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Function (music) In music, function (also referred to as harmonic function[1]) is a term used to denote the relationship of a chord [2] or a scale degree [3] to a tonal centre. Two main theories of tonal functions exist today: The German theory created by Hugo Riemann in his Vereinfachte Harmonielehre of 1893, which soon became an international ...
Musical form is the plan by which a short piece of music is extended. The term "plan" is also used in architecture, to which musical form is often compared. Like the architect, the composer must take into account the function for which the work is intended and the means available, practicing economy and making use of repetition and order. [10]
A function that is not monotonic. In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. [ 1 ][ 2 ][ 3 ] This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
For example, Seitenwechsel ("die Seiten wechseln" translates as "to exchange sides") mapped a triad on to its parallel minor or major, transforming C major to C minor and conversely. [7] Riemann's theory of transformations formed the basis for Neo-Riemannian theory , which expanded the idea of transformations beyond the basic tonal triads that ...
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences, i.e. sequences that are non- increasing, or non- decreasing. In its simplest form, it says that a non-decreasing bounded -above sequence of real numbers converges to ...
Musical symbols are marks and symbols in musical notation that indicate various aspects of how a piece of music is to be performed. There are symbols to communicate information about many musical elements, including pitch, duration, dynamics, or articulation of musical notes; tempo, metre, form (e.g., whether sections are repeated), and details about specific playing techniques (e.g., which ...
Cantor function. The graph of the Cantor function on the unit interval. In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure.
Free-space diagram of the red and the blue curve. In contrast to the definition in the text, which uses the parameter interval [0,1] for both curves, the curves are parameterized by arc length in this example. An important tool for calculating the Fréchet distance of two curves is the free-space diagram, which was introduced by Alt and Godau. [4]