Rank-3 scale: Difference between revisions

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A '''rank-'''''n'' '''scale''' is a scale whose intervals (in cents, or any other logarithmic [[interval size measure]]) generate a rank-''n'' group. Alternatively, a rank-''n'' scale is a finite set of notes of a rank-''n'' tuning, which is an infinite set of notes that can be generated by ''n'' generators, one of which is taken to be the period, at which any scale of the tuning repeats.

Rank-1 tunings and scales are [[equal ~~temperament~~]]s (ET). ~~ETs~~ are rank-1 because the generator achieves the octave by default. Thus, the octave is not counted as a generator.

Rank-1 tunings and scales are [[equal tuning]]s (ET). [[Edo]]s are rank-1 because the generator achieves the octave by default. Thus, the octave is not counted as a generator.

Rank-2 scales include [[MOS scales]] and other generated scales, [[MODMOS scale]]s, and other more complex scales that we are not as interested in.

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 '''''binary''''' and '''''ternary''''' being used for ''n'' = 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:

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 A '''rank-'''''n'' '''scale''' is a scale whose intervals (in cents, or any other logarithmic [[interval size measure]]) generate a rank-''n'' group. Alternatively, a rank-''n'' scale is a finite set of notes of a rank-''n'' tuning, which is an infinite set of notes that can be generated by ''n'' generators, one of which is taken to be the period, at which any scale of the tuning repeats.
-Rank-1 tunings and scales are [[equal temperament]]s (ET). ETs are rank-1 because the generator achieves the octave by default. Thus, the octave is not counted as a generator.
+Rank-1 tunings and scales are [[equal tuning]]s (ET). [[Edo]]s are rank-1 because the generator achieves the octave by default. Thus, the octave is not counted as a generator.
 Rank-2 scales include [[MOS scales]] and other generated scales, [[MODMOS scale]]s, and other more complex scales that we are not as interested in.
 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 '''''binary''''' and '''''ternary''''' being used for ''n'' = 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: