Rank-3 scale: Difference between revisions

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MOS scales consist of ''strict MOS'', which are the MOS scales as originally defined by Erv Wilson, and ''multi-MOS'' scales.

Strict MOS scales are the MOS scales that possess [[~~Scale~~ properties ~~simplified~~#Properties|Myhill’s property]], in which all generic intervals come in 2 sizes, and are also known as ''well-formed'' (WF) scales. The two sizes of each interval class in a WF or strict MOS scale differ by the ''chroma'' – the difference between the large and small steps of the scale. We will refer to these scales as WF scales for the remainder of this page. WF scales are defined (Carey & Clampitt, 1989) as generated scales for which the generator is of invariant generic interval size.

Strict MOS scales are the MOS scales that possess [[Glossary of scale properties#Properties|Myhill’s property]], in which all generic intervals come in 2 sizes, and are also known as ''well-formed'' (WF) scales. The two sizes of each interval class in a WF or strict MOS scale differ by the ''chroma'' – the difference between the large and small steps of the scale. We will refer to these scales as WF scales for the remainder of this page. WF scales are defined (Carey & Clampitt, 1989) as generated scales for which the generator is of invariant generic interval size.

Multi-MOS scales are MOS scales that are multiple periods of a WF scale. The interval class represented by any multiple of a period of a WF scale comes in only a single size, hence multi-MOS scales do not possess Myhill’s property.

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|-

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

|~~Product words~~ = 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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'''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|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, ~~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 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.

~~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.

== Pairwise well-formed scales==

Pairwise well-formed (PWF) scales, another generalization of WF scales into rank-3, are a subset of ~~product words~~.

Pairwise well-formed (PWF) scales, another generalization of WF scales into rank-3, are a subset of MOS pattern products.

If equating any pair of step sizes (tempering out their difference, if we involve mappings) of a rank-3 scale leads to 3 WF scales, the rank-3 scale is ''pairwise well-formed (PWF).''

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== Pairwise DE/MOS scales ==

A similar generalization, a larger subset of ~~product words~~, and a superset of PWF scales are ''pairwise DE'' (PDE) scales, defined for rank-3 scales such that equating any pair of steps (tempering out their difference, if we involve mappings), leads to a DE scale, or equivalently, an MOS scale. We may also call these ''pairwise MOS'' (PMOS) scales.

A similar generalization, a larger subset of MOS pattern products, and a superset of PWF scales are ''pairwise DE'' (PDE) scales, defined for rank-3 scales such that equating any pair of steps (tempering out their difference, if we involve mappings), leads to a DE scale, or equivalently, an MOS scale. We may also call these ''pairwise MOS'' (PMOS) scales.

Pairwise DE scales have MV3. Pairwise DE scales that are not PWF are not SV3; and at least one of the DE scales / MOS scales found by equating a pair of steps of such scales is a multi-MOS, which is DE / MV2, but does not demonstrate Myhill's property.

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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 18: / Line 18: @@
 MOS scales consist of ''strict MOS'', which are the MOS scales as originally defined by Erv Wilson, and ''multi-MOS'' scales.
-Strict MOS scales are the MOS scales that possess [[Scale properties simplified#Properties|Myhill’s property]], in which all generic intervals come in 2 sizes, and are also known as ''well-formed'' (WF) scales. The two sizes of each interval class in a WF or strict MOS scale differ by the ''chroma'' – the difference between the large and small steps of the scale. We will refer to these scales as WF scales for the remainder of this page. WF scales are defined (Carey & Clampitt, 1989) as generated scales for which the generator is of invariant generic interval size.
+Strict MOS scales are the MOS scales that possess [[Glossary of scale properties#Properties|Myhill’s property]], in which all generic intervals come in 2 sizes, and are also known as ''well-formed'' (WF) scales. The two sizes of each interval class in a WF or strict MOS scale differ by the ''chroma'' – the difference between the large and small steps of the scale. We will refer to these scales as WF scales for the remainder of this page. WF scales are defined (Carey & Clampitt, 1989) as generated scales for which the generator is of invariant generic interval size.
 Multi-MOS scales are MOS scales that are multiple periods of a WF scale. The interval class represented by any multiple of a period of a WF scale comes in only a single size, hence multi-MOS scales do not possess Myhill’s property.
@@ Line 46: / Line 46: @@
 |-
 |Can tessellate the entire lattice of pitch classes that it lives in
-|Product words = 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 65: / Line 66: @@
 '''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|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, 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 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.
-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.
 == Pairwise well-formed scales==
-Pairwise well-formed (PWF) scales, another generalization of WF scales into rank-3, are a subset of product words.
+Pairwise well-formed (PWF) scales, another generalization of WF scales into rank-3, are a subset of MOS pattern products.
 If equating any pair of step sizes (tempering out their difference, if we involve mappings) of a rank-3 scale leads to 3 WF scales, the rank-3 scale is ''pairwise well-formed (PWF).''
@@ Line 96: / Line 97: @@
 == Pairwise DE/MOS scales ==
-A similar generalization, a larger subset of product words, and a superset of PWF scales are ''pairwise DE'' (PDE) scales, defined for rank-3 scales such that equating any pair of steps (tempering out their difference, if we involve mappings), leads to a DE scale, or equivalently, an MOS scale. We may also call these ''pairwise MOS'' (PMOS) scales.
+A similar generalization, a larger subset of MOS pattern products, and a superset of PWF scales are ''pairwise DE'' (PDE) scales, defined for rank-3 scales such that equating any pair of steps (tempering out their difference, if we involve mappings), leads to a DE scale, or equivalently, an MOS scale. We may also call these ''pairwise MOS'' (PMOS) scales.
 Pairwise DE scales have MV3. Pairwise DE scales that are not PWF are not SV3; and at least one of the DE scales / MOS scales found by equating a pair of steps of such scales is a multi-MOS, which is DE / MV2, but does not demonstrate Myhill's property.
@@ Line 174: / Line 175: @@
 '''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]]