Rank 3 scale: Difference between revisions

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WF scales may be generalized into rank-3 via a generalization of Myhill's property into rank-3. We call a scale where each generic interval comes in 3 sizes ''trivalent''. Trivalent scales are clearly a subset of MV3 scales. Trivalent scales can only have odd numbers of notes.

=== Product words ===

=== [[Product word|Product words]] ===

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'', 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 33: / Line 33: @@
 WF scales may be generalized into rank-3 via a generalization of Myhill's property into rank-3. We call a scale where each generic interval comes in 3 sizes ''trivalent''. Trivalent scales are clearly a subset of MV3 scales. Trivalent scales can only have odd numbers of notes.
-=== Product words ===
+=== [[Product word|Product words]] ===
-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'', 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.