Rank-3 scale: 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.

* 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'' indicates that the rank of the group generated by the step sizes is disregarded, although an ''n''-ary ~~periodic~~ scale is still ''generically'' rank-''n'' (the group generated by the ''n'' step sizes X''i'' > 0, ''i'' = 1, ..., ''n'' has rank ''n'', not lower, for ''almost all'' choices of X''i'', in the same sense that almost all real numbers between 0 and 1 are irrational).

The term ''n-ary'' indicates that the rank of the group generated by the step sizes is disregarded, although an ''n''-ary scale is still ''generically'' rank-''n'' (the group generated by the ''n'' step sizes X''i'' > 0, ''i'' = 1, ..., ''n'' has rank ''n'', not lower, for ''almost all'' choices of X''i'', in the same sense that almost all real numbers between 0 and 1 are irrational).

== Rank-2 scales ==

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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.
 * 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'' indicates that the rank of the group generated by the step sizes is disregarded, although an ''n''-ary periodic 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'' indicates that the rank of the group generated by the step sizes is disregarded, 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).
 == Rank-2 scales ==