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'' 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 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 what we'll be calling here a ~~'''~~uniform map~~'''~~; ~~elsewhere you may find this~~ called a “[[generalized patent val]]”.<ref>See my ~~proposal to rename this object~~ here: https://en.xen.wiki/w/Talk:Patent_val</ref>)

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 what we'll be calling here a [[uniform map]]; it is more commonly called a "[[generalized patent val]]", but I am critical of that terminology.<ref>See my thoughts on that here: https://en.xen.wiki/w/Talk:Patent_val</ref>)

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.

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[[File:Shape_of_scale_of_movements_on_axes.png|thumb|left|200px|'''Figure 3e.''' the basic shape the scaled axes make between neighbor maps (maps with only 1 difference between their terms)]]

Our example ET will be 40. We'll start out at the map {{map|40 63 93}}. This map is a default of sorts for 40-ET, because it’s the map where all three terms are as close as possible to JI when prime 2 is exact (we'll be calling it a ~~'''~~simple map~~'''~~ here~~, though elsewhere you may find~~ it called a "[[patent val]]"<ref>See my ~~proposal to rename this object~~ here: https://en.xen.wiki/w/Talk:Patent_val</ref>).

Our example ET will be 40. We'll start out at the map {{map|40 63 93}}. This map is a default of sorts for 40-ET, because it’s the map where all three terms are as close as possible to JI when prime 2 is exact (we'll be calling it a [[simple map]] here; it has more commonly been called a "[[patent val]]", but I am critical of that terminology.<ref>See my thoughts on that here: https://en.xen.wiki/w/Talk:Patent_val</ref>).

From here, let’s move by a single step on the 5-axis by adding 1 to the 5-term of our map, from 93 to 94, therefore moving to the map {{map|40 63 94}}. This map is found directly to the left. This makes sense because the orientation of the 5-axis is horizontal, and the positive direction points out from the origin toward the left, so increases to the 5-term move us in that direction.

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 [[File:Near linings up rare2.png|600px|thumb|right|'''Figure 2d.''' Texture of ETs approximating prime harmonics. Where the ''numerals'' 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 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 what we'll be calling here a '''uniform map'''; elsewhere you may find this called a “[[generalized patent val]]”.<ref>See my proposal to rename this object here: https://en.xen.wiki/w/Talk:Patent_val</ref>)
+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 what we'll be calling here a [[uniform map]]; it is more commonly called a "[[generalized patent val]]", but I am critical of that terminology.<ref>See my thoughts on that here: https://en.xen.wiki/w/Talk:Patent_val</ref>)
 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.
@@ Line 206: / Line 206: @@
 [[File:Shape_of_scale_of_movements_on_axes.png|thumb|left|200px|'''Figure 3e.''' the basic shape the scaled axes make between neighbor maps (maps with only 1 difference between their terms)]]
-Our example ET will be 40. We'll start out at the map {{map|40 63 93}}. This map is a default of sorts for 40-ET, because it’s the map where all three terms are as close as possible to JI when prime 2 is exact (we'll be calling it a '''simple map''' here, though elsewhere you may find it called a "[[patent val]]"<ref>See my proposal to rename this object here: https://en.xen.wiki/w/Talk:Patent_val</ref>).
+Our example ET will be 40. We'll start out at the map {{map|40 63 93}}. This map is a default of sorts for 40-ET, because it’s the map where all three terms are as close as possible to JI when prime 2 is exact (we'll be calling it a [[simple map]] here; it has more commonly been called a "[[patent val]]", but I am critical of that terminology.<ref>See my thoughts on that here: https://en.xen.wiki/w/Talk:Patent_val</ref>).
 From here, let’s move by a single step on the 5-axis by adding 1 to the 5-term of our map, from 93 to 94, therefore moving to the map {{map|40 63 94}}. This map is found directly to the left. This makes sense because the orientation of the 5-axis is horizontal, and the positive direction points out from the origin toward the left, so increases to the 5-term move us in that direction.