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

[[File:Map and vector.png|500px|thumb|right|'''Figure 1a.''' Mapping example]]

So, 16/15 maps to one step in 12-EDO ''(see Figure 1a)''.

[[File:Meantone temper out.png|200px|frame|right|'''Figure 1b.''' meantone equates four fifths (3/2) with one major third (5/4)]]

But a more interesting way to think about this result involves treating {{monzo|-4 4 -1}} not as a single interval, but as the end result of moving by a combination of intervals. For example, moving up four fifths, 4 × {{monzo|-1 1 0}} = {{monzo|-4 4 0}}, and then moving down one pentave {{monzo|0 0 -1}}, gets you right back where you started in 12-EDO. Or, in other words, moving by one pentave is the same thing as moving by four fifths ''(see Figure 1b)''. One can make compelling music that [[Keenan's comma pump page|exploits such harmonic mechanisms]].

[[File:Approximation of logs.png|600px|thumb|left|'''Figure 1c.''' visualization of an ET as a logarithmic approximation. The curve of the blue line is the familiar logarithmic curve of the harmonic series (harmonic 4 was skipped because it's not prime). Each rectangular brick is one of our generators, or in other words, one of the same ET step. The goal is to choose a size of brick that allows us to build stacks which most closely matches the position of the blue line at all three of these primes' positions.]]

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 1c)''. 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 1d)'':

[[File:Near linings up rare2.png|600px|thumb|right|'''Figure 1d.''' Texture of ETs approximating prime harmonics. Where the ''numerals'' line up, all primes are well-approximated by a single step size (the boundaries between cells are midpoints between perfect approximations, or in other words, the point where the closest approximation switches over from one generator count to the next). Nudging one of the maps' vertical lines to the right would mean decreasing the generator size, flattening the tunings of all the primes, and vice versa, nudging it to the left would mean increasing the generator size, sharpening the tunings of all the primes. You can visualize this on Figure 1c. as shrinking or growing the height of the rectangular bricks. The positions of each map's vertical line, or in other words the tuning of its generator, has been optimized using some formula to distribute the detuning amongst the three primes; that's why you do not see any vertical line here for which the closest step counts for each prime are all on one side of it.]]

When we don’t enforce pure octaves, tuning becomes a more interesting problem. Approximating all three primes at once with the same generator is a balancing act. At least one of the primes will be tuned a bit sharp while at least one of them will be tuned a bit flat. In the case of {{val|12 19 28}}, the 5 is a bit sharp, and the 2 and 3 are each a tiny bit flat ''(as you can see in Figure 1c)''.

[[File:Why not just srhink every block.png|thumb|left|600px|'''Figure 1e.''' Visualization of pointlessness of tuning all primes sharp (or flat, as you could imagine)]]

If you think about it, you would never want to tune all the primes sharp at the same time, or all of them flat; if you care about this particular proportion of their tunings, why wouldn’t you shift them all in the same direction, toward accuracy, while maintaining that proportion? ''(see Figure 1e)''

[[File:17-ET mistunings.png|thumb|600px|right|'''Figure 1f.''' Detunings of various 17-ET maps, showing how the supposed "patent" val's total error can be improved upon by allowing flexible octaves and second-closest mappings of primes. Also shows that the pure octave 17c is improper insofar as all primes are detuned in one direction.]]

[[File:17-ET.png|thumb|400px|left|'''Figure 1g.''' Visualization of the 17-ETs on the GPV continuum, showing how the pure octave 17c is a "lie" insofar as there exists no generator that exactly reaches prime 2 in 17 steps while more closely approximating prime 5 as 40 steps than 39 steps.]]

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

This is 5-limit [[projective tuning space]], or PTS for short ''(see Figure 2a)''. This diagram was created by RTT pioneer [[Paul Erlich]]. It compresses a huge amount of valuable information into a small space. If at first it looks overwhelming or insane, do not despair. It may not be instantly easy to understand, but once you learn the tricks for navigating it from these materials, you will find it is very powerful. Perhaps you will even find patterns in it which others haven’t found yet.

[[File:JI scale 2.png|thumb|right|150px|'''Figure 2b.''' Just an example JI scale]]

[[File:PTS with axes.png|300px|left|thumb|'''Figure 2c.''' PTS, with axes]]

Of course, PTS looks nothing like this JI lattice ''(see Figure 2b)''. This diagram has a ton more information, and as such, Paul needed to get creative about how to structure it. It’s a little tricky, but we’ll get there. For starters, the axes are not actually shown on the PTS diagram; if they were, they would look like this ''(see Figure 2c)''.

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 2d)''.

[[File:PTS with finding 12-ET.png|400px|right|thumb|'''Figure 2d.''' arriving at 12-ET by moving in any of the 6 possible axis orders (Note: this is a visualization of an early guess at how things work. They're different and more complicated than this. Keep reading!)]]

The first difference to understand is that each axis’s steps have been scaled proportionally according to their prime ''(see Figure 2e)''. To illustrate this, let’s highlight an example ET and compare its position with the positions of three other ETs:

[[File:Shape_of_scale_of_movements_on_axes.png|thumb|left|200px|'''Figure 2e.''' the basic shape the scaled axes make between neighbor maps (maps with only 1 difference between their terms)]]

[[File:40-ET distances example.png|400px|right|thumb|'''Figure 2f.''' Distances between 40-ET's neighbors in PTS]]

Now let’s observe the difference in distances ''(see Figure 2f)''. Notice how the distance between the maps separated by a change in 5-term is the smallest, the maps separated by a change in 3-term have the medium-sized distance, and maps separated by a change in the 2-term have the largest distance. This tells us that steps along the 3-axis are larger than steps along the 5-axis, and steps along the 2-axis are larger still. The relationship between these sizes is that the 3-axis step has been divided by the binary logarithm of 3, written log₂3, which is approximately 1.585, while the 5-axis step has been divided by the binary logarithm of 5, written log₂5, and which is approximately 2.322. The 2-axis step can also be thought of as having been divided by the binary logarithm of its prime, but because log₂2 is exactly 1, and dividing by 1 does nothing, the scaling has no effect on the 2-axis.

Remember that near-just ETs have maps whose terms are in close proportion to log(2:3:5). ET maps use only integers, so they can only approximate this ideal, but a theoretical pure JI map would be {{val|log₂2 log₂3 log₂5}}. If we scaled this theoretical JI map by this scaling scheme, then, we’d get 1:1:1, because we’re just dividing things by themselves: log₂2/log₂2:log₂3/log₂3:log₂5/log₂5 = 1:1:1. This tells us that we should find this theoretical JI map at the point arrived at by moving exactly the same amount along the 2-axis, 3-axis, and 5-axis. Well, if we tried that, these three movements would cancel each other out: we’d draw an equilateral triangle and end up exactly where we started, at the origin, or in other words, at pure JI. Any other ET approximating but not exactly log(2:3:5) will be scaled to proportions not exactly 1:1:1, but approximately so, like maybe 1:0.999:1.002, and so you’ll move in something close to an equilateral triangle, but not exactly, and land in some interesting spot that’s not quite in the center. In other words, we scale the axes this way so that we can compare the maps not in absolute terms, but in terms of what direction and by how much they deviate from JI ''(see Figure 2g)''.

[[File:Scaling.png|600px|right|thumb|'''Figure 2g.''' a visualization of how scaling axes illuminates deviations from JI]]

Remember that 5-limit JI is 3D, but we’re viewing it on a 2D page. It’s not the case that its axes are flat on the page. They’re not literally occupying the same plane, 120° apart from each other. That’s just not how axes normally work, and it’s not how they work here either! The 5-axis is perpendicular to the 2-axis and 3-axis just like normal Cartesian space. Again, we’re looking only at the positive coordinates, which is to say that this is only the [https://en.wikipedia.org/wiki/Octant_(solid_geometry) +++ octant] of space, which comes to a point at the origin (0,0,0) like the corner of a cube. So you should think of this diagram as showing that cubic octant sticking its corner straight out of the page at us, like a triangular pyramid. So we’re like a tiny little bug, situated right at the tip of that corner, pointing straight down the octant’s interior diagonal, or in other words the line equidistant from three axes, the line which we understand represents theoretically pure JI. So we see that in the center of the page, represented as a red hexagram, and then toward the edges of the page is our peripheral vision. ''(See Figure 2h.)''

[[File:Understanding projection.png|600px|thumb|left|'''Figure 2h.''' Visualization of the projection process. (In real life, the cube is infinite in size. I made it smaller here to help make the shape clearer.)]]

Moreover, there’s a special relationship between the positions of n-ETs and 2n-ETs, and indeed between n-ETs and 3n-ETs, 4n-ETs, etc. To understand why, it’s instructive to plot it out ''(see Figure 2i)''.

[[File:Hiding vals.png|500px|thumb|right|'''Figure 2i.''' redundant maps hiding behind their simpler counterparts]]

[[File:Plot of 5 10 20 40 80.png|800px|thumb|left|'''Figure 2j.'''Plot of 40-ETs with 80-ETs halfway between each pair, including the contorted 40-ETs hiding behind 20-ET and 10-ET]]

I’ve circled every 40-ET visible in the chart ''(see Figure 2j)''. And you can see that halfway between each one, there’s an [[80edo|80-ET]] too. Well, sometimes it’s not actually printed on the diagram<ref>The reason is that Paul’s diagram, in addition to cutting off beyond 99-ET, also filters out maps that aren’t GPVs.</ref>, but it’s still there. You will also notice that if we also land right about on top of [[20edo|20-ET]] and [[10edo|10-ET]]. That’s no coincidence! Hiding behind that 20-ET is a redundant 40-ET whose terms are all 2x the 20-ET’s terms, and hiding behind the 10-ET is a redundant 40-ET whose terms are all 4x the 40-ET’s terms (and also a redundant 20-ET and a [[30edo|30-ET]], and [[50edo|50-ET]], [[60edo|60-ET]], etc. etc. etc.)

So if we take a look at a cross-section of projection again, but in terms of tuning space now ''(see Figure 2k)'', we can see how every point is about the same distance from us.

[[File:Tuning space version.png|300px|thumb|right|'''Figure 2k.''' Demonstration of projection in terms of ''tuning'' space (compare with Figure 2i).]]

[[File:Tuning projection.png|300px|thumb|left|'''Figure 2l.''' Demonstrating of tuning projection.]]

But perhaps even more interesting than this continuous tuning space that appears in PTS between points is the continuous tuning space that does not appear in PTS because it exists within each point, that is, exactly out from and deeper into the page at each point. In tuning space, as we’ve just established, there are no maps in front of or behind each other that get collapsed to a single point. But there are still many things that get collapsed to a single point like this, but in tuning space they are different tunings ''(see Figure 2l)''. For example, {{val|1200 1900 2800}} is the way we’d write 12-ET in tuning space. But there are other tunings represented by this same point in PTS, such as {{val|1200.12 1900.19 2800.28}} (note that in order to remain at the same point, we’ve maintained the exact proportions of all the prime tunings). That tuning might not be of particular interest. I just used it as a simple example to illustrate the point. A more useful example would be {{val|1198.440 1897.531 2796.361}}, which by some algorithm is the optimal tuning for 12-ET (minimizes damage across primes or intervals); it may not be as obvious from looking at that one, but if you check the proportions of those terms with each other, you will find they are still exactly 12:19:28.

[[File:Diagrams to understand PTS for RTT.png|thumb|left|400px|'''Figure 3a.''' How the tempered comma affects slope on PTS. A temperament defined by a comma with a 0 for a prime will be perpendicular to that prime's axis, because the tuning of that prime does not affect whether or not the comma is tempered out. Therefore the prime corresponding to the 0 in the comma is represented by x, which can be anything, while the proportion between the other two primes must remain fixed.]]

Those are the three simplest slopes to consider, i.e. the ones which are exactly parallel to the axes ''(see Figure 3a)''. But all the other temperament lines follow a similar principle. Their slopes are a manifestation of the prime factors in their defining comma. If having zero 5’s means you are perfectly horizontal, then having only one 5 means your slope will be close to horizontal, such as meantone {{monzo|-4 4 -1}} or [[helmholtz]] {{monzo|-15 8 1}}. Similarly, magic {{monzo|-10 -1 5}} and [[würschmidt]] {{monzo|17 1 -8}}, having only one 3 apiece, are close to parallel with the 3-axis, while porcupine {{monzo|1 -5 3}} and [[ripple]] {{monzo|-1 8 -5}}, having only one 2 apiece, are close to parallel with the 2-axis.

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 period to exactly 1200 cents, to establish a common basis for comparison. These are called [[Tour_of_Regular_Temperaments#Rank-2_temperaments|linear temperaments]]. 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 3b)''.

[[File:Generator sizes in PTS.png|800px|thumb|'''Figure 3b.''' Generator sizes of linear temperaments in PTS]]

[[File:Different nestings.png|400px|thumb|left|'''Figure 4a.''' How to write matrices in terms of either columns/vectors/commas or rows/covectors/maps.]]

We can extend our angle bracket notation (technically called [https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation bra-ket notation, or Dirac notation]) to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 4a)''. For example, we could have written our comma basis like this: {{val|{{monzo|-4 4 -1}} {{monzo|-10 -1 5}}}}. Starting from the outside, the {{val|}} tells us to think in terms of a row. It's just that this covector isn't a covector of numbers, like the ones we've gotten used to by now, but rather a covector of ''columns of'' numbers. So this row houses two such columns. Alternatively, we could have written this same matrix like {{monzo|{{val|-4 -10}} {{val|4 -1}} {{val|-1 5}}}}, but that would obscure the fact that it is the combination of two familiar commas (but that notation ''would'' be useful for expressing a matrix built out of multiple maps, as we will soon see).

So we could label the nodes with a list of approximations. For example, the node at {{monzo|-4 4}} would be ~5. We could label ~9/8 on {{monzo|-3 2}} just the same as we could label {{monzo|-3 2}} 9/8 in JI, however, here, we can also label that node ~10/9, because {{monzo|1 -2 1}} → 1×{{monzo|1 0}} + -2×{{monzo|0 1}} + 1×{{monzo|-4 4}} = {{monzo|1 0}} + {{monzo|0 -2}} + {{monzo|-4 4}} = {{monzo|-3 2}}. Cool, huh? Because conflating 9/8 and 10/9 is a quintessential example of the effect of tempering out the meantone comma ''(see Figure 4b)''.

[[File:Mapping to tempered vector.png|400px|thumb|right|'''Figure 4b.''' Converting from a JI interval vector to a tempered interval vector, with one less rank, conflating intervals related by the tempered out comma.]]

There’s a straightforward formula here: <span><math>d - n = r</math></span>, where <span><math>d</math></span> is dimensionality, <span><math>n</math></span> is nullity, and <span><math>r</math></span> is rank. We’ve seen every one of those words so far except '''nullity'''. [[Nullity]] simply means the count of commas tempered out, or in other words, the count of commas in a basis for the null-space ''(see Figure 4c)''.

[[File:Mapping and comma basis dnr.png|400px|thumb|right|'''Figure 4c.''' The relationship between dimensionality d, rank r, and nullity n]]

And these are no longer ''JI'' groups, of course, but you can even use irrationals, like the 2.π.5.7 group! The sky is the limit. Whatever you choose, though, this core structural rule <span><math>d - n = r</math></span> holds strong ''(see Figure 4d)''.

[[File:Temperaments by rnd.png|400px|thumb|left|'''Figure 4d.''' Some temperaments by dimension, rank, and nullity]]

[[File:Algebra notation.png|300px|thumb|right|'''Figure 5a.''' RTT bracket notation comparison.]]

[[File:Generating lattice (2) (2).png|thumb|left|400px|'''Figure 5b.''' Visualization of how primes 2 and 3 are capable of generating the entire tempered lattice for meantone, whether as generators 2/1 and 3/1, or 2/1 and 3/2]]

It’s easy to see why this is the case if you know how to visualize it on the tempered lattice. Let’s start with the happy case: primes 2 and 3. Prime 2 lets us move one step up (or down). Prime 3 lets us move one step right (or left). Clearly, with these two primes, we’d be able to reach any node in the lattice. We could do it with generators 2/1 and 3/1, in the most straightforward case. But we can also do it with 2/1 and 3/2: that just means one generator moves us down and to the right (or the opposite), and the other moves us straight up by one (or the opposite) ''(see Figure 5b)''. 2/1 and 4/3 works too: one moves us to the left and up two (or… you get the idea) and the other moves us straight up by one. Heck, even 3/2 and 4/3 work; try it yourself.

[[File:Generating lattice (2) (1).png|thumb|right|400px|'''Figure 5c.''' Visualization of how neither primes 2 and 5 or 3 and 5 are capable of generating the entire tempered lattice for meantone; they can only generate <span><math>\frac 14</math></span>th of it]]

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 5c)''.

[[File:RTT clean 3.png|800px|center|thumb|'''Figure 6a.''' Diagram of core RTT concepts.]]