Domain basis: Difference between revisions
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== General method to determine whether an interval subspace is a subspace of another == | == General method to determine whether an interval subspace is a subspace of another == | ||
[[ | [[Interval_basis#Examples|A couple subsections ago]], we provided a couple examples where we used natural language to explain — between two interval subspaces — which one was a subspace of the other. But we still need to describe a method to determine this in general. Let's do that next. | ||
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 <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. | 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 <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. | ||
For more information on merging interval bases, see [[ | For more information on merging interval bases, see [[Interval_basis#Merging]]. | ||
=== Example === | === Example === | ||
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== Canonicalize == | == Canonicalize == | ||
See [[ | See [[Interval basis#Canonicalization]]. | ||
== Notation == | == Notation == | ||
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Interval basis merging comes up in two key situations: | Interval basis merging comes up in two key situations: | ||
# Determining whether one interval subspace is a subspace of another: <math>B_1</math> is a subspace of <math>B_2</math> if <math>B_1|B_2 = B_2</math>. For more details, see: [[ | # Determining whether one interval subspace is a subspace of another: <math>B_1</math> is a subspace of <math>B_2</math> if <math>B_1|B_2 = B_2</math>. For more details, see: [[Interval basis#General method to determine whether an interval subspace is a subspace of another]]. | ||
# Comma-merging temperaments with different interval bases, in which case the comma-merged temperament's interval basis will be the merge of the all the input interval bases. | # Comma-merging temperaments with different interval bases, in which case the comma-merged temperament's interval basis will be the merge of the all the input interval bases. | ||
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== Applications == | == Applications == | ||
The intersection of interval bases comes up with doing a map-merge of temperaments. The resulting temperament's interval basis will be the intersection of all the input interval bases. For more information, see: [[ | The intersection of interval bases comes up with doing a map-merge of temperaments. The resulting temperament's interval basis will be the intersection of all the input interval bases. For more information, see: [[Temperament_merging_across_interval_bases#Map-merge]]. | ||
= Changing interval basis = | = Changing interval basis = | ||
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Given an interval, comma basis, or mapping — anything that has an associated interval basis — it is possible to change it from one interval basis to another. We can accomplish this using an '''interval rebase''', an object that works like a two-way bridge between two interval bases. | Given an interval, comma basis, or mapping — anything that has an associated interval basis — it is possible to change it from one interval basis to another. We can accomplish this using an '''interval rebase''', an object that works like a two-way bridge between two interval bases. | ||
Elsewhere, these have been called [[subgroup basis matrices]], but that terminology will not be used here, for the same reasons as are described in the last section of this article (here: [[ | Elsewhere, these have been called [[subgroup basis matrices]], but that terminology will not be used here, for the same reasons as are described in the last section of this article (here: [[Interval basis#Terminology: interval basis vs. subgroup]]) as well as the additional reason that such a name can easily be conflated with the interval basis itself. Sometimes the superspace interval basis is an identity matrix, in which case the interval rebase will be the same as the subspace interval basis, but this is not always the case. | ||
As discussed earlier, only certain interval basis changes are possible: (here: [[ | As discussed earlier, only certain interval basis changes are possible: (here: [[Interval basis#Application: determining whether it is possible to change the interval subspace]]). To quickly recap here, it is only possible to change between interval subspaces where one is a subspace of the other. So when we say a given interval rebase works like a two-way bridge, there's a more specific way to say what we mean: an interval rebase allows us to change either ''from the subspace to the superspace'', or ''from the superspace to the subspace''. Which direction we go just depends on which side we enter the bridge from: the right side or the left side. | ||
== Constructing an interval rebase == | == Constructing an interval rebase == | ||
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=== Example === | === Example === | ||
Let's construct the interval rebase <math>R</math> between 2.25/9.11/7 and 2.5/3.7.11. [[ | Let's construct the interval rebase <math>R</math> between 2.25/9.11/7 and 2.5/3.7.11. [[Interval basis#Example|As we proved earlier]], the former is a subspace of the latter. So this will be a 4×3 matrix. | ||
* The first column is easy. There's no change to prime 2 between these two interval bases. | * The first column is easy. There's no change to prime 2 between these two interval bases. | ||
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Setting aside the specialized use it has taken on in these RTT writings, a subgroup (or subspace) in the general mathematical sense is just a generic mathematical structure, like a matrix or vector. This article prefers to use specialized terminology for objects in our RTT application, so that we can clearly discuss them independently from the mathematical structures that represent them. Just like how we call certain objects represented by matrices "mappings" and certain objects represented by vectors "intervals", this article prefers using a specialized term for this RTT object — one that cannot be confused with a generic mathematical structure. | Setting aside the specialized use it has taken on in these RTT writings, a subgroup (or subspace) in the general mathematical sense is just a generic mathematical structure, like a matrix or vector. This article prefers to use specialized terminology for objects in our RTT application, so that we can clearly discuss them independently from the mathematical structures that represent them. Just like how we call certain objects represented by matrices "mappings" and certain objects represented by vectors "intervals", this article prefers using a specialized term for this RTT object — one that cannot be confused with a generic mathematical structure. | ||
A common need when dealing with interval subspaces is determining whether they are subspaces of other interval subspaces, as we discussed in the earlier section [[ | A common need when dealing with interval subspaces is determining whether they are subspaces of other interval subspaces, as we discussed in the earlier section [[Interval basis#Interval subspaces as subspaces of other interval subspaces]]. If the name for the specialized RTT object was simply "subspace" instead of "interval subspace", then each use of the word "subspace" could be unclear whether it was referring to the specialized RTT object or to the generic mathematical structure. Communicating about such things would become terribly confusing (as it is at present, in existing writings that use the term "subgroup" in both senses). | ||
Furthermore, interval subspaces are not the only subspaces in our RTT application. Comma bases, being bases, are just as much subspace bases: bases for comma subspaces. It may be argued that interval bases, being a more fundamental mathematical object, have more right to the generic "subspace basis" term than comma bases do. But there's another powerful argument that comma bases are the more basic concept that far more users of regular temperaments will ever need to understand, and so they should be the basis that gets the generic name "subspace". Neither argument can win, and why fight anyway. Why needlessly obfuscate the issue when we could simply choose less ambiguous terminology. | Furthermore, interval subspaces are not the only subspaces in our RTT application. Comma bases, being bases, are just as much subspace bases: bases for comma subspaces. It may be argued that interval bases, being a more fundamental mathematical object, have more right to the generic "subspace basis" term than comma bases do. But there's another powerful argument that comma bases are the more basic concept that far more users of regular temperaments will ever need to understand, and so they should be the basis that gets the generic name "subspace". Neither argument can win, and why fight anyway. Why needlessly obfuscate the issue when we could simply choose less ambiguous terminology. | ||