Theory

A monochord, or a string fixed rigidly at both ends, will exhibit various modes of vibration simultaneously. Each of these modes will have a wavelength associated with it. In vibrating strings, as well as other situations involving wave propagation, wavelength is related to frequency by the following formula:

$$v = f \lambda,$$ (1)

where $v$ is the velocity of propagation (this is typically constant for a given wave propagation medium) (in meters per second), $\lambda$ is the wavelength (in meters), and $f$ (in Hertz) is the frequency. For a constant wave velocity, it is important to note that this formula implies an inverse relationship between wavelength and frequency.

In a vibrating string held taught with a given tension $T$ (in newtons), and with a linear mass density $\mu$ (in kilograms per meter), the velocity of propagation $v$ is given by

$$v = \sqrt{\frac{T}{\mu}}.$$ (2)

For more information on these formulas, see Wikipedia's page on string vibration.

Finally, in a monochord of length $L$ (in meters), it turns out that there are infinitely many possible modes. If we let $n = 1, 2, ...$ be the mode number, each mode will have a certain frequency $f(n) = nf_0$, and a corresponding wavelength, $\lambda_n$. As a consequence, all modes must obey the following:

$$L = \frac{n \lambda_n}{2}.$$ (3)

In other words, for every mode, the monochord length must be a multiple of half of the mode's wavelength.