Bisection
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- For the numerical analysis algorithm, see bisection method.
Bisection is the general activity of dividing something into two parts. In geometry, the concept is limited to divisions into two equal parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are segment bisectors and angle bisectors.
A segment bisector passes through the midpoint of the segment. Particularly important is the perpendicular bisector of a segment, which, according to its name, meets the segment at right angles. The perpendicular bisector of a segment also has the property that each of its points is equidistant from the segment's endpoints.
An angle bisector divides the angle into two equal angles. An angle only has one bisector. Each point of an angle bisector is equidistant from the sides of the angle.
In classical geometry, the bisection is a simple ruler-and-compass construction, whose possibility depends on the ability to draw circles of equivalent radius and different centers.
The segment is bisected by drawing intersecting circles of equal radius, whose centers are the endpoints of the segment. The line determined by the points of intersection is the perpendicular bisector, and crosses our original segment at its center. Alternately, if a line and a point on it are given, we can find a perpendicular bisector by drawing a single circle whose center is that point. The circle intersects the line in two more points, and from here the problem reduces to bisecting the segment defined by these two points.
To bisect an angle, one draws a circle whose center is the vertex. The circle meets the angle at two points: one on each leg. Using each of these points as a center, draw two circles of the same size. The intersection of the circles (two points) determines a line that is the angle bisector.
The proof of the correctness of these two constructions is fairly intuitive, relying on the symmetry of the problem. It is interesting to note that the trisection of an angle (dividing it into three equal parts) is somewhat more difficult, and cannot be achieved with the ruler and compass alone.
In diatonic set theory a bisector divides the octave approximately in half and may be used in place of a generator to derive collections for which structure implies multiplicity is not true such as the ascending melodic minor, harmonic minor, and octatonic scales. Well formed generated collections generators and bisectors coincide, such as in the diatonic collection. The term was introduced by Jay Rahn (1977), who considers any division between one and two thirds as approximately half and who applied the term only the equally spaced collections. Clough and Johnson both adapt the term to apply to generic scale steps. Rahn also uses aliquant bisector for bisectors which may be used to generate every note in a collection, in which case the bisector and the number of notes must be coprime. Bisectors may be used to produce the diatonic, harmonic minor, and ascending melodic minor collections. (Johnson 2003, p.97, 101, 158n10-12)
Source
- Johnson, Timothy (2003). Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing. ISBN 1930190808.
- Rahn, Jay (1977). "Some Reccurent Features of Scales", In Theory Only 2, no. 11-12: 43-52