# How pure is the tone produced by a typical tuning fork in air?

When a musical instrument plays a note that note is usually just the highest amplitude contribution to the sound wave. That peak is known as the fundamental harmonic, and the Fourier transform of the sound wave also has peaks at higher frequency harmonics, known as overtones, and possibly at lower frequencies, called undertones. These peaks have finite width, and there's frequently some kind of overlap.

How pure is the tone produced by a typical tuning fork? Specifically, ignoring questions of the bandwidth of the peak, how much do overtones and undertones contribute to the sound of a tuning fork? The answer, of course, will be, "It depends on the tuning fork," but it is still interesting to ponder what is produced by typical tuning forks.

One way to see answers to this question is to enter "tuning fork spectrogram" (without quotes) into a search engine. The answer, qualitatively, is fairly pure, after the initial high pitch transients from striking it die out.

In the spectrograms presented in this work (Douglas Lyon, “The Discrete Fourier Transform, Part 5: Spectrogram”, Journal of Object Technology, Volume 9, no. 1 (January 2010),pp. 15-24), Figures 4 and 5, the tuning for clearly has both overtones and undertones present in the spectrogram. Qualitatively, though, Figure 4 (the one with a linear stretch) makes it appear that the fundamental is very dominant, probably at least a factor of 2 louder than the nearest harmonic (a difference of about 3 decibels). The downside is that this kind of quantitative statement based on the presented information could be wildly inaccurate (the numbers are guessed).

This video presents a spectrum for a recording of a tuning fork, but the horizontal axes are difficult to read.

The followup video shows animations of the different modes from a finite element analysis. This video is interesting primarily for it's demonstration of how many of the higher frequency modes are unbalanced, producing motion at the part of the tuning for that is held/mounted, leading to their increased damping speed.

Downloading the audio from the first YouTube video and feeding part of it (roughly time code 2:21.75 to 2:25.6) into Audacity gives me a spectral analysis of the tuning fork recording (Hamming window).

As you can see from the plot, the next highest peaks are about $46\operatorname{dB}$ down from the fundamental.

• Just pointing out that the recording had a big amount of background noise (likely from the recording gear used) and that makes a big difference in the spectrogram. The Tuning Fork's overtones from the harmonic series can be observed though: peaks around 523.3 Hz, 1046.6 Hz, 1569.9 Hz, etc... Sep 15, 2020 at 23:28