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. Logan Land

    MHB Find the following in x + iy form

    sinh(ln2+ ipi/3) so I have a general formula of sinh z= (e^z - e^-z)/2 so I obtained the following sinh= (e^(ln2 + ipi/3) - e^-(ln2 + ipi/3))/2 sinh= (e^(ln2) e^(ipi/3) - e^(-ln2) e^(ipi/3))/2 e^(ln2)=2 e^(-ln2)=1/2 e^(ipi/3)= (1/2)+(isqrt3/2) e^(-ipi/3)= (1/2)-(isqrt3/2) so when I plug...
  2. M

    Transforming predicate form to quantifiers

    Homework Statement Write the following statements in predicate form, using logical operators ^,∨, (NOT - negation but don't know where the symbol is :/) , and quantifiers ∀,∃. Below ℤ+ denotes all positive integers {1,2,3,...}. I need help with this first statement: For any x, y ∈ ℤ+ the...
  3. K

    MHB Show non-degenerate form of subspaces sum

    Let (E, d) be nonzero bilinear space over K and place conditions: d(x,y) = d(y,x) \\ d(x,y) = - d(y,x) for every x,y \in E. Show that: if E_1 and E_2 are singular (degenerate?) bilinear subspaces relative with d ( (E_1,d|(E_1 \times E_1) and (E_2,d|(E_2 \times E_2) are singular (degenerate?)...
  4. Logan Land

    Expressing complex numbers in the x + iy form

    Homework Statement ((1-i)/(sqrt2))^42 express in x+iy form Homework Equations z1/z1=(r1/r2)e^(i(theta1-theta2)) The Attempt at a Solution Ive found that (1-i) has r=sqrt2 so since r is sqrt2 and x=1 y=-1 so the angle is 7pi/4 so then I have (sqrt2e^(-i7pi/4)/sqrt2)^42 now from here is where I...
  5. 7

    Determine if all vectors of form (a,0,0) are subspace of R3

    I have the feeling that it is, but I am not really sure how to start the proof. I know I have to prove both closure axioms; u,v ∈ W, u+v ∈ W and k∈ℝ and u∈W then ku ∈ W. Do I just pick a vector arbitrarily say a vector v = (x,y,z) and go from there?
  6. David Carroll

    Calculators Graphing in polar form on the TI-81

    Greetings. I have been teaching myself Calculus. To do this I ordered a used Larson's 8th Edition Calculus and a used TI-81 graphing calculator. When I got to Chapter 10, I ran into a problem: the chapter introduces equations in polar form and when I whipped out my TI-81, I had no idea how to...
  7. F

    MHB How to find domain of function in implicit form

    What is the domain of y^2-2y=x^2-x-1? I don't know how to find it for implicit functions.
  8. binbagsss

    Levi-Civita Connection & Riemannian Geometry for GR

    Conventional GR is based on the Levi-Civita connection. From the fundamental theorem of Riemann geometry - that the metric tensor is covariantly constant, subject to the metric being symmetric, non-degenerate, and differential, and the connection associated is unique and torsion-free - the...
  9. J

    MHB Please help with one problem about writing ellipse in standard form?

    Write the equation for an ellipse with vertices (0,-3) and (0,3), minor axis of length 10. I know that the standard form of an ellipse is (x-h)^2/a^2 + (y-k)^2/b^2=1 Please help me ! Please! Thank you so much for your time, I appreciate it.
  10. Robsta

    Kinetic Theory Speed distribution (general form)

    Homework Statement A molecule has a velocity v and speed v. I've worked out (and understand) that the number of molecules in a gas with speeds between v and v+dv and moving at angles between Ө and Ө+dӨ to any chosen axis is: (1/2)nf(v)dvsin(Ө)dӨ The internet verifies this. f(v) is the speed...
  11. Coffee_

    Very basic form of the diffusion law for gasses

    I'm looking for a proof of the following statement at a level an early undergrad would understand: ##J=D \vec{\nabla} \vec{n}## where ##D=\frac{v_{th}l}{3}## with ##l## being the mean free path and ##v_{th}## the thermal agitation velocity, ##J## is the particle current density. I really did...
  12. B

    MHB Possible Echelon Form of Matrix

    Please check my solution.View image: Possible Echelon Form of Matrix
  13. B

    MHB Possible Echelon Form of a Matrix

    Please Check My Solution View image: Possible Echelon Form of Matrix
  14. evinda

    MHB What will be the form of the \$k^{th}\$ component of \$x^{(i)}\$?

    Hi! (Smile) Suppose that we index the components of the elements of $\mathbb{Z}_p$ by subscripts. Indexing the terms of the sequence by superscripts in parentheses$x^{(i)}$ is a term of the sequence, and $x^{(i)}_k$ its $k$-th component. So, if we have a sequence in $\mathbb{Z}_p$, it will be...
  15. bananabandana

    Hyperbola Focus Length Greater than Semi-Major Axis: Is it a Necessity?

    Homework Statement Why is it necessarily true that for a hyperbola, the focus length, ##f ## has got to be greater than the semi-major axis , ## a## - ## f >a ## ? Homework Equations - The Attempt at a Solution I needed to derive the cartesian equation of a hyperbola with centre at ##...
  16. gauss44

    Why does or doesn't a rainbow form?

    When light enters some pieces of glass from the air, such as a magnifying glass or window, rainbows usually don't form. But when light enters a prism, rainbows form. Why do rainbows form in the prism, but not in the magnifying glass or window?(This is my own personal curiosity and because I...
  17. Maged Saeed

    Question about indeterminate form

    why $$1^\infty$$ is indeterminate form?
  18. T

    Significance of Jordan Canonical Form

    I just finished a course on linear algebra which ended with Jordan Canonical Forms. There were many statements like "Jordan canonical forms are extremely useful," etc. However, we only learned a process to put things into Jordan canonical form, and that was it. What makes Jordan canonical...
  19. C

    General Solution to a certain form of ODE

    While fiddling around with some very simple linear ODEs, I "discovered" a formula that gives a solution to ODEs of the form: ##y'+y=ax^n ##. here it is: i'm sure that this was discovered before, but i was just wondering if it had any official name or something.
  20. binbagsss

    Form of Rienmann Tensor isotrpic & homogenous metric quick Question

    Context: Deriving the maximally symmetric- isotropic and homogenous- spatial metric I've seen a fair few sources state that the Rienamm tensor associated with the metric should take the form: * ##R_{abcd}=K(g_{ac}g_{bd}-g_{ad}g_{bc})## The arguing being that a maximally symmetric space has...
  21. P

    Norm of Vector Formed by Two Vectors

    If we have a vector $$\partial_v$$ and we want o find its norm, we easily say (According to the given metric) that the norm of that vector is:$$ g^{vv}\partial_v\partial_v$$. My question what if we have a vector that is combination of 2 vectors like: $$\phi =\partial_v + a\partial_x$$ where $a$...
  22. R

    How to turn these symmetric equations into the general form?

    I was solving this problem and I didn't want to do it the really long way by finding the equation of B(t) by first finding T(t) and N(t). So i took the cross product of r' and r'' so that they would be in the direction of B. Found the parametric equation of the plane but the book answer was in...
  23. B0b-A

    Do natural diamonds form from carbon-vapour or liquid-carbon

    What is the state of carbon just prior to it forming a diamond, deep in the Earth's crust ? Carbon vapour , or liquid carbon ?.Update: I think I've managed to answer my own question ... http://www-als.lbl.gov/images/stories/Science_Highlights/Highlights/108carbon1.png...
  24. perplexabot

    Matrix derivative of quadratic form?

    Homework Statement Find the derivative of f(X). f(X) = transpose(a) * X * b where: X is nxn a and b are n x 1 ai is the i'th element of a Xnm is the element in row n and column m let transpose(a) = aT let transpose(b) = bT Homework Equations I tried using the product rule...
  25. 3

    What Textbook Covers Hooke's Law in Tensor Form and Shear Stress?

    Hi! I'm studying physics and currently taking the first mechanics course. After dealing with rotation and gyroscopes, now we're working on things like shear stress, and Hooke's law in tensor form etc. I've got Kleppner/Kolenkow but shear stress, Hooke's law in tensor form and tensors in...
  26. L

    Binomial vs Geometric form for Taylor Series

    Homework Statement Sorry if this is a dumb question, but say you have 1/(1-x) This is the form of the geometric series, and is simply, sum of, from n = 0 to infiniti, X^n. I am also trying to think in terms of Binomial Series (i.e. 1 + px + p(p-1)x/2!...p(p-1)(p-2)(p-(n-1) / n!). 1/(1-x) is...
  27. DivergentSpectrum

    Is the Alternative Method for Integration by Parts Simpler?

    I have a question why everyone says ∫uv' dx=uv-∫u'v dx why don't they replace v' with v and v with ∫vdx and say ∫uv dx=u∫vdx-∫(u'∫vdx) dx i think this form is a lot simpler because you can just plug in and calculate, the other form forces you to think backwards and is unnecessarily complicated.
  28. D

    Geometry question form ax^2+bx+c

    I was tutoring a student and I came across the following question. I feel like I'm missing something obvious, but it seems like there are too many variables for an answer to be determined. The attached picture contains all of the question details.
  29. A

    MHB CTS and show the roots in this form

    I have to show the roots of x^{2}-8x-29=0 are c\pmd\sqrt{5} I used completing the square method. Once I used CTS I got the answer (x-4)^2-45=0 So I am not sure what is the next step to put it in the form of c\pmd\sqrt{5}
  30. caffeinemachine

    MHB Bilinear Form Non-Degenerate on a Subspace.

    I am trying to prove the following standard result:Let $V$ be a finite dimensional vector space over a field $F$ and $f:V\times V\to F$ be a symmetric bilinear form on $V$. Let $W$ be a subspace of $V$ such that $f$ is non-degenerate on $W$. Then $$V=W\oplus W^\perp$$(Here $W^\perp=\{v\in...
  31. TheFerruccio

    Integrating until symmetric bilinear form

    Homework Statement I am looking for some quick methods to integrate while leaving each step in its vector form without drilling down into component-wise integration, and I am wondering whether it is possible here. Suppose I have a square domain over which I am integrating two functions w and...
  32. I

    Find the following fourier series in trigonometric form

    Homework Statement Find the following Fourier series in trigonometric form. Homework Equations $$y(t)=a_0+\sum\limits_{n=1}^{\infty} a_n cos(n\omega_{0}t)+b_n sin(n\omega_{0}t)$$ The Attempt at a Solution The graph above is represented by the function: $$ x(t) = \left\{ \begin{array}{ll}...
  33. 2

    [CalcII/DiffEq] Closed form expression for f(x) which the series converges

    Homework Statement Find a closed form expression for the function f(x) which the power series Σn=0..∞ n(-1)nxn+1 converges to and determine the values of x for which f(x) equals the given power series. Homework Equations N/A The Attempt at a Solution I'm actually not sure how to start. First...
  34. PcumP_Ravenclaw

    Roots of unity, Roots of complex equations of the form z^n = 1

    Dear all, please see the page above, (Alan F, Beardon, Abstract Algebra and Geometry). On the page, Theorem 3.5.2 says that the set of Complex numbers from ## z^n = 1 ##, where ## |z| = 1 ## forms a group w.r.t multiplication. I want to know if... The inverse of all elements...
  35. V

    MHB Quadratic form, determine the surface

    Hi, I'm trying to solve this problem and I'm stuck. What I want to do is determine the kind of surface from this equation: x2-2y2-3z2-4xy-2xz-6yz = 11 Matrix representation: 1 -2 -1 -2 -2 -3 -1 -3 -3 I want to find the eigenvalues so I write the characteristic equation like...
  36. B

    Reduction of moisture content form propylene glycol

    Is it possible to reduce moisture content in Polyethylene glycol or propylene glycol by applying vacuum to around 15000 mTorr for 24 hours. If anyone has any idea please let me know
  37. twoski

    Can You Minimize Production Rules in Chomsky Normal Form?

    Homework Statement Grammar 'G' is any context-free grammar without any λ productions or unit productions. Let k be the max number of symbols on the right side of any production in P. Prove that there's an equivalent grammar in Chomsky Normal Form with no more than (k − 1)|P| + |T| production...
  38. K

    How Did the First Stars Form Without Supernova Compression?

    I was watching Nova Science Now and they were talking about how stars form when gas clouds are compacted when hit by supernova shockwaves, which then allows them to compress further by their own gravity and eventually they ignite. But I was left wondering how then the first stars were able to...
  39. M

    MHB Program that accepts inputs of the form 1^n 2^{n^2} 0

    Hey! :o I have to write a RAM program to accept all inputs of the form $1^n 2^{n^2} 0$. The instructions of the RAM machine are the following: I have done the following: Read x from input d=0, s=0 while x!=0 if x-1=0 d=d+1 else s=s+1 Read x from...
  40. L

    Need help using the atomic form factor

    Homework Statement I need to calculate interference peaks using the atomic form factor for several lattice types for a lab report. I was given the expected answers but told I should also be able to calculate them.Expected results for rocksalt (eg NaCl): h+k+l = 2n fh+k+l = 4(fa+fb) h+k+l =...
  41. E

    Will (NH4)HCO3 Form in Polar Solvents?

    I know aqueous Ammonium Bicarbonate forms when NH4+ and HCO3- ions are present in water after they've dissolved from their gaseous states of NH3(g) and CO2(g). This occurs in the reaction: NH3(g) + H2O(l) + CO2(g) => (NH4)HCO3(aq) If Ammonia gas and CO2 are present above a polar organic...
  42. E

    Exploring Alternate Forms of cos(nπ) in Fourier Series

    I saw somewhere that an alternate form of cos(n×π) was cos(n×π) = -1n+1 But to me this does not make sense. Am I wrong? For n = 0 cos(n×π) = 1 -1n+1 = -1 For n = 1 cos(n×π) = -1 -1n+1 = 1 etc. Is there another way to express cos(n×π) in an alternate form? PS. This is related to Fourier series.
  43. Math Amateur

    MHB Elements of the form a/b in an integral domain - simple question

    In Alaca and Williams' (A&W) book: Introductory Algebraic Number Theory , Theorem 1.2.1 reads as follows:Without any earlier definition or clarification, A&W refer to a/p and b/p (see text above) ... BUT ... how should we regard such elements? What exactly do they mean and how do we know they...
  44. J

    Closed Form Equations for Elasticity Properties for Anisotropic Materials

    Hi all, I'm wondering if anyone knows of a way to obtain elasticity properties (Ex, Ey, Ez, Gxy, Gxz, Gyz, vxy, vxz, vyz) from the terms of a 6x6 anisotropic stiffness or compliance matrix. I'm looking for a closed form solution, preferably. I would think that there should be a closed form...
  45. adjacent

    Calculate (4.3 x 10^8) + (2.5 x 10^7) in Standard Form | Homework Problem

    Homework Statement Calculate ##(4.3 \times 10^8)+(2.5 \times 10^7)## Give your answer in standard form The total marks for the question is 2 Homework EquationsThe Attempt at a Solution The answer is ##4.55 \times 10^8## because that's what my calculator gave. That will score me one mark. I am...
  46. L

    Complex numbers in trigonometry form

    Homework Statement Write down number 1+i and 1+i\sqrt{3} in trigonometry form.[/B]Homework Equations For complex number z=x+iy \rho=|z|=\sqrt{x^2+y^2} \varphi=arctg\frac{y}{x} And [/B]The Attempt at a Solution Ok. For z=1+i \rho=\sqrt{1+1}=\sqrt{2}...
  47. Nova

    Why do galaxies and solarsystems form disks?

    In galaxy formations/collisions and planetary dust from newborn solar systems, they begin with an irregularly shaped cloud of matter. But why, over their lifetime, do they form a disk shape? In the case of spiral galaxies and planet orbits.
  48. L

    Atomic Form Factor Homework: NaCl, LiF, GaP, Si

    Homework Statement I am trying to calculate the atomic form factor for various crystals (NaCl, LiF, GaP, Si) in an x-ray diffraction experiment. There seems to be very little information on this topic anywhere.Homework Equations [/B] I understand how to derive the geometric form factor Sk =...
  49. Hijaz Aslam

    Defining a vector in magnitude,angle form.

    I've learned that vectors can be defined in,basically, two ways. (1) Unit Vector Form (xi+yj+zk) (2)Magnitude angle form. The second form always adds a confusion in problems. For instance let's define a function A and B as: When we define -B why do we do it as follows: I mean why do we...
  50. SteliosVas

    How do I convert 2cis(-pi/3)cis(pi/6) into cartesian form?

    Homework Statement Convert 2cis(-pi/3)cis(pi/6) into cartesian form. Show all working to obtain full marks Homework Equations I know that the equation for it is 2((cos(theta) +isin(theta))+(cos(theta)+isin(theta))) The Attempt at a Solution Okay so cos of (-p/3) = 1/2 Sin of (-p/3) =...
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