Rank-3 scale: Difference between revisions

Line 59:

[[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3.

For all odd-cardinality MV3 scales apart from the scales abacaba, and its repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, and na nb nc scales, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1. This is the [[generator-offset property]] (GO).

For all odd-cardinality MV3 scales apart from the scales ''abacaba'', and its repetitions ''abacabaabacaba'' etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, and na nb nc scales, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1. This is the [[generator-offset property]] (GO).

All GO scales of odd cardinality are MV3. The only GO scale of even cardinality is abac.

All GO scales of odd cardinality are MV3. The only GO scale of even cardinality is ''abac''.

'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c (and their repeititions). Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).

'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form ''a...ba...c'' (and their repeititions). Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).

== Trivalent scales ==

@@ Line 59: / Line 59: @@
 [[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3.
-For all odd-cardinality MV3 scales apart from the scales abacaba, and its repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, and na nb nc scales, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1. This is the [[generator-offset property]] (GO).
+For all odd-cardinality MV3 scales apart from the scales ''abacaba'', and its repetitions ''abacabaabacaba'' etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, and na nb nc scales, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1. This is the [[generator-offset property]] (GO).
-All GO scales of odd cardinality are MV3. The only GO scale of even cardinality is abac.
+All GO scales of odd cardinality are MV3. The only GO scale of even cardinality is ''abac''.
-'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c (and their repeititions). Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).
+'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form ''a...ba...c'' (and their repeititions). Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).
 == Trivalent scales ==