The idea of pitch space goes back at least as far as the ancient Greek music theorists known as the Harmonists. To quote one of their number, Bacchius, "And what is a diagram? A representation of a musical system. And we use a diagram so that, for students of the subject, matters which are hard to grasp with the hearing may appear before their eyes." (Bacchius, in Franklin, Diatonic Music in Ancient Greece.) The Harmonists drew geometrical pictures so that the intervals of various scales could be compared visually ...

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Pitch space, Pitch space - History of pitch space, Pitch space - Fibered pitch space, Pitch space - External link

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In graph theory, certain vector spaces over the two-element field Z2 are associated with an undirected graph; this allows one to use the tools of linear algebra to study graphs. Let G be a finite simple undirected graph with edge set E. The power set of E becomes a Z2-vector space if we take the symmetric difference as addition. Every element of this vector space can be thought of as a linear combination of edges with coefficient from Z2. In yet another interpr ...

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Cycle space, Cycle space - The binary cycle space, Cycle space - The integral cycle space, Cycle space - The cycle space over a field or commutative ring

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Given a set X: the discrete topology on X is defined by letting every subset of X be open, and X is a discrete topological space if it is equipped with its discrete topology; the discrete uniformity on X is defined by letting every superset of the diagonal {(x,x) : x is in X} in X × X be an entourage, and X is a discrete uniform space if it is equipped with its discrete uniformity. the d ...

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Discrete space, Discrete space - Definitions, Discrete space - Properties, Discrete space - Uses, Discrete space - Indiscrete spaces, Discrete space - Quotation, Discrete space - Notes

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Spaced - Tim Bisley. (Played by Simon Pegg): Tim, an aspiring comic book artist, amateur skateboarder and passionate follower of cult fiction in many forms (including video games, science fiction and especially - at least initially - the original Star Wars trilogy), is a rather grumpy and short-tempered soul, mostly because his girlfriend Sarah broke his heart and dumped him after an affair with her boss, Duane Benzy. He's currently writing and illustrating a graphic novel about a boy who has been transfor ...

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Spaced, Spaced - Situation, Spaced - Main characters, Spaced - Tim Bisley, Spaced - Daisy Steiner, Spaced - Marsha Klein, Spaced - Brian Topp, Spaced - Mike Watt, Spaced - Twist Morgan, Spaced - Episode guide, Spaced - Series 1, Spaced - Series 2, Spaced - Style and references, Spaced - Quotes, Spaced - Trivia, Spaced - Future

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Spaced - Series 1. Beginnings - Daisy and Tim meet in a small cafe and bond over their mutual search for accommodation (he's been kicked out by girlfriend Sarah, she's squatting in a pit), and gradually form a friendship. Just when all seems lost, they stumble upon what seems to be a perfect place - trouble is, it's listed as being for a 'professional couple' only. Thus begins a complicated plan involving faked photos and memorizing every significant (and not-so-significant) fact about each other in ...

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Spaced, Spaced - Situation, Spaced - Main characters, Spaced - Tim Bisley, Spaced - Daisy Steiner, Spaced - Marsha Klein, Spaced - Brian Topp, Spaced - Mike Watt, Spaced - Twist Morgan, Spaced - Episode guide, Spaced - Series 1, Spaced - Series 2, Spaced - Style and references, Spaced - Quotes, Spaced - Trivia, Spaced - Future

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References to popular culture - particularly but not exclusively to science fiction and horror films, comic books and video games - abound in Spaced to the extent that the DVD of Series 2 includes the "Homage-o-meter", an alternative set of subtitles listing every reference and homage; for the "Definitive Collectors Edition" DVD boxed set, the Homage-o-meter was added to the first series as well. Providing the artwork for Tim's comic 'The Bear', drawings and doodles were 2000 AD's artists Jim Murray and Jason Brashill, who also ...

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Spaced, Spaced - Situation, Spaced - Main characters, Spaced - Tim Bisley, Spaced - Daisy Steiner, Spaced - Marsha Klein, Spaced - Brian Topp, Spaced - Mike Watt, Spaced - Twist Morgan, Spaced - Episode guide, Spaced - Series 1, Spaced - Series 2, Spaced - Style and references, Spaced - Quotes, Spaced - Trivia, Spaced - Future

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All Hilbert spaces are reflexive, as are the Lp spaces for 1 < p < ∞. More generally: all uniformly convex Banach spaces are reflexive according to the Milman-Pettis theorem. Montel spaces are reflexive. ...

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Reflexive space, Reflexive space - Definition, Reflexive space - Examples, Reflexive space - Properties, Reflexive space - Implications

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Hilbert spaces were named after David Hilbert, who studied them in the context of integral equations. The origin of the designation "der abstrakte Hilbertsche Raum" is John von Neumann in his famous work on unbounded Hermitian operators published in 1929. Von Neumann was perhaps the mathematician who most clearly recognized their importance as a result of his seminal work on the foundations of quantum mechanics begun with Hilbert and Lothar (Wolfgang) Nordheim and continued with Eugene Wigner. The name "Hilbert space" was soon adopted by oth ...

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Hilbert space, Hilbert space - Introduction, Hilbert space - Definition, Hilbert space - Examples, Hilbert space - Euclidean spaces, Hilbert space - Sequence spaces, Hilbert space - Lebesgue spaces, Hilbert space - Sobolev spaces, Hilbert space - Operations on Hilbert spaces, Hilbert space - Bases, Hilbert space - Orthogonal complements and projections, Hilbert space - Reflexivity, Hilbert space - Bounded operators, Hilbert space - Unbounded operators

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Locally ringed spaces have just enough structure to allow the meaningful definition of tangent spaces. Let X be locally ringed space with structure sheaf OX; we want to define the tangent space Tx at the point x ∈ X. Take the local ring (stalk) Rx at the point x, with maximal ideal mx. Then kx := Rx/mx is a field and mx/mx2 is a vector space over that field (the cotangent space). The tangent space RxSee also:

Ringed space, Ringed space - Definition, Ringed space - Examples, Ringed space - Morphisms, Ringed space - Tangent spaces, Ringed space - OX modules

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The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices which belong to one of 7 crystal systems. This results in a space group being some combination the translational symmetry of a unit cell including lattice centering, and the point group symmetry operations of reflection, rotation and rotoinversion (also called improper rotation). Furthermore one must consider the screw axis and glide plane symmetry operations. These are called compound symmetry operatioans and ...

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Space group, Space group - Space groups in crystallography, Space group - Glide planes and screw axes, Space group - Notation, Space group - Group theory, Space group - Space groups in various dimensions, Space group - Grouping space groups by point group, Space group - Further categorizing of space groups

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A topological space is a set X together with a collection T of subsets of X satisfying the following axioms: The empty set and X are in T. The union of any collection of sets in T is also in T. The intersection of any pair of sets in T is also in T. The set T is a topology on X. The sets in T are the open sets, and their complements in X are the closed sets. Th ...

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Topological space, Topological space - Definition, Topological space - Comparison of topologies, Topological space - Continuous functions, Topological space - Alternative definitions, Topological space - Examples of topological spaces, Topological space - Topological constructions, Topological space - Classification of topological spaces, Topological space - Topological spaces with algebraic structure, Topological space - Topological spaces with order structure

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A topological space is a set X together with a collection T of subsets of X satisfying the following axioms: The empty set and X are in T. The union of any collection of sets in T is also in T. The intersection of any pair of sets in T is also in T. The collection T is a topology on X. The sets in T are the open sets, and their complements in X are the closed sets. Th ...

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Topological space, Topological space - Definition, Topological space - Comparison of topologies, Topological space - Continuous functions, Topological space - Alternative definitions, Topological space - Examples of topological spaces, Topological space - Topological constructions, Topological space - Classification of topological spaces, Topological space - Topological spaces with algebraic structure, Topological space - Topological spaces with order structure

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An important concept is that of an orthonormal basis of a Hilbert space H: this is a family {ek}k ∈ B of H satisfying: Elements are normalized: Every element of the family has norm 1: ||ek|| = 1 for all k in B Elements are orthogonal: Every two different elements of B are orthogonal: <ek, ej> = 0 for all k, j in B with k ≠ j. Dense span: The linear span o ...

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Hilbert space, Hilbert space - Introduction, Hilbert space - Definition, Hilbert space - Examples, Hilbert space - Euclidean spaces, Hilbert space - Sequence spaces, Hilbert space - Lebesgue spaces, Hilbert space - Sobolev spaces, Hilbert space - Operations on Hilbert spaces, Hilbert space - Bases, Hilbert space - Orthogonal complements and projections, Hilbert space - Reflexivity, Hilbert space - Bounded operators, Hilbert space - Unbounded operators

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The terms "Hausdorff", "separated", and "preregular" can also be applied to such variants on topological spaces as uniform spaces, Cauchy spaces, and convergence spaces. The characteristic that unites the concept in all of these examples is that limits of nets and filters (when they exist) are unique (for separated spaces) or unique up to topological indistinguishability (for preregular spaces). As it turns out, uniform spaces, and more generally Cauchy spaces, are always preregular, so the Hausdorff condition in these cases reduces t ...

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Hausdorff space, Hausdorff space - Definitions, Hausdorff space - Examples and counterexamples, Hausdorff space - Properties, Hausdorff space - Preregularity versus regularity, Hausdorff space - Variants, Hausdorff space - Joke

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Analogous to the notion of complete metric space, one can also consider completeness in a uniform space. Instead of working with Cauchy sequences, one works with Cauchy nets or Cauchy filters. A Cauchy filter F on a uniform space X is a filter F such for every entourage U, there exists A∈F such that A×A ⊆ U. A uniform space is called ...

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Uniform space, Uniform space - History, Uniform space - Definition, Uniform space - Entourage definition, Uniform space - Uniform cover definition, Uniform space - Pseudometrics definition, Uniform space - Intuition, Uniform space - Examples, Uniform space - Uniformly continuous functions, Uniform space - Topology of uniform spaces, Uniform space - Completeness

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On the first day of April, 1985, Seattle area TV program Almost Live! set up a phony broadcast room and dressed actors as TV anchors to pull an April Fool's joke of legendary proportions. The mock newscast reported that the Space Needle had collapsed in a windstorm, featuring footage of downed parts of the needle and hysterical eyewitness accounts (also by actors). Local hospitals prepared for the i ...

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Space Needle, Space Needle - Architecture, Space Needle - History, Space Needle - Admission, Space Needle - Jumpers, Space Needle - Collapse of the Space Needle, Space Needle - Cultural references

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The Space Wolves are known for their fiercely anti-authoritarian behaviour. They strongly resist the central command structure of the Imperium, and refuse to follow the dictates of the Codex Astartes, which lays down the structure and tactics to be used by all Space Marines. Space Wolves - The Great Companies. Instead, of dividing into Chapters as per the Codex Astartes, the Space Wolves Legion split itself into twelve Great Companies, with a thirteenth named in honour of a large group of Space Wolves who had disappeared during the Great Crusade. Each of these is led by a Wolf Lord, who a ...

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Space Wolves, Space Wolves - History, Space Wolves - Leman Russ, Space Wolves - The Great Crusade, Space Wolves - After the Heresy, Space Wolves - Organization, Space Wolves - The Great Companies, Space Wolves - The Great Wolf, Space Wolves - Progression of a Space Wolf, Space Wolves - The Thirteenth Company, Space Wolves - Appearance, Space Wolves - Notable Members, Space Wolves - Background Inspiration

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More affordable space tourism is viewed as a money-making proposition by several companies, including Virgin Galactic, Blue Origin, Armadillo Aerospace, XCOR Aerospace, Rocketplane Limited, and others. Most are proposing vehicles that make suborbital flights peaking at an altitude of 100-160 kilometres. Passengers would experience three to six minutes of weightlessness, a view of a twinkle-free starfield, and a vista of the curved Earth below. Projected costs are expected to be in the range of $100,000-$200,000 per passenger, with costs dropping over time to $20,000 or less. In December of 2005, The U.S. Governme ...

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Space tourism, Space tourism - Early dreams, Space tourism - Subsidiary government flights, Space tourism - Commercial space flights, Space tourism - Space hotels

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Formally, a ringed space is a topological space X together with a sheaf of commutative rings OX on X. The sheaf OX is called the structure sheaf of X. A locally ringed space is a ringed space (X, OX) such that all stalks of OX are local rings (i.e. they have unique maximal ideals). Note that it is not required that OX(U) be a local ring for every open set U — in fact, ...

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Ringed space, Ringed space - Definition, Ringed space - Examples, Ringed space - Morphisms, Ringed space - Tangent spaces, Ringed space - OX modules

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Function space - Functional analysis. Schwartz space of smooth functions of rapid decrease and its dual, tempered distributions Lp space κ(R) continuous compact support with uniform norm C(R) bounded continuous (Bounded_function) C∞(R) functions which vanish at infinity C∞(R) Smooth_functions C∞0 smooth compact support uniform norm (and the one with ...

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Function space, Function space - List of function spaces, Function space - Functional analysis

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If we divide the octave into n parts, where n = rs is the product of two relatively prime integers r and s, we may represent every element of the tone space as the product of a certain number of "r" generators times a certain number of "s" generators; in other words, as the direct sum of two cyclic groups of orders r and s. We may now define a graph with n verticies on which the group acts, by adding an edge between to pitch classes whenever they differ by either an "r" generator or an "s" generator. The result is a graph of genus one, which is to say, a graph with a ...

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Modulatory space, Modulatory space - Circles of generators, Modulatory space - Toroidal modulatory spaces, Modulatory space - Chains of generators, Modulatory space - Cylindrical modulatory spaces, Modulatory space - Five-limit modulatory space, Modulatory space - Seven-limit modulatory space

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See also: Space and survival In 2001, the space news website SPACE.com asked Freeman Dyson, J. Richard Gott and Sid Goldstein for reasons why some humans should live in space. Their respective answers [1] were: To Spread Life and Beautify throughout the Universe To Ensure the Survival of Our Species To Make Money Save the Environment Provide entertainment value in ...

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Space colonization, Space colonization - Method, Space colonization - Materials, Space colonization - Energy, Space colonization - Transportation, Space colonization - Communication, Space colonization - Life support, Space colonization - Radiation protection, Space colonization - Self-replication, Space colonization - Population size, Space colonization - Location, Space colonization - Orbit, Space colonization - Asteroid, Space colonization - Terrestrial analogues, Space colonization - Mercury, Space colonization - Venus, Space colonization - Europa, Space colonization - Gas Giants, Space colonization - Space habitats, Space colonization - Spaceship, Space colonization - Justification, Space colonization - Advocacy, Space colonization - Objections

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