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>My main concern with the term "generalized patent val", or GPV, is that it gets things backwards: it posits the GPV as a type of patent val, when it makes more sense to think of it the other way around, with the patent val as a type of GPV.

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>)

~~ ~~

~~The present definition of GPV on the wiki bends over backwards~~ to ~~make its way work, essentially using non-integer EDOs, which are a contradiction in terms. Consider~~ this ~~alternative definition, however, which is more straightforward~~:

~~ ~~

~~''A GPV is any relatively near-just map found by uniformly multiplying the generators for JI (⟨log₂2 log₂3 log₂5~~ ... ]) by any value before rounding it to integers. For example, choosing 17.1, we find the map 17.1⟨1 1.585 2.322] = ⟨17.1 27.103 39.705] which rounds to ⟨17 27 40]. This is one of the many GPVs for 17-EDO, and every EDO has many possible GPVs. To find a GPV for n-EDO, choose any multiplier that rounds to n; another example for 17-EDO could be 16.9⟨1 1.585 2.322] = ⟨16.9 26.786 39.241] which rounds to ⟨17 27 39].''

~~ ~~

~~The key element in this definition is the uniform multiplier, and from it we draw our proposed replacement name for this structure: a uniform map. So the definition could be~~:

~~ ~~

''A uniform map is any relatively near-just map found by uniformly multiplying the generators for JI (⟨log₂2 log₂3 log₂5 ... ]) by any value before rounding it to integers. For example, choosing 17.1, we find the map 17.1⟨1 1.585 2.322] = ⟨17.1 27.103 39.705] which rounds to ⟨17 27 40]. This is one of the many uniform maps for 17-EDO, and every EDO has many possible uniform maps. To find a uniform map for n-EDO, choose any multiplier that rounds to n; another example for 17-EDO could be 16.9⟨1 1.585 2.322] = ⟨16.9 26.786 39.241] which rounds to ⟨17 27 39].''

~~ ~~

Any uniform map whose multiplier is an integer — or "integer uniform map" — is always the simple map for the corresponding EDO. And every simple map is also an integer uniform map. These are just two different helpful ways of thinking about the same structure; in contexts pertaining to tuning accuracy, "simple map" works great, and in contexts pertaining to other uniform maps, "integer uniform map" works great.</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: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>My main concern with the term "generalized patent val", or GPV, is that it gets things backwards: it posits the GPV as a type of patent val, when it makes more sense to think of it the other way around, with the patent val as a type of GPV.
+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>)
-<br><br>
-The present definition of GPV on the wiki bends over backwards to make its way work, essentially using non-integer EDOs, which are a contradiction in terms. Consider this alternative definition, however, which is more straightforward:
-<br><br>
-''A GPV is any relatively near-just map found by uniformly multiplying the generators for JI (⟨log₂2 log₂3 log₂5 ... ]) by any value before rounding it to integers. For example, choosing 17.1, we find the map 17.1⟨1 1.585 2.322] = ⟨17.1 27.103 39.705] which rounds to ⟨17 27 40]. This is one of the many GPVs for 17-EDO, and every EDO has many possible GPVs. To find a GPV for n-EDO, choose any multiplier that rounds to n; another example for 17-EDO could be 16.9⟨1 1.585 2.322] = ⟨16.9 26.786 39.241] which rounds to ⟨17 27 39].''
-<br><br>
-The key element in this definition is the uniform multiplier, and from it we draw our proposed replacement name for this structure: a uniform map. So the definition could be:
-<br><br>
-''A uniform map is any relatively near-just map found by uniformly multiplying the generators for JI (⟨log₂2 log₂3 log₂5 ... ]) by any value before rounding it to integers. For example, choosing 17.1, we find the map 17.1⟨1 1.585 2.322] = ⟨17.1 27.103 39.705] which rounds to ⟨17 27 40]. This is one of the many uniform maps for 17-EDO, and every EDO has many possible uniform maps. To find a uniform map for n-EDO, choose any multiplier that rounds to n; another example for 17-EDO could be 16.9⟨1 1.585 2.322] = ⟨16.9 26.786 39.241] which rounds to ⟨17 27 39].''
-<br><br>
-Any uniform map whose multiplier is an integer — or "integer uniform map" — is always the simple map for the corresponding EDO. And every simple map is also an integer uniform map. These are just two different helpful ways of thinking about the same structure; in contexts pertaining to tuning accuracy, "simple map" works great, and in contexts pertaining to other uniform maps, "integer uniform map" works great.</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.