Rank-3 scale: Difference between revisions

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== MV3 scales ==

== MV3 and SV3 scales ==

[[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. ~~We often speak of~~ strict-variety 3 (SV3) ~~instead~~, ~~meaning that~~ every interval class has ''exactly'' three sizes. ~~SV3~~ scales are ~~also called~~ '''trivalent'''.

[[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. An important subset strict-variety 3 (SV3) scales, which are a generalisation of strict MOS scales into rank-3, where-in every interval class has ''exactly'' three sizes. In academic literature these scales are instead described as '''trivalent'''.

'''Conjecture:''' For all odd-cardinality SV3 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.

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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 repetitions). 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 repetitions). Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).

== Product words ==

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-== MV3 scales ==
+== MV3 and SV3 scales ==
-[[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. We often speak of strict-variety 3 (SV3) instead, meaning that every interval class has ''exactly'' three sizes. SV3 scales are also called '''trivalent'''.
+[[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. An important subset strict-variety 3 (SV3) scales, which are a generalisation of strict MOS scales into rank-3, where-in every interval class has ''exactly'' three sizes. In academic literature these scales are instead described as '''trivalent'''.
 '''Conjecture:''' For all odd-cardinality SV3 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.
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 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 repetitions). 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 repetitions). Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).
 == Product words ==