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

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”.

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Until stated otherwise, this material will assume the [[5-limit|5 prime-limit]].

And if you’ve previously worked with EDOs, you may already be familiar with covectors. Covectors are a compact way to express EDOs in terms of the count of its steps it takes to reach its approximation of each prime harmonic, in order. For example, 12-EDO is {{map|12 19 28}}. The first term is the same as the name of the EDO, because the first prime harmonic is 2/1, or in other words: the octave. So this covector tells us that it takes 12 steps to reach 2/1 (the [[octave]]), 19 steps to reach 3/1 (the [[tritave]]), and 28 steps to each 5/1 (the [[pentave]]). Any or all of those intervals may be approximate.

Note the different direction of the brackets between covectors and vectors: covectors {{map|}} point left, vectors {{vector|}} point right.

So, 16/15 maps to one step in 12-EDO ''(see Figure 2a)''.

WolframAlpha's syntax is slightly different than what we use in RTT, but it's pretty alright for a free online tool capable of handling most of the math we need to do in RTT, so we're going to be supplementing several topics with WolframAlpha examples as we go. Here's the first:

So when I say 12:19:28 ≈ log(2:3:5) what I’m saying is that there is indeed some shared generator g for which log_g2 ≈ 12, log_g3 ≈ 19, and log_g5 ≈ 28 are all good approximations all at the same time, or, equivalently, a shared generator g for which g¹² ≈ 2, g¹⁹ ≈ 3, and g²⁸ ≈ 5 are all good approximations at the same time ''(see Figure 2c)''. And that’s a pretty cool thing to find! To be clear, with g = 1.059, we get g¹² ≈ 1.9982, g¹⁹ ≈ 2.9923, and g²⁸ ≈ 5.0291.

Why is this rare? Well, it’s like a game of trying to get these numbers to line up ''(see Figure 2d)'':

If the distance between entries in the row for 2 are defined as 1 unit apart, then the distance between entries in the row for prime 3 are 1/log₂3 units apart, and 1/log₂5 units apart for the prime 5. So, near-linings up don’t happen all that often!<ref>For more information, see: [[The_Riemann_zeta_function_and_tuning|The Riemann zeta function and tuning]].</ref> (By the way, any vertical line drawn through a chart like this is what we'll be calling here a '''uniform map'''; elsewhere you may find this called a “[[generalized patent val]]”.<ref>My main concern with the term "generalized patent val", or GPV, is that it gets things backwards: it posits the GPV as a type of patent val, when it makes more sense to think of it the other way around, with the patent val as a type of GPV.

The present definition of GPV on the wiki bends over backwards to make its way work, essentially using non-integer EDOs, which are a contradiction in terms. Consider this alternative definition, however, which is more straightforward:

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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.

=== a multitude of maps ===

The results of such inaccurate approximation are a bit chaotic. A ratio like 16/15 — where the factors of 3 and 5 are on the same side of the fraction bar and therefore cancel out each other’s error — fares relatively alright, if by “alright” we mean it gets tempered out despite being about 112¢ in JI. On the other hand, an interval like 27/25 where the factors of 3 and 5 are on opposite sides of the fraction bar and thus their errors compound, gets mapped to a whopping 4 steps, despite only being about 133¢ in JI.

If you did guess this, you are on the right track, but the full truth is a bit more complicated than that.

# the one which is one step away from it on the 5-axis,

[[File:Shape_of_scale_of_movements_on_axes.png|thumb|left|200px|'''Figure 3e.''' the basic shape the scaled axes make between neighbor maps (maps with only 1 difference between their terms)]]

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From here, let’s move by a single step on the 5-axis by adding 1 to the 5-term of our map, from 93 to 94, therefore moving to the map {{map|40 63 94}}. This map is found directly to the left. This makes sense because the orientation of the 5-axis is horizontal, and the positive direction points out from the origin toward the left, so increases to the 5-term move us in that direction.

In between where the colored lines touch the maps themselves and the page, we see a cluster of more maps, each of which starts with 6. In other words, these maps are about twice as far away from us as the others. Let’s consider {{map|6 10 14}} first. Notice that each of its terms is exactly 2x the corresponding term in {{map|3 5 7}}. In effect, {{map|6 10 14}} is redundant with {{map|3 5 7}}. If you imagine doing a mapping calculation or two, you can easily convince yourself that you’ll get the same answer as if you’d just done it with {{map|3 5 7}} instead and then simply divided by 2 one time at the end. It behaves in the exact same way as {{map|3 5 7}} in terms of the relationships between the intervals it maps, the only difference being that it needlessly includes twice as many steps to do so, never using every other one. So we don’t really care about {{map|6 10 14}}. Which is great, because it’s hidden exactly behind {{map|3 5 7}} from where we’re looking.

Now, to find a 6-ET with anything new to bring to the table, we’ll need to find one whose terms don’t share a common factor. That’s not hard. We’ll just take one of the ones halfway between the ones we just looked at. How about {{map|6 11 14}}, which is halfway between {{map|6 10 14}} and {{map|6 12 14}}. Notice that the purple line that runs through it lands halfway between the red and blue lines on the page. Similarly, {{map|6 10 15}} is halfway between {{map|6 10 14}} and {{map|6 10 16}}, and its yellow line appears halfway between the red and green lines on the page. What this is demonstrating is that halfway between any pair of n-ETs on the diagram, whether this pair is separated along the 3-axis or 5-axis, you will find a 2n-ET. We can’t really demonstrate this with 3-ET and 6-ET on the diagram, because those ETs are too inaccurate; they’ve been cropped off. But if we return to our 40-ET example, that will work just fine.

Also, check out the spot halfway between our two 17-ETs: there’s the 34-ET we briefly mused about earlier, which would solve 17’s problem of approximating prime 5 by subdividing each of its steps in half. We can confirm now that this 34-ET does a superb job at approximating prime 5, because it is almost vertically aligned with the JI red hexagram.

If you’re new to RTT, all of the other temperaments besides meantone, like “[[dicot]]”, “[[porcupine]]”, and “[[mavila]]”, are probably unfamiliar and their names may seem sort of random or bizarre looking. Well, you’re not wrong about the names being random and bizarre. But mathematically and musically, these temperaments are every bit as much real and of interest as meantone. One day you too may compose a piece or write an academic paper about porcupine temperament.

“[[Rank]]” has a slightly different meaning than dimension, but that’s not important yet. We’ll define rank, and discuss what exactly a rank-2 or -3 temperament means later. For now, it’s enough to know that each temperament line on this 5-limit PTS diagram is defined by tempering out a comma which has the same name. For now, we’re still focusing on visually how to navigate PTS. So the natural thing to wonder next, then, is what’s up with the slopes of all these temperament lines?

First of all, and this difference is superficial, it’s in a different format. We were expressing 12-ET’s generator 1.059 as a frequency multiplier; it’s like 2, 3, or 5, and this could be measured in Hz, say, by multiplying by 440 if A4 was our 1/1 (1.059 away from A is 466Hz, which is #A). But the meantone generators we’re looking at now in forms like <span><math>\frac 25</math></span>, <span><math>\frac 37</math></span>, or <span><math>\frac{5}{12}</math></span>, are expressed as fractional octaves, i.e. they’re in terms of pitch, something that could be measured in cents if we multiplied by 1200 (2/5 × 1200¢ = 480¢). We have a special way of writing fractional octaves, and that’s with a backslash instead of a slash, like this: 2\5, 3\7, 5\12.

Let’s put it this way. When we look at 12-ET in terms of itself, rather than in terms of any particular rank-2 temperament, its generator is 1\12. That’s the simplest, smallest generator which if we iterate it 12 times will touch every pitch in 12-ET. But when we look at 12-ET not as the end goal, but rather as a foundation upon which we could work with a given temperament, things change. We don’t necessarily need to include every pitch in 12-ET to realize a temperament it supports.

We mentioned that the generator value changes continuously as we move along a temperament line. So just to either side of 12-ET along the meantone line, the tuning of 2, 3, and 5 supports a generator size which in turn supports meantone, but it wouldn’t support augmented. And just to either side of 12-ET along the augmented line, the tuning of 2, 3, and 5 supports a generator which still supports augmented, but not meantone. 12-ET, we could say, is a convergence point between the meantone generator and the augmented generator. But it is not a convergence point because the two generators become identical in 12-ET, but rather because they can both be achieved in terms of 12-ET’s generator. In other words, 5\12 ≠ 4\12, but they are both multiples of 1\12.

=== periods and generators ===

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.

As we’ll soon see, there’s more than one way to generate a given rank-2 temperament. For example, meantone can be generated by an octave and a fourth. But it could equivalently be generated by an octave and a fifth. Or an octave and an [https://en.wikipedia.org/wiki/Augmented_unison augmented unison]. It could even be generated by cycling a fourth against a fifth. And so on.

Let’s bring up MOS theory again. We mentioned earlier that you might have been familiar with the scale tree if you’d worked with MOS scales before, and if so, the connection was scale cardinalities, or in other words, how many notes are in the resultant scales when you continuously iterate the generator until you reach points where there are only two scale step sizes. At these points scales tend to sound relatively good, and this is in fact the definition of a MOS scale. There’s a mathematical explanation for how to know, given a ratio between the size of your generator and period, the cardinalities of scales possible; we won’t re-explain it here. The point is that the scale tree can show you that pattern visually. And so if each temperament line in PTS is its own segment of the scale tree, then we can use it in a similar way.

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 join of rank-1 first.

I encourage you to spend some time playing around with [[Graham Breed]]'s [http://x31eq.com/temper/ online RTT tool]. For example, at http://x31eq.com/temper/net.html you can enter <code>12&19</code> in the "list of steps to the octave" field and <code>5</code> in the "limit" field and Submit, and you'll be taken to a results page for meantone.

Now how in the world could that matrix represent the same temperament as {{map|19 30 44}}? Well, they’re two different ways of describing it. {{map|19 30 44}}, as we know, tells us how many generator steps it takes to reach each prime approximation. This matrix, it turns out, is an equivalent way of stating the same information. This matrix is a minimal representation of the null-space of that mapping, or in other words, of all the commas it tempers out. (Don't worry about the word "mapping" just yet; for now, just imagine I'm writing "map". We'll explain the difference very soon.).

A good way to explain why we don’t need three of these commas is that if you had three of them, you could use any two of them to create the third, and then subtract the result from the third, turning that comma into a zero vector, or a vector with only zeroes, which is pretty useless, so we could just discard it.

[[File:Different nestings.png|400px|thumb|left|'''Figure 5a.''' How to write matrices in terms of either columns/vectors/commas or rows/covectors/maps.]]

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 this one a "basis", you ask? Well, it can be thought of as a basis too. It 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 may be 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 given mapping, i.e. it tempers out the same commas. In these contexts, it might make more sense to call such a mapping matrix a "mapping-row basis".

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Unfortunately, to the best of my Wolfram Alpha ability, I'm unable to make the entire process work in one go with Wolfram Alpha. Here is just the part where you take the null-space of the already anti-transposed comma basis:

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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:

So in this meantone mapping, the best approximation of the JI interval 10/9 is found by moving 1 step in each generator. We could write this in vector form as {{vector|1 1}}.

The critical thing here is that if {{vector|-4 4 -1}} is mapped to 0 steps by {{map|5 8 12}} individually and to 0 steps by {{map|7 11 16}} individually, then in total it comes out to 0 steps in the temperament, and thus is tempered out, or has vector {{vector|0 0}}.

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* 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..

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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.

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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:

* They both exist on the left half of tuning duality, on the side built up out of ETs which concerns rank (not the side built up out of commas, which uses vectors/columns, and concerns nullity).

In order to make these materials as accessible as possible, I have been doing what I can to lean away from RTT jargon and instead toward generic, previously established mathematical and/or musical concepts. That is why I have avoided the terms "monzo", "val", "breed", and why I will now avoid "wedgie". When established mathematical and/or musical concepts are unavailable, we can at least use unmistakable analogs built upon what we do have. In this case, if in RTT we use covectors to represent maps, then analogously, we can refer to the thing multicovectors represent in RTT as a '''multimap'''.

Alright, then, sounds great! But how do I convert a mapping-row 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 ∧.

# 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>.

And you've got your multimap!

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{{vector|{{covector|84 133 196}} {{covector|132 209 308}} {{covector|192 304 448}}}}<br>

|-

!RTT jargon

|list of vals

|wedgie, multival

I couldn’t have put this together without the help of:

* [[Paul Erlich]]

* [[Mike Battaglia]]

* [[Graham Breed]]

* [[Steve Martin]]

* [[Herman Miller]]

Happy tempering!