Fokker block: Difference between revisions

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=== Scale properties of Fokker blocks ===

By definition, a Fokker block is weakly epimorphic, which implies it is constant structure. Since the pitch classes are all of those contained in some parallelepiped, it is convex. A rank-''r'' Fokker block, meaning one which generates a group of rank ''r'', has {{nowrap|''r'' − 1}} abstract mos scales which can take at most two values for any interval class, by Myhill's property. Since the scale itself can be reconstituted from the {{nowrap|''r'' − 1}} abstract mosses, that means each interval class in the scale has at most 2(''r'' − 1) possible values; in other words, it has maximum variety less than or equal to 2(''r'' − 1).

By definition, a Fokker block is weakly epimorphic, which implies it is a weak constant structure (constant structure with negative steps allowed). Since the pitch classes are all of those contained in some parallelepiped, it is convex. A rank-''r'' Fokker block, meaning one which generates a group of rank ''r'', has {{nowrap|''r'' − 1}} abstract mos scales which can take at most two values for any interval class, by Myhill's property. Since the scale itself can be reconstituted from the {{nowrap|''r'' − 1}} abstract mosses, that means each interval class in the scale has at most 2(''r'' − 1) possible values; in other words, it has maximum variety less than or equal to 2(''r'' − 1).

The reconstitution can be obtained as follows: for every note of ''S''[''i''] except ''S''[0], S[''i''] will be either the rational number obtained by finding the monzo of the wedge products of the {{nowrap|''r'' − 1}} abstract mos vals for ''i'', taking the dual, and dividing by ''i''(''r'' − 1), or else the inverse of this number. Hence we may choose an ordering of the correct parity, and find the value associated to S[''i''] by {{nowrap|('''v'''1 ∧ '''v'''2 ∧ … ∧ '''v'''(''r'' − 1))°/''i'' (''r'' − 1)}}.

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 === Scale properties of Fokker blocks ===
-By definition, a Fokker block is weakly epimorphic, which implies it is constant structure. Since the pitch classes are all of those contained in some parallelepiped, it is convex. A rank-''r'' Fokker block, meaning one which generates a group of rank ''r'', has {{nowrap|''r'' − 1}} abstract mos scales which can take at most two values for any interval class, by Myhill's property. Since the scale itself can be reconstituted from the {{nowrap|''r'' − 1}} abstract mosses, that means each interval class in the scale has at most 2<sup style="white-space: nowrap;">(''r'' − 1)</sup> possible values; in other words, it has maximum variety less than or equal to 2<sup style="white-space: nowrap;">(''r'' − 1)</sup>.
+By definition, a Fokker block is weakly epimorphic, which implies it is a weak constant structure (constant structure with negative steps allowed). Since the pitch classes are all of those contained in some parallelepiped, it is convex. A rank-''r'' Fokker block, meaning one which generates a group of rank ''r'', has {{nowrap|''r'' − 1}} abstract mos scales which can take at most two values for any interval class, by Myhill's property. Since the scale itself can be reconstituted from the {{nowrap|''r'' − 1}} abstract mosses, that means each interval class in the scale has at most 2<sup style="white-space: nowrap;">(''r'' − 1)</sup> possible values; in other words, it has maximum variety less than or equal to 2<sup style="white-space: nowrap;">(''r'' − 1)</sup>.
 The reconstitution can be obtained as follows: for every note of ''S''[''i''] except ''S''[0], S[''i''] will be either the rational number obtained by finding the monzo of the wedge products of the {{nowrap|''r'' − 1}} abstract mos vals for ''i'', taking the dual, and dividing by ''i''<sup style="white-space: nowrap;">(''r'' − 1)</sup>, or else the inverse of this number. Hence we may choose an ordering of the correct parity, and find the value associated to S[''i''] by {{nowrap|('''v'''<sub>1</sub> ∧ '''v'''<sub>2</sub> ∧ … ∧ '''v'''<sub>(''r'' − 1)</sub>)°/''i''&#x200A;<sup>(''r'' − 1)</sup>}}.