Domain basis: Difference between revisions

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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.

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~~ [[Temperament merging across interval bases#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).

A common need when dealing with interval subspaces is determining whether they are subspaces of other interval subspaces (as discussed in [[Temperament merging across interval bases#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.

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For all these three reasons, the interval basis terminology makes no such exclusion.

= Interval basis operations =

* Merging: see [[Temperament merging across interval bases#Merging]].

* Intersecting: see [[Temperament merging across interval bases#Intersecting]].

* Changing: see [[Temperament merging across interval bases#Changing interval basis]].

= Footnotes =

@@ Line 290: / Line 290: @@
 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 [[Temperament merging across interval bases#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).
+A common need when dealing with interval subspaces is determining whether they are subspaces of other interval subspaces (as discussed in [[Temperament merging across interval bases#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.
@@ Line 303: / Line 303: @@
 For all these three reasons, the interval basis terminology makes no such exclusion.
+= Interval basis operations =
+* Merging: see [[Temperament merging across interval bases#Merging]].
+* Intersecting: see [[Temperament merging across interval bases#Intersecting]].
+* Changing: see [[Temperament merging across interval bases#Changing interval basis]].
 = Footnotes =