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). We can find the val V which tempers all of these commas; the ~~Fokker block then is a [[detemperament]]~~ of ~~that val~~. 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 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). We can find the val V which tempers all of these commas, which we can find through calculating the null space of the monzos of the commas. 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, the epimorph val maps ''c'' to 1 step. 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}}, 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 {{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''''n''", 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, {{nowrap|'''v'''''i''(''c''''j'') {{=}} δ(''i'', ''j'')}}, where δ(''i'', ''j'') is the {{w|Kronecker delta}}. Stated another way, '''v'''''i''(''c''''j'') is 0 unless {{nowrap|''i'' {{=}} ''j''}}, in which case {{nowrap|'''v'''''i''(''c''''i'') {{=}} 1}}.

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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). We can find the val V which tempers all of these commas; the Fokker block then is a [[detemperament]] of that val. 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 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). We can find the val V which tempers all of these commas, which we can find through calculating the null space of the monzos of the commas. 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, the epimorph val maps ''c'' to 1 step. 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}}, 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 {{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}}.