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

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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}}.

And how about something silly like {{monzo|1 0 0}}, the octave? Well, that maps to {{monzo|5 7}}. If the first generator was the exact same size as 5-ET’s generator, and the second generator was the exact same size as 7-ET’s generator, then 5 of the first and 7 of the second wouldn’t take us to the octave, it would take us to the double octave, 4/1. So maybe the first generator is the size of half a step of 5-ET, maybe? And the second generator is the size as half a step of 7-ET, maybe? I guess that would work out, since 120¢ + 85.714¢ = 205.714¢ and 10/9 is 182.404¢; I mean, it’s not super close, but it’s in the ballpark at least.

Previously we mentioned that any given rank-2 temperaments can be generated by a wide variety of combinations of intervals. In other words, the absolute size of the intervals is not the important part, in terms of their potential for generating the temperament; only their relative size matters. However, for us humans, it’s much easier to make sense of these things if we get them in a good old standard form, by locking one generator to the octave to establish a common basis for comparison, and the other generator to a size less than half of the octave (because anything past the halfway point and it would be the octave-complement of a smaller and therefore in some sense simpler interval). And there’s a way to find this form by transforming our matrix. In fact, it also uses Gaussian elimination, though in this case, we do it by columns. Our target this time is a bit harder to explain ahead of time, so this first time through, just watch, and we’ll review the result.

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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.

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. 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.

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 <math>p,q</math> we can make the two generators respectively <math>\frac{p}{5p+7q}</math> and <math>\frac{q}{5p+7q}</math> of an octave, e.g. <math>(p,q)=(1,0)</math> for 5-ET, <math>(0,1)</math> for 7-ET, <math>(1,1)</math> 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.

We’ve made it to a critical point here: we are now able to explain why RTT is called “regular” temperament theory. Regular here is a mathematical term, and I don’t have a straightforward definition of it for you, but it apparently refers to the fact that all intervals in the tuning are combinations of only these specified generators. So there you go.

@@ Line 862: / Line 862: @@
 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}}.
-And how about something silly like {{monzo|1 0 0}}, the octave? Well, that maps to {{monzo|5 7}}. If the first generator was the exact same size as 5-ET’s generator, and the second generator was the exact same size as 7-ET’s generator, then 5 of the first and 7 of the second wouldn’t take us to the octave, it would take us to the double octave, 4/1. So maybe the first generator is the size of half a step of 5-ET, maybe? And the second generator is the size as half a step of 7-ET, maybe? I guess that would work out, since 120¢ + 85.714¢ = 205.714¢ and 10/9 is 182.404¢; I mean, it’s not super close, but it’s in the ballpark at least.
 Previously we mentioned that any given rank-2 temperaments can be generated by a wide variety of combinations of intervals. In other words, the absolute size of the intervals is not the important part, in terms of their potential for generating the temperament; only their relative size matters. However, for us humans, it’s much easier to make sense of these things if we get them in a good old standard form, by locking one generator to the octave to establish a common basis for comparison, and the other generator to a size less than half of the octave (because anything past the halfway point and it would be the octave-complement of a smaller and therefore in some sense simpler interval). And there’s a way to find this form by transforming our matrix. In fact, it also uses Gaussian elimination, though in this case, we do it by columns. Our target this time is a bit harder to explain ahead of time, so this first time through, just watch, and we’ll review the result.
@@ Line 917: / Line 915: @@
 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.
-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. 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.
+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.
 We’ve made it to a critical point here: we are now able to explain why RTT is called “regular” temperament theory. Regular here is a mathematical term, and I don’t have a straightforward definition of it for you, but it apparently refers to the fact that all intervals in the tuning are combinations of only these specified generators. So there you go.