Rank-3 scale: Difference between revisions

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'''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~~ ==

== Scale pattern product ==

Two MOS scales can be combined into a rank-3 scale as a ''[[product word]]'', which reduces back to the two MOS scales when two of the three pairs of interval sizes are equated.

Two MOS scales can be combined into a rank-3 scale as a ''[[product word|scale pattern product]]'', which reduces back to the two MOS scales when two of the three pairs of interval sizes are equated.

When associated with a mapping, product words are the rank-3 ''[[Fokker blocks]]''. Fokker blocks have ''unison vectors'', which generalize the concept of the chroma of MOS scales to higher ranks. If these intervals are plotted onto a plane representing rank-3 octave equivalent pitch space, they tile the space into Fokker blocks which differ by combinations of these unison vectors. Rank-2 Fokker blocks are the MOS scales, so Fokker blocks can be considered a generalization of MOS scales into higher ranks.

@@ Line 65: / Line 65: @@
 '''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 ==
+== Scale pattern product ==
-Two MOS scales can be combined into a rank-3 scale as a ''[[product word]]'', which reduces back to the two MOS scales when two of the three pairs of interval sizes are equated.
+Two MOS scales can be combined into a rank-3 scale as a ''[[product word|scale pattern product]]'', which reduces back to the two MOS scales when two of the three pairs of interval sizes are equated.
 When associated with a mapping, product words are the rank-3 ''[[Fokker blocks]]''. Fokker blocks have ''unison vectors'', which generalize the concept of the chroma of MOS scales to higher ranks. If these intervals are plotted onto a plane representing rank-3 octave equivalent pitch space, they tile the space into Fokker blocks which differ by combinations of these unison vectors. Rank-2 Fokker blocks are the MOS scales, so Fokker blocks can be considered a generalization of MOS scales into higher ranks.