Harmonia

Meet Harmonia, a string held at both ends who can sing only in whole-number harmonies.

Pluck a string and waves run along it, reflect at the fixed ends and interfere. Only certain patterns survive as standing waves: those that fit a whole number n of half-wavelengths into the length L. A wavelength of 2L/n, with wave speed v, gives the frequencies f_n = nv/(2L). The lowest is the fundamental; the others are whole-number multiples of it, the harmonic series.

The wave speed depends on the string's tension and its mass per length, v = √(T/μ). Tighten a guitar string and the pitch rises; press it on a fret to shorten L and the pitch rises too. Halving the length doubles every frequency, an octave up.

Real instruments sound different from each other partly because of which harmonics are strong. Pianos, violins and guitars all use this physics.

Real strings are slightly stiff, so higher modes on a piano string are a little sharp of exact whole-number multiples. Piano tuners allow for this.