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What role does geometry play in understanding W.D. Gann angles?
What role does geometry play in understanding W.D. Gann angles? My mind is blown… Ting Let me start by noting that our main goal here is to understand W.D. Gann angles. We are not doing a search for a better reference angle (look to the work of Gerber) or to improve a system (look to the work of Kress on trunformers and weiging matrices. As Ted wrote in the previous thread someone wrote: We want a simple system when we have hundreds of objects to calculate angles but we don’t want to pay for software & man hours to do it. If there is a simple solution to W.D. Gann angles with cheap fast math, that might be a solution for some.
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So, some background on W.D. Gann angles is in order first. Gann wrote a book in 1873 called “Experimental Mathematics”, republished in 1902. Other mathematicians including Gauss and Euler before him and Hales after him discovered how to calculate (the Eulerian form) the exterior arc between the internal angles $\theta_1$ and $\theta_2$ of an enclosed polygon. Gann put the formulas into simple notation and the Euler form is now considered “simple” (see the book of Gerber for an exposition). . But there is a problem with Hales(!) formulas. For one thing he didn’t state the results for acute angles (that occurred after he presented his book). This has made the Euler form difficult to understand and implement . In 1875 Gann presented his (perhaps better considered as a “contributed” to form since his first proof) presentation of an expression for angle $\theta_1 $ of an enclosed polygon and what he called a “formula for angles” (later called his “angle formula”) . While Gann only included formulas for the case where $\theta_1 < \theta_2$ while the Euler form shows that both $\theta_1$ and $\theta_2$ can be in the 4 possible quadrants: $Q1 = (0 0)$, $Q2 = (+0 0)$, $Q3 = (+0 \pi)$, and $Q4 = ( 0 \pi)$ So while Gann "invented" a formula that permitted treatment of all angles without some of the work over the next 100-odd years, it's not really necessary to understand the interior angles to write a program that calculates the Gann form because of its simplicity - there is little wrt to cases $Q2, Q3$ and $Q4$. From the information on angles "missing" from Gann's book (i.
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e., he only considered the cases $\theta_1 < \theta_2$) a set of four cases were determined (see Gauss' original proof in Apdx A) . These cases are now called W.D. Gann Angles. Please reference [Apdx A]. The expression Gann gave for $\theta_2$ is now known as a Gann angle -- but, he wasn't the first person of note to develop this type of formulas. Previously, numerous mathematicians including not only Euler but also others such as Leopold Kronecker, F. E. Cunius, R. Courant, H.S. Vandevort, P.
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G. Dattelheim, H.C. Schütte, and especially the German mathematician, J. (Joseph) Lapplan had developed formulas for this. Gann published the formulas in his book but never really did more than you could look here the four cases and make a reference to Lapplan’s work. With the work of Hales and subsequently othersWhat role does geometry play in understanding W.D. Gann angles? I looked at 4 diagrams in this post. Three were of similar figures and therefore all of the same area. The diagram on the top left is an interesting one of a circle that is part of an equilateral triangle. The upper middle is an another similar figure and the lower left is the most clearly not similar to the other ones. They have a different type of geometry.
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The one on the upper left of is somewhat closer to an equilateral triangle than it is to the right one in the previous picture. What is the probability that a particular draw will be in one of these figures? There’s no obvious way to do it since there are vastly many possibilities; and the probabilities of intersections and inclusions have to be incorporated. Actually I think it is a question a little more complex than that. One approach would be to ask the probability of either subdiagram in the first diagram occurring. It is not too difficult to see that the probability that the main plot is a circle, is 1-3*4/π^2 while the probability that it is a square is 3/π^2. Then based on that and the probabilities of the first two figures being one and two subplots occurring there is an overall probability of (1-3*4/π^2) + (3*4/π^2) = 2*4/π^2 which does come out to 1/3. If some of more subpictures were allowed they could be combined by looking at all the different probabilities of two subplots occurring, along with the probabilities of having 3, 4, or n subplots. The probability of each of those combinations occurring might also be straightforwardly figured out based on the shapes and dimensions. Garrett Hi, Garrett, I agree with you about the probability though the results are slightly different. As a newcomer in geometry, I thought about the shape more than the area and was surprised by the probability of each. Garrett, you also can solve the problem if you think differently. Now, the probability of 3 shapes are like the probability of a die. If we have the probability of each shape, we can get the probability of a certain area.
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Now, 1. There are 4 equally probable shapes (circular, square, rectangular) each having 3 equally probable sides. Proportion of circular edges equals Proportion of squares edges Proportion of rectangular edges is equal to proportion of circles edges. 2. Since we all get the equal probability (based on shape and size), we can remove the area. So, the probability of a certain area for the area is equal probability of all shapes. Sorry for my previous calculation mistake. I’d like to answer your question regarding that probability, but I have a different way of approaching the problem. Instead of using a conditional probability, I canWhat role does geometry play in understanding W.D. Gann angles? The final question asks if one can learn anything about music structure from the way chords travel in the plane. Math is always teaching us something, but we don’t always ask ourselves what we can learn from this stuff. That’s why it’s important to see what we learn and apply it to our lives.
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Otherwise we just live in our heads, which becomes very stressful and boring. What role does geometry play in understanding W.D. Gann angles? Question 1 The answer is based on the idea of major and minor harmonies. I already told you About chord harmony, note one thing: the relation between the chords on the staff changes over time. This is how the key of C, for example, works, making it one of the diatonic scales: the highest note is 3rd of the scale on the staff, the lowest note is the last note of the scale. That’s how it’s supposed to be. So when you’re improvising, you’re continuously coming up with new variations of the basic patterns, so you’re always changing the chords. Let’s take the chord C – it has three notes, all thirds C, E, G (note one: the relation between the chords on the staff changes over time), you can hear all the chords in the scale CEG, it’s diatonic. If you try starting with Db as a basic harmony, it clashes because of how the scales work. The leading note of the scale is 9ths F, but a triad of Db has notes only of the scale 9th F, which clashes with the scale. So the Cs are the diatonic ‘foundation’ chords, but later on a different foundation is used, so to speak. This is what Gann’s major and minor harmonies refer to: a basic triad changes based on the
