Rank-3 scale: Difference between revisions

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|-

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

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

|MOS step 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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== MV3 and SV3 scales ==

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

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

SV3 scales are sometimes called [[trivalent scale]]s.<ref>Carey, N. (2007). [https://doi.org/10.1080/17459730701376743 ''Coherence and sameness in well-formed and pairwise well-formed scales'']. Journal of Mathematics and Music, 1(2), 79–98.</ref>

'''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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== Scale pattern product ==

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.

Two MOS scales can be combined into a rank-3 scale as a ''[[product word|step 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, 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.

When associated with a mapping, MOS step 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.

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'''''.

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'''Conjecture''': The only SN scales that are [[Balanced word|balanced]] are the ''Power SNS'', which are equivalent to the [[Fraenkel word|Fraenkel words]], and SNS wherein two step sizes occur only once.

== References ==

[[Category:Rank-3 scales| ]]

[[Category:Rank 3]]

[[Category:Pages with open problems]]

@@ Line 46: / Line 46: @@
 |-
 |Can tessellate the entire lattice of pitch classes that it lives in
-|MOS pattern products = rank-3 Fokker blocks (superset of Pairwise DE/MOS scales)
+|MOS step 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 57: / Line 57: @@
 == MV3 and SV3 scales ==
-[[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. An important subset are 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'''.
+[[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. An important subset are 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.
+SV3 scales are sometimes called [[trivalent scale]]s.<ref>Carey, N. (2007). [https://doi.org/10.1080/17459730701376743 ''Coherence and sameness in well-formed and pairwise well-formed scales'']. Journal of Mathematics and Music, 1(2), 79–98.</ref>
 '''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.
@@ Line 66: / Line 67: @@
 == Scale pattern product ==
-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.
+Two MOS scales can be combined into a rank-3 scale as a ''[[product word|step 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, 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.
+When associated with a mapping, MOS step 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.
 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'''''.
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 '''Conjecture''': The only SN scales that are [[Balanced word|balanced]] are the ''Power SNS'', which are equivalent to the [[Fraenkel word|Fraenkel words]], and SNS wherein two step sizes occur only once.
+== References ==
+<references />
 [[Category:Rank-3 scales| ]] <!--main article-->
 [[Category:Rank 3]]
 [[Category:Pages with open problems]]