Rank-3 scale: Difference between revisions

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Rank-3 scales described on this page are generalizations of MOS scales, and similar rank-2 scales, which will first be introduced.

The term ''n'''-ary scale''''' is used in certain academic scale theory literature for a scale with exactly ''n'' distinct step sizes, with '''''binary''''' and '''''ternary''''' being used for ''n'' = 2 and 3. This is more concrete terminology than the above, as it simply counts step sizes rather than impose an analysis in terms of rank. 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. This subtle difference between the terms also manifests in the fact that 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 scale''''' is used in certain academic scale theory literature for a scale with exactly ''n'' distinct step sizes, with '''''binary''''' and '''''ternary''''' being used for ''n'' = 2 and 3. This is more concrete terminology than the above, as it simply counts step sizes rather than impose an analysis in terms of rank. 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. This subtle difference between the terms also manifests in the fact that 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. Nonetheless, an ''n''-ary (periodic) scale is still ''generically'' rank-''n'', i.e. the group generated by the ''n'' step sizes X<sub>i</sub> > 0, i = 1, ..., ''n'' has rank ''n'' (not fewer) 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 ==

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 Rank-3 scales described on this page are generalizations of MOS scales, and similar rank-2 scales, which will first be introduced.
-The term ''n'''-ary scale''''' is used in certain academic scale theory literature for a scale with exactly ''n'' distinct step sizes, with '''''binary''''' and '''''ternary''''' being used for ''n'' = 2 and 3. This is more concrete terminology than the above, as it simply counts step sizes rather than impose an analysis in terms of rank. 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. This subtle difference between the terms also manifests in the fact that 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 scale''''' is used in certain academic scale theory literature for a scale with exactly ''n'' distinct step sizes, with '''''binary''''' and '''''ternary''''' being used for ''n'' = 2 and 3. This is more concrete terminology than the above, as it simply counts step sizes rather than impose an analysis in terms of rank. 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. This subtle difference between the terms also manifests in the fact that 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. Nonetheless, an ''n''-ary (periodic) scale is still ''generically'' rank-''n'', i.e. the group generated by the ''n'' step sizes X<sub>i</sub> > 0, i = 1, ..., ''n'' has rank ''n'' (not fewer) 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 ==