Domain basis: Difference between revisions
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So, for instance, a temperament in the 2.3.5 interval subspace cannot temper 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. | So, for instance, a temperament in the 2.3.5 interval subspace cannot temper 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 regular temperament mapping is a kind of function, and its [[Wikipedia:Domain_of_a_function|domain]] is an interval subspace. So, an interval basis may be used to label the columns of a mapping, with one formal prime per column. Here's [[ | A regular temperament mapping is a kind of function, and its [[Wikipedia:Domain_of_a_function|domain]] is an interval subspace. So, an interval basis may be used to label the columns of a mapping, with one formal prime per column. Here's [[Chromatic pairs#Slendric|slendric]], a temperament with a 2.3.7 interval basis: | ||
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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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== 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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== Wolfram implementation == | == Wolfram implementation == | ||
Functions for finding interval rebases have been implemented in the [[ | Functions for finding interval rebases have been implemented in the [[RTT library in Wolfram Language]] as <code>getRforM</code> and <code>getRforC</code>. Although it also simply contains <code>changeB</code> which you can use directly on any temperament and it will do this step under the hood for you. | ||
= Non-JI interval subspaces = | = Non-JI interval subspaces = | ||