Douglas Blumeyer's RTT How-To: Difference between revisions
Cmloegcmluin (talk | contribs) m →intersections and unions: link to Graham's app |
Cmloegcmluin (talk | contribs) m →rank-2 mappings: another no trailing 0 removal fix |
||
| Line 790: | Line 790: | ||
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}}. | 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}}, 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. | 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. | 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 838: | Line 838: | ||
</math> | </math> | ||
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}}. 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}}? 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. | 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. | ||
Hopefully manipulating these rows like this gives you some sort of feel for how what matters in a temperament mapping is not so much the absolute values but their relationship with each other. | Hopefully manipulating these rows like this gives you some sort of feel for how what matters in a temperament mapping is not so much the absolute values but their relationship with each other. | ||