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

If you’ve previously worked with JI, you may already be familiar with vectors. Vectors are a compact way to express JI intervals in terms of their prime factorization, or in other words, their harmonic building blocks. In JI, and in most contexts in RTT, vectors simply list the count of each prime, in order. For example, 16/15 is {{monzo|4 -1 -1}} because it has four 2’s in its numerator, one 3 in its denominator, and also one 5 in its denominator. You can look at each term as an exponent: 2⁴ × 3⁻¹ × 5⁻¹ = 16/15.

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 {{val|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 {{val|}} point left, vectors {{monzo|}} point right.

Covectors and vectors give us a way to bridge JI and EDOs. If the vector gives us a list of primes in a JI interval, and the covector tells us how many steps it takes to reach the approximation of each of those primes individually in an EDO, then when we put them together, we can see what step of the EDO should give the closest approximation of that JI interval. We say that the JI interval '''maps''' to that number of steps in the EDO. Calculating this looks like {{val|12 19 28}}{{monzo|4 -1 -1}}, and all that means is to multiply matching terms and sum the results (this is called the dot product).

For another example, can quickly find the fifth size for 12-EDO from its map, because 3/2 is {{monzo|-1 1 0}}, and so {{val|12 19 28}}{{monzo|-1 1 0}} = (12 × -1) + (19 × 1) = 7. Similarly, the major third — 5/4, or {{monzo|-2 0 1}} — is simply 28 - 12 - 12 = 4.

We can also see that the JI interval 81/80 maps to zero steps in 12-EDO, because {{val|12 19 28}}{{monzo|-4 4 -1}} = 0; we therefore say this JI interval '''vanishes''' in 12-EDO, or that it is '''[[Tempered out|tempered out]]'''. This type of JI interval is called a '''[[comma]]''', and this particular one is called the [[meantone comma]].

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 2b)''. One can make compelling music that [[Keenan's comma pump page|exploits such harmonic mechanisms]].

But we’re still only talking about JI and EDOs. If you’re familiar with meantone as a historical temperament, you may be aware already that it is neither JI nor an EDO. Well, we’ve got a ways to go yet before we get there.  

If you’ve seen one map before, it’s probably {{val|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.

Another glowing example is the map {{val|53 84 123}}, for which a good generator will give you g⁵³ ≈ 2.0002, g⁸⁴ ≈ 3.0005, g¹²³ ≈ 4.9974. This speaks to historical attention given to [[53edo|53-ET]]. So while 53:84:123 is an even better approximation of log(2:3:5) (and [https://en.xen.wiki/images/a/a2/Generalized_Patent_Vals.png you won’t find a better one] until 118:187:274), of course its integers aren’t as low, so that lessens its appeal.

And why is this cool? Well, if {{val|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 well-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 {{val|12 19 28}} approximates 2, 3, and 5 by 1.059¹² ≈ 1.998, 1.059¹⁹ ≈ 2.992, and 1.059²⁸ ≈ 5.029, respectively, then {{val|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 {{val|12 19 28}}, which checks out with our calculation we made in the previous section that the best approximation of 16/15 in {{val|12 19 28}} would be 1 step.

Well, you’ll notice that in the previous section, we did approximate the octave, using 1.998 instead of 2. But another thing {{val|12 19 28}} has going for it is that it excels at approximating 5-limit JI even if we constrain ourselves to pure octaves, locking g¹² to exactly 2: (¹²√2)¹⁹ ≈ 2.997 and (¹²√2)²⁸ ≈ 5.040. You can see that actually the approximation of 3 is even better here, marginally; it’s the damage to 5 which is lamentable.

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 2c)''.

Suppose we want to experiment with {{val|12 19 28}}’s map a bit. We’ll change one of the terms by 1, so now we have {{val|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.

If your goal is to evoke JI-like harmony, then, {{val|12 20 28}} is not your friend. Feel free to work out some other variations on {{val|12 19 28}} if you like, such as {{val|12 19 29}} maybe, but I guarantee you won’t find a better one that starts with 12 than {{val|12 19 28}}.

So the case is cut-and-dry for {{val|12 19 28}}, and therefore from now on I'm simply going to refer to this ET by "12-ET" rather than spelling out its map. But other ETs find themselves in trickier situations. Consider [[17edo|17-ET]]. One option we have is the map {{val|17 27 39}}, with a generator of about 1.0418, and prime approximations of 2.0045, 3.0177, 4.9302. But we have a second reasonable option here, too, where {{val|17 27 40}} gives us a generator of 1.0414, and prime approximations of 1.9929, 2.9898, and 5.0659. In either case, the approximations of 2 and 3 are close, but the approximation of 5 is way off. For {{val|17 27 39}}, it’s way small, while for {{val|17 27 40}} it’s way big. The conundrum could be described like this: any generator we could find that divides 2 into about 17 equal steps can do a good job dividing 3 into about 27 equal steps, too, but it will not do a good job of dividing 5 into equal steps; 5 is going to land, unfortunately, right about in the middle between the 39th and 40th steps, as far as possible from either of these two nearest approximations. To do a good job approximating prime 5, we’d really just want to subdivide each of these steps in half, or in other words, we’d want [[34edo|34-ET]].

Curiously, {{val|17 27 39}} is the map for which each prime individually is as closely approximated as possible when prime 2 is exact, so it is in a sense the naively best map for 17-ET, however, if that constraint is lifted, and we’re allowed to either temper prime 2 and/or choose the next-closest approximations for prime 5, the overall approximation can be improved; in other words, even though 39 steps can take you just a tiny bit closer to prime 5 than 40 steps can, the tiny amount by which it is closer is less than the improvements to the tuning of primes 2 and 3 you can get by using {{val|17 27 40}}. So again, the choice is not always cut-and-dry; there’s still a lot of personal preference going on in the tempering process.

So some musicians may conclude “17-ET is clearly not cut out for 5-limit music,” and move on to another ET. Other musicians may snicker maniacally, and choose one or the other map, and begin exploiting the profound and unusual 5-limit harmonic mechanisms it affords. {{val|17 27 40}}, like {{val|12 19 28}}, tempers out the meantone comma {{monzo|-4 4 -1}}, so even though fifths and major thirds are different sizes in these two ETs, the relationship that four fifths equals one major third is shared. {{val|17 27 39}}, on the other hand, does not work like that, but what it does do is temper out 25/24, {{monzo|-3 -1 2}}, or in other words, it equates one fifth with two major thirds.

If you’re enforcing pure octaves, the difference between {{val|17 27 39}} and {{val|17 27 40}} is nominal, or contextual. The steps in either case are identical: exactly ¹⁷√2, or 1200/17=70.588¢. You simply choose to think of 5 as being approximated by either 39 or 40 of those steps, or imply it in your composition. But when octaves are freed to temper, then the difference between these two maps becomes pronounced. When optimizing for {{val|17 27 39}}, the best step size is 70.225¢, but when optimizing for {{val|17 27 40}}, the best step size is more like 70.820¢.

If you’ve worked with 5-limit JI before, you’re probably aware that it is three-dimensional. You’ve probably reasoned about it as a 3D lattice, where one axis is for the factors of prime 2, one axis is for the factors of prime 3, and one axis is for the factors of prime 5. This way, you can use vectors, such as {{monzo|-4 4 -1}} or {{monzo|1 -2 1}}, just like coordinates.

PTS can be thought of as a projection of 5-limit JI map space, which similarly has one axis each for 2, 3, and 5. But it is no JI pitch lattice. In fact, in a sense, it is the opposite! This is because the coordinates in map space aren’t prime count lists, but maps, such as {{val|12 19 28}}. That particular map is seen here as the biggish, slightly tilted numeral 12 just to the left of the center point.

And the two 17-ETs we looked at can be found here too. {{val|17 27 40}} is the slightly smaller numeral 17 found on the line labeled “meantone” which the 12 is also on, thus representing the fact we mentioned earlier that they both temper it out. The other 17, {{val|17 27 39}}, is found on the other side of the center point, aligned horizontally with the first 17. So you could say that map space plots ETs, showing how they are related to each other.

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

Our example ET will be 40. We'll start out at the map {{val|40 63 93}}. This map is a default of sorts for 40-ET, because it’s the map where all three terms are as close as possible to JI when prime 2 is exact (sometimes unfortunately called a "[[patent val]]", which is related to the generalized patent val concept referenced earlier).

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 {{val|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.

Back from our starting point, let’s move by a single step again, but this time on the 3-axis, by adding 1 to the 3-term of our map, from 63 to 64, therefore moving to the map {{val|40 64 93}}. This map is found up and to the right. Again, this direction makes sense, because it’s the direction the 3-axis points.

Finally, let’s move by a single step on the 2-axis, from 40 to 41, moving to the map {{val|41 63 93}}, which unsurprisingly is in the direction the 2-axis points. This move actually takes us off the chart, way down here.

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 3g)''.

But if instead we picked some random alternate mapping of 12-ET, like {{val|12 23 25}}, looking at those integer terms directly, it may not be obvious how close to JI this map is. However, upon scaling them:

For simplicity, we’re looking at the octant cube here from the angle straight on to the 2-axis, so changes to the 2-terms don’t matter here. At the top is the origin; that’s the point at the center of PTS. Close-by, we can see the map {{val|3 5 7}}, and two closely related maps {{val|3 4 7}} and {{val|3 5 8}}. Colored lines have been drawn from the origin through these points to the black line in the top-right, which represents the page; this is portraying how if our eye is at that origin, where on the page these points would appear to be.

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 {{val|6 10 14}} first. Notice that each of its terms is exactly 2x the corresponding term in {{val|3 5 7}}. In effect, {{val|6 10 14}} is redundant with {{val|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 {{val|3 5 7}} instead and then simply divided by 2 one time at the end. It behaves in the exact same way as {{val|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 {{val|6 10 14}}. Which is great, because it’s hidden exactly behind {{val|3 5 7}} from where we’re looking.

The same is true of the map pair {{val|3 4 7}} and {{val|6 8 14}}, as well as of {{val|3 5 8}} and {{val|6 10 16}}. Any map whose terms have a common divisor other than 1 is going to be redundant in this sense, and therefore hidden. You can imagine that even further past {{val|3 5 7}} you’ll find {{val|9 15 21}}, {{val|12 20 28}}, and so on, and these are are called contorted maps<ref>On some versions of PTS which Paul prepared, these contorted ETs are actually printed on the page.</ref>. More on those later. What’s important to realize here is that Paul found a way to collapse 3 dimensions worth of information down to 2 dimensions without losing anything important. Each of these lines connecting redundant ETs have been [https://en.wikipedia.org/wiki/Projection_(mathematics) projected] onto the page as a single point. That’s why the diagram is called "projective" tuning space.

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 {{val|6 11 14}}, which is halfway between {{val|6 10 14}} and {{val|6 12 14}}. Notice that the purple line that runs through it lands halfway between the red and blue lines on the page. Similarly, {{val|6 10 15}} is halfway between {{val|6 10 14}} and {{val|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.

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 {{val|6 10 14}} in map space, but {{val|1200 2000 2800}} in tuning space.

The first key difference to notice is that we can normalize coordinates in tuning space, so that the first term of every coordinate is the same, namely, one octave, or 1200 cents. For example, note that while in map space, {{val|3 5 7}} is located physically in front of {{val|6 10 14}}, in tuning space, these two points collapse to literally the same point, {{val|1200 2000 2800}}. This can be helpful in a similar way to how the scaled axes of PTS help us visually compare maps’ proximity to the central JI spoke: they are now expressed closer to in terms of their deviation from JI, so we can more immediately compare maps to each other, as well as individually directly to the pure JI primes, as long as we memorize the cents values of those (they’re 1200, 1901.955, and 2786.314). For example, in map space, it may not be immediately obvious that {{val|6 9 14}} is halfway between {{val|3 5 7}} and {{val|3 4 7}}, but in tuning space it is immediately obvious that {{val|1200 1800 2800}} is halfway between {{val|1200 2000 2800}} and {{val|1200 1600 2800}}.

The other major difference is that tuning space is continuous, where map space is discrete. In other words, to find a map between {{val|6 10 14}} and {{val|6 9 14}}, you’re subdividing it by 2 or 3 and picking a point in between, that sort of thing. But between {{val|1200 2000 2800}} and {{val|1200 1800 2800}} you’ve got an infinitude of choices smoothly transitioning between each other; you’ve basically got knobs you can turn on the proportions of the tuning of 2, 3, and 5. Everything from from {{val|1200 1999.999 2800}} to {{val|1200 1901.955 2800}} to {{val|1200 1817.643 2800}} is along the way.

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 3l)''. 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.

So: we now know which point is {{val|12 19 28}}, and we know a couple of 17’s, 40’s and a 41. But can we answer in the general case? Given an arbitrary map, like {{val|7 11 16}}, can we find it on the diagram? Well, you may look to the first term, 7, which tells you it’s [[7edo|7-ET]]. There’s only one big 7 on this diagram, so it’s probably that. (You’re right). But that one’s easy. The 7 is huge.

What if I gave you {{val|43 68 100}}. Where’s [[43edo|43-ET]]? I’ll bet you’re still complaining: the map expresses the tempering of 2, 3, and 5 in terms of their shared generator, but doesn’t tell us directly which primes are sharp, and which primes are flat, so how could we know in which region to look for this ET?

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

Simply put, lines on PTS are '''temperaments'''. Specifically, they are [[abstract regular temperaments]]. If you are a student of historical temperaments, you may be familiar with e.g. [[quarter-comma meantone]]; to an RTT practitioner, this is actually a specific tuning of the meantone temperament. Meantone is an abstract temperament, which encompasses a range of other possible temperaments and tunings.  

Let’s grab some samples to confirm. We already know that 12-ET here looks like {{val|12 19}} (I’m dropping the 5 term for now). The 24-ET here looks like {{val|24 38}}, which is simply 2×{{val|12 19}}. The 36-ET here looks like {{val|36 57}} = 3×{{val|12 19}}. And so on. So that’s why we only see multiples of 12 along this line: because 12 and 19 are co-prime, so the only other maps which could have them in the same ratio would be multiples of them.

Let’s look at the other perfectly horizontal line on this diagram. It’s found about a quarter of the way down the diagram, and runs through the 10-ET and 20-ET we looked at earlier. This one’s called “[[blackwood]]”. Here, we can see that all of its ETs are all multiples of 5. In fact, [[5edo|5-ET]] itself is on this line, though we can only see a sliver of its giant numeral off the left edge of the diagram. Again, all of its maps have locked ratios between their mappings of prime 2 and prime 3: {{val|5 8}}, {{val|10 16}}, {{val|15 24}}, {{val|20 32}}, {{val|40 64}}, {{val|80 128}}, etc. You get the idea.

So what do these two temperaments have in common such that their lines are parallel? Well, they’re defined by commas, so why don’t we compare their commas. The compton comma is {{monzo|-19 12 0}}, and the blackwood comma is {{monzo|8 -5 0}}<ref>Yes, these are the same as the [[Pythagorean comma]] and [[Pythagorean diatonic semitone]], respectively.</ref>. What sticks out about these two commas is that they both have a 5-term of 0. This means that when we ask the question “how many steps does this comma map to in a given ET”, the ET’s mapping of 5 is irrelevant. Whether we check it in {{val|40 63 93}} or {{val|40 63 94}}, the result is going to be the same. So if {{val|40 63 93}} tempers out the blackwood comma, then {{val|40 63 94}} also tempers out the blackwood comma. And if {{val|24 38 56}} tempers out compton, then {{val|24 38 55}} tempers out compton. And so on.

Similar temperaments can be found which include only 2 of the 3 primes at once. Take “[[augmented]]”, for instance, running from bottom-left to top-right. This temperament is aligned with the 3-axis. This tells us several equivalent things: that relative changes to the mapping of 3 are irrelevant for augmented temperament, that the augmented comma has no 3’s in its prime factorization, and the ratios of the mappings of 2 and 5 are the same for any ET along this line. Indeed we find that the augmented comma is {{monzo|7 0 -3}}, or [[128/125]], which has no 3’s. And if we sample a few maps along this line, we find {{val|12 19 28}}, {{val|9 14 21}}, {{val|15 24 35}}, {{val|21 33 48}}, {{val|27 43 63}}, etc., for which there is no pattern to the 3-term, but the 2- and 5-terms for each are in a 3:7 ratio.

There are even temperaments whose comma includes only 3’s and 5’s, such as “[[bug]]” temperament, which tempers out [[27/25]], or {{monzo|0 3 -2}}. If you look on this PTS diagram, however, you won’t find bug. Paul chose not to draw it. There are infinite temperaments possible here, so he had to set a threshold somewhere on which temperaments to show, and bug just didn’t make the cut in terms of how much it distorts harmony from JI. If he had drawn it, it would have been way out on the left edge of the diagram, completely outside the concentric hexagons. It would run parallel to the 2-axis, or from top-left to bottom-right, and it would connect the 5-ET (the huge numeral which is cut off the left edge of the diagram so that we can only see a sliver of it) to the [[9edo|9-ET]] in the bottom left, running through the 19-ET and [[14edo|14-ET]] in-between. Indeed, these ET maps — {{val|9 14 21}}, {{val|5 8 12}}, {{val|19 30 45}}, and {{val|14 22 33}} — lock the ratio between their 3-terms and 5-terms, in this case to 2:3.

Those are the three simplest slopes to consider, i.e. the ones which are exactly parallel to the axes ''(see Figure 4a)''. 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.

Well, notice that meantone is not the only temperament which passes through 12-ET. Consider augmented temperament. Its generator at 12-ET is 400¢. What's key here is that all three of these generators — 100¢, 500¢, and 400¢ — are multiples of 100¢.  

19-ET. Its map is {{val|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 first critical step is to recall that temperaments are defined by commas, which can be expressed as vectors. So, we can represent meantone using the meantone comma, {{monzo|-4 4 -1}}, and magic using the magic comma {{monzo|-10 -1 5}}.

Now how in the world could that matrix represent the same temperament as {{val|19 30 44}}? Well, they’re two different ways of describing it. {{val|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.  

This was a bit tricky for me to get my head around, so let me hammer this point home: when you say "the null-space", you're referring to ''the entire infinite set of all commas that a mapping tempers out'', ''not only'' the two commas you see in any given basis for it. Think of the comma basis as one of many valid sets of instructions to find every possible comma, by adding or subtracting these two commas from each other<ref>To be clear, because what you are adding and subtracting in interval vectors are exponents (as you know), the commas are actually being multiplied by each other; e.g. {{monzo|-4 4 -1}} + {{monzo|10 1 -5}} = {{monzo|6 5 -6}}, which is the same thing as <span><math>\frac{81}{80} × \frac{3072}{3125} = \frac{15552}{15625}</math></span></ref>. The math term for adding and subtracting vectors like this, which you will certainly see plenty of as you explore RTT, is "linear combination". It should be visually clear from the PTS diagram that this 19-ET comma basis couldn't be listing every single comma 19-ET tempers out, because we can see there are at least four temperament lines that pass through it (there are actually infinity of them!). But so it turns out that picking two commas is perfectly enough; every other comma that 19-ET tempers out could be expressed in terms of these two!

Try one. How about the hanson comma, {{monzo|6 5 -6}}. Well that one’s too easy! Clearly if you go down by one magic comma to {{monzo|10 1 -5}} and then up by one meantone comma you get one hanson comma. What you’re doing when you’re adding and subtracting multiples of commas from each other like this is technically called “[https://en.wikipedia.org/wiki/Gaussian_elimination Gaussian elimination]”. Feel free to work through any other examples yourself.

When written with the {{val|}} notation, we’re expressing maps in “covector” form, or in other words, as the opposite of vectors. But we can also think of maps in terms of matrices. If vectors are like matrix columns, maps are like matrix rows. So while we have to write {{monzo|-4 4 -1}} vertically when in matrix form, {{val|19 30 44}} stays horizontal.

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]<ref>Dirac notation comes to RTT from quantum mechanics, not algebra.</ref>) to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 5a)''. 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).

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 a covector, like this: {{val|{{monzo|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside a vector, like this: {{monzo|{{val|19 30 44}}}}.

You will regularly see matrices across the wiki that use only square brackets on the outside, e.g. [{{val|5 8 12}} {{val|7 11 16}}] or [{{monzo|-4 4 -1}} {{monzo|-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.

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 {{monzo|{{val|19 30 44}}}}.

We already have the tools to check that each of these commas’ vectors is tempered out individually by the map {{val|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 basis matrices which represent the intersections of two commas are greater than the sum of their individual parts.  

We can confirm the relationship between an ET and its null-space by converting back and forth between them. There exists a mathematical function which — when input any one of these comma basis matrices — will output {{monzo|{{val|19 30 44}}}}, thus demonstrating the various bases' equivalence with respect to it. If the operation called "taking the null-space" is what gets you from {{monzo|{{val|19 30 44}}}} to one basis for the null-space, then ''this'' mathematical function is in effect ''undoing'' the null-space operation.  

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.  

And ta-da! You’ve found the mapping for which the comma basis we started with is a basis for the null-space, and it is {{monzo|{{val|19 30 44}}}}. Feel free to try this with any other combination of two commas tempered out by this map.

Now the null-space function, to take you from {{monzo|{{val|19 30 44}}}} back to the matrix, is pretty much the same thing, but a bit simpler. No need to transpose or reverse. Just start at the augmentation step:

So that’s not any of the commas we’ve looked at so far (it’s the [[19-comma]] and the [[acute limma]]). But it is clear to see that either of them would be tempered out by 19-ET (no need to map by hand — just look at these commas side-by-side with the map {{monzo|{{val|19 30 44}}}} and it should be apparent).

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

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 {{val|5 8 12}} and {{val|7 11 16}}, we get a matrix that looks like

This matrix represents meantone. In our angle bracket notation, we would write it as two covectors inside a vector (one column of two rows), like this: {{monzo|{{val|5 8 12}} {{val|7 11 16}}}}.

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, {{monzo|-4 4 -1}}. How are these two representations related?

Well, it’s actually quite simple! They’re related in the same way as {{monzo|{{val|19 30 44}}}} was related to {{val|{{monzo|-4 4 -1}} {{monzo|-10 -1 5}}}}: by the null-space operation. Specifically, {{val|{{monzo|-4 4 -1}}}} is a basis for the null-space of the mapping {{monzo|{{val|5 8 12}} {{val|7 11 16}}}}, because it is the minimal representation of all the commas tempered out by meantone temperament.

And there’s our {{val|{{monzo|4 -4 1}}}}. Feel free to try reversing the operation by working out the mapping from this if you like. And/or you could try working out that {{val|{{monzo|4 -4 1}}}} is a basis for the null-space of any other combination of ETs we found that could specify meantone, such as 7&12, or 12&19.

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 Gaussian addition and subtraction, just like we could manipulate commas to get from one to the other. For example, the map for 12-ET {{val|12 19 28}} is exactly what you get from summing the terms of 5-ET {{val|5 8 12}} and 7-ET {{val|7 11 16}}: {{val|5+7 8+11 12+16}} = {{val|12 19 28}}. Cool!

* {{monzo|{{val|19 30 44}}}} is the mapping for a rank-1 temperament.

* {{monzo|{{val|5 8 12}} {{val|7 11 16}}}} is the mapping for a rank-2 temperament.

* {{val|19 30 44}} asks us to imagine a generator g for which g¹⁹ ≈ 2, g³⁰ ≈ 3, and g⁴⁴ ≈ 5.

First we should show how to actually use rank-2 mappings. It’s actually not that complicated. It’s just like using a rank-1 mapping, except you have to find each of them separately, and then put them back together at the end. Let’s see how this plays out for 10/9, or {{monzo|1 -2 1}}.  

'''{{val|5 8 12}}:'''

* {{val|5 8 12}}{{monzo|1 -2 1}}

'''{{val|7 11 16}}:'''

* {{val|7 11 16}}{{monzo|1 -2 1}}

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 {{monzo|1 1}}.

If the familiar usage of vectors has been as prime count lists, we can now generalize that definition to things like this {{monzo|1 1}}: generator count lists. Since interval vectors are often called monzos, you’ll often see these called tempered monzos or [[Tmonzos_and_Tvals|tmonzos]] for short. There’s very little difference. We can use these vectors as coordinates in a lattice just the same as before. The main difference is that the nodes we visit on this lattice aren’t pure JI; they’re a tempered lattice.

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

So this is still meantone! But now it’s a bit more practical to think about. Because notice what happens to the octave, {{monzo|1 0 0}}. To approximate the octave, you simply move by one of the first generator, or {{monzo|1 0}}. The second generator has nothing to do with it. And how about the fifth, {{monzo|-1 1 0}}? Well, the first generator maps that to 0 steps, and the second generator maps that to 1 step, or {{monzo|0 1}}. So that tells us our second generator is the fifth. Which is… almost perfect! I would have preferred a fourth, which is the octave-complement of the fifth which is less than half of an octave. But it’s basically the same thing. Good enough.

* Both {{monzo|{{val|5 8 12}} {{val|7 11 16}}}} and {{monzo|{{val|1 1 0}} {{val|0 1 4}}}} are equivalent mappings, then. Converting between them we could call a change of basis. This makes more sense, of course, when speaking about converting between equivalent bases; I’ve been cautioned against referring to maps as “bases” despite the label seeming appropriate from an analogy standpoint.

* Note well: this is not to say that {{val|1 1 0}} or {{val|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. {{monzo|1 0 0}} maps to {{monzo|1 0}} — referring to {{val|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. {{monzo|-1 1 0}} maps to {{monzo|0 1}} — referring to {{val|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 {{val|1 1 0}} or {{val|0 1 4}} individually, we cannot say what intervals the generators are. What if the mapping was {{monzo|{{val|0 1 4}} {{val|1 2 4}}} instead? We'd still have the first generator mapping as {{val|1 1 0}}, but now that the second generator mapping is {{val|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 {{val|12 19 28}} was simply {{val|5 8 12}} + {{val|7 11 16}}? Well, if {{monzo|{{val|5 8 12}} {{val|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.

* Technically speaking, when we first learned how to map vectors with ETs, we could think of those outputs as vectors too, but they'd be 1-dimensional vectors, i.e. if 12-ET maps 16/15 to 1 step, we could write that as {{val|12 19 28}}{{monzo|4 -1 -1}} = {{monzo|1}}, where writing the answer as {{monzo|1}} expresses that 1 step as 1 of the only generator in this equal temperament.

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 {{monzo|1}}, {{monzo|0 1}}, and {{monzo|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.

The next step is to understand our primes in terms of this temperament’s generators. Meantone’s mapping is {{monzo|{{val|1 0 -4}} {{val|0 1 4}}}}. This maps prime 2 to one of the first generators and zero of the second generators. This can be seen plainly by slicing the first column from the matrix; we could even write it as the vector {{monzo|1 0}}. Similarly, this mapping maps prime 3 to zero of the first generator and one of the second generator, or in vector form {{monzo|0 1}}. Finally, this mapping maps prime 5 to negative four of the first generator and four of the second generator, or {{monzo|-4 4}}.

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 5b)''.

We don’t have a ton of options here! The PTS diagram for 3-limit JI could be a simple line. This axis would define the relative tuning of primes 2 and 3, which are the only harmonic building blocks available. Along this line we’ll find some points, which familiarly are ETs. For example, we find 12-ET. Its map here is {{val|12 19}}; no need to mention the 5-term because we have no vectors that will use it here. At this ET, being a rank-1 temperament, <span><math>r</math></span> = 1. So if <span><math>d</math></span> = 2, then solve for <span><math>n</math></span> and we find that it only tempers out a single comma (unlike the rank-1 temperaments in 5-limit JI, which tempered out two commas). We can use our familiar null-space function to find what this comma is:

Unsurprisingly, the comma is {{monzo|-19 12}}, the compton comma. Basically, any comma we could temper out in 3-limit JI is going to be obvious from the ET’s map. Another option would be the blackwood comma, {{monzo|-8 5}} tempered out in 5-ET, {{val|5 8}}. Exciting stuff! Okay, not really. But good to ground yourself with.

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.  

You could even choose a JI group with combinations of primes, such as the 2.5/3.7 group. Here, we still care about approximating primes 2, 3, 5, and 7, however there's something special about 3 and 5: we don't specifically care about approximating 3 or 5 individually, but only about approximating their combination. Note that this is different yet from the 2.15.7 group, where the combinations of 3 and 5 we care about approximating are when they're on the same side of the fraction bar.  

Alright, here’s where things start to get pretty fun. 7-limit JI is 4D. We can no longer refer to our 5-limit PTS diagram for help. Maps and vectors here will have four terms; the new fourth term being for prime 7. So the map for 12-ET here is {{val|12 19 28 34}}.

On the other side of duality, septimal meantone’s mapping 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 {{val|12 19 28 34}} and {{val|19 30 44 53}} would work. But so would {{val|19 30 44 53}} and {{val|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.

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

If you take the HNF of {{monzo|{{val|5 8 12}} {{val|7 11 16}}}}, that’s what you get. It’s also what you get if you take the HNF of {{monzo|{{val|12 19 28}} {{val|19 30 44}}}}, etc. That’s the power of normalization.

* They both use left angle brackets on the left and square brackets on the right, {{val|}}, to enclose their contents.

The main difference between the two is superficial, and has to do with the “multi” part of the name. A plain covector comes in only one type, but a multicovector can be a bicovector, tricovector, tetracovector, etc. Yes, a multicovector can even be a monocovector. Depending on its numeric prefix, a multicovector will be written with a different count of brackets on either side. For example, a bicovector uses two of each: {{multival|}}. A tricovector uses three of each: {{multival|rank=3|}}. A monocovector, written with one of each, like {{val|}}, is indistinguishable from a plain covector, and that’s okay.

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'''.  

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: {{multival|1 4 4}}. As you can see, it is a bicovector, or bimap, because it has two of each bracket.

# 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>.  

# Convert each of those combinations to a square <span><math>r×r</math></span> matrix by slicing a column for each prime out of the mapping and putting them together.  

# Take each matrix's determinant.  

These have no GCD, or in other words, their GCD is 1. So just set these inside a number brackets equal to the rank, and we’ve got {{multival|1 4 4}}.

This method even works on an equal temperament, e.g. {{val|12 19 28}}. The rank is 1 so each combination of primes has only a single prime: they’re <span><math>(2)</math></span>, <span><math>(3)</math></span>, and <span><math>(5)</math></span>. The square matrices are therefore <span><math>\begin{bmatrix}12\end{bmatrix} \begin{bmatrix}19\end{bmatrix} \begin{bmatrix}28\end{bmatrix}</math></span>. The determinant of a 1×1 matrix is defined as the value of its single term, so now we have 12 19 28. No GCD, and <span><math>r</math></span> = 1, so we set the answer inside one layer of brackets, so our monocovector is {{val|12 19 28}}. Yes, it looks the same as what we started with, which is fine.

In natural language, that’s each element of the first row times the determinant of the square matrix from the other two columns and the other two rows, summed but with an alternating pattern of negation beginning with positive. If you ever need to do determinants of matrices bigger than 3×3, see [https://www.mathsisfun.com/algebra/matrix-determinant.html this webpage]. Or, you can just use an online calculator.  

And so our results are <span><math>-2</math></span>, <span><math>3</math></span>, <span><math>1</math></span>, <span><math>-11</math></span>. There's no GCD to extract. We prefer for the first term to be positive; this doesn’t make a difference in how things behave, but is done because it normalizes things (we could have found the result where the first term came out positive by simply changing the order of the rows of our mapping, which doesn’t affect how the mapping works at all). And so we flip the signs<ref>If it helps you, you could think of this sign-flipping step as paired with the GCD extraction step, if you think of it like extracting a GCD of -1.</ref>, and our list ends up as <span><math>2</math></span>, <span><math>-3</math></span>, <span><math>-1</math></span>, <span><math>11</math></span>. Finally, set these inside triply-nested brackets, because it’s a trimap for a rank-3 temperament, and we get {{multival|rank=3|2 -3 -1 11}}.

You may have noticed that the multimap for meantone, {{multival|1 4 4}}, looks really similar to the meantone comma, {{monzo|-4 4 -1}}. This is not a coincidence.

Here’s the comma basis for meantone: {{val|{{monzo|-4 4 -1}}}}. Calculating the multicomma is almost the same as calculating the multimap. The only difference is that as a preliminary step you must transpose the matrix, or in other words, exchange rows and columns. In our bracket notation, that just looks like replacing {{val|{{monzo|-4 4 -1}}}} with {{monzo|{{val|-4 4 -1}}}}. Now we can see that this is just like our ET map example from the previous section: basically an identity operation, breaking the thing up into three 1×1 matrices <span><math>\begin{bmatrix}-4\end{bmatrix} \begin{bmatrix}4\end{bmatrix} \begin{bmatrix}-1\end{bmatrix}</math></span> which are their own determinants and then nesting back inside one layer of brackets because nullity is 1. So we have {{monzo|-4 4 -1}}. Except, be careful! We can’t skip the step where we extract the GCD, which in this case is -1 again, so the multicomma (a monocomma) is actually {{monzo|4 -4 1}}. By the way, we write this operation with a different symbol; it’s upside down from the exterior product, and so is called the interior product, and uses ∨, which is great because that’s also the logical operator for “or” which matches with the intersection of commas using the “|” operator which also means “or”.

# Set the result in the proper count of brackets: {{monzo|4 -4 1}}.

Ta-da! Both operations get us to the same result: {{monzo|4 -4 1}}.

The first thing to understand is that each term of the multimap pertains to a different combination of primes. We already know this: it’s how we calculated it from the mapping matrix. For example, in the multimap for meantone, {{multival|1 4 4}}, the 1 is for <span><math>(2,3)</math></span>, the first 4 is for <span><math>(2,5)</math></span>, and the second 4 is for <span><math>(3,5)</math></span>.

One day you might come across a multimap which has a term equal to zero. If you tried to interpret this term using the information here so far, you'd think it must generate <span><math>\frac 10</math></span>th of the tempered lattice. That's not easy to visualize or reason about. Does that mean it generates essentially infinity lattices? No, not really. More like the opposite. The question itself is somewhat undefined here. If anything, it's more like that combination of primes generates approximately ''none'' of the lattice. Because in this situation, the combination of primes whose multimap term is zero generates so little of the tempered lattice that it's completely missing an entire dimension of it, so it's an infinitesimal amount of it that it generates. For example, the 11-limit temperament 7&12&31 has multimap {{val|rank=3|0 1 1 4 4 -8 4 4 -12 -16}} and mapping {{monzo|{{val|1 0 -4 0 -12}} {{val|0 1 4 0 8}} {{val|0 0 0 1 1}}}}; we can see from this how primes <span><math>(2,3,5)</math></span> can only generate a rank-2 cross-section of the full rank-3 lattice, because while 2 and 3 do the trick of generating that rank-2 part (exactly as they do in 5-limit meantone), prime 5 doesn't bring anything to the table here so that's all we get.

=== finding approximate JI generators ===  

plus many many more. And of course I owe a big debt to [[Gene Ward Smith]].