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Spiral array model: pitch class, major/minor chord, and major/minor key helices. The model as proposed covers basic pitches, major chords, minor chords, major keys and minor keys, represented on five concentric helices. Starting with a formulation of the pitch helix, inner helices are generated as convex combinations of points on outer ones.
The value of the two products in the chord theorem depends only on the distance of the intersection point S from the circle's center and is called the absolute value of the power of S; more precisely, it can be stated that: | | | | = | | | | = where r is the radius of the circle, and d is the distance between the center of the circle and the ...
This formula is also true for other units of measurement such as in feet. The relationship of versine, chord and radius is derived from the Pythagorean theorem. Based on the diagram on the right: = We can replace OC with r (radius) minus v, OA with r and AC with L/2 (half a chord).
A circular segment (in green) is enclosed between a secant/chord (the dashed line) and the arc whose endpoints equal the chord's (the arc shown above the green area). In geometry, a circular segment or disk segment (symbol: ⌓) is a region of a disk [1] which is "cut off" from the rest of the disk by a straight line.
An inversion in their tangent point with respect to a circle of appropriate radius transforms the two touching given circles into two parallel lines, and the third given circle into another circle. Thus, the solutions may be found by sliding a circle of constant radius between two parallel lines until it contacts the transformed third circle.
Equal chords are subtended by equal angles from the center of the circle. A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle. If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (power of a point theorem).
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 ...
Hipparchus. The concepts of angle and radius were already used by ancient peoples of the first millennium BC.The Greek astronomer and astrologer Hipparchus (190–120 BC) created a table of chord functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions. [2]