Quintile

A quintile is the aspect of seventy-two degrees, a fifth of the circle. It is one of the minor aspects introduced by Johannes Kepler from his theory of astrological harmonies, set out in the Harmonices Mundi of 1619 together with the biquintile and the sesqui-quadrate, the earliest and most securely attested minor aspects in the record, since Kepler discusses them under his own name and his own reasoning rather than leaving them to be inferred from later practice.

Seventy-two degrees is the angle subtended by one side of the regular pentagon at the centre of its circumscribing circle, and Kepler's argument for admitting it rested on the claim that the pentagon, like the triangle and the square behind the older aspects, is a figure constructible by compass and straightedge and corresponds to a consonant musical interval, so that the geometry of workable aspects ought to follow harmonic ratios rather than the sign-based scheme of the older tradition. It is read as a creative and constructive minor aspect, indicating talent and a facility for pattern and form. Kepler's broader project in the same volume was to show that the same small set of ratios which made certain musical intervals sound consonant to the ear also governed which polygons could be built with compass and straightedge, and he treated the aspects as a third expression of that one underlying harmony, so the quintile stands or falls, in his own reasoning, together with his account of music and of geometry rather than as an isolated astrological rule.