Mathematical theory of Fokker blocks: Difference between revisions
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The {{nowrap|''n'' − 1}} vals '''u'''<sub>1</sub>, '''u'''<sub>2</sub>, …, '''u'''<sub style="white-space: nowrap;">(''n'' − 1)</sub> defined in the previous section gave us {{nowrap|''n'' − 1}} inequalities {{nowrap| ''a''<sub>''k''</sub> − ''P'' < '''u'''<sub>''k''</sub> (''q'') }} ≤ ''a''<sub>''k''</sub>, which apply to any ''q'' in the Fokker block. If we restrict ''q'' to {{nowrap| 1 ≤ ''q'' < 2 }}, and regard it as representing a pitch class, then it is associated to a lattice point in an {{nowrap|(''n'' − 1)}}-dimensional vector space, and in that space the {{nowrap|''n'' − 1}} inequalities define the boundaries of a parallelepiped. The Fokker blocks can be defined as the pitch classes lying within such a parallelepiped. By moving the parallelepiped around (in '''R'''<sup>2</sup>) in all ways which retain the same orientation and have the unison inside them, we obtain an arena. | The {{nowrap|''n'' − 1}} vals '''u'''<sub>1</sub>, '''u'''<sub>2</sub>, …, '''u'''<sub style="white-space: nowrap;">(''n'' − 1)</sub> defined in the previous section gave us {{nowrap|''n'' − 1}} inequalities {{nowrap| ''a''<sub>''k''</sub> − ''P'' < '''u'''<sub>''k''</sub> (''q'') }} ≤ ''a''<sub>''k''</sub>, which apply to any ''q'' in the Fokker block. If we restrict ''q'' to {{nowrap| 1 ≤ ''q'' < 2 }}, and regard it as representing a pitch class, then it is associated to a lattice point in an {{nowrap|(''n'' − 1)}}-dimensional vector space, and in that space the {{nowrap|''n'' − 1}} inequalities define the boundaries of a parallelepiped. The Fokker blocks can be defined as the pitch classes lying within such a parallelepiped. By moving the parallelepiped around (in '''R'''<sup>2</sup>) in all ways which retain the same orientation and have the unison inside them, we obtain an arena. | ||
=== Fourth definition of a Fokker block === | |||
The {{nowrap|''n'' − 1}} (tempered) abstract mos scales discussed with the second definition of Fokker blocks can be put into some chosen order, and the [[product word|step pattern product]] taken. This entails that every Fokker block leads to a step pattern product, and the process can be reversed, so that step pattern products of {{nowrap|''n'' − 1}} abstract mos scales (''not'' ignoring mode) lead to Fokker blocks. Given the {{nowrap|''n'' − 1}} vals obtained by taking the interior product with some interval ''q'', ''q'' can be recovered either by wedging the vals together and taking the [[the dual|dual]], or by taking the determinant of the {{nowrap| ''n'' × ''n'' }} matrix of vals whose first row consists of indeterminates, as in the [[#Preliminaries]] section. | The {{nowrap|''n'' − 1}} (tempered) abstract mos scales discussed with the second definition of Fokker blocks can be put into some chosen order, and the [[product word|step pattern product]] taken. This entails that every Fokker block leads to a step pattern product, and the process can be reversed, so that step pattern products of {{nowrap|''n'' − 1}} abstract mos scales (''not'' ignoring mode) lead to Fokker blocks. Given the {{nowrap|''n'' − 1}} vals obtained by taking the interior product with some interval ''q'', ''q'' can be recovered either by wedging the vals together and taking the [[the dual|dual]], or by taking the determinant of the {{nowrap| ''n'' × ''n'' }} matrix of vals whose first row consists of indeterminates, as in the [[#Preliminaries]] section. | ||