Fokker block: Difference between revisions

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

Above this, we add a row of ''n'' units:

[~~e2 e3 e5 e7~~]

[1 1 1 1]

[2 2 -1 -1]

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[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''~~.

The n-by-n grid arrangement that results is called a matrix. The determinant of a matrix is a special operation that tells you by how much the transformation defined by said matrix scales space. If the determinant of a matrix is 0, that means that it flattens space down to a lower dimension; in other words, all hypervolumes in the starting space become zero. This means that the commas are not independent, and we need to change one of our commas to something new in order to create a Fokker block.

~~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}}.

Taking the determinant of this matrix involves taking the determinant of smaller matrices for each prime, where the prime's column and the top row are excluded. For example, the determinant of [[2 2 -1][1 0 -2][6 -2 -1]] for the prime 5 in the example above. These can be called {{nowrap|''k''2, ''k''3, … ''k''''p''}}, and during the calculation of the determinant, we'll flip the signs of some of these to reach the values {{nowrap|''w''2, ''w''3, … ''w''''p''}} for each prime, which are integers and share their absolute value with {{nowrap|''k''2, ''k''3, … ''k''''p''}} mentioned previously. These integers ultimately tell us the mapping for each prime in the equal temperament which tempers out these commas; which leads to a natural interpretation as the [[vals and tuning space|val]] {{nowrap|'''v''' {{=}} {{val| w2 w3 … w''p'' }}}}. If, among the elements of this val, there exists a common divisor, then our generators are enfactored and more accurately describe a smaller edo, as they only hit edosteps that are multiples of some number. This ultimately means the system will have to be reinterpreted as a smaller edo or a new generator introduced. {{nowrap|''w''2}} may be negative, in which case we flip the sign of everything in the val so we're describing a positive edo. Ultimately, 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, which is a ''p''-limit interval ''c'' mapped to one step in our equal temperament (which is why we needed to fix the enfactoring problem earlier). 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 37: / Line 37: @@
 [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> }}:
+Above this, we add a row of ''n'' units:
-[e<sub>2</sub> e<sub>3</sub> e<sub>5</sub> e<sub>7</sub>]
+[1 1 1 1]
 [2 2 -1 -1]
@@ Line 47: / Line 47: @@
 [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''.
+The n-by-n grid arrangement that results is called a matrix. The determinant of a matrix is a special operation that tells you by how much the transformation defined by said matrix scales space. If the determinant of a matrix is 0, that means that it flattens space down to a lower dimension; in other words, all hypervolumes in the starting space become zero. This means that the commas are not independent, and we need to change one of our commas to something new in order to create a Fokker block.
-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}}.
+Taking the determinant of this matrix involves taking the determinant of smaller matrices for each prime, where the prime's column and the top row are excluded. For example, the determinant of [[2 2 -1][1 0 -2][6 -2 -1]] for the prime 5 in the example above. These can be called {{nowrap|''k''<sub>2</sub>, ''k''<sub>3</sub>, … ''k''<sub>''p''</sub>}}, and during the calculation of the determinant, we'll flip the signs of some of these to reach the values {{nowrap|''w''<sub>2</sub>, ''w''<sub>3</sub>, … ''w''<sub>''p''</sub>}} for each prime, which are integers and share their absolute value with {{nowrap|''k''<sub>2</sub>, ''k''<sub>3</sub>, … ''k''<sub>''p''</sub>}} mentioned previously. These integers ultimately tell us the mapping for each prime in the equal temperament which tempers out these commas; which leads to a natural interpretation as the [[vals and tuning space|val]] {{nowrap|'''v''' {{=}} {{val| w<sub>2</sub> w<sub>3</sub> … w<sub>''p''</sub> }}}}. If, among the elements of this val, there exists a common divisor, then our generators are enfactored and more accurately describe a smaller edo, as they only hit edosteps that are multiples of some number. This ultimately means the system will have to be reinterpreted as a smaller edo or a new generator introduced. {{nowrap|''w''<sub>2</sub>}} may be negative, in which case we flip the sign of everything in the val so we're describing a positive edo. Ultimately, 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, which is a ''p''-limit interval ''c'' mapped to one step in our equal temperament (which is why we needed to fix the enfactoring problem earlier). 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