In Search of the Missing Fundamental: by Richard K. Jones
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Step No. 6

Finding The Missing Fundamental Using Virtual Pitch

Electronic tuners that display the octave register as well as the frequency can be very useful for measuring the strength of mode 1,1 at each lug point. Since the spectrum of a well-adjusted timpano can approximate a harmonic series with a missing fundamental, when a tuner registers the actual pitch (at each lug point) as the one being an octave lower than the one actually being played, then the head has been balanced or tempered. E.g., when sounding a C3 @130 Hz on a 29” drum, if the tuner is registering C2 @65 Hz, this means that the upper partials are lined up well because the vibrations of the head are not producing the actual frequency of C2 @65 Hz . The Virtual Pitch of the spectrum is strong enough that the tuner detects a pitch of C2 @65 Hz (a series complete with a fundamental) even though the frequency C3 @130 Hz is being sounded. The tuner is interpolating the fundamental frequency of the pitch to be  C2 @65 Hz because it is detecting a series of harmonic overtones that belong to the frequency  C2 @65; in essence, the missing fundamental has been found.

The graphics below illustrate this phenomenon. A notated C3 was played on a 72cm Fiedler Timpano and measured with the Cleartune – Chromatic Tuner for iPhone & Android. The pitch that registered on the Cleartune App was a C2, which is one octave below notated and sounding pitch C3.

A written/notated C3 was registered as a C2 indicating that the tempering was such that the overtones produced enough harmonicity to create a spectrum where the virtual pitch of the missing fundamental C2 was strong enough to be detected by the Cleartune App.

Missing Fundamental Spectogram

The Spectogram below (click to enlarge) shows the relative frequencies for modes 1,1 through 6,1 as well as activity for the missing fundamental, which would be C2 65.41 Hz. This frequency (C2 65.41) is not a vibration that occurs in the head, but is visible in the spectrum because of the virtual pitch created by the near harmonic partials. This spectogram was generated with the SpectrumView Frequency Analysis iOS App on an iPad2.

Missing Fundamental Spectogram

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