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

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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.

~~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).~~ 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) (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]]

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.

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]]

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)''.

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.

=== big RTT examples chart ===

@@ Line 1,329: / Line 1,329: @@
 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.
-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). 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) (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]]
-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.
+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]]
+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)''.
 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.
 === big RTT examples chart ===