Domain basis: Difference between revisions

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We can say that an interval subspace <math>B_1</math> is a subspace of another interval subspace <math>B_2</math> if when we merge<ref>The technical mathematical term for this is "sumset", not "union" as we might expect; in many contexts, "union" is the dual operation to "intersection", but for vector spaces, the dual operation to intersection is "sumset" (see page 4 of https://www2.math.upenn.edu/~siegelch/Notes/linalg.pdf). The difference between union and sumset can be explained like this: if we had two planes in a volume, their union would be both the planes, but their sumset would be the volume.</ref> <math>B_1</math> and <math>B_2</math> we just get <math>B_2</math> again. In layperson's terms, if <math>B_1</math> brings nothing to the table that <math>B_2</math> hasn't already brought, then it is completely contained by <math>B_2</math> and therefore is a subspace of it.

In order to fully explain this definition, however, we must expand upon what is meant by "merging". If you are familiar with [[temperament merging]], it's a similar idea: concatenate, then ~~reduce~~ the result.

In order to fully explain this definition, however, we must expand upon what is meant by "merging". If you are familiar with [[temperament merging]], it's a similar idea: concatenate, then canonicalize the result.

=== Concatenate ===

The first step is concatenation. This is the easy part. Suppose we're merging 2.3.5 and 2.3.7; the concatenation of those two is quite simply 2.3.5.2.3.7. Yes, that result contains a lot of repetition. But that's what the next step — the ~~reduction~~ step — is there to solve.

The first step is concatenation. This is the easy part. Suppose we're merging 2.3.5 and 2.3.7; the concatenation of those two is quite simply 2.3.5.2.3.7. Yes, that result contains a lot of repetition. But that's what the next step — the canonicalization step — is there to solve.

=== ~~Reduce~~ ===

=== Canonicalize ===

==== Column Hermite normal form ====

~~The~~ reduction method we will use 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 interval subspaces, we need to temporarily convert them into matrix form, in order to use the Hermite normal form.

First, we reduce the matrix. And the reduction method we will use 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 interval subspaces, we need to temporarily convert them into matrix form, in order to use the 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 anti-transpose sandwich, [[Normal_lists#Defactored_Hermite_form_2|as described here]]).

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To make a popular culture reference, you may be starting to get an [[Wikipedia:Inception|Inception]] vibe here: we're breaking primes into deeper primes (perhaps we could call this "intervalception"?). Indeed, this might all seem dizzyingly abstract, but fortunately, we don't need to go any deeper than this. And we assure you that this matrix representation of the interval basis will be quite helpful for comparing different interval bases.

==== Vs. ~~reduction~~ in temperament merging: do not defactor ====

==== Vs. canonicalization in temperament merging: do not defactor ====

There's an important difference between the ~~reduction~~ step as it's done with temperament merging versus with interval basis merging. When merging temperaments, it's typical to fully [[defactor]] them as well as [[normal form|normalize]] them, because [[The_pathology_of_enfactoring|enfactored representations of temperaments are pathological]]. Enfactored interval bases, however, are ''not'' pathological; they represent meaningfully distinct interval bases.

There's an important difference between the canonicalization step as it's done with temperament merging versus with interval basis merging. When merging temperaments, it's typical to fully [[defactor]] them as well as [[normal form|normalize]] them, because [[The_pathology_of_enfactoring|enfactored representations of temperaments are pathological]]. Enfactored interval bases, however, are ''not'' pathological; they represent meaningfully distinct interval bases.

For example, if we were to defactor the 2.9.5 interval basis, we'd get 2.3.5. But 2.9.5 is a perfectly reasonable interval basis that we don't wish to conflate with 2.3.5.

@@ Line 147: / Line 147: @@
 We can say that an interval subspace <math>B_1</math> is a subspace of another interval subspace <math>B_2</math> if when we merge<ref>The technical mathematical term for this is "sumset", not "union" as we might expect; in many contexts, "union" is the dual operation to "intersection", but for vector spaces, the dual operation to intersection is "sumset" (see page 4 of https://www2.math.upenn.edu/~siegelch/Notes/linalg.pdf). The difference between union and sumset can be explained like this: if we had two planes in a volume, their union would be both the planes, but their sumset would be the volume.</ref> <math>B_1</math> and <math>B_2</math> we just get <math>B_2</math> again. In layperson's terms, if <math>B_1</math> brings nothing to the table that <math>B_2</math> hasn't already brought, then it is completely contained by <math>B_2</math> and therefore is a subspace of it.
-In order to fully explain this definition, however, we must expand upon what is meant by "merging". If you are familiar with [[temperament merging]], it's a similar idea: concatenate, then reduce the result.
+In order to fully explain this definition, however, we must expand upon what is meant by "merging". If you are familiar with [[temperament merging]], it's a similar idea: concatenate, then canonicalize the result.
 === Concatenate ===
-The first step is concatenation. This is the easy part. Suppose we're merging 2.3.5 and 2.3.7; the concatenation of those two is quite simply 2.3.5.2.3.7. Yes, that result contains a lot of repetition. But that's what the next step — the reduction step — is there to solve.
+The first step is concatenation. This is the easy part. Suppose we're merging 2.3.5 and 2.3.7; the concatenation of those two is quite simply 2.3.5.2.3.7. Yes, that result contains a lot of repetition. But that's what the next step — the canonicalization step — is there to solve.
-=== Reduce ===
+=== Canonicalize ===
 ==== Column Hermite normal form ====
-The reduction method we will use 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 interval subspaces, we need to temporarily convert them into matrix form, in order to use the Hermite normal form.
+First, we reduce the matrix. And the reduction method we will use 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 interval subspaces, we need to temporarily convert them into matrix form, in order to use the 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 anti-transpose sandwich, [[Normal_lists#Defactored_Hermite_form_2|as described here]]).
@@ Line 212: / Line 212: @@
 To make a popular culture reference, you may be starting to get an [[Wikipedia:Inception|Inception]] vibe here: we're breaking primes into deeper primes (perhaps we could call this "intervalception"?). Indeed, this might all seem dizzyingly abstract, but fortunately, we don't need to go any deeper than this. And we assure you that this matrix representation of the interval basis will be quite helpful for comparing different interval bases.
-==== Vs. reduction in temperament merging: do not defactor ====
+==== Vs. canonicalization in temperament merging: do not defactor ====
-There's an important difference between the reduction step as it's done with temperament merging versus with interval basis merging. When merging temperaments, it's typical to fully [[defactor]] them as well as [[normal form|normalize]] them, because [[The_pathology_of_enfactoring|enfactored representations of temperaments are pathological]]. Enfactored interval bases, however, are ''not'' pathological; they represent meaningfully distinct interval bases.
+There's an important difference between the canonicalization step as it's done with temperament merging versus with interval basis merging. When merging temperaments, it's typical to fully [[defactor]] them as well as [[normal form|normalize]] them, because [[The_pathology_of_enfactoring|enfactored representations of temperaments are pathological]]. Enfactored interval bases, however, are ''not'' pathological; they represent meaningfully distinct interval bases.
 For example, if we were to defactor the 2.9.5 interval basis, we'd get 2.3.5. But 2.9.5 is a perfectly reasonable interval basis that we don't wish to conflate with 2.3.5.