 No writings survive that are directly attributable to Pythagoras (c. 570–c. 490 BCE). His view of music must therefore be reconstructed from the works of other Pythagoreans, or from more or less sympathetic accounts provided by other authors (a useful collection of texts is found in Barker 1990: chapter 1). Central to Pythagorean philosophy is the notion of harmony as the unity of various parts. In this sense, the concept has a much broader application than to music, for harmony regulates the motion of celestial bodies as well as the human soul (on the concept of harmony in ancient Greek philosophical thought, see Lippman 1964: chapter 1). In the Pythagorean tradition, mathematics is the discipline that investigates harmony. Musical harmony is no exception to this, as the interval of the octave may be expressed as a numerical ratio (2:1). The octave may in turn be considered as the sum of a fifth (3:2) and a fourth (4:3). Pythagoras was generally credited with the discovery of the relations between these intervals. These are best illustrated using a monochord, an instrument constituted by a single string, the vibrating sections of which are determined by the position of a movable bridge. However, it is unclear whether the monochord was in fact used by Pythagoras (Barker 2014: 186), and it is also plausible to assume that instrument makers had some understanding of these relations before they became the object of intellectual speculation.