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Note that multicommas could just as well serve the purpose of a unique identifier for temperaments. There’s no particular reason to prefer the multimap to the multicomma. By convention, however, the multimap is the one that’s used.
Note that multicommas could just as well serve the purpose of a unique identifier for temperaments. There’s no particular reason to prefer the multimap to the multicomma. By convention, however, the multimap is the one that’s used.


And so here’s a table of terminology we’ve come across so far:
=== tempered lattice fractions generated by prime combinations ===
 
So we now understand how to get to multimaps. And we understand that they uniquely identify the temperament. But what about the individual terms — do they mean anything in and of themselves? It turns out: yes!
 
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>.
 
Now, let’s convert every term of the multimap 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 <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 6b.''' 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 6b)''. 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 6c.''' 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 6c)''.
 
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.
 
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.
 
== other topics (TBD) ==
 
=== summary diagrams and tables ===
 
[[File:RTT clean 3.png|800px|center|thumb|'''Figure 7a.''' Diagram of core RTT concepts.]]


{| class="wikitable"
{| class="wikitable"
|+ '''Table 5c.''' RTT terminology
|+ '''Table 6a.''' RTT terminology
!terminology category
!terminology category
!building block →
!building block →
Line 1,439: Line 1,465:
|}
|}


The cell in the "RTT structure" row and "multimap" column may seem a bit out of left field. But that's what we're going to talk about next.
=== tempered lattice fractions generated by prime combinations ===
So we now understand how to get to multimaps. And we understand that they uniquely identify the temperament. But what about the individual terms — do they mean anything in and of themselves? It turns out: yes!
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>.
Now, let’s convert every term of the multimap 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 <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 6b.''' 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 6b)''. 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 6c.''' 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 6c)''.
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.
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.
== other topics (TBD) ==
=== big RTT examples chart ===
[[File:RTT clean 3.png|800px|center|thumb|'''Figure 7a.''' Diagram of core RTT concepts.]]


=== tuning ===
=== tuning ===
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