WW Sawyer
' Figure 1 shows some of the ways in which a string can vibrate: A produces a certain note; B produces the first harmonic, an octave higher; and C fifth higher. In 1715, the English mathematician Brook Taylor gave this a mathematical form.
If we choose a suitable unit of time and a unit of length that makes the string have length , the displacement of the string at position
So far, these vibrations are separate possibilities. However the French composer,
Rameau, said that when a note was played he heard many of these harmonics at one and the same time. In 1726 he
published a theory of harmony (which I believe is still regarded as valid) based on this idea. The Swiss mathematician, Daniel Bernouilli, picked up this hint. He formed an arbitrary mixture of all the harmonics and thus in 1753 arrived at the formula y = a_{1} sin x cos t +
a_{2} sin 2x cos 2t + a_{3} sin 3x cos 3t + .....It is interesting that Bernouilli's argument is based purely on physics; there are no calculations in this part of his paper. He speaks of ' a melange of all these vibrations' and goes on to say, 'All musicians are agreed
that a vibrating string gives at the same time, besides its fundamental tone, others much higher; they notice above
all the mixture of the 12th and the major 17th'.His series contains an infinite number of constants a |