Rank-3 scale: Difference between revisions

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Rank-3 scales described on this page are generalizations of rank-2 scales (MOS scales and permutations thereof, and other scales that have a single generator), which will first be introduced.

~~== Terminology ==~~

The related term ''n'''-ary scale''''' is used in certain academic scale theory literature for a scale with exactly ''n'' distinct step sizes, with '''''unary''''', '''''binary''''' and '''''ternary''''' being used for ''n'' = 1, 2 and 3. To respect the subtlety of the notion of scale rank, certain abstract scale theorists in the xen community have taken to using the ''n-ary'' terminology. Examples of this subtlety are:

* 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'' 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).

== Rank-2 scales ==

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 Rank-3 scales described on this page are generalizations of rank-2 scales (MOS scales and permutations thereof, and other scales that have a single generator), which will first be introduced.
-== Terminology ==
-The related term ''n'''-ary scale''''' is used in certain academic scale theory literature for a scale with exactly ''n'' distinct step sizes, with '''''unary''''', '''''binary''''' and '''''ternary''''' being used for ''n'' = 1, 2 and 3. To respect the subtlety of the notion of scale rank, certain abstract scale theorists in the xen community have taken to using the ''n-ary'' terminology. Examples of this subtlety are:
-* 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'' 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).
 == Rank-2 scales ==