Step variety: Difference between revisions

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== Difference from scale rank ==

Certain abstract scale theorists in the xen community have taken to using the ''n-ary'' terminology in ~~order to respect~~ the subtlety of the notion of a scale's [[rank]]. Examples of this subtlety are:

Certain abstract scale theorists in the xen community have taken to using the ''n-ary'' terminology in view of the subtlety of the notion of a scale's [[rank]]. 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).

[[Category:Terms]][[Category:Scale]]

@@ Line 6: / Line 6: @@
 == Difference from scale rank ==
-Certain abstract scale theorists in the xen community have taken to using the ''n-ary'' terminology in order to respect the subtlety of the notion of a scale's [[rank]]. Examples of this subtlety are:
+Certain abstract scale theorists in the xen community have taken to using the ''n-ary'' terminology in view of the subtlety of the notion of a scale's [[rank]]. 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).
 [[Category:Terms]][[Category:Scale]]