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

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Why is this rare? Well, it’s like a game of trying to get these numbers to line up ''(see Figure 2d)'':

[[File:Near linings up rare2.png|600px|thumb|right|'''Figure 2d.''' Texture of ETs approximating prime harmonics. Where the ''numerals'' (and tick marks) line up, all primes are ~~well-~~approximated by a single step size (the boundaries between cells are midpoints between perfect approximations, or in other words, the point where the closest approximation switches over from one generator count to the next) (the numerals and tick marks are meant to be centered in each cell). Nudging one of the maps' vertical lines to the right would mean decreasing the generator size, flattening the tunings of all the primes, and vice versa, nudging it to the left would mean increasing the generator size, sharpening the tunings of all the primes. You can visualize this on Figure 2c. as shrinking or growing the height of the rectangular bricks. The positions of each map's vertical line, or in other words the tuning of its generator, has been optimized using some formula to distribute the deviations amongst the three primes; that's why you do not see any vertical line here for which the closest step counts for each prime are all on one side of it.]]

[[File:Near linings up rare2.png|600px|thumb|right|'''Figure 2d.''' Texture of ETs approximating prime harmonics. Where the ''numerals'' (and tick marks) line up, all primes are closely approximated by a single step size (the boundaries between cells are midpoints between perfect approximations, or in other words, the point where the closest approximation switches over from one generator count to the next) (the numerals and tick marks are meant to be centered in each cell). Nudging one of the maps' vertical lines to the right would mean decreasing the generator size, flattening the tunings of all the primes, and vice versa, nudging it to the left would mean increasing the generator size, sharpening the tunings of all the primes. You can visualize this on Figure 2c. as shrinking or growing the height of the rectangular bricks. The positions of each map's vertical line, or in other words the tuning of its generator, has been optimized using some formula to distribute the deviations amongst the three primes; that's why you do not see any vertical line here for which the closest step counts for each prime are all on one side of it.]]

If the distance between entries in the row for 2 are defined as 1 unit apart, then the distance between entries in the row for prime 3 are 1/log₂3 units apart, and 1/log₂5 units apart for the prime 5. So, near-linings up don’t happen all that often!<ref>For more information, see: [[The_Riemann_zeta_function_and_tuning|The Riemann zeta function and tuning]].</ref> (By the way, any vertical line drawn through a chart like this is called a ~~GPV~~, or “[[generalized patent val]]”.)

If the distance between entries in the row for 2 are defined as 1 unit apart, then the distance between entries in the row for prime 3 are 1/log₂3 units apart, and 1/log₂5 units apart for the prime 5. So, near-linings up don’t happen all that often!<ref>For more information, see: [[The_Riemann_zeta_function_and_tuning|The Riemann zeta function and tuning]].</ref> (By the way, any vertical line drawn through a chart like this is called a uniform map, or “[[generalized patent val]]”.)

And why is this cool? Well, if {{map|12 19 28}} approximates the harmonic building blocks well individually, then JI intervals built out of them, like 16/15, 5/4, 10/9, etc. should also be reasonably ~~well-~~approximated overall, and thus recognizable as their JI counterparts in musical context. You could think of it like taking all the primes in a prime factorization and swapping in their approximations. For example, if 16/15 = 2⁴ × 3⁻¹ × 5⁻¹ ≈ 1.067, and {{map|12 19 28}} approximates 2, 3, and 5 by 1.059¹² ≈ 1.998, 1.059¹⁹ ≈ 2.992, and 1.059²⁸ ≈ 5.029, respectively, then {{map|12 19 28}} maps 16/15 to 1.998⁴ × 2.992⁻¹ × 5.029⁻¹ ≈ 1.059, which is indeed pretty close to 1.067. Of course, we should also note that 1.059 is the same as our step of {{map|12 19 28}}, which checks out with our calculation we made in the previous section that the best approximation of 16/15 in {{map|12 19 28}} would be 1 step.

And why is this cool? Well, if {{map|12 19 28}} approximates the harmonic building blocks well individually, then JI intervals built out of them, like 16/15, 5/4, 10/9, etc. should also be reasonably closely approximated overall, and thus recognizable as their JI counterparts in musical context. You could think of it like taking all the primes in a prime factorization and swapping in their approximations. For example, if 16/15 = 2⁴ × 3⁻¹ × 5⁻¹ ≈ 1.067, and {{map|12 19 28}} approximates 2, 3, and 5 by 1.059¹² ≈ 1.998, 1.059¹⁹ ≈ 2.992, and 1.059²⁸ ≈ 5.029, respectively, then {{map|12 19 28}} maps 16/15 to 1.998⁴ × 2.992⁻¹ × 5.029⁻¹ ≈ 1.059, which is indeed pretty close to 1.067. Of course, we should also note that 1.059 is the same as our step of {{map|12 19 28}}, which checks out with our calculation we made in the previous section that the best approximation of 16/15 in {{map|12 19 28}} would be 1 step.

=== tuning & pure octaves ===

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

[[File:17-ET mistunings.png|thumb|600px|right|'''Figure 2f.''' Deviations from JI for 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.]]

[[File:17-ET mistunings.png|thumb|600px|right|'''Figure 2f.''' Deviations from JI for various 17-ET maps, showing how the simple map'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 {{map|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 {{map|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 {{map|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 {{map|17 27 39}}, it’s way small, while for {{map|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 2g.''' Visualization of ~~the~~ 17-ETs on the ~~GPV~~ continuum, ~~showing how for~~ the pure octave ~~17c there exists no generator that exactly reaches prime 2 in 17 steps while more closely approximating prime 5 as 40 steps than 39 steps~~. (If this diagram is unclear, please refer back to Figure 2d., which has the same type of information but with more thorough labelling.)]]

[[File:17-ET.png|thumb|300px|left|'''Figure 2g.''' Visualization of some 17-ETs on the uniform map continuum, along with their TE tunings, as well as the pure octave tuning. The pure octave tuning is always, by definition, inside the region for the simple map, denoted with the wart "p". (If this diagram is unclear, please refer back to Figure 2d., which has the same type of information but with more thorough labelling.)]]

Curiously, {{map|17 27 39}} is the map for which each prime individually is as closely approximated as possible when prime 2 is exact, so it is in a sense the naively best map for 17-ET, however, if that constraint is lifted, and we’re allowed to either temper prime 2 and/or choose the next-closest approximations for prime 5, the overall approximation can be improved; in other words, even though 39 steps can take you just a tiny bit closer to prime 5 than 40 steps can, the tiny amount by which it is closer is less than the improvements to the tuning of primes 2 and 3 you can get by using {{map|17 27 40}}. So again, the choice is not always cut-and-dry; there’s still a lot of personal preference going on in the tempering process.

@@ Line 118: / Line 118: @@
 Why is this rare? Well, it’s like a game of trying to get these numbers to line up ''(see Figure 2d)'':
-[[File:Near linings up rare2.png|600px|thumb|right|'''Figure 2d.''' Texture of ETs approximating prime harmonics. Where the ''numerals'' (and tick marks) line up, all primes are well-approximated by a single step size (the boundaries between cells are midpoints between perfect approximations, or in other words, the point where the closest approximation switches over from one generator count to the next) (the numerals and tick marks are meant to be centered in each cell). Nudging one of the maps' vertical lines to the right would mean decreasing the generator size, flattening the tunings of all the primes, and vice versa, nudging it to the left would mean increasing the generator size, sharpening the tunings of all the primes. You can visualize this on Figure 2c. as shrinking or growing the height of the rectangular bricks. The positions of each map's vertical line, or in other words the tuning of its generator, has been optimized using some formula to distribute the deviations amongst the three primes; that's why you do not see any vertical line here for which the closest step counts for each prime are all on one side of it.]]
+[[File:Near linings up rare2.png|600px|thumb|right|'''Figure 2d.''' Texture of ETs approximating prime harmonics. Where the ''numerals'' (and tick marks) line up, all primes are closely approximated by a single step size (the boundaries between cells are midpoints between perfect approximations, or in other words, the point where the closest approximation switches over from one generator count to the next) (the numerals and tick marks are meant to be centered in each cell). Nudging one of the maps' vertical lines to the right would mean decreasing the generator size, flattening the tunings of all the primes, and vice versa, nudging it to the left would mean increasing the generator size, sharpening the tunings of all the primes. You can visualize this on Figure 2c. as shrinking or growing the height of the rectangular bricks. The positions of each map's vertical line, or in other words the tuning of its generator, has been optimized using some formula to distribute the deviations amongst the three primes; that's why you do not see any vertical line here for which the closest step counts for each prime are all on one side of it.]]
-If the distance between entries in the row for 2 are defined as 1 unit apart, then the distance between entries in the row for prime 3 are 1/log₂3 units apart, and 1/log₂5 units apart for the prime 5. So, near-linings up don’t happen all that often!<ref>For more information, see: [[The_Riemann_zeta_function_and_tuning|The Riemann zeta function and tuning]].</ref> (By the way, any vertical line drawn through a chart like this is called a GPV, or “[[generalized patent val]]”.)
+If the distance between entries in the row for 2 are defined as 1 unit apart, then the distance between entries in the row for prime 3 are 1/log₂3 units apart, and 1/log₂5 units apart for the prime 5. So, near-linings up don’t happen all that often!<ref>For more information, see: [[The_Riemann_zeta_function_and_tuning|The Riemann zeta function and tuning]].</ref> (By the way, any vertical line drawn through a chart like this is called a uniform map, or “[[generalized patent val]]”.)
-And why is this cool? Well, if {{map|12 19 28}} approximates the harmonic building blocks well individually, then JI intervals built out of them, like 16/15, 5/4, 10/9, etc. should also be reasonably well-approximated overall, and thus recognizable as their JI counterparts in musical context. You could think of it like taking all the primes in a prime factorization and swapping in their approximations. For example, if 16/15 = 2⁴ × 3⁻¹ × 5⁻¹ ≈ 1.067, and {{map|12 19 28}} approximates 2, 3, and 5 by 1.059¹² ≈ 1.998, 1.059¹⁹ ≈ 2.992, and 1.059²⁸ ≈ 5.029, respectively, then {{map|12 19 28}} maps 16/15 to 1.998⁴ × 2.992⁻¹ × 5.029⁻¹ ≈ 1.059, which is indeed pretty close to 1.067. Of course, we should also note that 1.059 is the same as our step of {{map|12 19 28}}, which checks out with our calculation we made in the previous section that the best approximation of 16/15 in {{map|12 19 28}} would be 1 step.
+And why is this cool? Well, if {{map|12 19 28}} approximates the harmonic building blocks well individually, then JI intervals built out of them, like 16/15, 5/4, 10/9, etc. should also be reasonably closely approximated overall, and thus recognizable as their JI counterparts in musical context. You could think of it like taking all the primes in a prime factorization and swapping in their approximations. For example, if 16/15 = 2⁴ × 3⁻¹ × 5⁻¹ ≈ 1.067, and {{map|12 19 28}} approximates 2, 3, and 5 by 1.059¹² ≈ 1.998, 1.059¹⁹ ≈ 2.992, and 1.059²⁸ ≈ 5.029, respectively, then {{map|12 19 28}} maps 16/15 to 1.998⁴ × 2.992⁻¹ × 5.029⁻¹ ≈ 1.059, which is indeed pretty close to 1.067. Of course, we should also note that 1.059 is the same as our step of {{map|12 19 28}}, which checks out with our calculation we made in the previous section that the best approximation of 16/15 in {{map|12 19 28}} would be 1 step.
 === tuning & pure octaves ===
@@ Line 146: / Line 146: @@
 If your goal is to evoke JI-like harmony, then, {{map|12 20 28}} is not your friend. Feel free to work out some other variations on {{map|12 19 28}} if you like, such as {{map|12 19 29}} maybe, but I guarantee you won’t find a better one that starts with 12 than {{map|12 19 28}}.
-[[File:17-ET mistunings.png|thumb|600px|right|'''Figure 2f.''' Deviations from JI for 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.]]
+[[File:17-ET mistunings.png|thumb|600px|right|'''Figure 2f.''' Deviations from JI for various 17-ET maps, showing how the simple map'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 {{map|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 {{map|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 {{map|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 {{map|17 27 39}}, it’s way small, while for {{map|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 2g.''' Visualization of the 17-ETs on the GPV continuum, showing how for the pure octave 17c there exists no generator that exactly reaches prime 2 in 17 steps while more closely approximating prime 5 as 40 steps than 39 steps. (If this diagram is unclear, please refer back to Figure 2d., which has the same type of information but with more thorough labelling.)]]
+[[File:17-ET.png|thumb|300px|left|'''Figure 2g.''' Visualization of some 17-ETs on the uniform map continuum, along with their TE tunings, as well as the pure octave tuning. The pure octave tuning is always, by definition, inside the region for the simple map, denoted with the wart "p". (If this diagram is unclear, please refer back to Figure 2d., which has the same type of information but with more thorough labelling.)]]
 Curiously, {{map|17 27 39}} is the map for which each prime individually is as closely approximated as possible when prime 2 is exact, so it is in a sense the naively best map for 17-ET, however, if that constraint is lifted, and we’re allowed to either temper prime 2 and/or choose the next-closest approximations for prime 5, the overall approximation can be improved; in other words, even though 39 steps can take you just a tiny bit closer to prime 5 than 40 steps can, the tiny amount by which it is closer is less than the improvements to the tuning of primes 2 and 3 you can get by using {{map|17 27 40}}. So again, the choice is not always cut-and-dry; there’s still a lot of personal preference going on in the tempering process.