Rank-3 scale: Difference between revisions

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|-

|Can tessellate the entire lattice of pitch classes that it lives in

|~~Scale~~ pattern products = rank-3 Fokker blocks (superset of Pairwise DE/MOS scales)

|MOS pattern products = rank-3 Fokker blocks (superset of Pairwise DE/MOS scales)

|-

|[[Recursive structure of MOS scales|Recursive structure]], Uniquely defined by step signature and mapping (implies mirror-symmetric)

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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.

When associated with a mapping, MOS pattern products 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.

~~Product words~~ have maximum variety at most 4. The scale steps can be readily notated, sorted by size, as '''''L''''', '''''l''''', '''''S''''', '''''s''''', and they satisfy '''''L''''' - '''''l''''' = '''''S''''' - '''''s'''''.

MOS pattern products have maximum variety at most 4. The scale steps can be readily notated, sorted by size, as '''''L''''', '''''l''''', '''''S''''', '''''s''''', and they satisfy '''''L''''' - '''''l''''' = '''''S''''' - '''''s'''''.

Any Fokker block where the unison vectors are smaller than the smallest steps will be constant structures (CS). Not all Fokker blocks are CS.

@@ Line 46: / Line 46: @@
 |-
 |Can tessellate the entire lattice of pitch classes that it lives in
-|Scale pattern products = rank-3 Fokker blocks (superset of Pairwise DE/MOS scales)
+|MOS pattern products = rank-3 Fokker blocks (superset of Pairwise DE/MOS scales)
 |-
 |[[Recursive structure of MOS scales|Recursive structure]], Uniquely defined by step signature and mapping (implies mirror-symmetric)
@@ Line 68: / Line 68: @@
 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.
+When associated with a mapping, MOS pattern products 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.
-Product words have maximum variety at most 4. The scale steps can be readily notated, sorted by size, as '''''L''''', '''''l''''', '''''S''''', '''''s''''', and they satisfy '''''L''''' - '''''l''''' = '''''S''''' - '''''s'''''.
+MOS pattern products have maximum variety at most 4. The scale steps can be readily notated, sorted by size, as '''''L''''', '''''l''''', '''''S''''', '''''s''''', and they satisfy '''''L''''' - '''''l''''' = '''''S''''' - '''''s'''''.
 Any Fokker block where the unison vectors are smaller than the smallest steps will be constant structures (CS). Not all Fokker blocks are CS.