User:VectorGraphics/Mapping

The operation of "tempering" can be thought of as establishing a relationship or equality between two intervals. This can be seen as a "flattening" from a higher-dimensional interval space (such as just intonation) down to a lower-dimensional space, called a regular temperament; while an interval in 5-limit just intonation requires 3 numbers to represent (see Monzo notation), an interval in meantone temperament requires only 2 numbers to represent.

On this page, the higher-dimensional interval space will be assumed to be just intonation, but it can also be another regular temperament, or a different system entirely. A temperament mapping (or for short mapping) essentially tells you how to do that flattening, allowing you to find where a JI interval is in a tempered space.

Reading a mapping

A mapping can be seen as a list of monzos in the temperament's interval space telling you where to find each basis interval of the JI subgroup. Note that by convention, monzos are written vertically in a mapping.

For example, for meantone temperament, the mapping is:

[math]\displaystyle{ 2.3.5 \left[ \begin{array}{rr} 1 & 1 & 0 \\ 0 & 1 & 4 \\ \end{array} \right] }[/math]

This tells us that we're starting in the 5-limit, with 2, 3, and 5 as our generator intervals, and that we'll end up in a 2-dimensional space with generator intervals that we'll for now call A and B.

Then, the first column is the tempered monzo A.B [1 0], telling us that 2 "maps" to multiplying by "A" once, and by "B" zero times. So, we can label "A" as ~2 (note the tilde indicating a tempered interval), which is the meantone octave.

The second column is A.B [1 1], telling us that 3 maps to multiplying by "A" once, and by "B" once, so we can call "B" ~3/2. Here, ~3/2 is the meantone fifth, and is stacked to form the meantone chain of fifths.

The third column is A.B [0 4], telling us that 5 maps to multiplying by "A" zero times, and by "B" four times. This tells us that the 5th harmonic is found by stacking four fifths, which is characteristic of meantone.

Using a mapping

Because a mapping tells us the tempered monzos for each of the primes in a JI subgroup, you can use a mapping to find where any just interval is in a given temperament. For example, let's find where 25/24 is in meantone temperament. The monzo for 25/24 is 2.3.5 [-3 -1 2].

- The first entry tells us that you divide by 2 three times. The tmonzo for 2 is [1 0], so multiplying by -3 we can make our first "partial sum" [-3 0], telling us that you go down three tempered octaves.

- The next entry tells us that you divide by 3 once. The tmonzo for 3 is [1 1], so multiplying by -1 we can make our second "partial sum" [-1 -1], indicating going down one octave and one fifth.

- The final entry tells us that you multiply by 5 twice. The tmonzo for 5 is [0 4], so multiplying by 2 we can make our third "partial sum" [0 8], indicating going up eight fifths.

Adding these together, we receive [-3 0] + [-1 -1] + [0 8] = ~2.~3/2 [-4 7], which is precisely the tempered monzo for 25/24 in meantone temperament, indicating that 25/24 is the chromatic semitone (as meantone is a diatonic temperament).

Vals and mappings

A mapping to a 1-dimensional temperament is indistinguishable from a tuning map with only integer entries, and this common structure may be called a val. Thus, it can be said that the rows of any mapping are vals. A val can thus be interpreted as a tuning map (by treating the edostep as a logarithmic pitch unit) or as a mapping (by treating the edostep as a generator).

Mapping composition

If you apply one temperament, and then apply another temperament to the resulting tempered space, you will have two mappings to step through. Luckily, it is easy to combine these two mappings into a single mapping. This will have the same number of monzos in it as your first mapping (since you're starting with the same number of generators), and each monzo will have the same number of entries as in your second mapping (since you'll be ending with that number of generators).

For example, let's combine the mappings from just intonation to meantone, and then meantone to 12 equal temperament (which is a val, being interpreted as a mapping).

By convention, we place the first temperament on the right:

[math]\displaystyle{ \sim2.\sim3/2 \left[ \begin{array}{rr} 12 & 7 \\ \end{array} \right] 2.3.5 \left[ \begin{array}{rr} 1 & 1 & 0 \\ 0 & 1 & 4 \\ \end{array} \right] }[/math]

Multiple mappings for a temperament

In the above example, we wrote out the meantone mapping from the perspective of the two generators ~2/1 and ~3/2. What if we instead wanted to treat the generators as being ~2/1 and ~4/3? Or, what if we wanted to write it out from the perspective that the generators are ~2/1 and ~3/1? All of these will lead to different mappings, but will still represent the same temperament.

In the language of mathematics, you've simply changed the basis for your temperament, and the resulting temperamental spaces will be isomorphic to one another. This is just a fancy way of stating that they're the same temperament.

If we wanted to lay meantone out as having generators of 2/1 and 4/3, we arrive at the following mapping:

[math]\displaystyle{ 2.3.5 \left[ \begin{array}{rr} 1 & 2 & 4 \\ 0 & -1 & -4 \\ \end{array} \right] }[/math]

This is still rather intuitive: the 2/1 still maps to one period and no generators. The 3/1 is now reachable by two periods minus a generator, which is to say that it's just two octaves minus a perfect fourth. The 5/1 is a bit more complicated, but maps as four 4/3's down, plus four octaves—it is left as an exercise to the reader to prove that in a meantone system this will actually yield 5/1.

If you wanted your basis to be 2/1 and 3/1, you'd end up with the following mapping: (left as an exercise to the reader to derive):

[math]\displaystyle{ 2.3.5 \left[ \begin{array}{rr} 1 & 0 & -4 \\ 0 & 1 & 4 \\ \end{array} \right] }[/math]

The normal val list is a normalized form among the variety of writing the mapping matrices, and it is what appears in temperament pages on this wiki.

For more information on these sorts of operations, see Generator size manipulation.

Additional notes

• In the linear algebra formalism, mappings are represented by matrices, hence why you may see the term mapping matrix in use. The use of a mapping to find the tempered form of an interval is represented by matrix multiplication.