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>.

@@ 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>.