End

Pieces using sonata form typically use the recapitulation to conclude a piece, providing closure through the repetition of thematic material from the exposition in the tonic key. In all musical forms other techniques include "altogether unexpected digressions just as a work is drawing to its close, followed by a return...to a consequently more emphatic confirmation of the structural relations implied in the body of the work."

For example:

The slow movement of Bach's Brandenburg Concerto No. 2, where a "diminished-7th chord progression interrupts the final cadence."

The slow movement of Symphony No. 5 by Beethoven, where, "echoing afterthoughts", follow the initial statements of the first theme and only return expanded in the coda.

Varèse's Density 21.5, where partitioning of the chromatic scale into (two) whole tone scales provides the missing tritone of b implied in the previously exclusive partitioning by (three) diminished seventh chords.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Conclusion_(music)

End (graph theory)

In the mathematics of infinite graphs, an end of a graph represents, intuitively, a direction in which the graph extends to infinity. Ends may be formalized mathematically as equivalence classes of infinite paths, as havens describing strategies for pursuit-evasion games on the graph, or (in the case of locally finite graphs) as topological ends of topological spaces associated with the graph.

Ends of graphs may be used (via Cayley graphs) to define ends of finitely generated groups. Finitely generated infinite groups have one, two, or infinitely many ends, and the Stallings theorem about ends of groups provides a decomposition for groups with more than one end.

Definition and characterization

Ends of graphs were defined by Rudolf Halin (1964) in terms of equivalence classes of infinite paths. A ray in an infinite graph is a semi-infinite simple path; that is, it is an infinite sequence of vertices v₀, v₁, v₂, ... in which each vertex appears at most once in the sequence and each two consecutive vertices in the sequence are the two endpoints of an edge in the graph. According to Halin's definition, two rays r₀ and r₁ are equivalent if there is another ray r₂ (not necessarily different from either of the first two rays) that contains infinitely many of the vertices in each of r₀ and r₁. This is an equivalence relation: each ray is equivalent to itself, the definition is symmetric with regard to the ordering of the two rays, and it can be shown to be transitive. Therefore, it partitions the set of all rays into equivalence classes, and Halin defined an end as one of these equivalence classes.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/End_(graph_theory)

Podcasts:

End

ALBUMS

En förlorad vän released: 1984

En förlorad vän

Released 1984

End

ALBUMS

Open Music Contest #2: 'Jelly Good Follow' released: 2006

Open Music Contest #2: 'Jelly Good Follow'

Released 2006

End

ALBUMS

People Of The Stream's Mout released: 2014
III released: 2009
II released: 2003
End released:

People Of The Stream's Mout

Released 2014

III

Released 2009

II

Released 2003

End

ALBUMS

La naranja mecánika released: 2012

La naranja mecánika

Released 2012

end.

ALBUMS

The Dangerous Class released: 2009
A Girl Called Trouble released: 2008
The Sick Generation released: 2005
Percussions released: 2004
The Sounds of Disaster released: 2004
Science/Fiction released: 2002

The Dangerous Class

Released 2009

A Girl Called Trouble

Released 2008

The Sick Generation

Released 2005

Percussions

Released 2004

The Sounds of Disaster

Released 2004

Science/Fiction

Released 2002

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