Rank 3 scale: Difference between revisions

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

MOS scales are [[Scale properties simplified|symmetric]] (not ~~all symmetric~~ scales ~~are MOS)~~ and can be uniquely defined by their ''MOS signature'', i.e. the diatonic scale by 5L 2s.

MOS scales with odd number of steps are [[Scale properties simplified|symmetric]], and those with even number of steps are not.

MOS scales and can be uniquely defined by their ''MOS signature'', i.e. the diatonic scale by 5L 2s.

MOS scales consist of ''strict MOS'' and ''Multi-MOS'' scales.

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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.

MOS scales are ''[[~~Constant~~ structure~~|constant structures~~]]'', meaning that its generic interval classes are distinct, meaning that each specific interval is always subtended by (i.e. occurs as) the same number of steps. This is also known as the ''partitioning property''. ''[[MODMOS scales]]'' are also constant structures - which may be arrived at by raising or lower an interval or intervals of a MOS scale by a chroma. Constant structures exist in any rank. ETs are the rank-1 constant structures.

MOS scales are ''[[constant structure]]s'', meaning that its generic interval classes are distinct, meaning that each specific interval is always subtended by (i.e. occurs as) the same number of steps. This is also known as the ''partitioning property''. ''[[MODMOS scales]]'' are also constant structures - which may be arrived at by raising or lower an interval or intervals of a MOS scale by a chroma. Constant structures exist in any rank. ETs are the rank-1 constant structures.

== Rank-3 scales ==

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

~~'''Conjecture:''' Product words cannot be symmetric.~~

=== Pairwise well-formed (PWF) scales ===

@@ Line 12: / Line 12: @@
 MOS scales can be generated by stacking a single generator modulo a period. Not all generated scales are MOS.
-MOS scales are [[Scale properties simplified|symmetric]] (not all symmetric scales are MOS) and can be uniquely defined by their ''MOS signature'', i.e. the diatonic scale by 5L 2s.
+MOS scales with odd number of steps are [[Scale properties simplified|symmetric]], and those with even number of steps are not.
+MOS scales and can be uniquely defined by their ''MOS signature'', i.e. the diatonic scale by 5L 2s.
 MOS scales consist of ''strict MOS'' and ''Multi-MOS'' scales.
@@ Line 20: / 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.
-MOS scales are ''[[Constant structure|constant structures]]'', meaning that its generic interval classes are distinct, meaning that each specific interval is always subtended by (i.e. occurs as) the same number of steps. This is also known as the ''partitioning property''. ''[[MODMOS scales]]'' are also constant structures - which may be arrived at by raising or lower an interval or intervals of a MOS scale by a chroma. Constant structures exist in any rank. ETs are the rank-1 constant structures.
+MOS scales are ''[[constant structure]]s'', meaning that its generic interval classes are distinct, meaning that each specific interval is always subtended by (i.e. occurs as) the same number of steps. This is also known as the ''partitioning property''. ''[[MODMOS scales]]'' are also constant structures - which may be arrived at by raising or lower an interval or intervals of a MOS scale by a chroma. Constant structures exist in any rank. ETs are the rank-1 constant structures.
 == Rank-3 scales ==
@@ Line 42: / Line 44: @@
 Any Fokker block where the unison vectors are smaller than the smallest steps will be constant structures (CS). Not all Fokker blocks are CS.
-'''Conjecture:''' Product words cannot be symmetric.
 === Pairwise well-formed (PWF) scales ===