Rank 3 scale: Difference between revisions

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A rank-''n'' scale a scale whose intervals (in cents, or any other logarithmic [[interval size measure]]) generate a rank-''n'' group. Alternatively, a rank-''n'' scale is a finite set of notes of a rank-''n'' tuning, which is an infinite set of notes that can be generated by ''n'' generators, one of which is taken to be the period, at which any scale of the tuning repeats.

A rank ''n'' scale a scale whose intervals (in cents, or any other logarithmic [[interval size measure]]) generate a rank ''n'' group. Alternatively, a rank ''n'' scale is a finite set of notes of a rank ''n'' tuning, which is an infinite set of notes that can be generated by ''n'' generators, one of which is taken to be the period, at which any scale of the tuning repeats.

Rank-1 tunings and scales are [[ET]]<nowiki/>s. ET's are rank-1 because the generator achieves the octave by default. Thus, the octave is not counted as a generator.

Rank 1 tunings and scales are [[ET]]<nowiki/>s. ET's are rank 1 because the generator achieves the octave by default. Thus, the octave is not counted as a generator.

Rank-2 scales include [[MOS scales]] and other generated scales, [[MODMOS Scales|MODMOS scales]], and other more complex scales that we are not as interested in.

Rank 2 scales include [[MOS scales]] and other generated scales, [[MODMOS Scales|MODMOS scales]], and other more complex scales that we are not as interested in.

Rank-3 scales described on this page are generalizations of [[MOS scales]], and similar rank-2 scales, which will first be introduced.

Rank 3 scales described on this page are generalizations of [[MOS scales]], and similar rank 2 scales, which will first be introduced.

== Rank-2 scales ==

== Rank 2 scales ==

MOS scales are the MV2 ([[maximum variety]] 2) scales. MOS scales are DE ([[distributionally even]]), along with rank-1 scales, i.e., [[ET]]<nowiki/>s, which are MV1.

MOS scales are the MV2 ([[maximum variety]] 2) scales. MOS scales are DE ([[distributionally even]]), along with rank 1 scales, i.e., [[ET]]<nowiki/>s, which are MV1.

MOS scales can be generated by stacking a single generator modulo a period. Not all generated scales are MOS.

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''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.

Rank-3 scales are introduced from here, as generalizations of MOS scales.

Rank 3 scales are introduced from here, as generalizations of MOS scales.

==[[User:Inthar/MV3|MV3 scales]]==

Maximum variety 3, or MV3 scales are a generalization of MOS scales (the scales of MV2) into rank-3.

Maximum variety 3, or MV3 scales are a generalization of MOS scales (the scales of MV2) into rank 3.

For all MV3 scales apart from the scales abacaba, and it's repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1.

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== Trivalent scales ==

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.

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 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.

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.

Product words have max variety at most 4.

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== Pairwise well-formed (PWF) 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 product words.

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

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

PWF scales not only have MV3, but are trivalent.

When mappings are considered, PWF scales are rank-3 [[Gallery of wakalixes|''wakalixes'']] - Fokker blocks which are Fokker blocks in more than one way.

When mappings are considered, PWF scales are rank 3 [[Gallery of wakalixes|''wakalixes'']] - Fokker blocks which are Fokker blocks in more than one way.

Not all trivalent scales are PWF. Only a single scale - abcba - is trivalent and not 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 scales'', defined for rank-3 scales such that equating any pair of steps (tempering out their difference, if we involve mappings), leads to 3 DE scales, or equivalently, MOS scales. We may also call these pairwise MOS scales.

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

Pairwise-DE scales have MV3. Pairwise-DE scales that are not PWF are not trivalent; and 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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SN scales are generated iteratively by placing an instance of a new or the existing smallest step at the top or bottom of every larger step.

SN scales include MOS scales. MOS scales are the rank-2 SN scales, or 2-SN scales.

SN scales include MOS scales. MOS scales are the rank 2 SN scales, or 2-SN scales.

3-SN scales are generated from MOS scales, and 4-SN scales are generated from 3-SN scales, etc.

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'''Conjecture:''' abacaba and aabaabaac are the only SN scales with mean variety = 3.

[[Category: Rank 3]]

[[Category:Rank 3 scales| ]]

~~[[Category: Scale]]~~

~~[[Category: Theory]]~~

~~[[Category: Scale theory]]~~

@@ Line 1: / Line 1: @@
-A rank-''n'' scale a scale whose intervals (in cents, or any other logarithmic [[interval size measure]]) generate a rank-''n'' group. Alternatively, a rank-''n'' scale is a finite set of notes of a rank-''n'' tuning, which is an infinite set of notes that can be generated by ''n'' generators, one of which is taken to be the period, at which any scale of the tuning repeats.
+A rank ''n'' scale a scale whose intervals (in cents, or any other logarithmic [[interval size measure]]) generate a rank ''n'' group. Alternatively, a rank ''n'' scale is a finite set of notes of a rank ''n'' tuning, which is an infinite set of notes that can be generated by ''n'' generators, one of which is taken to be the period, at which any scale of the tuning repeats.
-Rank-1 tunings and scales are [[ET]]<nowiki/>s. ET's are rank-1 because the generator achieves the octave by default. Thus, the octave is not counted as a generator.
+Rank 1 tunings and scales are [[ET]]<nowiki/>s. ET's are rank 1 because the generator achieves the octave by default. Thus, the octave is not counted as a generator.
-Rank-2 scales include [[MOS scales]] and other generated scales, [[MODMOS Scales|MODMOS scales]], and other more complex scales that we are not as interested in.
+Rank 2 scales include [[MOS scales]] and other generated scales, [[MODMOS Scales|MODMOS scales]], and other more complex scales that we are not as interested in.
-Rank-3 scales described on this page are generalizations of [[MOS scales]], and similar rank-2 scales, which will first be introduced.
+Rank 3 scales described on this page are generalizations of [[MOS scales]], and similar rank 2 scales, which will first be introduced.
-== Rank-2 scales ==
+== Rank 2 scales ==
-MOS scales are the MV2 ([[maximum variety]] 2) scales. MOS scales are DE ([[distributionally even]]), along with rank-1 scales, i.e., [[ET]]<nowiki/>s, which are MV1.
+MOS scales are the MV2 ([[maximum variety]] 2) scales. MOS scales are DE ([[distributionally even]]), along with rank 1 scales, i.e., [[ET]]<nowiki/>s, which are MV1.
 MOS scales can be generated by stacking a single generator modulo a period. Not all generated scales are MOS.
@@ Line 22: / Line 22: @@
 ''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.
-Rank-3 scales are introduced from here, as generalizations of MOS scales.
+Rank 3 scales are introduced from here, as generalizations of MOS scales.
 ==[[User:Inthar/MV3|MV3 scales]]==
-Maximum variety 3, or MV3 scales are a generalization of MOS scales (the scales of MV2) into rank-3.
+Maximum variety 3, or MV3 scales are a generalization of MOS scales (the scales of MV2) into rank 3.
 For all MV3 scales apart from the scales abacaba, and it's repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1.
@@ Line 31: / Line 31: @@
 == Trivalent scales ==
-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.
+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 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.
+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.
 Product words have max variety at most 4.
@@ Line 43: / Line 43: @@
 == Pairwise well-formed (PWF) 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 product words.
-If equating any pair of step sizes (tempering out their difference, if we involve mappings) or a rank-3 scale leads to 3 WF scales, the rank-3 scale is ''Pairwise well-formed (PWF).''
+If equating any pair of step sizes (tempering out their difference, if we involve mappings) or a rank 3 scale leads to 3 WF scales, the rank 3 scale is ''Pairwise well-formed (PWF).''
 PWF scales not only have MV3, but are trivalent.
-When mappings are considered, PWF scales are rank-3 [[Gallery of wakalixes|''wakalixes'']] - Fokker blocks which are Fokker blocks in more than one way.
+When mappings are considered, PWF scales are rank 3 [[Gallery of wakalixes|''wakalixes'']] - Fokker blocks which are Fokker blocks in more than one way.
 Not all trivalent scales are PWF. Only a single scale - abcba - is trivalent and not PWF.
@@ Line 60: / Line 60: @@
 == Pairwise DE/MOS scales ==
-A similar generalization, a larger subset of product words, and a superset of PWF scales are ''pairwise-DE scales'', defined for rank-3 scales such that equating any pair of steps (tempering out their difference, if we involve mappings), leads to 3 DE scales, or equivalently, MOS scales. We may also call these pairwise MOS scales.
+A similar generalization, a larger subset of product words, and a superset of PWF scales are ''pairwise-DE scales'', defined for rank 3 scales such that equating any pair of steps (tempering out their difference, if we involve mappings), leads to 3 DE scales, or equivalently, MOS scales. We may also call these pairwise MOS scales.
 Pairwise-DE scales have MV3. Pairwise-DE scales that are not PWF are not trivalent; and 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 77: / Line 77: @@
 SN scales are generated iteratively by placing an instance of a new or the existing smallest step at the top or bottom of every larger step.
-SN scales include MOS scales. MOS scales are the rank-2 SN scales, or 2-SN scales.
+SN scales include MOS scales. MOS scales are the rank 2 SN scales, or 2-SN scales.
 -SN scales are generated from MOS scales, and 4-SN scales are generated from 3-SN scales, etc.
@@ Line 135: / Line 135: @@
 '''Conjecture:''' abacaba and aabaabaac are the only SN scales with mean variety = 3.
-[[Category: Rank 3]]
+[[Category:Rank 3 scales| ]] <!-- main article -->
-[[Category: Scale]]
-[[Category: Theory]]
-[[Category: Scale theory]]