Cross-domain temperament merging: Difference between revisions

A guiding principle when performing a cross-interval basis temperament merge is [[Temperament_merging#Application|described on the main page for temperament merging]]:

From this, we can make some helpful statements about cross-interval basis temperament merges.

''For a map-merge'', then, because the output temperament will deal only with tempered commas shared by every input temperament, then its interval basis only needs to include the formal primes that are present in all of the input interval bases. Here's why: a comma built using a formal prime that isn't shared by all input temperaments couldn't even be ''built'' in all input temperaments, yet alone tempered out by all of them. And so, to build the set of commas that are tempered out by all temperaments, we only need the building blocks that can be found in all of them. In mathematical terms, we must find the intersection of the input interval bases.

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.

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 formal primes of this interval basis, to help show the correspondence, as well as the rows with the primes these formal primes factor into:

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>.

# 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: [[Temperament merging across interval bases#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.

# Convert the interval bases to matrices, as with a merge.

# 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.

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>.

First, the two interval bases as matrices:

The notation for interval basis intersecting we'll use here is just the intersection symbol: <math>B_1∩B_2</math>.

[[File:Two-way bridge.png|400px|right|thumb|An interval rebase forms a two-way bridge between a superspace basis <math>B_L</math> and a subspace basis <math>B_s</math>. But mappings can only use it to go from the superspace to the subspace, and comma bases can only use it to go from the subspace to the superspace.]]

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.

# 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.

Let's construct the interval rebase <math>R</math> between 2.25/9.11/7 and 2.5/3.7.11. [[Temperament merging across interval bases#Example|As we proved earlier]], the former is a subspace of the latter. So this will be a 4×3 matrix.  

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>:

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>:

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>:

First, let's work through an example of a cross-interval basis comma-merge: meantone <math>T_1</math> with archytas <math>T_2</math>, where meantone is in the 5-limit standard interval basis 2.3.5, which we'll call <math>B_1</math>, and archytas is in the 2.3.7 interval basis, which we'll call <math>B_2</math>.

Then remove the columns of all zeros, go back to number list form, and make sure everything's greater than 1, which it is already. And so the merged interval basis <math>B_1|B_2</math> is 2.3.5.7.

Now we change the interval basis for each comma basis to <math>B_1|B_2</math>. Let's do meantone first. First we need to find our interval rebase <math>R_{B_1|B_2↔B_1}</math>.

Now, let's work through an example of a cross-interval basis map-merge: 22 equal temperament <math>T_1</math> with 17 equal temperament <math>T_2</math>, where 22-ET is in the interval basis 2.3.5.11, which we'll call <math>B_1</math>, and 17-ET is in the 2.9.7.11 interval basis, which we'll call <math>B_2</math>.

First we must intersect these two temperaments' interval bases. The first step of that is to convert them to matrices which have formal primes as columns and the merged set of the primes they factor into as the rows:

Canonicalize (HNF, remove all-zero columns, back to number list form, and make super). So that tells us that the intersected interval basis <math>B1∩B2</math> is 2.9.11.

Now we change the interval basis for each mapping to <math>B1∩B2</math>. Let's do 22-ET first. First we need to find our interval rebase <math>R_{1↔1∩2}</math>: