Temperament merging
This page gives a general introduction to this concept; for a more mathematical take on this, see Meet and join.
Temperament merging is a way to find new regular temperaments by merging others. There are two ways to merge temperaments: mapmerge, which works by merging the temperaments' mappings, and commamerge, which works by merging the temperaments' comma bases.
Merging
"Merging" in this context refers to concatenating the matrices in question and then canonicalizing them.
For mappings, the concatenation is vertical, while for commabases, the concatenation is horizontal:
[math]
\hspace{1cm}
\begin{array} {ccc}
\left[ \begin{array} {rrr}
12 & 19 & 28 \\
\end{array} \right] \\
\text{mapmerge} \\
\left[ \begin{array} {rrr}
19 & 30 & 44 \\
\end{array} \right] \\
↓ \\
\left[ \begin{array} {rrr}
12 & 19 & 28 \\
19 & 30 & 44 \\
\end{array} \right] \\
\text{which canonicalizes to} \\
\left[ \begin{array} {rrr}
1 & 0 & 4 \\
0 & 1 & 4 \\
\end{array} \right] \\
\end{array}
\hspace{1cm}
\hspace{1cm}
\left[ \begin{array} {rrr}
4 \\
4 \\
1 \\
\end{array} \right]
\text{commamerge}
\left[ \begin{array} {rrr}
7 \\
0 \\
3 \\
\end{array} \right]
→
\left[ \begin{array} {rrr}
4 & 7 \\
4 & 0 \\
1 & 3 \\
\end{array} \right]
\text{which canonicalizes to}
\left[ \begin{array} {rrr}
19 & 15 \\
12 & 8 \\
0 & 1 \\
\end{array} \right]
[/math]
Application
Mapmerging produces a temperament that tempers out only the commas that are tempered out by all of the input temperaments. Conversely, commamerging produces a temperament that tempers out every comma tempered out by any of the input temperaments.
Notation
The & ("ampersand") symbol is used (for example, on Graham Breed's temperament finding tool) to notate mapmerging, as in 12&19 = meantone; we can read this as "12ET and 19ET is meantone" or "12ET mapmerge 19ET is meantone". Here, 12 and 19 are wart notation for 12ET and 19ET.
The  ("pipe") symbol may be used to notate commamerging, as in meantoneporcupine = 7. We could read this as "meantone or porcupine" or "meantone commamerge porcupine is 7ET". As a mnemonic, because commas are represented by vectors, which are vertical columns, when they merge together into matrices, the pipe resembles the seam between them as they merge.
The & symbol is associated with the word "and", and in many programming languages, the  symbol is associated with the word "or". So a further mnemonic can be used to remember this pair of symbols: [math]𝓣_1 \& 𝓣_2[/math] is the merge that results in the temperament that tempers out the commas tempered out by [math]𝓣_1[/math] and [math]𝓣_2[/math], and [math]𝓣_1  𝓣_2[/math] is the merge that results in the temperament that tempers out the commas tempered out by [math]𝓣_1[/math] or [math]𝓣_2[/math].
Crossbreeding
Perhaps the most basic example of temperament merging is mapmerging equal temperaments (ETs). This is sometimes called "crossbreeding", because ET maps are sometimes called "breeds". And so meantone could be said to be a crossbreed of 12ET and 19ET, because 12&19 = meantone.
Multiple temperament merging
More than two temperaments may be merged at a time, such as 22&34d&37 to give ares.
Nonuniqueness
Note that while a given temperament merging expression unambiguously refers to a single temperament, a given temperament can be expressed by many possible different temperament merging expressions.
Canonicalization
The canonicalization step is important for eliminating any redundancies that may have been introduced by merging related temperaments, such as gradedeficiencies or enfactoring.
Gradedeficiencies
Sometimes when temperaments are merged, gradedeficiencies may occur. For example, commamerging septimal meantone and miracle temperaments:
[math]
\left[ \begin{array} {rrr}
4 & 13 \\
4 & 10 \\
1 & 0 \\
0 & 1 \\
\end{array} \right]

\left[ \begin{array} {rrr}
25 & 20 \\
7 & 5 \\
6 & 4 \\
0 & 1 \\
\end{array} \right]
→
\left[ \begin{array} {rrr}
4 & 13 & 25 & 20 \\
4 & 10 & 7 & 5 \\
1 & 0 & 6 & 4 \\
0 & 1 & 0 & 1 \\
\end{array} \right]
\text{which in normal form is}
\left[ \begin{array} {rrr}
\colorbox{pink}0 & 24 & 5 & 23 \\
\colorbox{pink}0 & 29 & 4 & 26 \\
\colorbox{pink}0 & 0 & 3 & 2 \\
\colorbox{pink}0 & 0 & 0 & 1 \\
\end{array} \right]
[/math]
Note that we've only put it into Hermite normal form; we've done this to illustrate that one of the vectors is now entirely zeros (highlighted in red). This means that the matrix was nullitydeficient, or in layperson's terms, contained redundant commas. In other words, these two temperaments tempered out some of the same commas, and so when we merged them, even though the input temperaments required 2 vectors each to represent, their merged result doesn't require all 4 vectors; it can be completely represented using only 3.
Once we fully canonicalize, though, the allzero row(s) are removed, and we end up with:
[math]
\left[ \begin{array} {rrr}
49 & 45 & 36 \\
31 & 27 & 21 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array} \right]
[/math]
Enfactoring
Sometimes when temperaments are merged, enfactoring may occur. For example:
[math]
\begin{array} {ccc}
\left[ \begin{array} {rrr}
5 & 8 & 12 \\
\end{array} \right] \\
\& \\
\left[ \begin{array} {rrr}
19 & 30 & 44 \\
\end{array} \right] \\
↓ \\
\left[ \begin{array} {rrr}
5 & 8 & 12 \\
19 & 30 & 44 \\
\end{array} \right] \\
\end{array}
[/math]
The greatest factor of this matrix is 2, because we can produce the row ⟨24 38 56] as a coprime linear combination of its rows (that's ⟨5 8 12] + ⟨19 30 44]), and the entries of this row have a GCD of 2, so in other words this matrix is 2enfactored. If we merely put it into Hermite normal form, we receive:
[math]
\left[ \begin{array} {rrr}
1 & 0 & 4 \\
0 & 2 & 8 \\
\end{array} \right]
[/math]
which is a 2enfactored meantone mapping, and it reveals the greatest factor as the GCD of the second row. But if we fully canonicalize it (defactor, and put into normal form), then we get:
[math]
\left[ \begin{array} {rrr}
1 & 0 & 4 \\
0 & 1 & 4 \\
\end{array} \right]
[/math]
which is simply the canonical mapping for meantone temperament.
Noncanonicalizing definition
By some definitions of the & operator, the defactoring part of canonicalization is not included — for example on Graham Breed's temperament finding tool. This allows for things like 5&19 to represent 2enfactored meantone, rather than meantone itself. Instead of a full canonicalization, then, this definition merely puts the result into normal form and removes any allzero rows or columns resulting from gradedeficiencies.
Parallel intersections
Every temperament mapping has a dual comma basis, and every comma basis has a dual mapping. Because of this duality, a special parallelism exists.
If we perform a mapmerge, then look at the resultant mapping's comma basis, we'll find that it is the intersection of the input temperaments' comma bases. In other words, and like stated earlier, it includes only the commas that were in both of their comma bases.
Similarly, if we perform a commamerge, then look at the resultant comma basis's mapping, we'll find that it is the intersection of the input temperaments' mappings, in a similar sense: it includes only the mappingrows that were in both of their mappings.
In this sense, mapmerging and commamerging should be understood not truly as operations on matrices, but as operations on bases. Another way to describe these such intersections would be as bases for the linear dependence of the given temperaments, as described here: Linear dependence#For a given set of basis matrices, how to compute a basis for their linearly dependent vectors.
Here we have a group of temperaments that are related by merges. Moving up in this diagram corresponds with mapmerges, and downward movement corresponds with commamerges. Temperaments lower on the chart support ones higher on the chart.
Vs. the wedge product
Temperament merging is closely related to the wedge product. For more information, see: Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Temperament merging.
Temperament merging across interval bases
It is possible to merge temperaments from different interval bases. For more information, see Temperament merging across interval bases.
Wolfram implementation
Temperament merging has been implemented as the functions mapMerge
and commaMerge
in the RTT library in Wolfram Language.