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

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</ol>

[[File:Approximation of logs.png|600px|thumb|left|'''Figure 1c.''' visualization of an ET as a logarithmic approximation]]

[[File:Approximation of logs.png|600px|thumb|left|'''Figure 1c.''' visualization of an ET as a logarithmic approximation. The curve of the blue line is the familiar logarithmic curve of the harmonic series (harmonic 4 was skipped because it's not prime). Each rectangular brick is one of our generators, or in other words, one of the same ET step. The goal is to choose a size of brick that allows us to build stacks which most closely matches the position of the blue line at all three of these primes' positions.]]

So when I say 12:19:28 ≈ log(2:3:5) what I’m saying is that there is indeed some shared generator g for which logg2 ≈ 12, logg3 ≈ 19, and logg5 ≈ 28 are all good approximations all at the same time, or, equivalently, a shared generator g for which g¹² ≈ 2, g¹⁹ ≈ 3, and g²⁸ ≈ 5 are all good approximations at the same time ''(see Figure 1c)''. And that’s a pretty cool thing to find! To be clear, with g = 1.059, we get g¹² ≈ 1.9982, g¹⁹ ≈ 2.9923, and g²⁸ ≈ 5.0291.

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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 1d)'':

[[File:Near linings up rare2.png|600px|thumb|right|'''Figure 1d.''' Texture of ETs approximating prime harmonics. Where the numerals line up, all primes are well-approximated by a single step size.]]

[[File:Near linings up rare2.png|600px|thumb|right|'''Figure 1d.''' Texture of ETs approximating prime harmonics. Where the ''numerals'' 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). 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. 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.)

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 </ol>
-[[File:Approximation of logs.png|600px|thumb|left|'''Figure 1c.''' visualization of an ET as a logarithmic approximation]]
+[[File:Approximation of logs.png|600px|thumb|left|'''Figure 1c.''' visualization of an ET as a logarithmic approximation. The curve of the blue line is the familiar logarithmic curve of the harmonic series (harmonic 4 was skipped because it's not prime). Each rectangular brick is one of our generators, or in other words, one of the same ET step. The goal is to choose a size of brick that allows us to build stacks which most closely matches the position of the blue line at all three of these primes' positions.]]
 So when I say 12:19:28 ≈ log(2:3:5) what I’m saying is that there is indeed some shared generator g for which log<sub>g</sub>2 ≈ 12, log<sub>g</sub>3 ≈ 19, and log<sub>g</sub>5 ≈ 28 are all good approximations all at the same time, or, equivalently, a shared generator g for which g¹² ≈ 2, g¹⁹ ≈ 3, and g²⁸ ≈ 5 are all good approximations at the same time ''(see Figure 1c)''. And that’s a pretty cool thing to find! To be clear, with g = 1.059, we get g¹² ≈ 1.9982, g¹⁹ ≈ 2.9923, and g²⁸ ≈ 5.0291.
@@ 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 1d)'':
-[[File:Near linings up rare2.png|600px|thumb|right|'''Figure 1d.''' Texture of ETs approximating prime harmonics. Where the numerals line up, all primes are well-approximated by a single step size.]]
+[[File:Near linings up rare2.png|600px|thumb|right|'''Figure 1d.''' Texture of ETs approximating prime harmonics. Where the ''numerals'' 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). 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. 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.)