Douglas Blumeyer's RTT How-To: Difference between revisions

In this first section, you will learn about maps — one of the basic building blocks of temperaments — and the effect maps have on musical intervals.  

=== mappings and comma bases ===

If a comma basis is the name for the matrix made out of commas, then we could say a “'''mapping'''” is the name for the matrix made out of maps. Why isn't it '''map basis''', you ask? Well, the answer is: it is that too. You can totally call it that, and when you use one or the other name just depends on the context. When you use the word "mapping" for it, you're treating it like a function, or a machine: it takes in intervals, and spits out new forms of intervals. That's how we've been using it here. But in other places, you're thinking of this matrix as a basis for the infinite space of possible maps that could be combined to produce a matrix which works the same way as a mapping, i.e. it tempers out the same commas. In these contexts, it might make more sense to call such a mapping matrix a "map basis".

So, yes, that's right: maps are similar to commas insofar as — once you have more than one of them in your matrix — the possibilities for individual members immediately go infinite (technically speaking, though, while a comma basis is a basis of the null-space of the temperament, a map basis is a ''row-''basis of the ''row-''space of the temperament).

We already have the tools to check that each of these commas’ vectors is tempered out individually by the map {{map|19 30 44}}; we learned this bit in the very first section: all we have to do is make sure that the comma maps to zero steps in this ET. But that's not a special relationship between 19-ET and any of these commas ''individually''; each of these commas are tempered out by many different ETs, not just 19-ET. The special relationship 19-ET has is to a null-space which can be expressed in basis form as the meet of ''two'' commas (at least in the 5-limit; more on this later). In this way, the comma bases which represent the meet of two commas are greater than the sum of their individual parts.

And ta-da! You’ve found a map basis from a comma basis, and it is {{vector|{{map|19 30 44}}}}. In other words, you've found a map basis which is a row-basis for the mapping row-space of the same temperament for which the comma basis we started with is a basis for the null-space. Feel free to try this with any other combination of two commas tempered out by this map.

So why why did we need to do those extra reversals at the beginning and end? Besides, I never said we ''must'' find get the zeroes in the top half on the top right of the augmented matrix when doing column Gaussian elimination, so wasn't rearranging the columns pointless? Well, the reason I told you to do it was because if you're going to adapt this process to a math program like Wolfram Alpha or perhaps even general computer code, you ''will'' need that step, because the way the null-space algorithm is implemented, it ''will'' try to get those zeroes on the right. So I understand this is a bit of a hand-wavy answer, and perhaps one day someone else can edit this with harder facts. But based on observation, if you do not do the reversing, you end up with an incorrect answer, and in particular, it has got zeroes on the wrong side of the matrix than you would expect.  

So that’s not any of the commas we’ve looked at so far (it’s the [[19-comma]] and the [[acute limma]]). But it is clear to see that either of them would be tempered out by 19-ET (no need to map by hand — just look at these commas side-by-side with the map {{vector|{{map|19 30 44}}}} and it should be apparent). We're done!

So we can now convert back and forth between a map basis and a comma basis. We could imagine drawing a diagram with a line of duality down the center, with a temperament's map basis on the left, and its comma basis on the right. Either side ultimately gives the same information, but sometimes you want to come at it in terms of the maps, and sometimes in terms of the commas.

So far we've looked at how to meet comma vectors to form a comma basis. Next, let's look at the other side of duality, and see how to form a map basis out of joining maps. In many ways, the approaches are similar; the line of duality is a lot like a mirror in that way.

Well, it’s actually quite simple! They’re related in the same way as {{vector|{{map|19 30 44}}}} was related to {{map|{{vector|-4 4 -1}} {{vector|-10 -1 5}}}}: by the null-space operation. Specifically, {{map|{{vector|-4 4 -1}}}} is a basis for the null-space of the temperament with map basis {{vector|{{map|5 8 12}} {{map|7 11 16}}}}, because it is the minimal representation of all the commas tempered out by meantone temperament.

And there’s our {{map|{{vector|4 -4 1}}}}. Feel free to try reversing the operation by working out the map basis from this if you like. And/or you could try working out that {{map|{{vector|4 -4 1}}}} is a basis for the null-space of any other combination of ETs we found that could specify meantone, such as 7&12, or 12&19.

supposed to be a map basis, or mapping, for meantone? What does that even mean?

* Both {{vector|{{map|5 8 12}} {{map|7 11 16}}}} and {{vector|{{map|1 1 0}} {{map|0 1 4}}}} are equivalent mappings, then. In other words, they are both map bases of the same mapping row-space. Converting between them we could call a change of basis.

* Note well: this is not to say that {{map|1 1 0}} or {{map|0 1 4}} ''are'' the generators for meantone. They are generator ''mappings'': when assembled together, they collectively describe behavior of the generators, but they are ''not'' themselves the generators. This situation can be confusing; it confused me for many weeks. I thought of it this way: because the first generator is 2/1 — i.e. {{vector|1 0 0}} maps to {{vector|1 0}} — referring to {{map|1 1 0}} as the octave or period seems reasonable and is effective when the context is clear. And similarly, because the second generator is 3/2 — i.e. {{vector|-1 1 0}} maps to {{vector|0 1}} — referring to {{map|0 1 4}} as the fifth or the generator seems reasonable as is effective when the context is clear. But it's critical to understand that the first generator "being" the octave here is ''contingent upon the definition of the second generator'', and vice versa, the second generator "being" the fifth here is ''contingent upon the definition of the first generator''. Considering {{map|1 1 0}} or {{map|0 1 4}} individually, we cannot say what intervals the generators are. What if the mapping was {{vector|{{map|0 1 4}} {{map|1 2 4}}} instead? We'd still have the first generator mapping as {{map|1 1 0}}, but now that the second generator mapping is {{map|1 2 4}}, the two generators must be the fourth and the fifth. In summary, neither mapping row describes a generator in a vacuum, but does so in the context of all the other mapping rows.

The particular case I’d like to focus our attention on here is the rank-2 case. This is the first situation we’ve been able to achieve which boasts both an infinitude of matrices made from comma vectors which can represent the temperament by its comma basis, as well as an infinitude of matrices made from ET maps which can represent a temperament by its map basis. These are not contradictory. Let’s look at an example: septimal meantone.

On the other side of duality, septimal meantone’s map basis has two rows, corresponding to its two generators. We don’t have PTS for 7-limit JI handy, but because septimal meantone includes, or extends plain meantone, we can still refer to 5-limit PTS, and pick ETs from the meantone line there. The difference is that this time we need to include their 7-term. So the join of {{map|12 19 28 34}} and {{map|19 30 44 53}} would work. But so would {{map|19 30 44 53}} and {{map|31 49 72 87}}. We have an infinitude of options on this side of duality too, but here it’s not because our nullity is greater than 1, but because our rank is greater than 1.

Another form you might want the map basis in is the type Graham Breed's temperament finder puts them in, where all values in a map basis row may be negative, but this is in the service of the generator being less than half the size of the period. For example, for meantone, we'd want the fourth instead of the fifth, and we can see that

It’s often the case that a temperament’s nullity is greater than 1 or its rank is greater than 1, and therefore we have an infinitude of equivalent ways of expressing the comma basis or the map basis. This can be problematic, if we want to efficiently communicate about and catalog temperaments. It’s good to have a standardized form in these cases. The approach RTT takes here is to get these matrices into '''“canonical” form'''. In plain words, this just means: we have a function which takes in a matrix and spits out a matrix of the same shape, and no matter which matrix we input from a set of matrices which we consider all to be equivalent to each other, it will spit out the same result. This output is what we call the “canonical” matrix, and it can therefore uniquely identify a temperament.

I've seen many specialized matrix forms used in RTT for problems like this, such as [https://en.wikipedia.org/wiki/Smith_normal_form Smith normal form], [https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form], [http://home.iitk.ac.in/~rksr/html/03CANONICALFACTORIZATIONS.htm Hermite Canonical Form], and maybe others. What we will be using here is "[[IRREF|integer canonical form]]".<ref>Historically, Hermite Normal Form was used as if it gave a unique identifier for each temperament, when in fact it did not. It failed to remove possibly-hidden common factors in the columns of comma bases and rows of map bases, leading to situations which were named "[[torsion]]" and "[[contorsion]]", respectively. Hermite Canonical Form goes too far it reduces the matrix's diagonals to all 1's or 0's, which loses important information for temperaments with non-octave periods. Integer canonical form removes such common factors without compromising non-octave periods, therefore truly giving a unique identifier, and also eliminating the need to deal with torsion or contorsion.</ref>.  

To find the integer canonical form, we can combine the two processes we already know how to do: null-space for getting from a map basis to a comma basis, and anti-null-space to get from a comma basis to a map basis. Basically, to achieve canonical form of one type of basis, we convert it into the other type of basis, then back, and voilà: canonicalization.  

For this, we need to up our game to Wolfram computable notebooks. You can click "Make Your Own Copy" in the top left of the screen if you want to modify this notebook to make your own calculations.  

Why care about multimaps? Well, a key reason is that they can serve the same purpose as the canonical form of a temperament’s map basis: the process for converting a map basis to a multimap will convert any equivalent map basis to the same exact multimap. In other words, a multimap can serve as a unique identifier for its temperament.

Alright, then, sounds great! But how do I convert a map basis to a multimap? The process is doable. It’s closely related to the wedge product<ref>The wedge product is also known as the exterior product. Exterior and interior products (∧ and ⨼) should not be confused with outer and inner products, which in RTT are ordinary matrix products between a vector and a covector. The outer product gives a matrix and the inner product a scalar. e.g. [1 2⟩⟨3 4] = ⟨[3 4⟩ [6 8⟩] and ⟨3 4][1 2⟩ = 11. In our application, the inner product is the same as the dot product used for maps and intervals.</ref> (hence the name “wedgie”), which we write it with the symbol ∧.

# Convert each of those combinations to a square <span><math>r×r</math></span> matrix by slicing a column for each prime out of the map basis and putting them together.

Here's the meantone map basis again, with some color applied which should help identify the combinations:

So now each of those combinations becomes a square matrix, made out of bits from the map basis, which again is:

This method even works on an equal temperament, e.g. {{map|12 19 28}}. The rank is 1 so each combination of primes has only a single prime: they’re <span><math>(2)</math></span>, <span><math>(3)</math></span>, and <span><math>(5)</math></span>. The square matrices are therefore <span><math>\begin{bmatrix}12\end{bmatrix} \begin{bmatrix}19\end{bmatrix} \begin{bmatrix}28\end{bmatrix}</math></span>. The determinant of a 1×1 matrix is defined as the value of its single term, so now we have 12 19 28. <span><math>r</math></span> = 1, so we set the answer inside one layer of brackets, so our monocovector is {{map|12 19 28}}.  

If the map basis is

Now for canonical form. We need the first term to be positive; this doesn’t make a difference in how things behave, but is done because it canonicalizes things (we could have found the result where the first term came out positive by simply changing the order of the rows of our map basis, which doesn’t affect how it works as a mapping at all, or mean there's anything different about the temperament). And so we change the signs<ref>If it helps you, you could think of this sign-changing step as paired with the GCD extraction step, if you think of it like extracting a GCD of -1.</ref>, and our list ends up as <span><math>2</math></span>, <span><math>-3</math></span>, <span><math>-1</math></span>, <span><math>11</math></span>. There's no GCD to extract. So now we have {{multicovector|rank=3|2 -3 -1 11}}.

As for getting from a multimap back to the map basis, you can solve a system of equations for that. Though it’s not easy and there may not be a unique solution. And you probably will never find yourself with a multimap but without a corresponding map basis anyway.

# We can calculate a multicomma from a comma basis much in the same way we can calculate a multimap from a map basis

Here’s the comma basis for meantone: {{map|{{vector|-4 4 -1}}}}. Calculating the multicomma is almost the same as calculating the multimap. The only difference is that as a preliminary step you must transpose the matrix, or in other words, exchange rows and columns. In our bracket notation, that just looks like replacing {{map|{{vector|-4 4 -1}}}} with {{vector|{{map|-4 4 -1}}}}. Now we can see that this is just like our ET map example from the previous section: basically an identity operation, breaking the thing up into three 1×1 matrices <span><math>\begin{bmatrix}-4\end{bmatrix} \begin{bmatrix}4\end{bmatrix} \begin{bmatrix}-1\end{bmatrix}</math></span> which are their own determinants and then nesting back inside one layer of brackets because nullity is 1. So we have {{vector|-4 4 -1}}.  

If we want the canonical multicomma, extracting any GCD is necessary. But unlike with the canonical multimap, we can't necessarily assume the leading term's sign will be positive. Since the canonical state is defined in terms of the canonical multimap, if we want to know the signs of the canonical multicomma, we'll have to find the canonical multimap. We can either separately find the canonical multimap, then take its dual, and that'd be our canonical multicomma. Or, we can take the dual of the multicomma we've found here, and if that has a negative leading term, then we just need to change the signs on both the multimap and the multicomma so they remain each other's duals but  

# Match up the terms of your multimap with these symbols. If the symbol is <span><math>+</math></span>, do nothing. If the symbol is <span><math>-</math></span>, change the sign (positive to negative, or negative to positive; you could think of it like multiplying by either +1 or -1).  

The first thing to understand is that each term of the canonical multimap pertains to a different combination of primes. We already know this: it’s how we calculated it from the map basis. For example, in the canonical multimap for meantone, {{multicovector|1 4 4}}, the 1 is for <span><math>(2,3)</math></span>, the first 4 is for <span><math>(2,5)</math></span>, and the second 4 is for <span><math>(3,5)</math></span>.

One day you might come across a canonical multimap which has a term equal to zero. If you tried to interpret this term using the information here so far, you'd think it must generate <span><math>\frac 10</math></span>th of the tempered lattice. That's not easy to visualize or reason about. Does that mean it generates essentially infinity lattices? No, not really. More like the opposite. The question itself is somewhat undefined here. If anything, it's more like that combination of primes generates approximately ''none'' of the lattice. Because in this situation, the combination of primes whose canonical multimap term is zero generates so little of the tempered lattice that it's completely missing one entire dimension of it, so it's an infinitesimal amount of it that it generates. For example, the 11-limit temperament 7&12&31 has canonical multimap {{map|rank=3|0 1 1 4 4 -8 4 4 -12 -16}} and map basis {{vector|{{map|1 0 -4 0 -12}} {{map|0 1 4 0 8}} {{map|0 0 0 1 1}}}}; we can see from this how primes <span><math>(2,3,5)</math></span> can only generate a rank-2 cross-section of the full rank-3 lattice, because while 2 and 3 do the trick of generating that rank-2 part (exactly as they do in 5-limit meantone), prime 5 doesn't bring anything to the table here so that's all we get.

|map (often an ET)

|mapping, map basis

|comma

|map, mapping row, map basis row