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

=== intersections and unions ===

We’ve seen how 12-ET is found at the convergence of meantone and augmented temperaments, and therefore supports both at the same time. In fact, no other ET can boast this feat. Therefore, we can even go so far as to describe 12-ET as the intersection of the meantone line and the augmented line. Using the pipe operator “|” to mean “intersection”, then, we could call 12-ET “meantone|augmented”. In other words, we express a rank-1 temperament in terms of two rank-2 temperaments.

For another rank-1 example, we could call 7-ET “meantone|dicot”, because it is the intersection between meantone and dicot temperaments. It’s not merely at that intersection, it is the intersection.

For example, we could choose 7-ET and 12-ET. Looking at either 12-ET or 7-ET, we can see that many, many temperament lines pass through them individually. Even more pass through them which Paul chose (via a complexity threshold) not to show. But there’s only one line which runs through both 7-ET and 12-ET, and that’s the meantone line. So of all the commas that 7-ET tempers out, and all the commas that 12-ET tempers out, there’s only a single one which they have in common, and that’s the meantone comma. Therefore we could give meantone temperament another name, and that’s “7&12”; in this case we use the ampersand operator, not the pipe, because this combination is a union.<ref>This union and intersection terminology is from the perspective of tuning space. It is possible to think of it completely the other way around, using projective tone space. In this space, ETs are the lines and temperaments/commas are the points. So there, you could think of intersecting maps or unioning commas. However, we’re standardizing around this way of thinking, for consistency with the operators for wedgies. More on that later.</ref>

When specifying a rank-1 temperament in terms of two rank-2 temperaments, an obvious constraint is that the two rank-2 temperaments cannot be parallel. When specifying a rank-2 temperament in terms of two rank-1 temperaments, it seems like things should be more open-ended. Indeed, however, there is a special additional constraint on either method, and they’re related to each other. Let’s look at rank-2 as the union of rank-1 first.

7&12 is valid for meantone. So is 5&7, and 7&12. 12&19 and 19&7 are both fine too, and so are 5&17 and 17&12. Yes, these are all literally the same thing (though you may connote a meantone generator size on the meantone line somewhere between these two ETs). So how could we mess this one up, then? Well, here’s our first counterexamples: 5&19, 7&17, and 17&19. And what problem do all these share in common? The problem is that between 5 and 19 on the meantone line we find 12, and 12 is a smaller number than 19 (or, if you prefer, on PTS, it is printed as a larger numeral). It’s the same problem with 17&19, and with 7&17 the problem is that 12 is smaller than 17. It’s tricky, but you have to make sure that between the two ETs you union there’s not a smaller ET (which you should be unioning with instead). The reason why is out of scope to explain here, but we’ll get to it eventually.

From the PTS diagram, we can visually pick out rank-1 temperaments at the intersection of rank-2 temperaments as well as rank-2 temperaments as the unions of rank-1 temperaments. But we can also understand these results through covectors and vectors. And we're going to need to learn how, because PTS can only take us so far. 5-limit PTS is good for humans because we live in a physically 3-dimensional world (and spend a lot of time sitting in front of 2D pages on paper and on computer screens), but as soon as you want to start working in 7-limit harmony, which is 4D, visual analogies will begin to fail us, and if we’re not equipped with the necessary mathematical abstractions, we’ll no longer be able to effectively navigate.

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 intersection of the meantone and magic temperament lines. But how would we mathematically, non-visually make this connection?

The intersection of two vectors can be represented as a matrix. If a vector is like a list of numbers, a matrix is a table of them. Technically, vectors are vertical lists of numbers, or columns, so when we put meantone and magic together, we get a matrix that looks like this:

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 intersection of any two of these commas will capture the same exact null-space of {{vector|{{map|19 30 44}}}}.

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 intersection of ''two'' commas (at least in the 5-limit; more on this later). In this way, the comma bases which represent the intersections of two commas are greater than the sum of their individual parts.

So far we've looked at how to intersect 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 unioning maps. In many ways, the approaches are similar; the line of duality is a lot like a mirror in that way.

When we union 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

Our hypothesis might be: this represents the entirety of 5-limit JI. If two rank-1 temperaments — each of which can be described as tempering out 2 commas — when unioned result in a rank-2 temperament — which is defined as tempering out 1 comma — then when we union three rank-1 temperaments, we should expect to get a rank-3 temperament, which tempers out 0 commas. The rank-1 temperaments appear as 0D points in PTS but are understood to be a 1D line coming straight at us; the rank-2 temperaments appear as 1D points in PTS but are understood to be 2D planes coming straight at us; the rank-3 temperament appear as the 2D plane of the entire PTS diagram but is understood to be the entire 3D space.

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 union 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.

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 (hence the name “wedgie”), but better known as the '''exterior product'''. We write it with the symbol ∧, which is great because that’s the logical operator for “and”, forming a clear association with the unioning of two ETs.

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 both more and less and different ways from the strict definition of the exterior 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.