Fokker block: Difference between revisions
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Suppose we have {{nowrap|''n'' − 1}} commas, which we will assume are greater than 1, and we form an {{nowrap| ''n'' × ''n'' }} matrix, the top row of which are ''n'' indeterminate elements {{monzo| ''e''<sub>2</sub> ''e''<sub>3</sub> ''e''<sub>5</sub> … ''e''<sub>''p''</sub> }}, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get {{nowrap|''w''<sub>2</sub>''e''<sub>2</sub> + ''w''<sub>3</sub>''e''<sub>3</sub> + … + ''w''<sub>''p''</sub>''e''<sub>''p''</sub>}} where the ''w''<sub>2</sub>, ''w''<sub>3</sub> … ''w''<sub>''p''</sub> are integers. We interpret this as the [[vals and tuning space|val]] {{nowrap|'''v''' {{=}} {{val| w<sub>2</sub> w<sub>3</sub> … w<sub>''p''</sub> }}}}. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a torsion problem, and we discard the comma set. Otherwise, if {{nowrap|''w''<sub>2</sub> < 0}} we reverse sign, and we have a val ''V'' which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 225/224, 100/99, 176/175, and 385/384, the above procedure gives us {{nowrap|''V'' {{=}} {{val| 22 35 51 62 76 }}}}, and we will be looking at a 22-note scale in the 11-limit. We may call ''V'' the ''epimorph val'', and the {{nowrap|''n'' − 1}} commas, which form a basis for the kernel of ''V'', the ''chroma basis''. | Suppose we have {{nowrap|''n'' − 1}} commas, which we will assume are greater than 1, and we form an {{nowrap| ''n'' × ''n'' }} matrix, the top row of which are ''n'' indeterminate elements {{monzo| ''e''<sub>2</sub> ''e''<sub>3</sub> ''e''<sub>5</sub> … ''e''<sub>''p''</sub> }}, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get {{nowrap|''w''<sub>2</sub>''e''<sub>2</sub> + ''w''<sub>3</sub>''e''<sub>3</sub> + … + ''w''<sub>''p''</sub>''e''<sub>''p''</sub>}} where the ''w''<sub>2</sub>, ''w''<sub>3</sub> … ''w''<sub>''p''</sub> are integers. We interpret this as the [[vals and tuning space|val]] {{nowrap|'''v''' {{=}} {{val| w<sub>2</sub> w<sub>3</sub> … w<sub>''p''</sub> }}}}. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a torsion problem, and we discard the comma set. Otherwise, if {{nowrap|''w''<sub>2</sub> < 0}} we reverse sign, and we have a val ''V'' which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 225/224, 100/99, 176/175, and 385/384, the above procedure gives us {{nowrap|''V'' {{=}} {{val| 22 35 51 62 76 }}}}, and we will be looking at a 22-note scale in the 11-limit. We may call ''V'' the ''epimorph val'', and the {{nowrap|''n'' − 1}} commas, which form a basis for the kernel of ''V'', the ''chroma basis''. | ||
Now choose a uniformizing step for the Fokker block, by which is meant a ''p''-limit interval ''c'' such that {{nowrap|''V''(''c'') {{=}} 1}}; that is, if '''m''' is the monzo for ''c'', then {{nowrap|{{ | Now choose a uniformizing step for the Fokker block, by which is meant a ''p''-limit interval ''c'' such that {{nowrap|''V''(''c'') {{=}} 1}}; that is, if '''m''' is the monzo for ''c'', then {{nowrap|{{vmp|''V''|'''m'''}} {{=}} 1}}. Precisely which interval with this property we choose does not 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 {{nowrap| ''n'' × ''n'' }} matrix whose last row is the monzo for the step ''c'', and whose other rows are the monzos of the {{nowrap|''n'' − 1}} chromas. Because we have chosen ''c'' so that {{nowrap|''V''(''c'') {{=}} 1}}, the determinant of this matrix will be ±1. It is therefore a {{w|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 style="white-space: nowrap;">(''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, {{nowrap|'''v'''<sub>''i''</sub>(''c''<sub>''j''</sub>) {{=}} δ(''i'', ''j'')}}, where δ(''i'', ''j'') is the {{w|Kronecker delta}}. Stated another way, '''v'''<sub>''i''</sub>(''c''<sub>''j''</sub>) is 0 unless {{nowrap|''i'' {{=}} ''j''}}, in which case {{nowrap|'''v'''<sub>''i''</sub>(''c''<sub>''i''</sub>) {{=}} 1}}. | ||
These unimodular matrices define a {{w|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 | These unimodular matrices define a {{w|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 | ||