Explain harmonics and overtones.

Answer: Harmonics are the frequencies that are integral multiples of the fundamental frequency; overtones are the frequencies actually present in a note that are higher than the fundamental.

• Fundamental: the lowest frequency \(n_1\) with which a string or air column can vibrate.
• Harmonics: frequencies \(n_1, 2n_1, 3n_1, \dots\) that are integral multiples of the fundamental. The fundamental itself is the first harmonic, \(2n_1\) the second harmonic, and so on.
• Overtones: the higher frequencies actually produced along with the fundamental. The first frequency above the fundamental is the first overtone, the next the second overtone, and so on.

Thus all overtones are harmonics, but all harmonics need not be overtones. For a string or a pipe open at both ends, all harmonics are present (first overtone = second harmonic). For a pipe closed at one end, only odd harmonics \(n_1, 3n_1, 5n_1, \dots\) are present, so the first overtone is the third harmonic.