Mathematical theory of Fokker blocks: Difference between revisions

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== Preliminaries ==

While the idea generalizes easily to [[just intonation subgroups]] and tempered groups, for ease of exposition we will suppose that we are in a [[~~Harmonic~~ limit|''p''-limit]] situation.

While the idea generalizes easily to [[just intonation subgroups]] and tempered groups, for ease of exposition we will suppose that we are in a [[harmonic limit|''p''-limit]] situation.

Suppose n is equal to the number of primes up to and including p, and that we have {{nowrap|''n'' − 1}} commas. Call the n-1 commas ''c''1, ''c''2, … , ''c''''n-1''. We can pick some uniformizing step ''c''''n'' which allows us to find ''n'' vals '''v'''1, '''v'''2, … , '''v'''''n'' such that '''v'''i tempers out all ''c''''k'' except ''c''''i'', which it maps to 1 step. If ''q'' is a ''p''-limit rational number, we may write it as

Suppose ''n'' is equal to the number of primes up to and including ''p'', and that we have {{nowrap|(''n'' − 1)}} commas. Call the {{nowrap|(''n'' − 1)}} commas ''c''1, ''c''2, …, ''c''(''n'' - 1). We can pick some uniformizing step ''c''''n'' which allows us to find ''n'' vals '''v'''1, '''v'''2, …, '''v'''''n'' such that '''v'''''i'' tempers out all ''c''''k'' except ''c''''i'', which it maps to 1 step. If ''q'' is a ''p''-limit rational number, we may write it as

<math>q = c_1^{\vec v_1(q)} c_2^{\vec v_2(q)} \cdots c_n^{\vec v_n(q)}.</math>.

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The {{nowrap|''n'' − 1}} vals '''u'''1, '''u'''2, …, '''u'''(''n'' − 1) defined in the previous section gave us {{nowrap|''n'' − 1}} inequalities {{nowrap| ''a''''k'' − ''P'' < '''u'''''k'' (''q'') }} ≤ ''a''''k'', 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'''2) in all ways which retain the same orientation and have the unison inside them, we obtain an arena.

==== Fourth definition of a Fokker block ====

=== 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.

@@ Line 4: / Line 4: @@
 == Preliminaries ==
-While the idea generalizes easily to [[just intonation subgroups]] and tempered groups, for ease of exposition we will suppose that we are in a [[Harmonic limit|''p''-limit]] situation.
+While the idea generalizes easily to [[just intonation subgroups]] and tempered groups, for ease of exposition we will suppose that we are in a [[harmonic limit|''p''-limit]] situation.
-Suppose n is equal to the number of primes up to and including p, and that we have {{nowrap|''n'' − 1}} commas. Call the n-1 commas ''c''<sub>1</sub>, ''c''<sub>2</sub>, … , ''c''<sub>''n-1''</sub>. We can pick some uniformizing step ''c''<sub>''n''</sub> which allows us to find ''n'' vals '''v'''<sub>1</sub>, '''v'''<sub>2</sub>, … , '''v'''<sub>''n''</sub> such that '''v'''<sub>i</sub> tempers out all ''c''<sub>''k''</sub> except ''c''<sub>''i''</sub>, which it maps to 1 step. If ''q'' is a ''p''-limit rational number, we may write it as
+Suppose ''n'' is equal to the number of primes up to and including ''p'', and that we have {{nowrap|(''n'' − 1)}} commas. Call the {{nowrap|(''n'' − 1)}} commas ''c''<sub>1</sub>, ''c''<sub>2</sub>, …, ''c''<sub>(''n'' - 1)</sub>. We can pick some uniformizing step ''c''<sub>''n''</sub> which allows us to find ''n'' vals '''v'''<sub>1</sub>, '''v'''<sub>2</sub>, …, '''v'''<sub>''n''</sub> such that '''v'''<sub>''i''</sub> tempers out all ''c''<sub>''k''</sub> except ''c''<sub>''i''</sub>, which it maps to 1 step. If ''q'' is a ''p''-limit rational number, we may write it as
 <math>q = c_1^{\vec v_1(q)} c_2^{\vec v_2(q)} \cdots c_n^{\vec v_n(q)}.</math>.
@@ Line 35: / Line 35: @@
 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'') }} ≤&nbsp;''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 ====
+=== 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.