Temperament addition: Difference between revisions

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Meet and join does not ''necessarily'' involve linear dependence. Linear dependence only matters for meet and join when you attempt to do it using ''exterior'' algebra, that is, by using the wedge product, rather than the ''linear'' algebra approach, which is just to concatenate the vectors as a matrix and reduce. For more information on this, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#The linearly dependent exception to the wedge product]].

=~~Negation~~=

=<math>g_{\text{min}}=1</math>=

[[File:Very simple illustration of temperament sum vs diff.png|500px|thumb|left|Equivalences of temperament arithmetic depending on negativity. ]]

As stated above, temperament arithmetic is simplest for temperaments which can be represented by single vectors, or in other words, temperaments that are <math>g_{\text{min}}=1</math>, and for other temperaments, the computation gets a little trickier. Here we'll look at how to handle the simple case of <math>g_{\text{min}}=1</math>.

As shown in the [[Temperament_arithmetic#Introductory_examples|introductory examples]], <math>g_{\text{min}}=1</math> examples are as easy as entry-wise addition or subtraction. But there's just a couple tricks to it.

==Getting to the side of duality with <math>g_{\text{min}}=1</math>==

We may be looking at a temperament representation which itself does not consist of a single vector, but its dual does. For example, the meantone mapping {{ket|{{map|1 0 -4}} {{map|0 1 4}}}} and the porcupine mapping {{ket|{{map|1 2 3}} {{map|0 3 5}}}} each consist of two vectors. So these representations cannot be summed or differenced using simple entry-wise addition or subtraction. But their duals can! If we simply find a comma basis for each of these mappings, we get {{bra|{{vector|4 -4 1}}}} and {{bra|{{vector|1 -5 3}}}}. In this form, the temperaments can be entry-wise added, to {{bra|{{vector|5 -9 4}}}} as we saw earlier. And if in the end we're still after a mapping, since we started with mappings, we can take the dual of this comma basis, to find the mapping {{ket|{{map|1 1 1}} {{map|0 4 9}}}}.

==Negation==

[[File:Very simple illustration of temperament sum vs diff.png|500px|thumb|left|Equivalences of temperament arithmetic depending on negativity.]]

There's just one other trick to it, and that's that we have to be mindful of negation.

The temperament difference can be understood as being the same operation as the temperament sum except with one of the two temperaments negated.

For single vectors (and multivectors), negation is as simple as changing the sign of every entry; for matrices, negation is accomplished by choosing a single vector and changing the sign of every entry in it. In the case of comma bases, a vector is a column, whereas in a mapping a vector (technically a row vector, or covector) is a row.

For single vectors (and multivectors), negation is as simple as changing the sign of every entry.

Suppose you have a matrix representing temperament <math>T_1</math> and another matrix representing <math>T_2</math>. If you want to find both their sum and difference, you can calculate both <math>T_1 + T_2</math> and <math>T_1 + -T_2</math>. There's no need to also find <math>-T_1 + T_2</math>; this will merely give the negation of <math>T_1 + -T_2</math>. The same goes for <math>-T_1 + -T_2</math>, which is the negation of <math>T_1 + T_2</math>.

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But a question remains: which result between <math>T_1 + T_2</math> and <math>T_1 + -T_2</math> is actually the sum and which is the difference? This seems like an obvious question to answer, except for one key problem: how can we be certain that <math>T_1</math> or <math>T_2</math> wasn't already in negated form to begin with? We need to establish a way to check for matrix negativity.

The check is ~~related to canonicalization of varianced multivectors as are used~~ in ~~exterior algebra for RTT~~. ~~Essentially we take~~ the ~~minors of~~ the ~~matrix, and then look at their leading or~~ trailing entry (~~leading in~~ the ~~case of~~ a covariant ~~matrix~~, ~~like a~~ mapping~~; trailing in~~ the ~~case of a contravariant matrix, like a comma basis~~)~~: if this entry is~~ positive~~, so is the temperament, and vice versa~~.

The check is that the vectors must be in [[canonical form]]. For a contravariant vector, such as the kind that represent commas, canonical form means that the trailing entry (the final non-zero entry) must be positive. For a covariant vector, such as the kind that represent mapping-rows, canonical form means that the leading entry (the first non-zero entry) must be positive.

~~=Beyond <math>g_~~{~~\text~~{~~min~~}}=1~~</math>=~~

Sometimes the canonical form of a vector is not the most popular form. For instance, the meantone comma is usually expressed in positive form, that is, with its numerator greater than its denominator, so that its cents value is positive, or in other words, it's the meantone comma upwards in pitch, not downwards. But the prime count vector for that form, 81/80, is {{vector|-4 4 -1}}, and as we can see, its trailing entry -1 is negative. So the canonical form of meantone is actually {{vector|4 -4 1}}.

As stated above, temperament arithmetic is simplest for temperaments which can be represented by single vectors, or in other words, temperaments that are <math>g_{\text{min}}=1</math>, and for other temperaments, the computation gets a little trickier. Here we'll look at how to handle it.

=<math>g_{\text{min}}>1</math>=

As stated above, temperament arithmetic is simplest for temperaments which can be represented by single vectors, or in other words, temperaments that are <math>g_{\text{min}}=1</math>, and for other temperaments, the computation gets a little trickier. Here we'll look at how to handle the trickier cases of <math>g_{\text{min}}>1</math>.

==Addability==

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Throughout this section, we will be using <span style="color: #3C8031;">a green color on linearly dependent objects and values</span>, and <span style="color: #B6321C;">a red color on linearly independent objects and values</span>, to help differentiate between the two.

===Negation===

For matrices, negation is accomplished by choosing a single vector and changing the sign of every entry in it. In the case of comma bases, a vector is a column, whereas in a mapping a vector (technically a row vector, or covector) is a row.

For matrices, the check for negation is related to canonicalization of multivectors as are used in exterior algebra for RTT. Essentially we take the minors of the matrix, and then look at their leading or trailing entry (leading in the case of a covariant matrix, like a mapping; trailing in the case of a contravariant matrix, like a comma basis): if this entry is positive, so is the temperament, and vice versa.

===Example===

@@ Line 195: / Line 195: @@
 Meet and join does not ''necessarily'' involve linear dependence. Linear dependence only matters for meet and join when you attempt to do it using ''exterior'' algebra, that is, by using the wedge product, rather than the ''linear'' algebra approach, which is just to concatenate the vectors as a matrix and reduce. For more information on this, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#The linearly dependent exception to the wedge product]].
-=Negation=
+=<math>g_{\text{min}}=1</math>=
-[[File:Very simple illustration of temperament sum vs diff.png|500px|thumb|left|Equivalences of temperament arithmetic depending on negativity. ]]
+As stated above, temperament arithmetic is simplest for temperaments which can be represented by single vectors, or in other words, temperaments that are <math>g_{\text{min}}=1</math>, and for other temperaments, the computation gets a little trickier. Here we'll look at how to handle the simple case of <math>g_{\text{min}}=1</math>.
+As shown in the [[Temperament_arithmetic#Introductory_examples|introductory examples]], <math>g_{\text{min}}=1</math> examples are as easy as entry-wise addition or subtraction. But there's just a couple tricks to it.
+==Getting to the side of duality with <math>g_{\text{min}}=1</math>==
+We may be looking at a temperament representation which itself does not consist of a single vector, but its dual does. For example, the meantone mapping {{ket|{{map|1 0 -4}} {{map|0 1 4}}}} and the porcupine mapping {{ket|{{map|1 2 3}} {{map|0 3 5}}}} each consist of two vectors. So these representations cannot be summed or differenced using simple entry-wise addition or subtraction. But their duals can! If we simply find a comma basis for each of these mappings, we get {{bra|{{vector|4 -4 1}}}} and {{bra|{{vector|1 -5 3}}}}. In this form, the temperaments can be entry-wise added, to {{bra|{{vector|5 -9 4}}}} as we saw earlier. And if in the end we're still after a mapping, since we started with mappings, we can take the dual of this comma basis, to find the mapping {{ket|{{map|1 1 1}} {{map|0 4 9}}}}.
+==Negation==
+[[File:Very simple illustration of temperament sum vs diff.png|500px|thumb|left|Equivalences of temperament arithmetic depending on negativity.]]
+There's just one other trick to it, and that's that we have to be mindful of negation.
 The temperament difference can be understood as being the same operation as the temperament sum except with one of the two temperaments negated.
-For single vectors (and multivectors), negation is as simple as changing the sign of every entry; for matrices, negation is accomplished by choosing a single vector and changing the sign of every entry in it. In the case of comma bases, a vector is a column, whereas in a mapping a vector (technically a row vector, or covector) is a row.
+For single vectors (and multivectors), negation is as simple as changing the sign of every entry.
 Suppose you have a matrix representing temperament <math>T_1</math> and another matrix representing <math>T_2</math>. If you want to find both their sum and difference, you can calculate both <math>T_1 + T_2</math> and <math>T_1 + -T_2</math>. There's no need to also find <math>-T_1 + T_2</math>; this will merely give the negation of <math>T_1 + -T_2</math>. The same goes for <math>-T_1 + -T_2</math>, which is the negation of <math>T_1 + T_2</math>.
@@ Line 207: / Line 219: @@
 But a question remains: which result between <math>T_1 + T_2</math> and <math>T_1 + -T_2</math> is actually the sum and which is the difference? This seems like an obvious question to answer, except for one key problem: how can we be certain that <math>T_1</math> or <math>T_2</math> wasn't already in negated form to begin with? We need to establish a way to check for matrix negativity.
-The check is related to canonicalization of varianced multivectors as are used in exterior algebra for RTT. Essentially we take the minors of the matrix, and then look at their leading or trailing entry (leading in the case of a covariant matrix, like a mapping; trailing in the case of a contravariant matrix, like a comma basis): if this entry is positive, so is the temperament, and vice versa.
+The check is that the vectors must be in [[canonical form]]. For a contravariant vector, such as the kind that represent commas, canonical form means that the trailing entry (the final non-zero entry) must be positive. For a covariant vector, such as the kind that represent mapping-rows, canonical form means that the leading entry (the first non-zero entry) must be positive.
-=Beyond <math>g_{\text{min}}=1</math>=
+Sometimes the canonical form of a vector is not the most popular form. For instance, the meantone comma is usually expressed in positive form, that is, with its numerator greater than its denominator, so that its cents value is positive, or in other words, it's the meantone comma upwards in pitch, not downwards. But the prime count vector for that form, 81/80, is {{vector|-4 4 -1}}, and as we can see, its trailing entry -1 is negative. So the canonical form of meantone is actually {{vector|4 -4 1}}.
-As stated above, temperament arithmetic is simplest for temperaments which can be represented by single vectors, or in other words, temperaments that are <math>g_{\text{min}}=1</math>, and for other temperaments, the computation gets a little trickier. Here we'll look at how to handle it.
+=<math>g_{\text{min}}>1</math>=
+As stated above, temperament arithmetic is simplest for temperaments which can be represented by single vectors, or in other words, temperaments that are <math>g_{\text{min}}=1</math>, and for other temperaments, the computation gets a little trickier. Here we'll look at how to handle the trickier cases of <math>g_{\text{min}}>1</math>.
 ==Addability==
@@ Line 599: / Line 613: @@
 Throughout this section, we will be using <span style="color: #3C8031;">a green color on linearly dependent objects and values</span>, and <span style="color: #B6321C;">a red color on linearly independent objects and values</span>, to help differentiate between the two.
+===Negation===
+For matrices, negation is accomplished by choosing a single vector and changing the sign of every entry in it. In the case of comma bases, a vector is a column, whereas in a mapping a vector (technically a row vector, or covector) is a row.
+For matrices, the check for negation is related to canonicalization of multivectors as are used in exterior algebra for RTT. Essentially we take the minors of the matrix, and then look at their leading or trailing entry (leading in the case of a covariant matrix, like a mapping; trailing in the case of a contravariant matrix, like a comma basis): if this entry is positive, so is the temperament, and vice versa.
 ===Example===