Step variety: Difference between revisions
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* Equal tunings contain MOS scales and ternary scales, but the group generated by the step sizes in these tunings of the scales must be rank 1. | * Equal tunings contain MOS scales and ternary scales, but the group generated by the step sizes in these tunings of the scales must be rank 1. | ||
* Certain chroma-altered MOS scales, which are contained in the group generated by the period and the generator of the unaltered MOS are ternary. An example is harmonic minor in any non-edo diatonic tuning, a chroma-alteration of the diatonic MOS with step pattern msmmsLs. | * Certain chroma-altered MOS scales, which are contained in the group generated by the period and the generator of the unaltered MOS are ternary. An example is harmonic minor in any non-edo diatonic tuning, a chroma-alteration of the diatonic MOS with step pattern msmmsLs. | ||
The term ''n-ary'' disregards the rank of the group generated by the step sizes, although an ''n''-ary scale is still ''generically'' rank-''n'' (the group generated by the ''n'' step sizes X<sub>''i''</sub> > 0, ''i'' = 1, ..., ''n'', has rank ''n'', not lower, for ''almost all'' choices of X<sub>''i''</sub>, in the same sense that almost all real numbers between 0 and 1 are irrational). | The term ''n-ary'' disregards the rank of the group generated by the step sizes, although an ''n''-ary scale is still, in a probabilistic sense, ''generically'' rank-''n'' (the group generated by the ''n'' step sizes X<sub>''i''</sub> > 0, ''i'' = 1, ..., ''n'', has rank ''n'', not lower, for ''almost all'' choices of X<sub>''i''</sub>, in the same sense that almost all real numbers between 0 and 1 are irrational). | ||
== Mathematical facts == | == Mathematical facts == | ||
=== Counting scales of given arity and size === | === Counting scales of given arity and size === | ||