Domain basis: Difference between revisions

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The notation used for merging here is the same as comma-merge: <math>B_1|B_2</math><ref>Using ∩ for intersection, which seems obvious. But the merge notation is tricky. We could use ∪, of course. But technically speaking, it's not a union, but a sumset, and the notation for that is unfortunately just the plus sign +, which could be confusing. Furthermore, in the context of merging temperaments, we don't use either of those symbols. Actually, we use two different symbols there, depending on what we're merging! We use & if it's maps, and | if it's commas. At least, that's the notation used on the [[Meet and join]] and [[Temperament merging]] pages. And because intersections also arise for temperament matrices like mappings and comma bases, this article has gone with consistent notation for interval bases. Interval bases concatenate horizontally, like comma bases, so we use | and consider it a "basis-merge" symbol, i.e. it works on both comma bases and interval bases.</ref>.

== Applications ==

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: [[User:Cmloegcmluin/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.

= Intersecting =

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 The notation used for merging here is the same as comma-merge: <math>B_1|B_2</math><ref>Using ∩ for intersection, which seems obvious. But the merge notation is tricky. We could use ∪, of course. But technically speaking, it's not a union, but a sumset, and the notation for that is unfortunately just the plus sign +, which could be confusing. Furthermore, in the context of merging temperaments, we don't use either of those symbols. Actually, we use two different symbols there, depending on what we're merging! We use & if it's maps, and | if it's commas. At least, that's the notation used on the [[Meet and join]] and [[Temperament merging]] pages. And because intersections also arise for temperament matrices like mappings and comma bases, this article has gone with consistent notation for interval bases. Interval bases concatenate horizontally, like comma bases, so we use | and consider it a "basis-merge" symbol, i.e. it works on both comma bases and interval bases.</ref>.
+== Applications ==
+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: [[User:Cmloegcmluin/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.
 = Intersecting =