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| Line 109: |
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| For additional information about non-standard just intonation (JI) interval subspaces, as well as a gateway to browse temperaments within popular interval subspaces of this sort, see [[Just intonation subgroup|this page]]. | | For additional information about non-standard just intonation (JI) interval subspaces, as well as a gateway to browse temperaments within popular interval subspaces of this sort, see [[Just intonation subgroup|this page]]. |
|
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| = Interval subspaces as subspaces of other interval subspaces =
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|
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| In the same way that an ''sub''space is a part of the full space, a subspace can be seen as a part of another larger subspace. So we can say an interval subspace is itself ''a subspace of'' another interval subspace.
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|
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| Any given interval subspace will be a subspace of infinitely many other interval subspaces.
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|
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| == Examples ==
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|
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| [[File:Interval subspaces 2.3 vs 2.3.7.png|300px|thumb|right|'''Figure 1.''' The interval subspace 2.3 can be clearly seen to be a subspace of 2.3.7. The latter is simply many copies of the former, separated by 7's.]]
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|
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| For example, the 2.3 interval subspace is a subspace of the 2.3.7 interval subspace; this is clearly apparent, because the 2.3.7 interval subspace is the same as the 2.3 interval subspace except with the addition of a new formal prime, 7 (see Figure 1).
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|
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| [[File:Interval subspaces 2.9 vs 2.3.png|300px|thumb|right|'''Figure 2.''' Here we can see how the interval subspace 2.9 is a subspace of 2.3. The latter is simply two copies of the former, offset by 3.]]
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|
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| For a perhaps less obvious example, the 2.9.5 interval subspace is a subspace of the 2.3.5 interval subspace; this may be surprising, because 2.9.5 is the one with a ''larger'' formal prime, but what this actually means is that it spans a ''smaller'' subspace, because while 2.3.5 contains all intervals with ''any'' power of 3, 2.9.5 contains only ''half'' of those, specifically those with ''even'' powers of 3, i.e., powers of 9 (see Figure 2).
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|
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| Sometimes, neither interval subspace is a subspace of the other. Consider 2.3.5 and 2.3.7: the former lacks a 7, and the latter lacks a 5.
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|
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| == Application: determining whether it is possible to change the interval subspace ==
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| Understanding which interval subspaces are subspaces of each other is important when changing the interval subspace for an interval or temperament. This is because only certain changes are possible: specifically, it is only possible to change between interval subspaces where one is a subspace of the other. Otherwise, the interval subspaces are incomparable.
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|
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| We then have further constraints, depending on which type of object we're changing the interval subspace for:
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| * For objects that are made of intervals — such as individual intervals themselves, or comma bases — we can only change in the direction from the subspace to the ''super''space. This is because unless the target interval subspace completely contains the original interval subspace, there's no guarantee that we'll still have all the formal prime building blocks that we need to represent our intervals.
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| * For objects that are made of [[map]]s, i.e. mappings, the opposite is true: we can only change from the superspace to the subspace. Think of it this way: maps are functions, and they only claim to know what to do with inputs from within their given domain, and their domain is the interval subspace. So we can restrict their behavior just fine, because there's no question about what they do with inputs that don't happen to use every available building block from their domain. But there's no unambiguous way to say what they should do with any inputs that use building blocks from outside that domain, if we were to try to expand it.
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|
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| These two types of objects are in fact the only two types of objects we need to worry about in RTT. The technical term for the difference between these two types of objects is [[variance]]. There are only two variances: contravariant, and covariant. Intervals are contravariant, and maps are covariant.
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|
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| So from these two opposing bulleted facts above, we can conclude that any for pair of interval subspaces where neither one is a subspace of the other, there would be no way for us to express ''either intervals or maps'' from one in the other. And that's why we could say that they're incomparable interval subspaces.
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|
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| == General method to determine whether an interval subspace is a subspace of another ==
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| [[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.
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|
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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 <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.
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|
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| For more information on merging interval bases, see [[Interval basis#Merging]].
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|
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| === Example ===
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| For instance, we can demonstrate how 2.25/9.11/7 is a subspace of 2.5/3.7.11 using this approach. If you've really got a knack for this stuff, you may be able to eyeball even this somewhat intense example, but it's obviously good to have a rigorous method like this to fall back on, if only to convince ourselves that we've got the right answer (or to automate things with code, as has been done with these methods in the [[RTT library in Wolfram Language]]).
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|
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| So, first, we do the first step of merging interval bases: concatenate them. That gets us 2.25/9.11/7.2.5/3.7.11.
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| The next step of merging is to canonicalize. To begin that, we convert our interval basis to a matrix <math>B</math>. Here, we've labeled the columns with the number-list representation of the interval basis, to help show the correspondence, as well as the rows with the basis elements:
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| <math>
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|
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| \begin{array} {ccc}
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|
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| \begin{array} {ccc}
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| \\
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| \end{array} \\
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|
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| \begin{array} {rrr}
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| \scriptsize{2} \\
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| \scriptsize{3} \\
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| \scriptsize{5} \\
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| \scriptsize{7} \\
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| \scriptsize{11} \\
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| \end{array} \\
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|
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| \end{array}
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|
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|
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| \begin{array} {ccc}
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|
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| \begin{array} {ccc}
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| \scriptsize{2} & \scriptsize{25/9} & \scriptsize{11/7} & \scriptsize{2} & \scriptsize{5/3} & \scriptsize{7} & \scriptsize{11}
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| \end{array} \\
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|
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| \left[ \begin{array} {rrr}
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| 1 & 0 & 0 & 1 & 0 & 0 & 0 \\
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| 0 & -2 & 0 & 0 & -1 & 0 & 0 \\
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| 0 & 2 & 0 & 0 & 1 & 0 & 0 \\
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| 0 & 0 & -1 & 0 & 0 & 1 & 0 \\
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| 0 & 0 & 1 & 0 & 0 & 0 & 1 \\
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| \end{array} \right] \\
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|
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| \end{array}
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|
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| </math>
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|
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|
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| And now we reduce that big resultant matrix, using column-style Hermite normal form (the labels have been updated match the results, too):
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|
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|
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| <math>
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|
| |
| \begin{array} {ccc}
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|
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| \begin{array} {ccc}
| |
| \\
| |
| \end{array} \\
| |
|
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| \begin{array} {rrr}
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| \scriptsize{2} \\
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| \scriptsize{3} \\
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| \scriptsize{5} \\
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| \scriptsize{7} \\
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| \scriptsize{11} \\
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| \end{array} \\
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|
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| \end{array}
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|
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| \begin{array} {lll}
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|
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| \begin{array} {lll}
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| & \scriptsize{2} & \scriptsize{5/3} & \scriptsize{7} & \scriptsize{11} \\
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| \end{array} \\
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|
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| \left[ \begin{array} {rrr}
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| 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
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| 0 & -1 & 0 & 0 & 0 & 0 & 0 \\
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| 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
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| 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
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| 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
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| \end{array} \right] \\
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|
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| \end{array}
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|
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| </math>
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|
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|
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| The columns with all zeroes are not useful and so we can throw those away.
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|
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| And so when we convert this back to the typical list of numbers form, we have 2.5/3.7.11 again. So this tells us that 2.25/9.11/7 is totally contained by 2.5/3.7.11, and so it's a subspace of it.
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| = Canonical form = | | = Canonical form = |
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| There are no zero columns to eliminate, we've already got the thing as a list of numbers since we've updated the labels on the matrix columns, and all of those numbers are super already, so we're done! The answer is 2.5/3.7/3. | | There are no zero columns to eliminate, we've already got the thing as a list of numbers since we've updated the labels on the matrix columns, and all of those numbers are super already, so we're done! The answer is 2.5/3.7/3. |
|
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| = Operations =
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|
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| == Merging ==
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|
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| If you happen to already be familiar with [[temperament merging]], merging<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> interval bases follows a similar pattern: concatenate, then canonicalize the result.
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|
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| === But first: a gentle introduction ===
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|
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| Many times, it's easy to eyeball the merge of two interval bases. The basic idea is to just take everything that's in either one basis or the other. So 2.3.7 merged with 2.3.5 should just be 2.3.5.7, easy. Sometimes it can get kind of tricky, though. Like, what's the merge of 2.3.7/5 and 2.9.21/5? Not so obvious now. Hint: it's not 2.3.9.7/5.21/5!
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|
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| === Concatenate ===
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|
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| 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.
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|
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| === Canonicalize ===
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|
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| See [[Interval basis#Canonical form]].
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|
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| === Notation ===
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|
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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>.
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|
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| === Applications ===
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|
| |
| 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: [[Interval basis#General method to determine whether an interval subspace is a subspace of another]].
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| # 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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|
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| == Intersecting ==
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|
| |
| Finding the intersection of interval bases is surprisingly tricky<ref>This approach was found by Sintel here: https://math.stackexchange.com/questions/1560411/basis-for-the-intersection-of-two-integer-lattices/2472784#2472784</ref>:
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|
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| # Convert the interval bases to matrices, as with a merge.
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| # Create a [[Wikipedia:Block_matrix|block matrix]] by stacking two copies of one interval basis on the left side, and then setting one copy of the other interval basis on the right side, with the bottom-right quadrant of this block matrix filled in with all zeros.
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| # HNF this.
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| # The results we want are in the bottom-right. We take only half-columns, from the bottom half, and only half-columns where their corresponding top-half are all zeros (which will only happen to columns that are sorted to the right side).
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| # Canonicalize.
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|
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| The reason this works is that wherever the corresponding top-half columns are all zeros, this was achieved through linear combinations of vectors from both interval bases, which means the information below them represents vectors that are in both of them. In other words, if <math>(x, x) + (y, 0) = (0, z)</math> and <math>x</math> is in <math>B_1</math> and <math>y</math> is in <math>B_2</math>, then we must have <math>x + y = 0</math> and <math>z = x</math><ref>credit this explanation to Tom Price on Discord</ref>. We're sort of abusing HNF as a way to solve a system, kind of like [[Douglas Blumeyer's RTT How-To#Null-space|when we calculate the null-space]]<ref>Credit this explanation to Sintel on Discord</ref>.
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|
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| === But first: a gentle introduction ===
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|
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| As with the interval basis merge, it is sometimes practical to eyeball the answer. The basic idea is just to take only formal primes that in both of the interval bases. So the intersection of 2.3.5 and 2.3.7 is plainly just 2.3. But other times the answer may not be so clear. Such as: What is the intersection of 2.5/3.9/7 and 2.9.5? Hint: It's not just 2!
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|
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| === Example ===
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|
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| Let's find the intersection of 2.5/3 and 2.9.5.
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|
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| First, the two interval bases as matrices:
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|
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|
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| <math>
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|
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| \begin{array} {ccc}
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|
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| \begin{array} {ccc} \\ \end{array} \\
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|
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| \begin{array} {rrr}
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| \scriptsize{2} \\
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| \scriptsize{3} \\
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| \scriptsize{5} \\
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| \end{array}
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|
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| \end{array}
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|
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| \begin{array} {ccc}
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|
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| \begin{array} {ccc}
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| \scriptsize{2} & \scriptsize{5/3} \\
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| \end{array} \\
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|
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| \left[ \begin{array} {rrr}
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| 1 & 0 \\
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| 0 & -1 \\
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| 0 & 1 \\
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| \end{array} \right]
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|
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| \end{array}
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|
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|
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| \hspace{1cm}
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|
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|
| |
| \begin{array} {ccc}
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|
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| \begin{array} {ccc} \\ \end{array} \\
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|
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| \begin{array} {rrr}
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| \scriptsize{2} \\
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| \scriptsize{3} \\
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| \scriptsize{5} \\
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| \end{array}
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|
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| \end{array}
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|
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| \begin{array} {ccc}
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|
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| \begin{array} {ccc}
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| \scriptsize{2} & \scriptsize{9} & \scriptsize{5} \\
| |
| \end{array} \\
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|
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| \left[ \begin{array} {rrr}
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| 1 & 0 & 0 \\
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| 0 & 2 & 0 \\
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| 0 & 0 & 1 \\
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| \end{array} \right]
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|
| |
| \end{array}
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|
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| </math>
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|
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|
| |
| Now we must combine them into one big block matrix. Two copies of the first on the left, one copy of the other in the top-right, and zeros to pad out:
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|
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|
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| <math>
| |
| \left[ \begin{array} {rr|rrr}
| |
| 1 & 0 & 1 & 0 & 0 \\
| |
| 0 & -1 & 0 & 2 & 0 \\
| |
| 0 & 1 & 0 & 0 & 1 \\
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| \hline
| |
| 1 & 0 & 0 & 0 & 0 \\
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| 0 & -1 & 0 & 0 & 0 \\
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| 0 & 1 & 0 & 0 & 0 \\
| |
| \end{array} \right]
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| </math>
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|
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|
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| Now, HNF that (column-style, so zeros end up in the top right):
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|
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|
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| <math>
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| \left[ \begin{array} {rrrrr}
| |
| 1 & 0 & 0 & 0 & 0 \\
| |
| 0 & 1 & 0 & 0 & 0 \\
| |
| 0 & 0 & 1 & 0 & 0 \\
| |
| 0 & 0 & 0 & 1 & 0 \\
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| 0 & 1 & 0 & 0 & 2 \\
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| 0 & -1 & 0 & 0 & -2 \\
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| \end{array} \right]
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| </math>
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|
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|
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| Now we're looking for any half-columns in the bottom half wherever there are all zeroes in the corresponding top half. Such zeroes are highlighted red here, and what we're looking for is highlighted in yellow:
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|
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|
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| <math>
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| \left[ \begin{array} {rrrrr}
| |
| 1 & 0 & 0 & \colorbox{pink}0 & \colorbox{pink}0 \\
| |
| 0 & 1 & 0 & \colorbox{pink}0 & \colorbox{pink}0 \\
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| 0 & 0 & 1 & \colorbox{pink}0 & \colorbox{pink}0 \\
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| \hline
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| 0 & 0 & 0 & \colorbox{yellow}1 & \colorbox{yellow}0 \\
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| 0 & 1 & 0 & \colorbox{yellow}0 & \colorbox{yellow}2 \\
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| 0 & -1 & 0 & \colorbox{yellow}0 & \colorbox{yellow}{-2} \\
| |
| \end{array} \right]
| |
| </math>
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|
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|
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| So let's focus in on that result:
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|
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|
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| <math>
| |
|
| |
| \begin{array} {ccc}
| |
|
| |
| \begin{array} {ccc} \\ \end{array} \\
| |
|
| |
| \begin{array} {rrr}
| |
| \scriptsize{2} \\
| |
| \scriptsize{3} \\
| |
| \scriptsize{5} \\
| |
| \end{array}
| |
|
| |
| \end{array}
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|
| |
| \begin{array} {ccc}
| |
|
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| \begin{array} {ccc}
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| \scriptsize{2} & \scriptsize{9/25} \\
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| \end{array} \\
| |
|
| |
| \left[ \begin{array} {rrr}
| |
| 1 & 0 \\
| |
| 0 & 2 \\
| |
| 0 & -2 \\
| |
| \end{array} \right]
| |
|
| |
| \end{array}
| |
|
| |
| </math>
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|
| |
|
| |
| Canonicalization time. That's already in matrix form, and HNF even (yes, we use HNF again here). No all zeros columns. Converted back to a list of numbers we at first have 2.9/25. But the last step is to take the undirected value, which reciprocates 9/25 to its super form which is 25/9. So the intersection of 2.5/3 and 2.9.5 is 2.25/9.
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|
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| === Notation ===
| |
|
| |
| The notation for interval basis intersecting we'll use here is just the intersection symbol: <math>B_1∩B_2</math>.
| |
|
| |
| === 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: [[Temperament merging across interval bases#Map-merge]].
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|
| |
| = Changing interval basis =
| |
|
| |
| 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: [[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: [[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 ==
| |
|
| |
| Here are the steps:
| |
| # Set up a matrix with <math>d_L</math> rows, where <math>d_L</math> is the dimensionality of the superspace, and <math>d_s</math> columns, where <math>d_s</math> is the dimensionality of the subspace<ref>We're borrowing <math>L</math> and <math>s</math> from [[MOS]] scale theory; there's no direct conceptual connection here, nor any need to understand anything about such scale theory at this moment, but if you happen to be familiar with the conventional use of <math>L</math> for "Large" and <math>s</math> for "small" in that other xenharmonic topic, then this variable choice may be particularly helpful for you.</ref>.
| |
| # Label the rows with the superspace interval basis.
| |
| # Label the columns with the subspace interval basis.
| |
| # Fill in each entry with the count of formal primes from the superspace basis for this row that could be used to build the formal primes in the subspace interval basis for this column.
| |
|
| |
| === Example ===
| |
|
| |
| 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 second column isn't so tricky, if you can recognize that 25/9 is simply 5/3 squared. So we need a 2 in the cell connecting those two formal primes, and zeroes elsewhere.
| |
| * The third column isn't so tricky either. It's just one 11 in the numerator, so that's a +1, and one 7 in the denominator, so that's a -1.
| |
|
| |
| And here's the final result:
| |
|
| |
|
| |
| <math>
| |
|
| |
| \begin{array} {ccc}
| |
|
| |
| \begin{array} {ccc}
| |
| \\
| |
| \end{array} \\
| |
|
| |
| \begin{array} {rrr}
| |
| \scriptsize{2} \\
| |
| \scriptsize{5/3} \\
| |
| \scriptsize{7} \\
| |
| \scriptsize{11} \\
| |
| \end{array} \\
| |
|
| |
| \end{array}
| |
|
| |
|
| |
| \begin{array} {lll}
| |
|
| |
| \begin{array} {lll}
| |
| & \scriptsize{2} & \scriptsize{25/9} & \scriptsize{11/7} \\
| |
| \end{array} \\
| |
|
| |
| \left[ \begin{array} {rrr}
| |
| 1 & 0 & 0 \\
| |
| 0 & 2 & 0 \\
| |
| 0 & 0 & -1 \\
| |
| 0 & 0 & 1 \\
| |
| \end{array} \right] \\
| |
|
| |
| \end{array}
| |
|
| |
| </math>
| |
|
| |
|
| |
| == Using the interval rebase ==
| |
|
| |
| For intervals and comma bases, which can only be changed from a subspace to a superspace, we left-multiply by the interval rebase; this process is identical to the process used when mapping intervals with ordinary temperament mappings, except replacing the mapping with the interval rebase.
| |
|
| |
| As for changing such temperament mapping matrices themselves — which can only be changed the other way, from a superspace to a subspace — we instead ''right''-multiply by the interval rebase. So, strangely, this is also identical to the process used when mapping intervals with ordinary temperament mappings, except replacing the ''intervals'' with the interval rebase.
| |
|
| |
| === Examples ===
| |
|
| |
| Suppose we have the interval rebase <math>R_{L↔s}</math> between 2.3.5.7 <math>B_L</math> and 2.9/7.5/3 <math>B_s</math>. The superspace is 2.3.5.7, so that's the rows, and 2.9/7.5/3 is the subspace, so that's the columns. And so here's our <math>R_{L↔s}</math>:
| |
|
| |
|
| |
| <math>
| |
|
| |
| \begin{array} {ccc}
| |
|
| |
| \begin{array} {ccc}
| |
| \\
| |
| \end{array} \\
| |
|
| |
| \begin{array} {rrr}
| |
| \scriptsize{2} \\
| |
| \scriptsize{3} \\
| |
| \scriptsize{5} \\
| |
| \scriptsize{7} \\
| |
| \end{array} \\
| |
|
| |
| \end{array}
| |
|
| |
|
| |
| \begin{array} {lll}
| |
|
| |
| \begin{array} {lll}
| |
| & \scriptsize{2} & \scriptsize{5/3} & \scriptsize{9/7} \\
| |
| \end{array} \\
| |
|
| |
| \left[ \begin{array} {rrr}
| |
| 1 & 0 & 0 \\
| |
| 0 & -1 & 2 \\
| |
| 0 & 1 & 0 \\
| |
| 0 & 0 & -1 \\
| |
| \end{array} \right] \\
| |
|
| |
| \end{array}
| |
|
| |
| </math>
| |
|
| |
|
| |
| First, let's use this to convert a comma basis <math>C_s</math> (that's in the <math>B_s</math> interval subspace) to the <math>B_L</math>interval subspace, by doing <math>R_{L↔s}.C</math>:
| |
|
| |
|
| |
| <math>
| |
|
| |
| \begin{array} {ccc}
| |
|
| |
| \\
| |
|
| |
| \begin{array} {rrr}
| |
| \\
| |
| \end{array} \\
| |
|
| |
| \begin{array} {rrr}
| |
| \scriptsize{2} \\
| |
| \scriptsize{3} \\
| |
| \scriptsize{5} \\
| |
| \scriptsize{7} \\
| |
| \end{array}
| |
|
| |
| \end{array}
| |
|
| |
| \begin{array} {ccc}
| |
|
| |
| C_L \\
| |
|
| |
| \begin{array} {ccc}
| |
| \\
| |
| \end{array} \\
| |
|
| |
| \left[ \begin{array} {rrr}
| |
| -8 & -4 \\
| |
| -11 & -3 \\
| |
| 11 & 5 \\
| |
| 0 & -1 \\
| |
| \end{array} \right]
| |
|
| |
| \end{array}
| |
|
| |
|
| |
| \hspace{0.5cm}
| |
| \large{←} \normalsize{}
| |
| \hspace{0.5cm}
| |
|
| |
|
| |
| \begin{array} {ccc}
| |
|
| |
| \\
| |
|
| |
| \begin{array} {ccc}
| |
| \\
| |
| \end{array} \\
| |
|
| |
| \begin{array} {rrr}
| |
| \scriptsize{2} \\
| |
| \scriptsize{3} \\
| |
| \scriptsize{5} \\
| |
| \scriptsize{7} \\
| |
| \end{array}
| |
|
| |
| \end{array}
| |
|
| |
| \begin{array} {ccc}
| |
|
| |
| R_{L↔s} \\
| |
|
| |
| \begin{array} {ccc}
| |
| \scriptsize{2} & \scriptsize{5/3} & \scriptsize{9/7} \\
| |
| \end{array} \\
| |
|
| |
| \left[ \begin{array} {rrr}
| |
| 1 & 0 & 0 \\
| |
| 0 & -1 & 2 \\
| |
| 0 & 1 & 0 \\
| |
| 0 & 0 & -1 \\
| |
| \end{array} \right]
| |
|
| |
| \end{array}
| |
|
| |
|
| |
| \hspace{0.5cm}
| |
| \large{×} \normalsize{}
| |
| \hspace{0.5cm}
| |
|
| |
|
| |
| \begin{array} {ccc}
| |
|
| |
| \\
| |
|
| |
| \begin{array} {ccc}
| |
| \\
| |
| \end{array} \\
| |
|
| |
| \begin{array} {rrr}
| |
| \scriptsize{2} \\
| |
| \scriptsize{5/3} \\
| |
| \scriptsize{9/7} \\
| |
| \end{array} \\
| |
|
| |
| \begin{array} {rrr}
| |
| \\
| |
| \end{array} \\
| |
|
| |
| \end{array}
| |
|
| |
| \begin{array} {ccc}
| |
|
| |
| C_s \\
| |
|
| |
| \begin{array} {ccc}
| |
| \\
| |
| \end{array} \\
| |
|
| |
| \left[ \begin{array} {rrr}
| |
| -8 & -4 \\
| |
| 11 & 5 \\
| |
| 0 & 1 \\
| |
| \end{array} \right] \\
| |
|
| |
| \begin{array} {rrr}
| |
| \\
| |
| \end{array} \\
| |
|
| |
| \end{array}
| |
|
| |
| </math>
| |
|
| |
|
| |
| And now let's use <math>R_{L↔s}</math> to convert a mapping the other way, from <math>B_L</math> to <math>B_s</math>, by doing <math>M_L.R_{L↔s}</math>:
| |
|
| |
|
| |
| <math>
| |
|
| |
| \begin{array} {ccc}
| |
|
| |
| M_L \\
| |
|
| |
| \begin{array} {ccc}
| |
| \scriptsize{2} & \scriptsize{3} & \scriptsize{5} & \scriptsize{7} \\
| |
| \end{array} \\
| |
|
| |
| \left[ \begin{array} {rrr}
| |
| 12 & 19 & 28 & 34
| |
| \end{array} \right]
| |
|
| |
| \end{array}
| |
|
| |
|
| |
| \hspace{0.5cm}
| |
| \large{×} \normalsize{}
| |
| \hspace{0.5cm}
| |
|
| |
|
| |
| \begin{array} {ccc}
| |
|
| |
| \\
| |
|
| |
| \begin{array} {ccc}
| |
| \\
| |
| \end{array} \\
| |
|
| |
| \begin{array} {rrr}
| |
| \scriptsize{2} \\
| |
| \scriptsize{3} \\
| |
| \scriptsize{5} \\
| |
| \scriptsize{7} \\
| |
| \end{array}
| |
|
| |
| \end{array}
| |
|
| |
| \begin{array} {ccc}
| |
|
| |
| R_{L↔s} \\
| |
|
| |
| \begin{array} {ccc}
| |
| \scriptsize{2} & \scriptsize{5/3} & \scriptsize{9/7} \\
| |
| \end{array} \\
| |
|
| |
| \left[ \begin{array} {rrr}
| |
| 1 & 0 & 0 \\
| |
| 0 & -1 & 2 \\
| |
| 0 & 1 & 0 \\
| |
| 0 & 0 & -1 \\
| |
| \end{array} \right]
| |
|
| |
| \end{array}
| |
|
| |
|
| |
| \hspace{0.5cm}
| |
| \large{→} \normalsize{}
| |
| \hspace{0.5cm}
| |
|
| |
|
| |
| \begin{array} {ccc}
| |
|
| |
| M_s \\
| |
|
| |
| \begin{array} {ccc}
| |
| \scriptsize{2} & \scriptsize{5/3} & \scriptsize{9/7} \\
| |
| \end{array} \\
| |
|
| |
| \left[ \begin{array} {rrr}
| |
| 12 & 9 & 4
| |
| \end{array} \right] \\
| |
|
| |
| \begin{array} {rrr}
| |
| \\
| |
| \end{array} \\
| |
|
| |
| \end{array}
| |
|
| |
| </math>
| |
|
| |
|
| |
| == Wolfram implementation ==
| |
|
| |
| 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 = |
| Line 951: |
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. | | 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 [[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). | | 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). |
|
| |
|
| 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. |