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The relation between local and global truncation errors is slightly different from in the simpler setting of one-step methods. For linear multistep methods, an additional concept called zero-stability is needed to explain the relation between local and global truncation errors.
Removing two notes for every one kept creates a new truncated mode of limited transposition. C F ♯ Keeping two notes for every one removed creates another truncated mode of limited transposition. C E F ♯ A ♯ Only Messiaen's mode 7 and mode 3 are not truncated modes: the other modes may be constructed from them or from one or more of their ...
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In music, a transformation consists of any operation or process that may apply to a musical variable (usually a set or tone row in twelve tone music, or a melody or chord progression in tonal music), or rhythm in composition, performance, or analysis.
In both the global and local cases, the concept of a strict extremum can be defined. For example, x ∗ is a strict global maximum point if for all x in X with x ≠ x ∗, we have f(x ∗) > f(x), and x ∗ is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x ∗ with x ≠ x ∗, we ...
It is used in tonal and atonal music, and is a common method of localized development and closure. Fragmentation is related to Arnold Schoenberg 's concept of liquidation , [ 1 ] a common compositional technique that describes the reduction of a large-scale musical idea to its essential form (such as a contour line, a specific harmonic motion ...
Music geography is the study of music production and consumption as a reflection of the landscape and geographical spaces surrounding it. It became evident that individuals associate music with space. [1] Historically, music was purely an oral tradition that was replaced by the introduction of radio broadcasting in the 1920s. [2]
Heavy layering or multitracking; in the context of pop and rock music, this may refer to the addition of elements such as chorused vocals or backing strings. Radio versions of songs pushed to be more "pop" through the use of loud drum beats or other instrumentation changes. Heavy use of pitch correction, time correction, or quantization.