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

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The first thing to understand is that each term of the multicovector 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 multicovector for meantone, {{multival|1 4 4}}, the 1 is for <math>(2,3)</math>, the first 4 is for <math>(2,5)</math>, and the second 4 is for <math>(3,5)</math>.

Now, let’s convert every term of the multicovector by taking its absolute value and its inverse. In this case, ~~all~~ of our terms ~~are~~ already positive, so that has no effect. But taking the inverse converts us to 1/1, 1/4, 1/4. These values tell us what fraction of the tempered lattice we can generate using the corresponding combination of primes.

Now, let’s convert every term of the multicovector by taking its absolute value and its inverse. In this case, each of our terms is already positive, so that has no effect. But taking the inverse converts us to <math>\frac 11</math>, <math>\frac 14</math>, <math>\frac 14</math>. These values tell us what fraction of the tempered lattice we can generate using the corresponding combination of primes.

What does that mean? Who cares? The motivation here is that it’s a good thing to be able to generate the entire lattice. We may be looking for JI intervals we could use as generators for our temperament, and if so, we need to know what primes to build them out of so that we can make full use of the temperament. So this tells us that if we try to build generators out of primes 2 and 3, we will succeed in generating 1/1, or all, of the tempered lattice. Whereas if we try to build the generators out of primes 2 and 5, or 3 and 5, we will fail; we will only be able to generate 1/4 of the lattice. In other words, prime 5 is the bad seed here; it messes things up.

What does that mean? Who cares? The motivation here is that it’s a good thing to be able to generate the entire lattice. We may be looking for JI intervals we could use as generators for our temperament, and if so, we need to know what primes to build them out of so that we can make full use of the temperament. So this tells us that if we try to build generators out of primes 2 and 3, we will succeed in generating <math>\frac 11</math> or in other words all of the tempered lattice. Whereas if we try to build the generators out of primes 2 and 5, or 3 and 5, we will fail; we will only be able to generate <math>\frac 14</math> of the lattice. In other words, prime 5 is the bad seed here; it messes things up.

[[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]]

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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 1/~~4th~~ of it]]

[[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 <math>\frac 14</math>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)''.

One day you might come across a multicovector 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 <math>\frac 10</math>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 multicovector 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 multicovector {{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 <math>(2,3,5)</math> 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.

We’ll look in more detail later at how exactly to best find these generators, once you know which primes to make them out of.

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 The first thing to understand is that each term of the multicovector 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 multicovector 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>.
-Now, let’s convert every term of the multicovector by taking its absolute value and its inverse. In this case, all of our terms are already positive, so that has no effect. But taking the inverse converts us to 1/1, 1/4, 1/4. These values tell us what fraction of the tempered lattice we can generate using the corresponding combination of primes.
+Now, let’s convert every term of the multicovector by taking its absolute value and its inverse. In this case, each of our terms is already positive, so that has no effect. But taking the inverse converts us to <span><math>\frac 11</math></span>, <span><math>\frac 14</math></span>, <span><math>\frac 14</math></span>. These values tell us what fraction of the tempered lattice we can generate using the corresponding combination of primes.
-What does that mean? Who cares? The motivation here is that it’s a good thing to be able to generate the entire lattice. We may be looking for JI intervals we could use as generators for our temperament, and if so, we need to know what primes to build them out of so that we can make full use of the temperament. So this tells us that if we try to build generators out of primes 2 and 3, we will succeed in generating 1/1, or all, of the tempered lattice. Whereas if we try to build the generators out of primes 2 and 5, or 3 and 5, we will fail; we will only be able to generate 1/4 of the lattice. In other words, prime 5 is the bad seed here; it messes things up.
+What does that mean? Who cares? The motivation here is that it’s a good thing to be able to generate the entire lattice. We may be looking for JI intervals we could use as generators for our temperament, and if so, we need to know what primes to build them out of so that we can make full use of the temperament. So this tells us that if we try to build generators out of primes 2 and 3, we will succeed in generating <span><math>\frac 11</math></span> or in other words all of the tempered lattice. Whereas if we try to build the generators out of primes 2 and 5, or 3 and 5, we will fail; we will only be able to generate <span><math>\frac 14</math></span> of the lattice. In other words, prime 5 is the bad seed here; it messes things up.
 [[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]]
@@ Line 1,441: / Line 1,441: @@
 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 1/4th of it]]
+[[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)''.
+One day you might come across a multicovector 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 multicovector 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 multicovector {{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.
 We’ll look in more detail later at how exactly to best find these generators, once you know which primes to make them out of.