The second half of the statement is wrong, but the first half is right. 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.
An extreme example is present on modern pianos, where the high rigidity of the loud, heavy piano strings can cause tuners to stretch the lowest and highest notes as much as a half-semitone so that their harmonics are in tune with the note the next octave down or up. In other words, the first harmonic on the lowest note of a piano can be as much as 1/2 of a note sharp.
And when your oscillator is no longer one-dimensional, most harmonics aren't even close to integer multiples. The harmonics of bells, cymbals and drums are all over the place. That's what gives them their percussive sound. (Edit: some of these modes of vibration aren't harmonics in the linear sense.)
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]
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...
> And when your oscillator is no longer one-dimensional, most harmonics aren't even close to integer multiples. The harmonics of bells, cymbals and drums are all over the place. That's what gives them their percussive sound.
But those aren't harmonics, they're inharmonic partials.
Actually, they are: https://en.wikipedia.org/wiki/Harmonics