Comment by Anechoic
14 years ago
Harmonics in real-world instruments are not usually exact multiples of the fundamental. A simple diffeq model of a rigid oscillator will show you this mathematically.
That is absolutely incorrect, mathematically speaking, harmonics are by definition "integral multiples of the fundamental." (Fundamentals of Acoustics, Kinsler & Frey).
People from a musical, non-signals background tend to use 'harmonics' as a synonym for 'overtones' or 'partial tones', which is where the confusion arises, I suspect.
There's a measure -- inharmonicity[1] -- of how far the actual overtones of a particular instrument differ from their theoretical fundamental multiples.
[I suspect you already know this. This reply is probably for others' benefit]
[1] https://en.wikipedia.org/wiki/Inharmonicity
Then one would be forced to conclude that many instruments have no harmonics at all, which is obviously not what 'harmonic' is referring to in this thread of discussion. Why be pedantic when it's obvious what everyone is talking about?
Anyway, it's not as though mathematical literature requires you to use a term exactly one way. I had a diff eqs textbook that used the word 'harmonic' in exactly the way I used it above when I made reference to diff eqs...
> I had a diff eqs textbook that used the word 'harmonic' in exactly the [incorrect] way I used it above
So you had a textbook with a mistake in it. What book was it?