Fokker block: Difference between revisions
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Now choose a uniformizing step for the Fokker block, by which is meant a ''p''-limit interval ''c'' such that V (''c'') = 1; that is, if m is the monzo for ''c'', then ⟨V|m⟩ = 1. Precisely which interval with this property we choose doesn't actually matter, so if our chromas are 225/224, 100/99, 176/175 and 385/384, we could for instance choose 22/21, 25/24, 28/27, 33/32, 36/35, 45/44 or 49/48. Having selected a step, form the ''n'' by ''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, the determinant of this matrix will be ±1. It is therefore a [[Wikipedia: Unimodular matrix|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 [[Wikipedia: Kronecker delta|Kronecker delta]]. Stated another way, v<sub>''i''</sub> (''c''<sub>''j''</sub>) is 0 unless ''i'' equals ''j'', in which case v<sub>''i''</sub> (''c''<sub>''i''</sub>) = 1. | Now choose a uniformizing step for the Fokker block, by which is meant a ''p''-limit interval ''c'' such that V (''c'') = 1; that is, if m is the monzo for ''c'', then ⟨V|m⟩ = 1. Precisely which interval with this property we choose doesn't actually matter, so if our chromas are 225/224, 100/99, 176/175 and 385/384, we could for instance choose 22/21, 25/24, 28/27, 33/32, 36/35, 45/44 or 49/48. Having selected a step, form the ''n'' by ''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, the determinant of this matrix will be ±1. It is therefore a [[Wikipedia: Unimodular matrix|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 [[Wikipedia: Kronecker delta|Kronecker delta]]. Stated another way, v<sub>''i''</sub> (''c''<sub>''j''</sub>) is 0 unless ''i'' equals ''j'', in which case v<sub>''i''</sub> (''c''<sub>''i''</sub>) = 1. | ||
These unimodular matricies define a [[Wikipedia: Change of basis|change of basis]] for the ''p''-limit | These unimodular matricies define a [[Wikipedia: Change of basis|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. To determine the exponents, we use v<sub>1</sub>, v<sub>2</sub>, … , v<sub>''n''</sub>, so that if ''q'' is a ''p''-limit rational number, we may write it as | ||
<math>q = c_1^{v_1(q)} c_2^{v_2(q)} \cdots c_n^{v_n(q)}.</math> | <math>q = c_1^{v_1(q)} c_2^{v_2(q)} \cdots c_n^{v_n(q)}.</math> | ||