Rank-3 scale: Difference between revisions

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A '''rank-'''''n'' '''scale'''~~, known as an ''n'''''-ary scale''' in certain academic scale theory literature,~~ 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.

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

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

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.

== Rank-2 scales ==

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-A '''rank-'''''n'' '''scale''', known as an ''n'''''-ary scale''' in certain academic scale theory literature, 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.
+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-2, or binary, scales include [[MOS scales]] and other generated scales, [[MODMOS scale]]s, and other more complex scales that we are not as interested in.
+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, or ternary, scales described on this page are generalizations of MOS scales, and similar rank-2 scales, which will first be introduced.
+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.
 == Rank-2 scales ==