Domain basis: Difference between revisions

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So, for instance, a temperament in the 2.3.5 domain cannot map the intervals 7/6 or 11/8, because there is no way to represent either of those intervals using only the primes 2, 3, and 5. It could, however, temper 6/5, 5/4, 10/9, or 9/8, etc., because those intervals ''can'' be represented using only those three primes.

A basis may be used to label the columns of a mapping, with one basis element per column. Here's [[~~Chromatic pairs~~#Slendric|slendric]], a temperament of the 2.3.7 domain:

A basis may be used to label the columns of a mapping, with one basis element per column. Here's [[Gamelismic clan#Slendric|slendric]], a temperament of the 2.3.7 domain:

<math>

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</math>

Note for comparison that a comma basis is also a type of basis. In the same way that a domain basis is a minimal representation of all the ''intervals'' in the temperament, a comma basis is a minimal representation of all the ''commas'' in the temperament{{mdash}}to be precise, the subspace of all commas that are made to vanish.

Note for comparison that a comma basis is also a type of basis. In the same way that a domain basis is a minimal representation of all the ''intervals'' in the temperament, a comma basis is a minimal representation of all the ''commas'' in the temperament—to be precise, the subspace of all commas that are made to vanish.

In the case of a comma basis, both the basis vectors and all of the spanned vectors are commas. But in the case of a domain basis, neither of these things is true. The basis vectors constitute an identity matrix, which is why they're our "mother of all bases"; at the point one hits basis identity matrix bedrock like this, the only place to go is defining what the entries of these vectors actually stand for, which in our case is prime bases of exponents.

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== Basis matrix conversion ==

The reduction method we will use as part of canonicalization is the [[Hermite normal form]]. If you are previously familiar with it, you may be surprised to see it here, because you may realize that it is defined for matrices, not lists of numbers. So far, when we've looked at subspaces{{mdash}}or at least looked at the bases that represent them{{mdash}}we've simply notated them as lists of numbers, such as 2.3.7. And in most contexts this number list notation is sufficient. However, in order to merge domains, we need to temporarily convert their bases them into matrix form, in order to use the Hermite normal form.

The reduction method we will use as part of canonicalization is the [[Hermite normal form]]. If you are previously familiar with it, you may be surprised to see it here, because you may realize that it is defined for matrices, not lists of numbers. So far, when we've looked at subspaces—or at least looked at the bases that represent them—we've simply notated them as lists of numbers, such as 2.3.7. And in most contexts this number list notation is sufficient. However, in order to merge domains, we need to temporarily convert their bases them into matrix form, in order to use the Hermite normal form.

Well, let's get to the matrix-ifying!

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We can do this by factorizing the basis elements in just the same way we factor intervals into [[prime-count vector]]s, such as 5/4 factorizing to {{vector|-2 0 1}}. This is also the same way we represent comma intervals within the other key RTT basis: the comma basis.

But here, we're going one step deeper down! Now we're breaking down our basis elements{{mdash}}the building blocks of our intervals{{mdash}}into ''their own building blocks''. And these, finally, are just actual prime numbers.

But here, we're going one step deeper down! Now we're breaking down our basis elements—the building blocks of our intervals—into ''their own building blocks''. And these, finally, are just actual prime numbers.

Then, each resulting vector becomes a column of our desired matrix.

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== Column Hermite normal form ==

To be exact, we want to use the ''column-style'' Hermite normal form, sometimes called column Hermite normal form for short. All this means is that we put the HNF call in an antitranspose sandwich, [[Normal_lists#~~Defactored_Hermite_form_2~~|as described here]]).

To be exact, we want to use the ''column-style'' Hermite normal form, sometimes called column Hermite normal form for short. All this means is that we put the HNF call in an antitranspose sandwich, [[Normal_lists#Antitransposed defactored Hermite form|as described here]]).

== Vs. canonical form for temperaments themselves: do not defactor ==

@@ Line 52: / Line 52: @@
 So, for instance, a temperament in the 2.3.5 domain cannot map the intervals 7/6 or 11/8, because there is no way to represent either of those intervals using only the primes 2, 3, and 5. It could, however, temper 6/5, 5/4, 10/9, or 9/8, etc., because those intervals ''can'' be represented using only those three primes.
-A basis may be used to label the columns of a mapping, with one basis element per column. Here's [[Chromatic pairs#Slendric|slendric]], a temperament of the 2.3.7 domain:
+A basis may be used to label the columns of a mapping, with one basis element per column. Here's [[Gamelismic clan#Slendric|slendric]], a temperament of the 2.3.7 domain:
 <math>
@@ Line 82: / Line 82: @@
 </math>
-Note for comparison that a comma basis is also a type of basis. In the same way that a domain basis is a minimal representation of all the ''intervals'' in the temperament, a comma basis is a minimal representation of all the ''commas'' in the temperament{{mdash}}to be precise, the subspace of all commas that are made to vanish.
+Note for comparison that a comma basis is also a type of basis. In the same way that a domain basis is a minimal representation of all the ''intervals'' in the temperament, a comma basis is a minimal representation of all the ''commas'' in the temperament&mdash;to be precise, the subspace of all commas that are made to vanish.
 In the case of a comma basis, both the basis vectors and all of the spanned vectors are commas. But in the case of a domain basis, neither of these things is true. The basis vectors constitute an identity matrix, which is why they're our "mother of all bases"; at the point one hits basis identity matrix bedrock like this, the only place to go is defining what the entries of these vectors actually stand for, which in our case is prime bases of exponents.
@@ Line 143: / Line 143: @@
 == Basis matrix conversion ==
-The reduction method we will use as part of canonicalization is the [[Hermite normal form]]. If you are previously familiar with it, you may be surprised to see it here, because you may realize that it is defined for matrices, not lists of numbers. So far, when we've looked at subspaces{{mdash}}or at least looked at the bases that represent them{{mdash}}we've simply notated them as lists of numbers, such as 2.3.7. And in most contexts this number list notation is sufficient. However, in order to merge domains, we need to temporarily convert their bases them into matrix form, in order to use the Hermite normal form.
+The reduction method we will use as part of canonicalization is the [[Hermite normal form]]. If you are previously familiar with it, you may be surprised to see it here, because you may realize that it is defined for matrices, not lists of numbers. So far, when we've looked at subspaces&mdash;or at least looked at the bases that represent them&mdash;we've simply notated them as lists of numbers, such as 2.3.7. And in most contexts this number list notation is sufficient. However, in order to merge domains, we need to temporarily convert their bases them into matrix form, in order to use the Hermite normal form.
 Well, let's get to the matrix-ifying!
@@ Line 149: / Line 149: @@
 We can do this by factorizing the basis elements in just the same way we factor intervals into [[prime-count vector]]s, such as 5/4 factorizing to {{vector|-2 0 1}}. This is also the same way we represent comma intervals within the other key RTT basis: the comma basis.
-But here, we're going one step deeper down! Now we're breaking down our basis elements{{mdash}}the building blocks of our intervals{{mdash}}into ''their own building blocks''. And these, finally, are just actual prime numbers.
+But here, we're going one step deeper down! Now we're breaking down our basis elements&mdash;the building blocks of our intervals&mdash;into ''their own building blocks''. And these, finally, are just actual prime numbers.
 Then, each resulting vector becomes a column of our desired matrix.
@@ Line 185: / Line 185: @@
 == Column Hermite normal form ==
-To be exact, we want to use the ''column-style'' Hermite normal form, sometimes called column Hermite normal form for short. All this means is that we put the HNF call in an antitranspose sandwich, [[Normal_lists#Defactored_Hermite_form_2|as described here]]).
+To be exact, we want to use the ''column-style'' Hermite normal form, sometimes called column Hermite normal form for short. All this means is that we put the HNF call in an antitranspose sandwich, [[Normal_lists#Antitransposed defactored Hermite form|as described here]]).
 == Vs. canonical form for temperaments themselves: do not defactor ==