# How to compute fundamental frequency from a list of overtones?

Given a list of overtones (F1, F2, F3, etc), how do I compute the fundamental frequency? Can I do something like F2/F1=F1/F0? Is it the correct method to use?

• It's the GCD of the overtones, but where did the overtones come from? If they are measured from an FFT, there will be error which ruins the GCD. Also for certain sources (plucked string instruments) there will be inharmonicity to consider, and what exactly you then mean by "fundamental". – endolith Apr 29 '13 at 17:44

The frequencies of the harmonics are integer multiples of the fundamental frequency $f_0$, i.e. $f_n = (n+1)f_0$. The fundamental frequency $f_0$ is the greatest common divisor of the harmonics $f_n$. If you are sure that there is no other unknown harmonic between two known harmonics, e.g. you know that you have the fourth and the fifth harmonic, then $f_0$ is of course the difference between the two. But if you just have a collection of harmonics and you don't know anything else about them, then you need to determine $f_0$ as the gcd of $f_n$.

• I don't quite believe $f_n = n f_0$. What happens if $n=0$? $f_0 = 0. f_0 = 0$! :-) I think you mean $f_{n-1} = n f_0$ for $n=1\ldots$. – Peter K. Apr 28 '13 at 23:06
• $n=0$ is simply an unfortunate choice ;) OK, of course you're right, even though I also believe that the concept is so simple that even my sloppy (and incorrect!) notation won't cause any confusion. Anyway, thanks for clearing it up! – Matt L. Apr 29 '13 at 6:52

Nope. Difference between overtones is a good point to start, i,e F3-F2, F2-F1. The differences should be all the same or multiple of each other. The smallest one is often the fundamental. It gets more tricky of the spectrum is "sparse", i.e. a lot of the harmonics are missing. Then you need to find a largest possible divisor that turns all frequencies into integers or, to be precise, so that the ratio of frequency to fundamental is within the measurement accuracy of the nearest integer.

Look up the Harmonic Product Spectrum algorithm, which given a sufficient number of actual overtones, is a bit more robust against missing overtones and added noise spectra, than just subtracting all successive tones frequency pairs.