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In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. [1] A graphical illustration of a zero-dimensional space is a point. [2]
Spatial music is composed music that intentionally exploits sound localization. Though present in Western music from biblical times in the form of the antiphon , as a component specific to new musical techniques the concept of spatial music ( Raummusik , usually translated as " space music ") was introduced as early as 1928 in Germany.
Space music appears in many film soundtracks and is commonly played in planetariums. [21] According to Hill space music is an eclectic music produced almost exclusively by independent labels and it occupies a small niche in the marketplace, supported and enjoyed by a relatively small audience of loyal enthusiastic listeners. [22]
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 ...
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.
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 main difference with a sound sculpture is that a sound installation has a three-dimensional space and the axes with which the different sound objects are being organized are not exclusively internal to the work, but also external. [12] A work of art is an installation only if it makes a dialog with the surrounding space. [13]
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 ...