Present day scales

The Pythagorean research into the sounds produced by vibrating strings was continued by Marin Mersenne (1588 – 1648). He found that increasing the mass and decreasing the tension of a string produced lower notes in a given length of string, a result of some relevance to the tuning and development of stringed instruments. Mersenne’s book Harmonium Liber (1636) was the first published account of the laws of vibrating strings. (Pythagoras preferred not to publish anything!) Mersenne also showed how to measure the frequency of a musical note, extending the work of Galileo who had discovered that the pitch of a note depends on the rate at which the sound source vibrates.

A complete mathematical analysis of the behaviour of vibrating strings was eventually achieved by d’Alembert (1717 – 1783) using calculus and trigonometric functions.

With the development by Stevin in the 16th century of the mathematically defined equal temperament scale, which divides the octave into 12 equal semitones, and the perfection of the tuning fork in the 19th century, it was possible to determine exactly the pitch of a particular note (for example A = 440 vibrations per second). Thus a piano could be tuned by varying the tension of the strings, which have varying masses and lengths. The age of the piano tuner had arrived.