Question

Explain harmonics and overtones.

Hint:

{Harmonics} are a complete musical scale (1x, 2x, 3x... of the base note). {Overtones} are just the notes "over" the base one (the 2nd, 3rd, 4th... possible notes). They are often the same, but not always.

Answer 1

In the context of standing waves produced in a medium (like a string or an air column), harmonics and overtones describe the different frequencies at which the medium can naturally vibrate.
Fundamental Frequency (\(n_0\)): This is the lowest possible frequency at which a system can vibrate. It is also called the first harmonic.
Harmonics: Harmonics are all the integer multiples of the fundamental frequency. The series of frequencies is given by \(n_0, 2n_0, 3n_0, 4n_0, \ldots\). The first harmonic is the fundamental frequency itself, the second harmonic is \(2n_0\), the third is \(3n_0\), and so on. All possible vibrational modes are harmonics.
Overtones: Overtones are any frequencies that can be produced by the system that are higher than the fundamental frequency. The first overtone is the next possible frequency above the fundamental, the second overtone is the next frequency after that, and so on.
Relationship:
For a system where all harmonics are present (e.g., a string fixed at both ends, an open organ pipe), the first overtone corresponds to the second harmonic, the second overtone corresponds to the third harmonic, and the \(p^{th}\) overtone corresponds to the \((p+1)^{th}\) harmonic.
For a system where only odd harmonics are present (e.g., a closed organ pipe), the first overtone corresponds to the third harmonic, the second overtone corresponds to the fifth harmonic, etc.