What is Form: Definition and 1000 Discussions

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. C

    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. C

    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 Margenau on quadratic form

    Looking at the proof of the Schwarz inequality in Margenau and Murphy, you will see what I attached. Gamma is asserted to be positive (OK). Given that the usual "quadratic form" solution would read "-(B+B*) .....". The sign does not seem correct to me as shown. In a fact B+B* = 2Re(B) and...
  9. S

    I Does antineutrino capture preferentially form neutrons?

    Besides the energetic preference (lower threshold, and more phase space above)? Antineutrino capture is a weak process, so it can and does change quark flavour. p+ν=n+e+ is actually uud+ν=udd+e+ that is u+ν=d+e+ But given enough energy (like cosmic ray neutrinos), do antineutrinos also get...
  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

    A Converting this vector into polar form

    In the following%3A%20https://pubs.rsc.org/en/content/articlehtml/2013/sm/c3sm00140g?casa_token=3O_jwMdswQQAAAAA%3AaSRtvg3XUHSnUwFKEDo01etmudxmMm8lcU4dIUSkJ52Hzitv2c_RSQJYsoHE1Bm2ubZ3sdt6mq5S-w'] paper, the surface velocity for a moving, spherical particle is given as (eq 1)...
  12. Y

    A Experimental nucleon form factors from electron-nucleus scattering

    I am looking for experimental nucleon form factors from electron-nucleus scattering. Is there any compliations or tables? In 'The proton charge radius ', H. Gao and M. Vanderhaeghen, Rev. Mod. Phys. 94, 015002 (2022), p. 24, there is 'world data on the proton and also the neutron'. In 'Form...
  13. Omega0

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    Please help me in understanding the history of physics regarding the atomic or non-atomic, say continous, structure of physics. In my years at school I grew up with physics of ultimate simplifications. Everything was a point "particle", like the moon revolving around the earth. Like the apple...
  14. E

    A Can four electrons form a completely antisymmetric joint spin WF?

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  15. C

    Writing a logarithm in a form not involving logarithms

    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

    B Why the expansion after the "Big Bang" is showed in cylindrical form?

    Everywhere on the net, there is no image that the expansion has a departing point, through an expansion in spherical form... Why is that?
  19. C

    I Congruence for Symmetric and non-Symmetric Matrices for Quadratic Form

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  20. chwala

    Express the given matrix in the form ##L_1DU_1##

    $$\begin{bmatrix} 4& 3 \\ 6 & -2 & \\ \end{bmatrix}= \begin{bmatrix} 1& 0 \\ a& 1& \\ \end{bmatrix}⋅ \begin{bmatrix} b& c \\ 0& d& \\ \end{bmatrix}$$ ##b=4, ab=6,⇒b=1.5, d=-6.5, c=3## $$\begin{bmatrix} 4& 3 \\ 6 & -2 & \\ \end{bmatrix} = \begin{bmatrix} 1& 0\\ \dfrac{3}{2} & 1 & \\...
  21. A

    I Explicit non-local form for the vector potential?

    Hello everyone, I was looking at the light matter interaction Hamiltonian and I worked out a simple calculation where I was surprised to see that I had to introduce an explicitly non-local vector potential if I want to go further: $$\langle\psi|...
  22. M

    Prove that there are infinitely many primes of the form ## 8k-1 ##

    Proof: Suppose for the sake of contradiction that the only primes of the form ## 8k-1 ## are ## p_{1}, p_{2}, ..., p_{n} ## where ## N=16p_{1}^2p_{2}^2\dotsb p_{n}^2-2 ##. Then ## N=(4p_{1}p_{2}\dotsb p_{n})^2-2 ##. Note that there exists at least one odd prime divisor ## p ## of ## N ## such...
  23. A

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  24. T

    I Closed Form for Complex Gamma Function

    Hey all, I was wondering if there was an equivalent closed form expression for ##\Gamma(\frac{1}{2}+ib)## where ##b## is a real number. I came across the following answer...
  25. O

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    When I observe a qubit's state, decoherence happens such that I find the qubit in a particular state. After I cease observing a qubit's state, what physical process causes a fresh superposition of states to develop? Is zero-point energy at least a contributor?
  26. abdulbadii

    Is anhydrous sodium hypochlorite stable in solid form?

    Can there be NaClO sold in solid form? If yes how common is it and its comparison in practical use to the most common liquid form?
  27. N

    I Why does the answer key sometimes have a different form compared to my solution?

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  28. N

    [Mathematical logic] prenex normal form and skolem normal form

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  29. M

    A Sampling Electrons from a 2D Projection: Is There a Functional Form?

    Hello! I have some electrons produced from a 3D gaussian source isotropically inside a uniform electric field. The electric field guides them towards a position sensitive detector and I end up with an image like the one below (with more electrons on the edge and fewer as you move towards the...
  30. A

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    Hi, In my calculus book,I found this vector form of line equation in space (bold means vector): Given point (x1,x2,x3) lies on line L & v=<a,b,c>, then equation of line is : r = <x1,x2,x3> + t <a,b,c> with t any number. Now, my question if I plug any number for t, then result will be vector...
  31. Astronuc

    I Four neutrons form a transient isolated entity - a tetraneutron?

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  32. Purplepixie

    MHB Closed form solution to sum of sine positive zero-crossings

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  33. H

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  34. tworitdash

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  35. tworitdash

    MATLAB Closed form not the same as the discrete form

    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

    Find the equation of given curve in the form ##e^{3y}=f(x)##

    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

    CO2 allows volcanoes to form persistent lava lakes at the surface

    Antarctica's only active volcano shows how CO2 allows volcanoes to form persistent lava lakes at the surface https://phys.org/news/2022-05-antarctica-volcano-co2-volcanoes-persistent.html
  38. M

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

    Proof: Suppose ## n>1 ## is an integer not of the form ## 6k+3 ##. Then we have ## n=6k ## for some ## k\in\mathbb{Z} ##. Thus ## n^{2}+2^{n}=(6k)^{2}+2^{6k} ## ## =36k^{2}+2^{6k} ## ## =2(18k^{2}+2^{6k-1}) ##...
  39. K

    I Form of potential operator of two interacting particles

    Considering two interacting particles in 3d, the corresponding Hilbert space ##H## is the tensor product of the two individual Hilbert spaces of the two particles. If the particle interaction is given by a potential ##V(\mathbf r_1 -\mathbf r_2)## ,what is the corresponding potential operator...
  40. M

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

    Proof: ## 17=2^4+1 ## Now I'm stuck. How should I find the correct answers to this problem?
  41. B

    Help me use my library of textbooks to form a study plan

    Hi everyone! Hope your week is going well. I'm an ex-physics and math student, now getting my PhD in mathematical biology, and I've recently come back to the subjects because I miss them and feel like it'd be fun to get proficient in some of this again. I've been mostly working on building my...
  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

    Find ##z## in the form ##a+bi## under Complex Numbers

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  45. M

    The only prime of the form n^2-4 is 5?

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  46. M

    The only prime of the form n^3-1 is 7?

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  47. wrobel

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  48. C

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  49. D

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  50. S

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