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

There's still a ton more to say here, though, and I hope to get to completing this material soon.

What’s tempering, you ask, and why temper? I won’t be answering those questions in depth here. Plenty has been said about the “what” and “why” elsewhere<ref>And curiously little about the history.</ref>. These materials are about the “how”.

As for whether ''determining'' a middle path tuning is any harder than determining an ED or JI tuning, I think it would be fair to say that in the exact same way that a middle path tuning — once attained — combines the strengths of ED and of JI, determining a middle path tuning combines the challenges of determining good ED tunings and of determining good JI tunings. You have been warned.

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.

It’s hard to get too far with RTT before you understand '''vectors''' and '''covectors''', so let’s start there.

Wolfram Language's syntax is a bit different than what we use in RTT, which can take a little getting used to. As you can see here, both our vector and covector use the same curly brackets instead of angle and square brackets. This is a very simple example, though, and the actual difference between RTT and Wolfram Language syntax gets more complicated than just replacing all brackets with curlies; but we won't have to worry about that for a while.

[[File:Meantone temper out.png|200px|frame|right|'''Figure 2b.''' meantone equates four fifths (3/2) with one major third (5/4)]]

One thing we can easily begin to do now, though, is this: refer to EDOs instead as ETs, or equal temperaments. The two terms are [[EDO_vs_ET|roughly synonymous]], but have different implications and connotations. To put it briefly, the difference can be found in the names: 12 '''E'''qual '''D'''ivisions of the '''O'''ctave suggests only that your goal is equally dividing the octave, while 12 '''E'''qual '''T'''emperament suggests that your goal is to temper and that you have settled on a single equal step to accomplish that. Because we’re learning about temperament theory here, it would be more appropriate and accurate to use the local terminology. 12-ET it is, then.

If you’ve seen one map before, it’s probably {{map|12 19 28}}. That’s because this map is the foundation of conventional Western tuning: [[12edo|12 equal temperament]]. A major reason it stuck is because — for its low complexity — it can closely approximate all three of the 5 prime-limit harmonics 2, 3, and 5 at the same time.

And why is this cool? Well, if {{map|12 19 28}} approximates the harmonic building blocks well individually, then JI intervals built out of them, like 16/15, 5/4, 10/9, etc. should also be reasonably closely approximated overall, and thus recognizable as their JI counterparts in musical context. You could think of it like taking all the primes in a prime factorization and swapping in their approximations. For example, if 16/15 = 2⁴ × 3⁻¹ × 5⁻¹ ≈ 1.067, and {{map|12 19 28}} approximates 2, 3, and 5 by 1.059¹² ≈ 1.998, 1.059¹⁹ ≈ 2.992, and 1.059²⁸ ≈ 5.029, respectively, then {{map|12 19 28}} maps 16/15 to 1.998⁴ × 2.992⁻¹ × 5.029⁻¹ ≈ 1.059, which is indeed pretty close to 1.067. Of course, we should also note that 1.059 is the same as our step of {{map|12 19 28}}, which checks out with our calculation we made in the previous section that the best approximation of 16/15 in {{map|12 19 28}} would be 1 step.

Now, because the octave is the [[interval of equivalence]] in terms of human pitch perception, it’s a major convenience to enforce pure octaves, and so many people prefer the first term to be exact. In fact, I’ll bet many readers have never even heard of or imagined impure octaves, if my own anecdotal experience is any indicator; the idea that I could temper octaves to optimize tunings came rather late to me.

This matter of choosing the exact generator for a map is called '''tuning''', and if you’ll believe it, we won’t actually talk about that in detail again until much later. Temperament — the second ‘T’ in “RTT” — is the discipline concerned with choosing an interesting map, and tuning can remain largely independent from it. The temperament is only concerned with the fact that — no matter what exact size you ultimately make the generator — it is the case e.g. that 12 of them make a 2, 19 of them make a 3, and 28 of them make a 5. So, for now, whenever we show a value for g, assume we’ve given a computer a formula for optimizing the tuning to approximate all three primes equally well. As for us humans, let’s stay focused on tempering.

Suppose we want to experiment with the {{map|12 19 28}} map a bit. We’ll change one of the terms by 1, so now we have {{map|12 20 28}}. Because the previous map did such a great job of approximating the 5-limit (i.e. log(2:3:5)), though, it should be unsurprising that this new map cannot achieve that feat. The proportions, 12:20:28, should now be about as out of whack as they can get. The best generator we can do here is about 1.0583 (getting a little more precise now), and 1.0583¹² ≈ 1.9738 which isn’t so bad, but 1.0583¹⁹ = 3.1058 and 1.0583²⁸ = 4.8870 which are both way off! And they’re way off in the opposite direction — 3.1058 is too big and 4.8870 is too small — which is why our tuning formula for g, which is designed to make the approximation good for every prime at once, can’t improve the situation: either sharpening or flattening helps one but hurts the other.

At this point you should have a pretty good sense for why choosing a map makes an important impact on how your music sounds. Now we just need to help you find and compare maps! Or, similarly, how to find and compare intervals to temper. To do this, we need to give you the ability to navigate tuning space.

In this section, we will be going into potentially excruciating detail about how to read the projective tuning space diagram featured prominently in Paul Erlich's Middle Path paper. For me personally, attaining total understanding of this diagram was critical before the linear algebra stuff (that we'll discuss afterwards) started to mean much to me. But other people might not work that way, and the extent of detail I go into in this section is not necessary to become competent with RTT (in fact, to my delight, one of the points I make in this section was news to Paul himself). So if you're already confident about reading the PTS diagram, you may try skipping ahead.

[[File:Pts-2-3-5-e2-twtop-tlin.jpg|center|thumb|800px|'''Figure 3a.''' 5-limit projective tuning space]]

And so it makes sense that {{map|17 27 40}} and {{map|17 27 39}} are aligned horizontally, because the only difference between their maps is in the 5-term, and the 5-axis is horizontal.

You might guess that to arrive at that tilted numeral 12, you would start at the origin in the center, move 12 steps toward the bottom right (along the 2-axis), 19 steps toward the top right (not along, but parallel to the 3-axis), and then 28 steps toward the left (parallel to the 5-axis). And if you guessed this, you’d probably also figure that you could perform these moves in any order, because you’d arrive at the same ending position regardless ''(see Figure 3d)''.

We can imagine that if we hadn’t scaled the steps, as in our initial naive guess, we’d have ended up nowhere near the center of the diagram. How could we have, if the steps are all the same size, but we’re moving 28 of them to the left, but only 12 and 19 of them to the bottom left and top right? We’d clearly end up way, way further to the left, and also above the horizontal midline. And this is where pretty much any near-just ET would get plotted, because 3 being bigger than 2 would dominate its behavior, and 5 being larger still than 3 would dominate its behavior.

The truth about distances between related ETs on the PTS diagram is actually slightly even more complicated than that, though; as we mentioned, the scaled axes are only the first difference from our initial guess. In addition to the effect of the scaling of the axes, there is another effect, which is like a perspective effect. Basically, as ETs get more complex, you can think of them as getting farther and farther away; to suggest this, they are printed smaller and smaller on the page, and the distances between them appear smaller and smaller too.

Just as there are 2n-ETs halfway between n-ETs, there are 3n-ETs a third of the way between n-ETs. Look at these two [[29edo|29-ET]]s here. The [[58edo|58-ET]] is here halfway between them, and two [[87edo|87-ET]]s are here each a third of the way between.

So far, we’ve been describing PTS as a projection of map space, which is to say that we’ve been thinking of maps as the coordinates. We should be aware that tuning space is a slightly different structure. In tuning space, coordinates are not maps, but tunings, specified in cents, octaves, or some other unit of pitch. So a coordinate might be {{map|6 10 14}} in map space, but {{map|1200 2000 2800}} in tuning space.

The key point here is that, as we mentioned before, the problems of tuning and tempering are largely separate. PTS projects all tunings of the same temperament to the same point. This way, issues of tuning are completely hidden and ignored on PTS, so we can focus instead on tempering.

We’ve shown that ETs with the same number that are horizontally aligned differ in their mapping of 5, and ETs with the same number that are aligned on the 3-axis running bottom left to top right differ in their mapping of 3. These basic relationships can be extrapolated to be understood in a general sense. ETs found in the center-left map 5 relatively big and 2 and 3 relatively small. ETs found in the top-right map 3 relatively big and 2 and 5 relatively small. ETs found in the bottom-right map 2 relatively big and 3 and 5 relatively small. And for each of these three statements, the region on the opposite side maps things in the opposite way.

Or maybe you know which commas {{map|43 68 100}} tempers out, so you can find it along the line for that comma’s temperament.

We're about to take our first look at temperaments beyond mere equal temperaments. By the end of this section, you'll be able to explain the musical meaning of the patterns in the numerals along lines in PTS, the labels of these lines, as well as what's happening at their intersections and what their slopes mean. In other words, pretty much all of the major remaining visual elements on PTS should make sense to you.

So we understand the shape of projective tuning space. And we understand what points are in this space. But what about the magenta lines, now?

Think of it like this: for meantone, a change to the mapping of 5 doesn’t make near as much of a difference to the outcome as does a change to the mapping of 2 or 3, therefore, changes along the 5-axis don’t have near as much of an effect on that line, so it ends up roughly parallel to it.

Patterns, patterns, everywhere. PTS is chock full of them. One pattern we haven’t discussed yet is the pattern made by the ETs that fall along each temperament line.

Okay, so it’s easy to see how all this follows from here. But where the heck did I get <math>\frac 25</math> and <math>\frac 37</math> in the first place? I seemed to pull them out of thin air. And what the heck is this value?

[[File:Generator sizes in PTS.png|800px|thumb|'''Figure 4b.''' Generator sizes of linear temperaments in PTS. Don't worry too much about the valid ranges yet; we'll discuss that part later. And I didn’t break down what’s happening along the blackwood, compton, augmented, dimipent, and some other lines which are labelled on the original PTS diagram. In some cases, it’s just because I got lazy and didn’t want to deal with fitting more numbers on this thing. But in the case of all those that I just listed, it’s because although they are rank-2, those temperaments all have non-octave periods and are therefore not linear, and so it doesn't make enough sense to compare their generators here. You'll learn about periods in the next section.]]

Here’s a diagram that shows how the generator size changes gradually across each line in PTS. It may seem weird how the same generator size appears in multiple different places across the space. But keep in mind that pretty much any generator is possible pretty much anywhere here. This is simply the generator size pattern you get when you lock the octave to exactly 1200 cents, to establish a common basis for comparison. This is what enables us to produce maps of temperaments such as the one found at [[Map_of_linear_temperaments|this Xen wiki page]], or this chart here ''(see Figure 4b)''.

Earlier we mentioned the term “rank”. I warned you then that it wasn’t actually the same thing as dimensionality, even though we could use dimensionality in the PTS to help differentiate rank-2 from rank-1 temperaments. Now it’s time to learn the true meaning of rank: it’s how many generators a temperament has. So, it ''is'' the dimensionality of the ''tempered'' lattice; but it's still important to stay clear about the fact that it's different from the dimensionality of the original system from which you are tempering.

The structure when you stop iterating the meantone generator with five notes is called meantone[5]. If you were to use the entirety of 12-ET as meantone then that’d be meantone[12]. But you can also realize meantone[12] in 19-ET; in the former you have only one step size, but in the latter you have two. You can’t realize meantone[19] in 12-ET, but you could also realize it in 31-ET.

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 meeting of the meantone line and the augmented line. Using the pipe operator “|” to mean “meet”, then, we could call 12-ET “meantone|augmented”, read "meantone meet augmented". In other words, we express a rank-1 temperament in terms of two rank-2 temperaments.

And the related constraint for rank-1 from two rank-2 is that you can’t choose two temperaments whose names are printed smaller on the page than another temperament between them. More on that later.

From the PTS diagram, we can visually pick out rank-1 temperaments at the meetings of rank-2 temperaments as well as rank-2 temperaments as the joinings 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 meeting of the meantone and magic temperament lines. But how would we mathematically, non-visually make this connection?

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 {{ket|{{map|19 30 44}}}}.

It's great to have Wolfram and other such tools to compute these things for us, once we understand them. But I think it’s a very good idea to work through these operations by hand at least a couple times, to demystify them and give you a feel for them.

So we can now convert back and forth between a mapping-row-basis and a comma-basis. We could imagine drawing a diagram with a line of duality down the center, with a temperament's mapping-row-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.

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

Let’s consider some facts:

That looks like an identity matrix! Well, in this case the best interpretation can be found by checking its mapping of 2/1, 3/1, and 5/1, or in other words {{vector|1}}, {{vector|0 1}}, and {{vector|0 0 1}}. Each prime is generated by a different generator, independently. And if you think about the implications of that, you’ll realize that this is simply another way of expressing the idea of 5-limit JI! Because the three generators are entirely independent, we are capable of exactly generating literally any 5-limit interval. Which is another way of confirming our hypothesis that no commas are tempered out.

Let’s make sure we establish what exactly the tempered lattice is. This is something like the JI lattice we looked at very early on, except instead of one axis per prime, we have one axis per generator. As we saw just a moment ago, these two situations are not all that different; the JI lattice could be viewed as a tempered lattice, where each prime is a generator.

And so we can see that tempering has reduced the dimensionality of our lattice by 1. Or in other words, the dimensionality of our lattice was always the rank; it’s just that in JI, the rank was equal to the dimensionality. And what’s happened by reducing this rank is that we eliminated one of the primes in a sense, by making it so we can only express things in terms of it via combinations of the other remaining primes.

Let’s review what we’ve seen so far. 5-limit JI is 3-dimensional. When we have a rank-3 temperament of 5-limit JI, 0 commas are tempered out. When we have a rank-2 temperament of 5-limit JI, 1 comma is tempered out. When we have a rank-1 temperament of 5-limit JI, 2 commas are tempered out.<ref>Probably, a rank-0 temperament of 5-limit JI would temper 3 commas out. All I can think a rank-0 temperament could be is a single pitch, or in other words, everything is tempered out. So perhaps in some theoretical sense, a comma-basis in 5-limit made out of 3 vectors, thus a square 3×3 matrix, as long as none of the lines are parallel, should minimally represent every interval in the space.</ref>

[[File:Mapping and comma basis dnr.png|400px|thumb|right|'''Figure 5c.''' The relationship between dimensionality d, rank r, and nullity n]]

So far we’ve only been dealing with RTT in terms of prime limits, which is by far the most common and simplest way to use it. But nothing is stopping you from using other JI groups. What is a JI group? Well, I'll explain in terms of what we already know: prime limits. Prime limits are basically the simplest type of JI group. A prime limit is shorthand for the JI group consisting of all the primes up to that prime which is your limit; for example, the 7-limit is the same thing as the JI group "2.3.5.7". So JI groups are just sets of harmonics, and they are notated by separating the selected harmonics with dots.

On the other side of duality, septimal meantone’s mapping-row-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.

Recently we reduced

Canonicalization used to be achieved in RTT through the use of the "wedgie", an object that involves more advanced math. So while you may see "wedgies" around on the wiki and elsewhere, don't worry — you don't need to worry about them in order to do RTT. If you want to learn more anyway, I've gathered up everything I figured out about those here: [[VEA]].

{| class="wikitable"