Start Date: 07/05/2020
Course Type: Common Course |
Course Link: https://www.coursera.org/learn/musicianship-harmony
Explore 1600+ online courses from top universities. Join Coursera today to learn data science, programming, business strategy, and more.After a tremendous response from learners on Coursera, Berklee Online has created a Developing Your Musicianship specialization, and this course is the third course in the series. If you have a general understanding of music theory or if you have completed Developing Your Musicianship I and II, this course will continue to help you understand musical concepts, enabling you to create and perform contemporary music. Taught by Berklee College of Music professor George W. Russell, Jr., the course includes four lessons that delve into an intermediate level of harmony and ear training. The course will introduce you to new key signatures, and explore how they are constructed. You will continue to train your ear, learning to differentiate between the various intervals and chords that were explored in Developing Your Musicianship I and II. You will learn how to borrow chords from parallel tonalities (modal interchange), and how to write more common chord progressions. The course culminates with an assignment that asks you to compose and perform a composition using popular chord progressions. As with Developing Your Musicianship I and II, this course is designed to share the joy of creating music.
Welcome to Musicianship: Musicianship: Harmonic Function, Modal Interchange, and Tensions (also known as Developing Your Musicianship III)! Here we will cover all the details about the course and what you'll need to know to get the most out of your course experience.
Musicianship: Tensions, Harmonic Function, and Modal Interchange This course is about how to use basic harmonic and dissonant analysis in your playing, as well as in other areas of your life. We'll learn what these terms mean, what they mean in practice, and what they mean for your listeners. This is the first course in the Professional Musicianship Specialization. This course is important, because this is the area of music that you learn the most -- and least -- in school. This course will take you from beginner to advanced player, and from basic to advanced player. You'll learn more about musical concepts that are fundamental than you would in any other subject--music theory, music notation, or performance art. You'll also learn more about how to play and arrange music with other musicians. You’ll focus on how to use musical concepts to form a coherent and meaningful musical composition which you can then execute into your finished recording.Tensions, Harmonic Function, and Modal Interchange An Introduction to Harmonic Analysis Other Topics in Harmony Composition for Musicianship Musicianship: Chord Charts and Chord Progressions This course is about how to use basic harmonic and dissonant analysis in your playing, as well as in other areas of your life. We'll learn what they mean, what they mean in practice, and what they mean for your listeners. This is the second course in the Professional Musicianship Specialization
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Harmonic function | The singular points of the harmonic functions above are expressed as "charges" and "charge densities" using the terminology of electrostatics, and so the corresponding harmonic function will be proportional to the electrostatic potential due to these charge distributions. Each function above will yield another harmonic function when multiplied by a constant, rotated, and/or has a constant added. The inversion of each function will yield another harmonic function which has singularities which are the images of the original singularities in a spherical "mirror". Also, the sum of any two harmonic functions will yield another harmonic function. |
Harmonic function | in a weak sense (or, equivalently, in the sense of distributions). A weakly harmonic function coincides almost everywhere with a strongly harmonic function, and is in particular smooth. A weakly harmonic distribution is precisely the distribution associated to a strongly harmonic function, and so also is smooth. This is Weyl's lemma. |
Harmonic function | The uniform limit of a convergent sequence of harmonic functions is still harmonic. This is true because every continuous function satisfying the mean value property is harmonic. Consider the sequence on (−∞, 0) × R defined by formula_12. This sequence is harmonic and converges uniformly to the zero function; however note that the partial derivatives are not uniformly convergent to the zero function (the derivative of the zero function). This example shows the importance of relying on the mean value property and continuity to argue that the limit is harmonic. |
Harmonic function | denotes the characteristic function of the ball with radius "r" about the origin, normalized so that formula_16, the function "u" is harmonic on Ω if and only if |
Harmonic function | A "C" function that satisfies Δ"f" ≥ 0 is called subharmonic. This condition guarantees that the maximum principle will hold, although other properties of harmonic functions may fail. More generally, a function is subharmonic if and only if, in the interior of any ball in its domain, its graph lies below that of the harmonic function interpolating its boundary values on the ball. |
Harmonic function | The real and imaginary part of any holomorphic function yield harmonic functions on R (these are said to be a pair of harmonic conjugate functions). Conversely, any harmonic function "u" on an open subset Ω of R is "locally" the real part of a holomorphic function. This is immediately seen observing that, writing "z" = "x" + "iy", the complex function "g"("z") := "u" − i "u" is holomorphic in Ω because it satisfies the Cauchy–Riemann equations. Therefore, "g" has locally a primitive "f", and "u" is the real part of "f" up to a constant, as "u" is the real part of formula_13 . |
Harmonic function | Harmonic functions can be defined on an arbitrary Riemannian manifold, using the Laplace–Beltrami operator Δ. In this context, a function is called "harmonic" if |
Harmonic function | If "f" is a harmonic function on "U", then all partial derivatives of "f" are also harmonic functions on "U". The Laplace operator Δ and the partial derivative operator will commute on this class of functions. |
Harmonic function | A function (or, more generally, a distribution) is weakly harmonic if it satisfies Laplace's equation |
Harmonic function | In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function "f" : "U" → R (where "U" is an open subset of R) which satisfies Laplace's equation, i.e. |
Harmonic function | Harmonic functions that arise in physics are determined by their singularities and boundary conditions (such as Dirichlet boundary conditions or Neumann boundary conditions). On regions without boundaries, adding the real or imaginary part of any entire function will produce a harmonic function with the same singularity, so in this case the harmonic function is not determined by its singularities; however, we can make the solution unique in physical situations by requiring that the solution goes to 0 as you go to infinity. In this case, uniqueness follows by Liouville's theorem. |
Harmonic function | then "f" extends to a harmonic function on Ω (compare Riemann's theorem for functions of a complex variable). |
Harmonic function | Let "u" be a non-negative harmonic function in a bounded domain Ω. Then for every connected set |
Weakly harmonic function | for all formula_4 with compact support in formula_2 and continuous second derivatives, where Δ is the Laplacian. This is the same notion as a weak derivative, however, a function can have a weak derivative and not be differentiable. In this case, we have the somewhat surprising result that a function is weakly harmonic if and only if it is harmonic. Thus weakly harmonic is actually equivalent to the seemingly stronger harmonic condition. |
Harmonic function | The following principle of removal of singularities holds for harmonic functions. If "f" is a harmonic function defined on a dotted open subset formula_31 of R, which is less singular at "x" than the fundamental solution, that is |
Harmonic function | The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion. This solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as "harmonics". Fourier analysis involves expanding periodic functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit n-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and over time "harmonic" was used to refer to all functions satisfying Laplace's equation. |
Weakly harmonic function | In mathematics, a function formula_1 is weakly harmonic in a domain formula_2 if |
Positive harmonic function | The formula clearly defines a positive harmonic function with "f"(0) = 1. |
Harmonic function | If "f" is a harmonic function defined on all of R which is bounded above or bounded below, then "f" is constant (compare Liouville's theorem for functions of a complex variable). |
Harmonic function | One generalization of the study of harmonic functions is the study of harmonic forms on Riemannian manifolds, and it is related to the study of cohomology. Also, it is possible to define harmonic vector-valued functions, or harmonic maps of two Riemannian manifolds, which are critical points of a generalized Dirichlet energy functional (this includes harmonic functions as a special case, a result known as Dirichlet principle). This kind of harmonic maps appear in the theory of minimal surfaces. For example, a curve, that is, a map from an interval in R to a Riemannian manifold, is a harmonic map if and only if it is a geodesic. |