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

19-ET. Its map is {{map|19 30 44}}. We also now know that we could call it “meantone|magic”, because we find it at the meeting of the meantone and magic temperament lines. But how would we mathematically, non-visually make this connection?

Sometimes a comma-basis may have only a single comma. That’s okay. A single vector can become a matrix. To disambiguate this situation, you could put the vector inside row brackets, like this: {{map|{{vector|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside column brackets, like this: {{vector|{{map|19 30 44}}}}.

And now you wonder why it's not just "map basis". Well, that's answerable too. It's because "map" is the analogous term to an "interval", but we're looking for the analogous term to a "comma". A comma is an interval which is tempered out. So we need a word that means a map which tempers out, and that term is "mapping-row".

One last note back on the bracket notation before we proceed: you will regularly see matrices across the wiki that use only square brackets on the outside, e.g. [{{map|5 8 12}} {{map|7 11 16}}] or [{{vector|-4 4 -1}} {{vector|-10 -1 5}}]. That's fine because it's unambiguous; if you have a list of rows, it's fairly obvious you've arranged them vertically, and if you've got a list of columns, it's fairly obvious you've arranged them horizontally. I personally prefer the style of using angle brackets at both levels — for slightly more effort, it raises slightly less questions — but using only square brackets on the outside should not be said to be wrong<ref>Besides, in most contexts the null-space of a linear mapping is thought of as a list of vectors, rather than a matrix, but it’s generally more helpful for us here to think of it smooshed together as a matrix.</ref>.

There’s nothing special about the pairing of meantone and magic. We could have chosen meantone|hanson, or magic|negri, etc. A matrix formed out of the meet of any two of these particular commas will capture the same exact null-space of {{vector|{{map|19 30 44}}}}.

And interestingly enough, as you'll soon see, the process is almost the same to take the null-space as it is to undo it.

=== the other side of duality ===

When we join two maps, we put them together into a matrix, just like how we put two vectors together into a matrix. But again, where vectors are vertical columns, maps are horizontal rows. So when we combine {{map|5 8 12}} and {{map|7 11 16}}, we get a matrix that looks like

Again, we find ourselves in the position where we must reconcile a strange new representation of an object with an existing one. We already know that meantone can be represented by the vector for the comma it tempers out, {{vector|-4 4 -1}}. How are these two representations related?

We can work this one out by hand too:

|}

It’s worth noting that, just as 2 commas were exactly enough to define a rank-1 temperament, though there were an infinitude of equivalent pairs of commas we could choose to fill that role, there’s a similar thing happening here, where 2 maps are exactly enough to define a rank-2 temperament, but an infinitude of equivalent pairs of them. We can even see that we can convert between these maps using Gauss-Jordan addition and subtraction, just like we could manipulate commas to get from one to the other. For example, the map for 12-ET {{map|12 19 28}} is exactly what you get from summing the terms of 5-ET {{map|5 8 12}} and 7-ET {{map|7 11 16}}: {{map|5+7 8+11 12+16}} = {{map|12 19 28}}. Cool!

</math>

=== rank-2 mappings ===

* We’ve made it to a critical point here: we are now able to explain why RTT is called “regular” temperament theory. Regular here is a mathematical term, and I don’t have a straightforward definition of it for you, but it apparently refers to the fact that all intervals in the tuning are combinations of only these specified generators. So there you go.

* 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.

* This also gives us a new way to think about the scale tree patterns. Remember how earlier we pointed out that {{map|12 19 28}} was simply {{map|5 8 12}} + {{map|7 11 16}}? Well, if {{vector|{{map|5 8 12}} {{map|7 11 16}}}} is a way of expressing meantone in terms of its two generators, you can imagine that 12-ET is the point where those two generators converge on being the same exact size<ref>For real numbers <span><math>p,q</math></span> we can make the two generators respectively <span><math>\frac{p}{5p+7q}</math></span> and <span><math>\frac{q}{5p+7q}</math></span> of an octave, e.g. <span><math>(p,q)=(1,0)</math></span> for 5-ET, <span><math>(0,1)</math></span> for 7-ET, <span><math>(1,1)</math></span> for 12-ET, and many other possibilities.</ref>. If they become the same size, then they aren’t truly two separate generators, or at least there’s no effect in thinking of them as separate. And so for convenience you can simply combine their mappings into one. You could imagine gradually increasing the size of one generator and decreasing the size of the other until they were both 100¢. As long as you maintain the correct proportion, you'll stay along the meantone line..

Because we're starting in 4D here, if we temper out one comma, we still have a rank-3 temperament, with 3 independent generators. Temper out two commas, and we have a rank-2 temperament, with 2 generators (remember, one of them is the period, which is usually the octave). And we’d need to temper out 3 commas here to pinpoint a single ET.

=== canonical form ===

The latter is sometimes called the “musician’s form” of the temperament, because it’s easy to reason about from a musical perspective. But it turns out there’s not a particularly clean function for consistently getting to it, or even defining it.

<math>

maps the fourth (4/3, {{vector|2 -1 0 }}) to {{vector|0 1}}.

To be clear, canonical form isn’t necessary to avoid ambiguity: you will never find a comma-basis that could represent more than one temperament.

For example, the canonical form of meantone is:

If you take the integer canonical form of {{vector|{{map|5 8 12}} {{map|7 11 16}}}}, that’s what you get. It’s also what you get if you take the integer canonical form of {{vector|{{map|12 19 28}} {{map|19 30 44}}}}, etc. That’s the power of 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.

So we mentioned that a multimap is an alternative way to represent a temperament. Let's look at an example now. Meantone's multimap looks like this: {{multicovector|1 4 4}}. As you can see, it is a bicovector, or bimap, because it has two of each bracket.

First I’ll list the steps. Don’t worry if it doesn’t all make sense the first time. We’ll work through an example and go into more detail as we do. To be clear, what we're doing here is different from the strict definition of the wedge product as you may see it elsewhere; I'm specifically here describing the process for finding the multimap in the form you're going to be interested in for RTT purposes.

# Take each combination of <span><math>r</math></span> primes where <span><math>r</math></span> is the rank, sorted in [https://en.wikipedia.org/wiki/Lexicographic_order lexicographic order], e.g. if we're in the 7-limit, we'd have <span><math>(2,3,5)</math></span>, <span><math>(2,3,7)</math></span>, <span><math>(2,5,7)</math></span>, and <span><math>(3,5,7)</math></span>.

# Take each matrix's determinant.

# Set the results inside <span><math>r</math></span> brackets.

We have rank <span><math>r</math></span> = 2, so we’re looking for every combination of two primes. That’s out of the three total primes we have in the 5-limit: 2, 3, and 5. So those combinations are <span><math>(2,3)</math></span>, <span><math>(2,5)</math></span>, and <span><math>(3,5)</math></span>. Those are already in lexicographic order, or in other words, just like how alphabetic order works, but generalized to work for size of numbers too (so that 11 comes after 2, not before).

<math>

</math>

<math>

Let’s try a slightly harder example now: a rank-3 temperament, and in the 7-limit. There are four different ways to take 3 of 4 primes: <span><math>(2,3,5)</math></span>, <span><math>(2,3,7)</math></span>, <span><math>(2,5,7)</math></span>, and <span><math>(3,5,7)</math></span>.

<math>

|}

=== multicommas ===

To understand why, we have to cover a few key points:

# Just as a vector is the dual of a covector, we also have a '''multivector''' which is the dual of a multicovector. Analogously, we call the thing the multivector represents a '''multicomma'''.

# We can convert between multimaps and multicommas using an operation called “taking '''[[the dual]]'''”<ref>This operation uses the same process as is used for finding the complement in exterior algebra, however, whereas exterior algebra does not convert between vectors and covectors (it can be used on either one, staying within that category), with RTT's dual you switch which type it is as the last step. More details can be found below. The dual we use in RTT is #2 in the table. It essentially combines elements from both #1 and #3.

So we now understand how to get to canonical multimaps. And we understand that they uniquely identify the temperament. But what about the individual terms — do they mean anything in and of themselves? It turns out: yes!

Now, let’s convert every term of the canonical multimap by taking its absolute value and its inverse. In this case, each of our terms is already positive, so that has no effect. But taking the inverse converts us to <span><math>\frac 11</math></span>, <span><math>\frac 14</math></span>, <span><math>\frac 14</math></span>. These values tell us what fraction of the tempered lattice we can generate using the corresponding combination of primes.

But now try it with only 5 and one other of primes 2 or 3. Prime 5 takes you over 4 in both directions. But if you have only prime 2 otherwise, then you can only move up or down from there, so you’ll only cover every fourth vertical line through the tempered lattice. Or if you only had prime 3 otherwise, then you could only move left and right from there, you’d only cover every fourth horizontal line ''(see Figure 6c)''.

We’ll look in more detail later at how exactly to best find these generators, once you know which primes to make them out of.

!RTT application

|map, mapping-row (often an ET)

|multimap

|multicomma