It turns out I need to do some correcting with my blog post about the bell’s strike note. A commenter (thanks BDF!) asked if other musical instruments besides bells had a similar phantom-like note, and that prompted me to find this surprising tidbit.
When we hear musical sounds that conform to the natural harmonic series, we hear a virtual pitch that coincides with the fundamental, or the lowest note in the series. This phenomenon is the same as with the strike note in the bell—it is produced in our brain from hearing the upper partials. This virtual pitch coincides with the fundamental, but it is not the same thing. In fact, when the fundamental and even other lower partials are subtracted from the sound (leaving the middle and upper partials to be heard), we can still hear the virtual pitch that coincides with the fundamental.
Furthermore, this phenomenon of our brain creating a primary pitch out of a complex series of partials is the rule, rather than the exception. So, the bell is not exceptional in this regard. It is still exceptional since 1) the strike tone does not always coincide with one of the partials and 2) we have not devised a theory that adequately explains the placement of the strike note relative to the other partials.
Another interesting tidbit that I should have shared the last time I presented bell tuning: the frequencies of the natural harmonic series follow a mathematic ratio of 1:2:3:4:5:6, etc. This means that if the first frequency were 100 Hz (cycles per second), the next partial would be 200 Hz, then 300 Hz, then 400 Hz, and so on. That’s probably an easier way to conceptualize the natural harmonic series as opposed to seeing it on the staves!