BACKGROUND

Although primarily a philosopher, in the modern world, Pythagoras is most of all famous as a mathematician because of the theorem named after him. During ancient times, he was known as a cosmologist, because of the striking view of a universe ascribed to him in the later tradition, in which the heavenly bodies produce “the music of the spheres” by their movements. A famous discovery attributed to Pythagoras in the later tradition (7th century B.C.), was that of the central musical concords (the octave, fifth and fourth) corresponding to the whole number ratios 2: 1, 3 : 2 and 4 : 3 respectively (e.g., Nicomachus, Handbook 6 = Iamblichus, On the Pythagorean Life 115). The acusmata (hearings) reported by Aristotle, report on the so-called tetraktys (literally "the four"), which stands for "the harmony in which the Sirens sing” (Iamblichus, On the Pythagorean Life, 82, probably derived from Aristotle), referring to the first four numbers, which when added together equal the number ten (regarded as the perfect number in fifth-century Pythagoreanism). Allegedly, all these notions were originally demonstrated by Pythagoras himself making use of his famous monochord.

The monochord, also known as single-stringed sonometer, is an ancient musical and scientific laboratory instrument (traced back to the Pythagorean Era) involving one (mono-) string (chord), upon which points may be delineated to signify where the string must be stopped to give certain notes. The string is fixed at both ends and stretched over a sound box. One or more movable bridges (tasta) are then manipulated to demonstrate mathematical relationships among frequencies produced, allowing for comparisons. Given its increased usability and ergonomic design it was quickly mainstreamed, with Music, Numeracy, and Astronomy left inexorably linked to it at the time. 

This one-string zither finds its oldest known written trace in Division of the Kanōn (probably by Euclid). It is described as a long hollow wooden box along the top of which is stretched one string, rigidly attached to the box at one end, with provision at the other for changing the tension. As such, it could be used for studying audible changes in pitch (fractions above/below some fixed original frequency), as well as showing that frequency varies inversely as the length (or as the square root of the mass per unit length) of the vibrating string, or directly as the square root of its tension. Although somewhat simple in its conceptualization, what distinguishes this case study from the rest is the intangibility aspect. No concrete evidence about the actual dimensions or materials has been found so far, however a lot can be inferred from the way ancient Greeks managed to transform auditory experience to visual perception; namely acoustical axioms, pure mathematical propositions, and a series of corollaries, relating notes within the Greater Perfect System.