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In analyses, this is represented by two crossing lines with double arrowheads indicating the exchanged pitches. A common exchange of this sort involves a progression of a third using a passing tone, the exchange notated by the interval succession 10-8-6 (if this is with the bass, the third chord is a first inversion of the first).
In music, voice crossing is the intersection of melodic lines in a composition, leaving a lower voice on a higher pitch than a higher voice (and vice versa). Because this can cause registral confusion and reduce the independence of the voices, [ 1 ] it is sometimes avoided in composition and pedagogical exercises.
Download as PDF; Printable version; In other projects Wikidata item; Appearance. move to sidebar hide. Crossing the Lines may refer to: Crossing the Lines (project ...
Eli Terry was using interchangeable parts using a milling machine as early as 1800. Ward Francillon, a horologist, concluded in a study that Terry had already accomplished interchangeable parts as early as 1800. The study examined several of Terry's clocks produced between 1800–1807. The parts were labelled and interchanged as needed.
In geometry, an arrangement of lines is the subdivision of the Euclidean plane formed by a finite set of lines. An arrangement consists of bounded and unbounded convex polygons , the cells of the arrangement, line segments and rays , the edges of the arrangement, and points where two or more lines cross, the vertices of the arrangement.
If r and t cross, add that crossing point as a potential future event in the event queue. If p is the crossing point of two segments s and t (with s below t to the left of the crossing), swap the positions of s and t in T. After the swap, find the segments r and u (if they exist) that are immediately below and above t and s, respectively.
The crossing number of a reduced, alternating diagram is the crossing number of the knot. This last is one of the celebrated Tait conjectures. An alternating knot diagram is in one-to-one correspondence with a planar graph. Each crossing is associated with an edge and half of the connected components of the complement of the diagram are ...
A variation of this concept, the rectilinear crossing number, requires all edges to be straight line segments, and may differ from the crossing number. In particular, the rectilinear crossing number of a complete graph is essentially the same as the minimum number of convex quadrilaterals determined by a set of n points in general position.