The theory of Virtual Pitch was developed in 1969-1970 by Prof. Ernst Terhardt (Technische Universität München).22 Terhardt’s theory extends Giuseppe Tartini’s terzo suono (third sound), J. P. Rameau’s theory of fundamental bass, August Seebeck’s theory of periodicity pitch and J. F. Schouten Pitch of the Residue theory; all deal with the phenomenon of the missing fundamental and the way pitch can be inferred from patterns of partials.
Virtual pitch is a phenomenon in which the auditory system infers a pitch that is not necessarily present as a physical spectral component at the inferred fundamental frequency. This is in contrast to a spectral pitch, which corresponds directly to a tone that physically exists in the spectrum. In its simplest form, when given a series of pitches that are part of a harmonic overtone series, one may hear an approximate fundamental frequency if the pitches are a close enough match to the harmonics of the fundamental. The perception of virtual pitch can vary from listener to listener based on how the individual perceives pitch. It also depends on the loudness (sound pressure level, SPL) of the sounds, and the mixture of the partials. Complex sounds almost never fuse completely into a single pitch; instead they produce a spectrum, and which pitch from the spectrum is most prominent can depend on the musical context. This is still an active area of research in the psychological acoustics community.
Dr. Robert Ehleehle, Professor of Music, University of Northern Colorado demonstrates Prof. Ernst Terhardt’s classic virtual pitch experiment.
Virtual Pitch Demonstration
courtesy of theflyingrobby
Does virtual pitch exist in timpani pitch?
Timpani pitch (at its best) will always be a compromise of true harmonic pitch. No matter how hard one tries to temper or balance a head, it can never produce a true harmonic series. Humans are hard-wired to recognize sounds as being pitched only if they have numerous strong harmonic overtones. The lack of numerous strong harmonic overtones in timpani pitch can make tuning problematic for some people. (see Seebeck vs. Ohm) Contemporary theories on pitch perception suggest that listeners can differ in how they weight spectral and holistic (virtual/F0) cues.23 There are of course, combinations of the two. The spectral listener tends to break down a sound and hear it more as a group of individual frequencies rather than a single sound. The holistic listener tends to group all of the frequencies into a single sound. The predominantly spectral listener could easily be confused by the lack of strong harmonic overtones and the myriad of non-harmonic overtones found in timpani pitch. The predominantly holistic listener might mistake a dark or bright sound as being a flat pitch or sharp pitch respectively. Most seasoned timpanists are both spectral and holistic listeners and switch seamlessly between the two. They have to know which overtones to ignore and which ones to listen to when dealing with timpani pitch. This is especially true when it comes to the process of tempering heads.
Most people hear the actual pitch of the instrument as being that of the principal tone, mode 1,1, which is the first strong pitch-related partial (and not the membrane’s true fundamental mode (0,1)). Since human perception of virtual pitch depends to a great extent on the loudness (sound pressure level, SPL) and the strength and duration of the overtones, the missing fundamental may not be perceived if the partials are not harmonic enough or strong enough. Some timpanists describe an indistinct sound an octave below the principal tone when the drum is struck with a soft mallet at certain spots.24 Timpani virtual pitch may or may not also be perceived by the human ear due to one’s propensity for being a spectral listener and not a holistic listener (see above). Once the virtual pitch of a well tuned timpano has been reached it can also be destroyed when mode 2,1 (the fifth) becomes too predominant in the spectrum (see Pleading the Fifth).
Even if the missing fundamental cannot be perceived by the human ear, the concept of virtual pitch is useful to the timpanist when using electronic tuners for tempering timpani. 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, and the tuner is registering C2 @65 Hz, this means that the upper partials are lined up well since the vibrations of the head are not producing that actual frequency. The virtual pitch of the spectrum is strong enough that the tuner detects a pitch with fundamental and harmonic series. (Depending on the tuner’s algorithm, this can be an extremely useful indicator of harmonic alignment, even though it is not a direct measurement of a physical membrane vibration at C2.) 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.
Tempering a Head
Using a different electronic tuner app on an iPhone (insTuner), on the same 72cm Fiedler timpano, when the head was tempered at a low C2, the insTuner registered the pitch as being that of C1.
Missing Fundamental Spectrogram
When sounding a C3 @130 Hz on a 72cm Fiedler timpano, the Spectrogram 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 spectrogram was generated with the SpectrumView Frequency Analysis iOS App on an iPad2.
How to Use Virtual Pitch When Tempering Timpani
For a detailed process on how to temper timpani, please read Chapter 5: Tempering Timpani. If your timpani heads are relatively clear, but just need some fine tuning, please read the section Step 6 Fine-Tuning. This step will guide you through the process of fine tuning timpani using an electronic tuner to define the Virtual Pitch of a timpano’s overtone series.
This page applied Virtual Pitch in a practical way: a well-cleared timpano can produce a spectrum in which the ear, and sometimes an electronic tuner, infers a pitch an octave below the principal tone, even though that frequency is not strongly present as a physical component of the membrane’s vibration. This demonstrates that timpani pitch is not simply “one frequency,” but an inference drawn from a small set of partials and their relationships.
The next question is: what makes that inference reliable? Modern pitch research reframes the older debates by distinguishing pitch derived from spectrally resolved low-numbered harmonics versus pitch derived from timing (periodicity) cues when harmonics are less well resolved. This distinction is directly relevant to timpani because the pitch-bearing part of the spectrum is dominated by a limited set of partials, the preferred diametric modes, and the strength of the perceived pitch depends on how well those preferred-mode partials are aligned, how strongly they dominate the spectrum, and how much masking/inharmonic energy competes with them.
In other words, Virtual Pitch explains how a missing-fundamental pitch can be inferred; the final page explains why certain timpani spectra support that inference better than others, by connecting modern pitch perception to the preferred modes and their quasi-harmonic structure.