Here we are concerned with one of the most ancient branches of mathematics, the theory of the vibrating string, which has its roots in the ideas of the Greek mathematician Pythagoras. - Norbert Wiener, I am a Mathematician (1956) [7, p.160,]
The earliest acoustical studies on record were those of Pythagoras, who lived from 572 until 497 B.C. Pythagoras experimented with stretched strings of varying lengths, thicknesses and with varying tension, that when plucked, would produce a musical tone in the same manner as a modern day guitar or piano (Figure 2.1).
He showed that the simplest and most obvious of all such relationships - that between a note and its octave - is always obtained with the two segments of a stretched string when it is divided by a movable bridge so that the ratio of the lengths of the segments is 2:1. [24, p.50,]
Figure 2.1: The vibrating string of Pythagoras
What Pythagoras found was that consonant sounding musical intervals would fall where small whole number ratios could be used to describe the lengths of the two sections of divided string - specifically, ratios composed of the counting numbers of 1 to 10. Moreover, he discovered these results were independent of the tension and width of the string; those factors did change the musical pitch of the system, but the musical interval between the two sections would only be affected by the movable bridge.
Three intervals in particular sounded especially pleasing to Pythagoras: 1:2, 2:3 and 3:4 - what Western culture would later label as the perfect octave, perfect fifth and perfect fourth, respectively. What is perhaps more amazing is that cultures around the world already had these same intervals established as focal points of their tonalities. In fact, as early as 1000 B.C.E these intervals had evolved out of Saamagaayana chants sung by monks in northern India, and were grouped under the term Swarita or level sound potency., implying that their culture's view of the intervals was nearly coincident with our own. 
Table 2.1 provides a list of the complete set of whole number ratio consonant intervals:
These ratios formed the earliest connection between mathematics and music - a connection often remarked on by mathematicians and musicians alike. Interestingly, the case can be made that music is actually responsible for a great amount of development in several areas of mathematics and physics. As Stewart points out:
Many have professed to detect an affinity between mathematics and music. Be that as it may, an amazing amount of important mathematics can be derived from the problem of a vibrating violin string.[22, p.37,]
Beginning in the 1700's, several important advances in mathematics were brought to light due to a resurgence in interest on the violin string problem. Newton's calculus was being absorbed in the mathematical community, supplying mathematicians with the tools to solve increasingly nasty problems, most of which were differential equations.