Fokker block: Difference between revisions

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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~~, which we will assume are greater than 1 (i.e. ascending intervals)~~. Call the n-1 commas ''c''1, ''c''2, … , ''c''''n-1.'' ~~There is a (nonunique)~~ 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 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>.

Note that this is only possible if the lattice spanned by the commas has full rank and is [[torsion]]-free.

==== Proof of the existence of these vals ====

Having selected a step, form the ''n'' × ''n'' matrix whose last row is the monzo for the step ''c'', and whose other rows are the monzos of the ''n'' − 1 chromas. Because we have chosen ''c'' so that ''V''(''c'') = 1, each point ''S'' in the lattice ~~is findable~~ by stacking row vectors (stacking ''c'' as many times as ''S'' is mapped to by ''V'' and adjusting by the commas of ''V'') and so the determinant of this matrix will be ±1. It is ~~therefore~~ a unimodular matrix, ~~that is, a square matrix with coefficients~~ which ~~are integers and with determinant ±1. Such a matrix~~ is invertible~~, and~~ the inverse matrix is also unimodular. If we call ''c'' "''cn''", and label the chromas ''c''1, ''c''2, … , ''c''(''n'' − 1); and if we consider the columns of the inverse matrix to be vals and call them '''v'''1, '''v'''2, … , '''v'''''n'', then by the definition of the inverse of a matrix, '''v'''''i''(''cj'') = δ(''i'', ''j''), where δ(''i'', ''j'') is the Kronecker delta. Stated another way, '''v'''''i''(''cj'') is 0 unless ''i'' = ''j'', in which case '''v'''''i''(''ci'') = 1.

Having selected a step, form the ''n'' × ''n'' matrix whose last row is the monzo for the step ''c'', and whose other rows are the monzos of the ''n'' − 1 chromas. Because we have chosen ''c'' so that ''V''(''c'') = 1, each point ''S'' in the lattice can be found by stacking row vectors (stacking ''c'' as many times as ''S'' is mapped to by ''V'' and adjusting by the commas of ''V'') and so the determinant of this matrix will be ±1. Such a matrix is called a unimodular matrix, which is invertible in the integers. The inverse matrix is also unimodular. If we call ''c'' "''cn''", and label the chromas ''c''1, ''c''2, … , ''c''(''n'' − 1); and if we consider the columns of the inverse matrix to be vals and call them '''v'''1, '''v'''2, … , '''v'''''n'', then by the definition of the inverse of a matrix, '''v'''''i''(''cj'') = δ(''i'', ''j''), where δ(''i'', ''j'') is the Kronecker delta. Stated another way, '''v'''''i''(''cj'') is 0 unless ''i'' = ''j'', in which case '''v'''''i''(''ci'') = 1.

These unimodular matrices define a change of basis for the ''p''-limit JI group: just as every ''p''-limit interval can be written as a product of primes up to ''p'' with integer exponents, every such interval is a product of ''c''1, ''c''2, … , ''cn'' with integer exponents.

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 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, which we will assume are greater than 1 (i.e. ascending intervals). Call the n-1 commas ''c''<sub>1</sub>, ''c''<sub>2</sub>, … , ''c''<sub>''n-1.''</sub> There is a (nonunique) 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 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>.
+Note that this is only possible if the lattice spanned by the commas has full rank and is [[torsion]]-free.
 ==== Proof of the existence of these vals ====
-Having selected a step, form the ''n'' × ''n'' matrix whose last row is the monzo for the step ''c'', and whose other rows are the monzos of the ''n'' − 1 chromas. Because we have chosen ''c'' so that ''V''(''c'') = 1, each point ''S'' in the lattice is findable by stacking row vectors (stacking ''c'' as many times as ''S'' is mapped to by ''V'' and adjusting by the commas of ''V'') and so the determinant of this matrix will be ±1. It is therefore a unimodular matrix, that is, a square matrix with coefficients which are integers and with determinant ±1. Such a matrix is invertible, and the inverse matrix is also unimodular. If we call ''c'' "''c<sub>n</sub>''", and label the chromas ''c''<sub>1</sub>, ''c''<sub>2</sub>, … , ''c''<sub>(''n'' − 1)</sub>; and if we consider the columns of the inverse matrix to be vals and call them '''v'''<sub>1</sub>, '''v'''<sub>2</sub>, … , '''v'''<sub>''n''</sub>, then by the definition of the inverse of a matrix, '''v'''<sub>''i''</sub>(''c<sub>j</sub>'') = δ(''i'', ''j''), where δ(''i'', ''j'') is the Kronecker delta. Stated another way, '''v'''<sub>''i''</sub>(''c<sub>j</sub>'') is 0 unless ''i'' = ''j'', in which case '''v'''<sub>''i''</sub>(''c<sub>i</sub>'') = 1.
+Having selected a step, form the ''n'' × ''n'' matrix whose last row is the monzo for the step ''c'', and whose other rows are the monzos of the ''n'' − 1 chromas. Because we have chosen ''c'' so that ''V''(''c'') = 1, each point ''S'' in the lattice can be found by stacking row vectors (stacking ''c'' as many times as ''S'' is mapped to by ''V'' and adjusting by the commas of ''V'') and so the determinant of this matrix will be ±1. Such a matrix is called a unimodular matrix, which is invertible in the integers. The inverse matrix is also unimodular. If we call ''c'' "''c<sub>n</sub>''", and label the chromas ''c''<sub>1</sub>, ''c''<sub>2</sub>, … , ''c''<sub>(''n'' − 1)</sub>; and if we consider the columns of the inverse matrix to be vals and call them '''v'''<sub>1</sub>, '''v'''<sub>2</sub>, … , '''v'''<sub>''n''</sub>, then by the definition of the inverse of a matrix, '''v'''<sub>''i''</sub>(''c<sub>j</sub>'') = δ(''i'', ''j''), where δ(''i'', ''j'') is the Kronecker delta. Stated another way, '''v'''<sub>''i''</sub>(''c<sub>j</sub>'') is 0 unless ''i'' = ''j'', in which case '''v'''<sub>''i''</sub>(''c<sub>i</sub>'') = 1.
 These unimodular matrices define a change of basis for the ''p''-limit JI group: just as every ''p''-limit interval can be written as a product of primes up to ''p'' with integer exponents, every such interval is a product of ''c''<sub>1</sub>, ''c''<sub>2</sub>, … , ''c<sub>n</sub>'' with integer exponents.