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. Lolligirl

    MHB Finishing converting to Chomsky Normal Form from a CFG?

    Hello everyone! Here is a question I am working on: Consider the context-free grammar G = ( { S, B, E }, { 0, 1, i, e, s }, R, S ), where R is: S --> iBSE | s B --> 0 | 1 E --> lambda | eS Convert this to Chomsky Normal Form, showing all steps. Alrighty, so I removed the lambda and got: S0...
  2. B

    What form for the particular integral of this 2nd order Diff Eqn

    Hi Folks, I have the following forced torsional vibration problem of the form ##\displaystyle J_0 \ddot{\theta}+k_t\theta=\frac{a_0}{2}+\sum_{n=1}^{\infty} (a_n \cos n w t+b_n \sin n w t)## I assume the solution of the CF is in the form ##\theta=A \cos nwt+B\sin nwt## but I am not sure what...
  3. R

    Why is mild steel appear in black or bright form?

    Moderator note: The OP was warned that he needed to do more research before asking questions like this. So mild steel appear to have black mild steel or bright mild steel... What are the differences between them... And bright mild steel appear to have higher quality than black mild steel... In...
  4. SteliosVas

    Converting to Polar and Cartesian form

    Mod note: This post with template not used and no effort shown received a warning. Okay I am totally confused in this. This is not a homework question but rather one I saw online and was wondering for example how to solve it The question was -3-i/-8+6i to be expressed into Cartesian form...
  5. T

    Proof of equivalence between nabla form and integral form of Divergence

    Does anybody knows how you can reach one form of the divergence formula from the other? Or in general, why is the equivalence true?
  6. B

    Differential form of gauss's law.

    I don't understand what charge density is meant in the equation: div E = constant times charge density. I have the derivation in front of me and the last step follows from accepting that the rate of change of the integral of the field divergence per change in volume is the same as the rate of...
  7. skate_nerd

    Small question about atomic form factor calculation

    Homework Statement This problem just has me find an atomic form factor for some arbitrary basis atom in a bravais lattice where the electron wave function is given (it has a dependence on the Bohr radius in an exponential). I calculated the form factor (a very long, nasty integral that...
  8. V

    Prove that all integrable functions form a vector space

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  9. 1

    Converting an Equation to Neutral Form

    Homework Statement Hi. I have convert the following equation (which is in ionic form) into neutral form: 4S2O32-+O2+2H2O=2S4O62-+4OH- The question tells me to use calcium as a counterion for all anions. Homework Equations N/A The Attempt at a Solution I basically just added calcium to the...
  10. B

    Proton Charge Distribution and Form Factor Problem

    Homework Statement Hi all - I have been trying to evaluate part II of this problem for a long time now... For a simplified model of a proton's charge distribution, Find the constant of proportionality required to normalise ρ correctly. Show that Homework Equations N/A The Attempt at a...
  11. P

    Non-canonical form into canonical transformation 1-d partial dif.

    Homework Statement Problem 29. Use the subtraction trick U(tilda) = U−U1 to reduce the following problems with non-canonical boundary conditions to the canonical ones and write down the equations in terms of the variable ˜u (do not solve them). Note that there are infinitely many u1’s that...
  12. F

    MHB How to derive a recurrence relation from explicit form

    I am given a formula in explicit form and as a recurrence relation. It is asked to derive the recurrence relation from the explicit form. How is this done?
  13. E

    Complex form of Fourier series

    Let function $f(t)$ is represented by Fourier series, $$\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos{\frac{2n\pi t}{b-a}}+b_n\sin{\frac{2n\pi t}{b-a}}),$$ $$a_0=\frac{2}{b-a}\int_{a}^{b}f(t)dt,$$ $$a_n=\frac{2}{b-a}\int_{a}^{b}f(t)cos\frac{2n\pi t}{b-a}dt,$$...
  14. C

    Integrating (x^4+x^2)^0.5 form 3^0.5 to -3^0.5

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  15. M

    Writing 3rd Order Tensor Symmetric Part in Tensor Form

    Can some one write for me the Symmetric part of a third order tensor (as a tensor form) Thanks .
  16. TheFerruccio

    Weak Form with Weighted Residuals

    Homework Statement Given a strong form boundary value problem, derive the weak form using weighted residuals. Homework Equations ##(2-x)u''(x)-u'(x)+u(x)=f(x)## for ##x\in(0,1)## with $$u(0)=u(1)=0$$ The Attempt at a Solution I must multiply both sides of this equation by an arbitrary test...
  17. bartekac

    Acoustic waves - factors determining their form

    Hi, I am not a native English Speaker, so some words might not be appropriately used below, but I will try my best to explain what I was thinking about. In general, I have never learned how acoustic waves emerge microscopically. The application of the theoretical knowledge I acquired was always...
  18. stevendaryl

    Infinite Sum of Powers: Is There a Closed Form for the Series?

    This isn't quite a calculus question, but it didn't seem right for any of the other mathematics forums, either. Does anybody if there is a closed form for the following infinite series: \sum_n x^{n^2} for 0 < x < 1
  19. V

    Proof That Universe Can Form Spontaneously From Nothingness

    "A Mathematical Proof That The Universe Could Have Formed Spontaneously From Nothing Cosmologists assume that natural quantum fluctuations allowed the Big Bang to happen spontaneously. Now they have a mathematical proof" Okay, but "natural quantum fluctuations" is STILL something it seems...it...
  20. B

    What Are the Units of b in a Damped Harmonic Oscillator Equation?

    Homework Statement Hello everyone, first time poster. I am in an Analytical Mechanics class and we are covering Dimensionless Form and Free Fall and am struggling with some of the concepts. The question has multiple parts, but looks like this. The equation for a damped harmonic oscillator is...
  21. M

    Segment of a circle (Exact form answer)

    Homework Statement Find the circle segment area that has the boundaries of line segment AB and the minor arc ACB. Give the area in an exact form in terms of surds and Pi. (see attachments for annotated picture & original question). Homework Equations Equation 1: Area of a segment = Sector...
  22. J

    What is the work required to push three protons to form an nucleus?

    Hello there, Imagine that a nucleus consists of three atoms arranged in a equilateral triangle with the length of each side, ##R=2 \rm fm##. Our protons starts infinitely far away. What is the work required to push these protons together in order to overcome the electric force between them? I...
  23. A

    Osmotic power - a new form of renewable energy

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  24. A

    Specific heat capacity of liquid form of permanent gases.

    Hi, The permanent gases like Nitrogen, Helium etc. have more specific heat capacity as liquid than their gases. Seemingly degree of freedom should reduce in liquid form, and therefore, specific heat capacity must reduce in liquid form. But this isn't the case. I remember reading somewhere...
  25. M

    What would be general formula of a circle in form of variables ?

    General formula of a line is ax+b=0 Similarly can we have a general formula of a circle ?
  26. P

    Complex Numbers converting from Polar form to Acos(wt + x)

    Homework Statement "Put each of the following into the form Acos(ωt+θ)..." (a.) 4ejt+4e-jt Homework Equations Euler's Identity: ejθ = cos(θ)+jsin(θ) Phasor Analysis(?): Mcos(ωt+θ) ←→ Mejθ j = ej π/2 Trignometric Identities The Attempt at a Solution I attempted to use phasor analysis to...
  27. J

    Limit gives indeterminate form

    Homework Statement limx->∞ e^5x/ln(2x) Homework Equations The Attempt at a Solution
  28. mesa

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    For certain types of functions I have been using the term 'general form' as a way of stating that some function of some variable(s) is true for more than one value of the variable (usually an infinite number). Here is a simple example, 1=cos(2∏x) There have been (at times) where the use of...
  29. B

    Lie derivative of contraction and of differential form

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  30. J

    Can a Unitary Operator Be Expressed in Exponential Form?

    Homework Statement I'm working on this problem: Let \hat{U} an unitary operator defined by: \hat{U}=\frac{I+i\hat{G}}{I-i\hat{G}} with \hat{G} hermitian. Show that \hat{U} can be written as: \hat{U}=Exp[i\hat{K}] where \hat{K} is hermitian. Homework Equations...
  31. A

    How can electrons form a wave?

    an air wave takes place in volumes of air, water takes place in volumes of water... but u can have an electromagnetic wave go through empty space using only one particle. how does this work? this to me does not seem analogous to the waves that i am used to. does this "wave" instead refer to the...
  32. M

    Relating integral of powers of Sin b/w 0 and pi/2 to factorial form

    Our integral \int\limits_0^{\pi/2} \sin^{2a+1}(x)\,dx Has a Factorial Form: {(2^a a!)}^2 \over (2a+1)! What is the process behind going from that integral to that factorial form? My approach which is not very insightful: I used mathematica to calculate the integral to return...
  33. R

    Converting an explicit series to a recursive form

    Let e_i be a unit vector with one 1 in the i-th element. Is the following expression has a recursive presentation? $$y_N = \sum_{k=0}^N {\frac{{{X^k} e_i}}{\|{{{X^k} e_i}\|}_2}} $$ where X is a n \times n square matrix, and {\| \cdot \|}_2 is a vector norm defined as {\|z\|}_2 =...
  34. T

    Question on How energy equation is presented in Diff form

    Hi guys , I'm currently reading a book and I faced this with DEs: In the 5 attached images you'll find the textbook paragraph I'm asking about . I think I understood the first part under the title of : meaning of differential equation but I can't figure out how did he proceed with the same...
  35. JonnyMaddox

    What is the generalization of the tensor product definition for a three form?

    Hello, the tensor product definition of a two form is \alpha^{1} \wedge \beta^{1} := \alpha \otimes \beta - \beta \otimes \alpha \alpha \wedge \beta(v,w) = \beta(v)\alpha(w)-\beta(w)\alpha(v) But what is the definition in this sense for a three form?
  36. C

    Significance of standard form of 1st and 2nd order transfer functions?

    I foolishly skipped most of my analogue electronics classes, and inevitably failed the exam. I'm now trying to revise for the resit but I'm so far behind that I just cannot understand any of the lecture slides, and I'm getting very stressed. The part of the module I am revising at the moment...
  37. ChrisVer

    Scalar Propagator form with width

    I would like to ask when can someone add the width in a scalar particle's propagator. In general the scalar propagator can be: \frac{1}{k^{2}-m^{2}+i \epsilon} (\epsilon \rightarrow 0) However I read somewhere that if necessary one can include a width for the propagator...
  38. B

    Fermilab Muon g-2: How Do Muons Form?

    I was reading about Fermilab moving their new storage ring to the Muon Campus for the Muon g-2 experiment. I was curious about how the produce the Muons. I understand that protons hit a graphite target producing pions that quickly decay into Muons. How much energy are is required? How much...
  39. C

    Maurer-Cartan form involved in Lie bracket

    The Maurer-Cartan one-form ##\Theta = g^{-1} dg## is though of as a lie algebra valued form. It arises in connection with Yang-Mill's theory where the gauge potential transforms as $$A \mapsto g Ag^{-1} - g^{-1} dg.$$ However, one also defines for lie-algebra valued differential forms...
  40. N

    How to rewrite the provided sum in another form?

    Homework Statement How to get from Sum of 2(cos((3pi)/(2^(k+1)))sin(pi/(2^(k+1)))) from k = 1 to infinity to Sum of sin((4pi)/(2^(k+1))) - sin((2pi)/(2^(k+1))) from k = 1 to infinity The two expressions are equivalent. I need help getting from the first expression to the second.
  41. Math Amateur

    MHB How Can Pi Be Represented as a Continued Fraction?

    Iam reading Julian Havil's book, The Irrationals: A Story of the Numbers You Can't Count On. In Chapter 3 Havil is writing about progress in the eighteenth century in determining the nature of \pi and e through the use of continued fractions. He writes (pages 92 - 93): Can someone please...
  42. S

    Variation in tube wall temperature as steam condenses to form liquid

    I'm having a bit of difficulty conceptually understanding a problem. Hoping someone here can clear it up for me. In a simple circular condenser tube (which is cooled by water or air) what happens the tube wall temperature as the steam inside is condensed? Intuitively I would have thought that...
  43. Greg Bernhardt

    What are the commonly encountered indeterminate forms in algebraic functions?

    Definition/Summary An algebraic function of a pair of numbers is an indeterminate form at a particular pair of values (which may include infinity) if the function does not tend to a unique limit at that pair of values. For example, \infty\ -\ \infty is an indeterminate form because the...
  44. gfd43tg

    Standard form of linear program

    Homework Statement To preface, I believe this question is equally applicable to either the math or engineering homework forum, but I am having more trouble with the math part than the actual programming. I can't do any programming (with the understanding that linear programming doesn't...
  45. C

    Homogenous Differential Equations (Converting Back into Explicit Form

    Homework Statement x*e^(y/x) + y dx = xdy, y(1) = 0 Homework Equations The Attempt at a Solution To solve, I divide everything by x dx to put everything in terms of v. e^v + v = dy/dx dy/dx = x dv/dx + v e^v + v = x dv/dx + v e^v = x dv/dx e^v / dv = x/dx Flip both sides. e^-v dv =...
  46. resurgance2001

    Understanding One Forms in General Relativity: A Beginner's Guide

    Hi I am a new(ish) student of general relativity. I am currently reading 'Relativitiy DeMystified' However this guys explanation of one forms is completely mystifying to me. He says that basis vectors e_a = ∂_a = {\frac{∂a}{∂X^a}} And then says that this type of basis is...
  47. W

    Is Every Diff. Form on a Submanifold the Restriction of a Form in R^n?

    (From another site) I think the answer is no, for ##i: S \rightarrow \mathbb R^n ## the inclusion/restriction, and ##w## any form, we have that ##i_{*}dw=d(i_{*}w )## , but the homology of ##\mathbb R^n## is trivial(i.e., every closed form is exact), so that we can write ##w= d(\alpha)##, for...
  48. Z

    Closed form integral of abs(cos(x))

    Hi everyone. Recently, I came across a closed form solution to ∫|cos(x)|dx as sin(x-∏*floor(x/∏+1/2)) + 2*floor(x/∏+1/2) I have no idea how to reach this solution but checking this for definite integral from 0 to 3∏/4 or ∏ seems to work. Using |cos(x)| as cos(x)*sgn(cos(x)) doesn't...
  49. TheSodesa

    Form some third degree equation that has the roots -3, -1 and 2.

    Homework Statement What the title says. There's a b part to the problem, but of course I can't move on to it until I understand what is going on here. Homework Equations A third degree polynomial is of the form f(x) = ax3 + bx2 + cx + d This information was not given in the...
  50. A

    Why do clouds cast shadows in space?

    I am guessing this is due to diffraction due to the small size of the molecules of air. But then, we never see diffraction effects (interference) for any arrangement or configuration of molecules in space?
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