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

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[[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 detuning 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]]”~~; I think the association with "[[patent val]]" is confused, and "patent" isn't a good word for it in the first place, and I would prefer to characterize it as a “generatable map” myself~~.)

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]]”.)

And why is this cool? Well, if {{val|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 {{val|12 19 28}} approximates 2, 3, and 5 by 1.059¹² ≈ 1.998, 1.059¹⁹ ≈ 2.992, and 1.059²⁸ ≈ 5.029, respectively, then {{val|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 {{val|12 19 28}}, which checks out with our calculation we made in the previous section that the best approximation of 16/15 in {{val|12 19 28}} would be 1 step.

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 [[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 detuning 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]]”; I think the association with "[[patent val]]" is confused, and "patent" isn't a good word for it in the first place, and I would prefer to characterize it as a “generatable map” myself.)
+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]]”.)
 And why is this cool? Well, if {{val|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 {{val|12 19 28}} approximates 2, 3, and 5 by 1.059¹² ≈ 1.998, 1.059¹⁹ ≈ 2.992, and 1.059²⁸ ≈ 5.029, respectively, then {{val|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 {{val|12 19 28}}, which checks out with our calculation we made in the previous section that the best approximation of 16/15 in {{val|12 19 28}} would be 1 step.