Form Definition and 1000 Threads

Sonata form (also sonata-allegro form or first movement form) is a musical structure consisting of three main sections: an exposition, a development, and a recapitulation. It has been used widely since the middle of the 18th century (the early Classical period).
While it is typically used in the first movement of multi-movement pieces, it is sometimes used in subsequent movements as well—particularly the final movement. The teaching of sonata form in music theory rests on a standard definition and a series of hypotheses about the underlying reasons for the durability and variety of the form—a definition that arose in the second quarter of the 19th century. There is little disagreement that on the largest level, the form consists of three main sections: an exposition, a development, and a recapitulation; however, beneath this general structure, sonata form is difficult to pin down to a single model.
The standard definition focuses on the thematic and harmonic organization of tonal materials that are presented in an exposition, elaborated and contrasted in a development and then resolved harmonically and thematically in a recapitulation. In addition, the standard definition recognizes that an introduction and a coda may be present. Each of the sections is often further divided or characterized by the particular means by which it accomplishes its function in the form.
After its establishment, the sonata form became the most common form in the first movement of works entitled "sonata", as well as other long works of classical music, including the symphony, concerto, string quartet, and so on. Accordingly, there is a large body of theory on what unifies and distinguishes practice in the sonata form, both within and between eras. Even works that do not adhere to the standard description of a sonata form often present analogous structures or can be analyzed as elaborations or expansions of the standard description of sonata form.

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  1. chwala

    Integration of functions of form ##\dfrac{1}{ax+b}##

    This is a bit confusing...conflicting report from attached wolfram and symbolab. Which approach is correct?
  2. M

    Using Conjunctive Normal form to find when wff is true

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  3. chwala

    Conceptual question on equations of the form ##x=ay^2+by+c##

    Now i just need some clarification; we know that quadratic equations are equations of the form ##y=ax^2+bx+c## with ##a,b## and ##c## being constants and ##x## and ##y## variables. Now my question is... can we also view/look at ##x=y^2+2y+1## as quadratic equations having switched the...
  4. M

    General form of Newton II -- Not understanding this step in the derivation

    For this, Does someone please know how do we derive equation 9.9 from 9.8? Do we take the limits as t approach's zero for both sides? Why not take limit as momentum goes to zero? Many thanks!
  5. B

    B Mapping wave forms to sphere, does wave form y=0 have a reflection?

    Zero does not have an inverse. And y=0 does not have an inverse. Does the wave form y=0 for all x have an inverse?
  6. chwala

    Find in the form, ##x+iy## in the given complex number problem

    This is the question as it appears on the pdf. copy; ##z=2\left[\cos \dfrac{3π}{4} + i \sin \dfrac{3π}{4}\right]## My approach; ##\dfrac{3π}{4}=135^0## ##\tan 135^0=-\tan 45^0=\dfrac{-\sqrt{2}}{\sqrt{2}}## therefore, ##z=-\sqrt{2}+\sqrt{2}i## There may be a better approach.
  7. codebpr

    A Finding a suitable form factor for a given set of conditions

    This is basically a physics problem but I will try my best to highlight the mathematics behind it. Suppose I have two functions: $$T(z,B)=\frac{\text{z}^3 e^{-3 A(\text{z})-B^2 \text{z}^2}}{4 \pi \int_0^{\text{z}} \xi ^3 e^{-3 A(\xi )-B^2 \xi ^2} \, d\xi },$$ $$\phi(z,B)=\int_0^z...
  8. fsonnichsen

    I What Is the Correct Sign for the Quadratic Form in Margenau's Proof?

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  9. S

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  10. M

    Prove that there are infinitely many primes of the form ## 6k+1 ##?

    Proof: Suppose that the only prime numbers of the form ## 6k+1 ## are ## p_{1}, p_{2}, ..., p_{n} ##, and let ## N=4p_{1}^{2}p_{2}^{2}\dotsb p_{n}^{2}+3 ##. Since ## N ## is odd, ## N ## is divisible by some prime ## p ##, so ## 4p_{1}^{2}\dotsb p_{n}^{2}\equiv -3\pmod {p} ##. That is, ##...
  11. R

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  12. Y

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  13. Omega0

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  14. E

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  15. C

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    logyx + logxy = 3/2 Solution $$\begin{align*}\log_{ y }{ x } + \log_{ x }{ y } &= \frac{ 3 }{ 2 } \\ \log_{ x }{ y } &= \frac{ \log_{ y }{ y } }{ \log_{ y }{ x } } \\ \log_{ y }{ x } + \frac{ 1 }{ \log_{ y }{ x } } &= \frac{ 3 }{ 2 } \\ \left(\log_{ y }{ x } \right)^ { 2 } + 1 &=...
  16. fluidistic

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  17. heroslayer99

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  18. crakedhead

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  19. C

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  20. chwala

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  21. A

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  22. M

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  23. A

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  24. T

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  25. O

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  26. abdulbadii

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  27. N

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  28. N

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  29. M

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  30. A

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  31. Astronuc

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  32. Purplepixie

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  33. H

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  34. tworitdash

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  35. tworitdash

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    clear; lambda = 3e-2; x = 4 * pi/lambda * linspace(eps, 15, 100000); T = 5e-3; t = [0:0.001e-3:T] ; % 0.1:1e-3:0.1+T]; u = 3; a = 4*pi/lambda * u; for i = 1:length(x) Z(i) = sum(-((cos(a.*t) - cos(x(i).*t)).^2 + (sin(a.*t) - sin(x(i).*t)).^2)); end % Z1 = csc((a+x)/2) .*...
  36. chwala

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    For part (a); $$\int e^{3y} \,dy=\int 3x^2\ln x \,dx$$$$\frac{e^{3y}}{3}=x^3\ln x-\frac{x^3}{3}+k$$$$\frac{e^{3}}{3}=e^3-\frac{e^3}{3}+k$$$$\frac{e^{3y}}{3}=x^3\ln x-\frac{x^3}{3}-\frac{e^3}{3}$$$$e^{3y}=3x^3 \ln x-x^3-e^3$$ You may check my working...i do not have the solution.
  37. Astronuc

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  38. M

    Prove n^2+2^n Composite if n not 6k+3

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  39. K

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  40. M

    How to Find an Answer to 2^n + 1 Prime Question

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  41. B

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  42. M

    Any integer of the form ## 8^n+1 ##, where n##\geq##1, is composite?

    Proof: Suppose ##a=8^n+1 ## for some ##a \in\mathbb{Z}## such that n##\geq##1. Then we have ##a=8^n+1 ## =## (2^3)^n+1 ## =## (2^n+1)(2^{2n} -2^n+1) ##. This means ## 2^n+1\mid 2^{3n} +1 ##. Since ##2^n+1>1## and ##2^{2n} -2^n+1>1## for all...
  43. M

    Every integer of the form n^4+4, with n>1, is composite?

    Proof: Suppose a=n^4+4 for some a##\in\mathbb{Z}## such that n>1. Then we have a=n^4+4=(n^2-2n+2)(n^2+2n+2). Note that n^2-2n+2>1 and n^2+2n+2>1 for n>1. Therefore, every integer of the form n^4+4, with n>1, is composite.
  44. chwala

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  45. M

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  46. M

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  47. wrobel

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  49. D

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  50. S

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