by the following [[Scala]] file: | ||
<pre> | |||
Ptolemy's diatonic scale approximated with perfect balance | |||
7 | |||
212.52434916042787 | |||
406.31587841801996 | |||
521.6495827358298 | |||
721.9824694633705 | |||
894.3281503142967 | |||
1088.9892227955181 | |||
2/1 | |||
</pre> | |||
[[Category:Scale]] | [[Category:Scale]] | ||
[[Category:Theory]] | [[Category:Theory]] | ||