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Zero-dimensional Polish spaces are a particularly convenient setting for descriptive set theory. Examples of such spaces include the Cantor space and Baire space. Hausdorff zero-dimensional spaces are precisely the subspaces of topological powers where = {,} is given the discrete topology.
The first two, the "dependency locality theory" and the "expectancy theory" refer to syntactic processing in language, whereas the third one, the "tonal pitch space theory", relates to the syntactic processing in music. The language theories contribute to the concept that in order to conceive the structure of a sentence, resources are consumed.
In 2005, Robert F. Port and Adam P. Leary published an argument against the existence of a fixed phonetic inventory. They presented the idea of a phonetic space as unrealistic in terms of the broadness of languages present and more specifically that languages are not consistent in distinctness, discreteness, or temporal patterns, even within the same language. [6]
An open cover of a topological space X is a family of open sets U α such that their union is the whole space, U α = X. The order or ply of an open cover A {\displaystyle {\mathfrak {A}}} = { U α } is the smallest number m (if it exists) for which each point of the space belongs to at most m open sets in the cover: in other words U α 1 ∩ ...
The simplest pitch space model is the real line. A fundamental frequency f is mapped to a real number p according to the equation = + (/) This creates a linear space in which octaves have size 12, semitones (the distance between adjacent keys on the piano keyboard) have size 1, and middle C is assigned the number 60, as it is in MIDI. 440 Hz is the standard frequency of 'concert A', which ...
In geometry, a point is an abstract idealization of an exact position, without size, in physical space, [1] or its generalization to other kinds of mathematical spaces.As zero-dimensional objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional curves, two-dimensional surfaces, and higher-dimensional objects consist; conversely ...
Examples of musical lattices include the Tonnetz of Euler (1739) and Hugo Riemann and the tuning systems of composer-theorists Ben Johnston and James Tenney. Musical intervals in just intonation are related to those in equal tuning by Adriaan Fokker's Fokker periodicity blocks. Many multi-dimensional higher-limit tunings have been mapped by Erv ...
For example, motion between a C major and E minor triad, in either direction, is executed by an "L" transformation. Extended progressions of harmonies are characteristically displayed on a geometric plane, or map, which portrays the entire system of harmonic relations.