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

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If your goal is to evoke JI-like harmony, then, {{val|12 20 28}} is not your friend. Feel free to work out some other variations on {{val|12 19 28}} if you like, such as {{val|12 19 29}} maybe, but I guarantee you won’t find a better one that starts with 12 than {{val|12 19 28}}.

[[File:17-ET mistunings.png|thumb|600px|right|'''Figure 2f.''' Detunings of various 17-ET maps, showing how the supposed "patent" val's total error can be improved upon by allowing tempered octaves and second-closest mappings of primes. It also shows ~~the concerning case that~~ pure octave 17c has no primes tuned flat.]]

[[File:17-ET mistunings.png|thumb|600px|right|'''Figure 2f.''' Detunings of various 17-ET maps, showing how the supposed "patent" val's total error can be improved upon by allowing tempered octaves and second-closest mappings of primes. It also shows how pure octave 17c has no primes tuned flat.]]

So the case is cut-and-dry for {{val|12 19 28}}, and therefore from now on I'm simply going to refer to this ET by "12-ET" rather than spelling out its map. But other ETs find themselves in trickier situations. Consider [[17edo|17-ET]]. One option we have is the map {{val|17 27 39}}, with a generator of about 1.0418, and prime approximations of 2.0045, 3.0177, 4.9302. But we have a second reasonable option here, too, where {{val|17 27 40}} gives us a generator of 1.0414, and prime approximations of 1.9929, 2.9898, and 5.0659. In either case, the approximations of 2 and 3 are close, but the approximation of 5 is way off. For {{val|17 27 39}}, it’s way small, while for {{val|17 27 40}} it’s way big. The conundrum could be described like this: any generator we could find that divides 2 into about 17 equal steps can do a good job dividing 3 into about 27 equal steps, too, but it will not do a good job of dividing 5 into equal steps; 5 is going to land, unfortunately, right about in the middle between the 39th and 40th steps, as far as possible from either of these two nearest approximations. To do a good job approximating prime 5, we’d really just want to subdivide each of these steps in half, or in other words, we’d want [[34edo|34-ET]].

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 If your goal is to evoke JI-like harmony, then, {{val|12 20 28}} is not your friend. Feel free to work out some other variations on {{val|12 19 28}} if you like, such as {{val|12 19 29}} maybe, but I guarantee you won’t find a better one that starts with 12 than {{val|12 19 28}}.
-[[File:17-ET mistunings.png|thumb|600px|right|'''Figure 2f.''' Detunings of various 17-ET maps, showing how the supposed "patent" val's total error can be improved upon by allowing tempered octaves and second-closest mappings of primes. It also shows the concerning case that pure octave 17c has no primes tuned flat.]]
+[[File:17-ET mistunings.png|thumb|600px|right|'''Figure 2f.''' Detunings of various 17-ET maps, showing how the supposed "patent" val's total error can be improved upon by allowing tempered octaves and second-closest mappings of primes. It also shows how pure octave 17c has no primes tuned flat.]]
 So the case is cut-and-dry for {{val|12 19 28}}, and therefore from now on I'm simply going to refer to this ET by "12-ET" rather than spelling out its map. But other ETs find themselves in trickier situations. Consider [[17edo|17-ET]]. One option we have is the map {{val|17 27 39}}, with a generator of about 1.0418, and prime approximations of 2.0045, 3.0177, 4.9302. But we have a second reasonable option here, too, where {{val|17 27 40}} gives us a generator of 1.0414, and prime approximations of 1.9929, 2.9898, and 5.0659. In either case, the approximations of 2 and 3 are close, but the approximation of 5 is way off. For {{val|17 27 39}}, it’s way small, while for {{val|17 27 40}} it’s way big. The conundrum could be described like this: any generator we could find that divides 2 into about 17 equal steps can do a good job dividing 3 into about 27 equal steps, too, but it will not do a good job of dividing 5 into equal steps; 5 is going to land, unfortunately, right about in the middle between the 39th and 40th steps, as far as possible from either of these two nearest approximations. To do a good job approximating prime 5, we’d really just want to subdivide each of these steps in half, or in other words, we’d want [[34edo|34-ET]].