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

Line 129:

[[File:17-ET mistunings.png|thumb|600px|right|'''Figure 1f.''' Mistunings of various 17-ET maps, showing how the supposed "patent" val's total error can be improved upon by allowing flexible octaves and second-closest mappings of primes. Also shows that the pure octave 17c is improper insofar as all primes are mistuned in one direction.]]

So the case is cut-and-dry for 12-ET. 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].

So the case is cut-and-dry for 12-ET. 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]].

[[File:17-ET.png|thumb|400px|left|'''Figure 1g.''' Visualization of the 17-ETs on the GPV continuum, showing how the pure octave 17c is a "lie" insofar as there exists no generator that exactly reaches prime 2 in 17 steps while more closely approximating prime 5 as 40 steps than 39 steps.]]

Line 142:

At this point you should have a pretty good sense for why choosing a map makes an important impact on how your music sounds. Now we just need to help you find and compare maps! Or, similarly, how to find and compare intervals to temper. To do this, we need to give you the ability to navigate tuning space.

== projective tuning space ==

@@ Line 129: / Line 129: @@
 [[File:17-ET mistunings.png|thumb|600px|right|'''Figure 1f.''' Mistunings of various 17-ET maps, showing how the supposed "patent" val's total error can be improved upon by allowing flexible octaves and second-closest mappings of primes. Also shows that the pure octave 17c is improper insofar as all primes are mistuned in one direction.]]
-So the case is cut-and-dry for 12-ET. 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].
+So the case is cut-and-dry for 12-ET. 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]].
 [[File:17-ET.png|thumb|400px|left|'''Figure 1g.''' Visualization of the 17-ETs on the GPV continuum, showing how the pure octave 17c is a "lie" insofar as there exists no generator that exactly reaches prime 2 in 17 steps while more closely approximating prime 5 as 40 steps than 39 steps.]]
@@ Line 142: / Line 142: @@
 At this point you should have a pretty good sense for why choosing a map makes an important impact on how your music sounds. Now we just need to help you find and compare maps! Or, similarly, how to find and compare intervals to temper. To do this, we need to give you the ability to navigate tuning space.
 == projective tuning space ==