Fokker block: Difference between revisions

Line 26:

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).

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''.

~~Now, we can list out the monzos of the commas like this (with the example of 12edo in the 2.3.5.7 subgroup):~~

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 column 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}}.

~~[2 2 -1 -1]~~

~~[1 0 2 -2]~~

~~[6 -2 0 -1]~~

~~Above this, we add a row of ''n'' indeterminate elements {{monzo| ''e''2 ''e''3 ''e''5 … ''e''''p'' }}:~~

~~[e2 e3 e5 e7]~~

~~[2 2 -1 -1]~~

~~[1 0 2 -2]~~

~~[6 -2 0 -1]~~

The n-by-n grid arrangement that results is called a matrix. If we take the determinant of this matrix (a special operation that returns a number), we get {{nowrap|''w''2''e''2 + ''w''3''e''3 + … + ''w''''p''''e''''p''}} where the ''w''2, ''w''3 … ''w''''p'' are integers. We can then interpret this as the [[vals and tuning space|val]] {{nowrap|'''v''' {{=}} {{val| w2 w3 … w''p'' }}}}. 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 [[Saturation, torsion, and contorsion|torsion]] problem, and we discard the comma set. Otherwise, if {{nowrap|''w''2 < 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|{{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''''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}}.

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''1, ''c''2, … , ''c''''n'' with integer exponents. To determine the exponents, we use '''v'''1, '''v'''2, … , '''v'''''n'', so that if ''q'' is a ''p''-limit rational number, we may write it as

@@ Line 26: / Line 26: @@
 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).
+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''.
-Now, we can list out the monzos of the commas like this (with the example of 12edo in the 2.3.5.7 subgroup):
+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 column 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}}.
-[2 2 -1 -1]
-[1 0 2 -2]
-[6 -2 0 -1]
-Above this, we add a row of ''n'' indeterminate elements {{monzo| ''e''<sub>2</sub> ''e''<sub>3</sub> ''e''<sub>5</sub> … ''e''<sub>''p''</sub> }}:
-[e<sub>2</sub> e<sub>3</sub> e<sub>5</sub> e<sub>7</sub>]
-[2 2 -1 -1]
-[1 0 2 -2]
-[6 -2 0 -1]
-The n-by-n grid arrangement that results is called a matrix. If we take the determinant of this matrix (a special operation that returns a number), 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 can then 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 [[Saturation, torsion, and contorsion|torsion]] problem, and we discard the comma set. Otherwise, if {{nowrap|''w''<sub>2</sub> &lt; 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|{{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'',&nbsp;''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