Chord

Chord, in Geometry, is the line joining any two points on a curve. The limiting position of this line, as one point is brought nearer and nearer to the other, is the tangent at this point. In the case of a circle any chord is bisected by the perpendicular on to it from the centre, and diminishes in length as its distance from the centre increases. Hence the greatest chords of a circle are its diameters, all passing through the centre, and all of equal length. Given the radius of a circle and the length of chord subtending any angle at the centre, we may construct this angle. Thus, a scale of chords is sometimes used for marking off angles, though the protractor (q.v.) is better for the purpose. If two circles intersect, they possess a common chord, the line joining their two points of intersection. All tangents to the two circles from any point on the common chord are of equal length. If three circles intersect, there will be three such common chords or radical axes as they are termed. These three axes will invariably be found to pass through one point. In any conic section the mid-points of chords parallel to any diameter lie on a line through the centre of the conic parallel to the tangents at the extremities of the given diameter.

In the theory of sound a chord is understood to mean the simultaneous production of simple sounds in harmony with each other. The vibration numbers for the constituent sounds must be in simple ratios, or else discordant effects are produced. Thus the notes C, E, and G, forming the common chord, have their vibration numbers in the ratio 4:5:6. In the minor key the ratios are slightly more complex, the faint suggestion of discord producing a favourable effect. There are very many different names given to particular chords, such as common chord, diminished, augmented, chord of the dominant, of the sixth, etc., etc.