The Problem the Ear Solves ↑ menu
When a timpano vibrates, it produces a spectrum of partials, not all of which are equally useful to the ear. The auditory system’s ability to extract pitch from a complex tone depends critically on whether individual harmonics are resolved by the cochlea. Harmonics below about the 10th are typically resolvable as distinct spectral components; above that threshold, the auditory filters become too broad and adjacent harmonics fall within the same critical band, forcing the ear to rely on temporal cues instead.[1]
This distinction matters enormously for the timpanist. When a timpano is tuned to a given note, say, C3 (130.8 Hz), the principal vibrational mode of the membrane, the (1,1) mode, does not correspond to the first position in a harmonic series. In the Duff/Benade interpretation used here, it can be treated as the second harmonic of an implied harmonic pattern whose nominal fundamental lies an octave below: C2 at 65.4 Hz. That lower frequency is not a strong physical component of the drum’s sustained spectrum; under favorable conditions, it may be inferred by the auditory system from the relationships among the audible partials.
This is a missing-fundamental or virtual-pitch interpretation of timpani pitch. It is an important psychoacoustic possibility in the instrument’s tone production, but it should not be treated as an automatic or exclusive explanation: many listeners may also anchor the heard note to the principal (1,1) mode itself, depending on the drum, the stroke, the room, and the listener.
The Duff/Benade Mode Ratios and Their Harmonic Correspondence ↑ menu
In 1973, Arthur Benade measured the frequency ratios of the vibrational modes of a timpano belonging to Cloyd Duff, principal timpanist of the Cleveland Orchestra, after the head had been cleared to Duff’s exacting standard. The ratios he reported for the first six preferred diametric modes, with (1,1) as the reference (1.0×), are:[2]
| Mode | Duff Ratio | Nearest Integer Harmonic | Deviation from Integer |
|---|---|---|---|
| (1,1) | 1.000 | 2nd | (reference) |
| (2,1) | 1.504 | 3rd | +0.7% |
| (3,1) | 2.000 | 4th | 0.0% |
| (4,1) | 2.494 | 5th | −1.2% |
| (5,1) | 2.979 | 6th | −0.7% |
| (6,1) | 3.462 | 7th | −1.1% |
The (1,1) mode at 1.0× is the second harmonic of the implied missing fundamental. Modes (2,1) through (6,1) occupy harmonic positions 3 through 7. The near-perfect alignment of (3,1) to the 4th harmonic (exactly 2.0×) is particularly striking, an octave above the (1,1) mode, as close to perfectly harmonic as any real membrane can achieve.
All six preferred diametric modes in this Duff/Benade example fall within the seventh harmonic of the implied fundamental. That low-numbered, near-harmonic grouping is a strong reason a well-cleared timpano can support a clear pitch percept, even though the strength, decay time, and audibility of each mode still matter.
Why No (0,1) Mode? The Question of the Missing Fundamental ↑ menu
In a theoretical ideal membrane, the (0,1) mode, a circular node with no diametrical nodes, would be the lowest in frequency and would correspond to the first harmonic (the fundamental). In practice, the bowl, bearing edge, head tensioning, air loading, damping, and normal off-center playing technique mean that the (0,1) mode may be present in the attack but does not behave as the useful sustained pitch carrier on a timpano. The (1,1) mode, with one diametrical node line, is the lowest usable mode; it sits at the second harmonic position.
The ear’s pitch extraction system can work with this arrangement when the relevant partials are sufficiently strong, low-numbered, and close to harmonic. When presented with a series such as harmonics 2 through 7 of an implied fundamental, the auditory system may infer a virtual pitch an octave below the (1,1) mode. This is related to the missing-fundamental effect familiar from many complex tones, including some bandwidth-limited speech and musical sounds. Shepard tones involve octave ambiguity and pitch-class circularity, so they should be treated as a related but distinct psychoacoustic phenomenon rather than a direct equivalent.
This point is worth dwelling on: the timpano need not produce a strong physical fundamental at the implied octave below. Instead, a well-cleared drum can produce a spectrum whose preferred diametric modes approximate harmonic positions 2 through 7 of an implied fundamental. The ear may use that pattern to strengthen or supplement the pitch percept, while other listeners or playing conditions may leave the perceived pitch more closely anchored to the principal (1,1) mode.
Resolved Harmonics and Pitch Identification ↑ menu
A 2025 study by Albera and colleagues at the University of Turin investigated how accurately listeners can identify musical pitch from harmonic complexes containing only two or four consecutive harmonics, with no physical fundamental present. Using synthetic tones presented to 30 amateur musicians with normal hearing, they found that correct identification rates ranged from 8% to 100%, depending on which harmonics were presented. Crucially, identification was most reliable when the harmonics were low-numbered and close to the fundamental; correct identification reached 88% to 100% with harmonics 2 through 5, and 82% to 96% with harmonics 3 through 6. In the two-harmonic condition, performance declined as the first, or lowest, presented harmonic moved upward; the authors note a marked decline once that lowest presented harmonic was roughly above the 1500–2000 Hz region.[3]
This finding reinforces a foundational principle in psychoacoustics: pitch salience from harmonic complexes drops dramatically when low-numbered harmonics are removed or poorly represented.[4] The auditory system then has fewer reliable spectral cues to work with. For the timpanist, this means that the presence, tuning, amplitude, and decay behavior of the preferred diametric modes, all of them falling within the seventh harmonic in the Duff/Benade example, are not merely matters of timbre. They are important contributors to whether the drum produces a clear, stable pitch percept.
Two Modes of Pitch Perception: Spectral and Virtual ↑ menu
Contemporary psychoacoustic research distinguishes between two perceptual strategies for pitch extraction from complex tones:
Spectral pitch (sometimes called analytical listening) arises when the ear resolves individual harmonic components and matches them to spectral templates. Each resolved harmonic contributes its own pitchlike sensation, and the perceived pitch results from a weighting or higher-level inference across the resolved partials.[5]
Virtual pitch (sometimes called synthetic listening or periodicity pitch) arises when the ear detects the repetition rate of the waveform, even in the absence of a physical component at that frequency, and infers a fundamental pitch. This is the modern descendant of Seebeck’s periodicity pitch.[6]
A line of research from the Heidelberg group, including Schneider, Seither-Preisler, and colleagues, has reported stable, measurable listener preferences for one strategy or the other, along with associated anatomical and cortical patterns. Listeners who favor spectral or overtone-based listening tend to show rightward hemispheric dominance and stronger responses in right-hemisphere auditory regions; listeners who favor virtual or fundamental-based pitch tend to show leftward dominance and stronger early responses in left-hemisphere regions. These preferences appear to be relatively stable individual traits and are not strongly predicted by musical training or expertise alone.[7]
A 2025 MEG study by Saus, Seither-Preisler, and Schneider added a finer-grained oscillatory dimension to this picture. The two listening modes appear to differ in their characteristic cortical oscillations: spectral listeners tend to show elevated theta-band power and reduced gamma-band power, while fundamental listeners show the opposite pattern. The two effects (anatomical lateralization and oscillatory signature) appear to be related but distinct markers of the same underlying perceptual tendency.[7]
What This Means for the Preferred Diametric Modes ↑ menu
The Duff/Benade data give us a precise picture of why the first six diametric modes are perceptually preferred:
All six are low-numbered harmonics in the implied pattern. Modes (1,1) through (6,1) occupy harmonic positions 2 through 7 of the implied missing fundamental. They fall within the usual low-order range in which harmonics are more likely to be spectrally resolved, though actual resolvability depends on absolute frequency, level, masking, and listener hearing. This keeps the main pitch-bearing modes away from the region where adjacent high-numbered harmonics are typically much less separable.
The (3,1) mode is very nearly harmonic in this measurement. At 2.0× the (1,1) frequency, (3,1) is an octave above the principal. This octave relationship supplies a strong internal cue that can stabilize either a principal-tone pitch or a virtual-pitch interpretation, depending on listening strategy and context.
The (2,1) and (4,1) modes are close to the 3rd and 5th harmonics. At 1.504× and 2.494× respectively, they correspond closely to ratios of 3/2 and 5/4 relative to the (1,1)-as-second-harmonic interpretation. These are among the most consonant interval relationships in Western tonal practice, and in the Duff-cleared drum they appear as measurable structural features of the modal spectrum.
The (5,1) and (6,1) modes extend the series through the 6th and 7th harmonics, adding further spectral coherence to the pitch percept while remaining within the resolvable range.
This harmonic architecture, modes near the 2nd through 7th harmonics of an implied missing fundamental, should be understood as an idealized interpretation of the Duff/Benade measurement, not as a universal law for every timpano. Duff’s technique of precise subsidiary tension adjustment appears to bring important partials closer to near-integer ratios. Benade observed that on Duff’s properly cleared drum, “the frequency alignment of the principal tone, fifth, octave, and tenth” (Modes (1,1), (2,1), (3,1), and (4,1)) was “remarkably close to harmonic.”[2] When the head is well cleared, the timpano’s spectrum can approximate the kind of low-order harmonic pattern to which the auditory system is especially sensitive.
The Duff Clearing Process as Perceptual Engineering ↑ menu
Duff described the process as “clearing,” persuading the partials to more closely match the ideal. Benade, quoting Duff directly, explained it this way: “It is not sufficient merely to get the overall skin tension correct for the desired pitch of the kettledrum, one must also make small additional changes in the tension produced by the various screws around the periphery of the drum.” These small adjustments compensate for the inherent irregularity of the membrane and for eccentricity in the kettle rim. When everything is in perfect adjustment, the drum “is said to have been cleared.” Benade described hearing Duff clear a well-tuned drum as “a revelation,” the tone “ringing with smoothness and clarity.”[2]
What Duff was doing, in acoustic terms, can be described as increasing the harmonic coherence of the timpano’s preferred modal spectrum. In the Duff/Benade measurement, that means bringing the (2,1), (3,1), (4,1), (5,1), and (6,1) modes closer to harmonic positions 3 through 7 of the implied missing fundamental, so that the auditory system has a stronger set of low-order cues from which to extract pitch.
This helps explain why a well-cleared timpano can produce a surprisingly definite pitch for a struck membrane instrument. The ear receives a cluster of near-harmonic, low-order partials and may use them to reinforce the heard note, whether the listener experiences that note mainly as the principal (1,1) pitch, as a virtual pitch related to the missing octave below, or as some weighted combination of both. The result is powerful, but not automatic: mallet choice, strike location, damping, room acoustics, modal amplitudes, and decay times all affect the final percept.
The Individual Listener Factor ↑ menu
One significant finding from recent psychoacoustic research is that pitch perception strategy (spectral versus virtual) varies across individuals and appears to be a relatively stable trait. A 2026 MEG study by Andermann and colleagues found that listeners with stronger f0-based pitch processing showed measurably worse behavioral performance on some pitch-discrimination tasks when the fundamental was absent. Critically, this individual difference was not predicted by musical training or expertise.[8]
This has a practical implication: two listeners in the same hall hearing the same drum may weight the available cues differently. One may emphasize the virtual pitch implied by the harmonic pattern; another with a stronger spectral tendency may attend more to the resolved partials themselves, especially the principal and its octave-related partners. Both may arrive at a usable pitch percept, but via different auditory routes. This may contribute to why players sometimes disagree about whether a particular drum is “in tune,” although differences in seating position, room response, mallet, dynamics, and damping can also be important.
Summary ↑ menu
In the Duff/Benade measurement, the preferred diametric modes of the timpano, (1,1), (2,1), (3,1), (4,1), (5,1), and (6,1), fall near harmonic positions 2 through 7 of an implied missing fundamental. This near-harmonic grouping is one plausible acoustic goal of expert clearing: it brings the most important pitch-bearing modes closer to ratios that support reliable auditory pitch extraction. Because these modes lie in a low-order harmonic range, they are more likely to provide useful resolved spectral cues than a set of higher, more crowded partials. The (3,1) mode provides a strong octave relationship above the (1,1) mode; the (2,1) and (4,1) modes provide close fifth- and major-third-related cues within the implied harmonic pattern. The result is not a string-like or wind-like spectrum, and the missing fundamental is not guaranteed to dominate perception. Rather, a well-cleared timpano can present the ear with a compact, low-order, near-harmonic modal pattern that supports a definite musical pitch in a way that is unusually effective for a struck membrane instrument. Modern psychoacoustic research on resolved harmonics, individual pitch-processing strategies, and the neural correlates of spectral versus virtual pitch perception helps explain why this works, while also showing why different listeners and playing conditions can produce different pitch impressions.
References ↑ menu
- [1] Resolution threshold for low-numbered harmonics (cochlear filter bandwidth and critical-band arguments). See ref. [4] and the broader psychoacoustic literature on resolved and unresolved harmonics for the modern statement. The “about the 10th harmonic” boundary is a useful rule of thumb, not a fixed physiological cutoff.
- [2] Benade, A. H. (1973). Measurements on Duff’s timpano, reported in “Defining a Standard,” The Quantum World of Timpani Pitch, Pauken, 2021. https://quantum.pauken.org/defining-a-standard/. The primary measurement record is also reproduced in WTT Chapter 3 (Missing Fundamental): https://wtt.pauken.org/chapter-3/missing-fundamental. This is a historically valuable but limited source chain: it reports a particular Duff-cleared instrument and should not be treated by itself as a complete statistical survey of timpani acoustics. The Duff “It is not sufficient merely to get the overall skin tension…” and “a revelation to listen to an expert such as Duff clearing a good drum” passages are Benade’s paraphrases of Duff’s teaching, captured in Benade’s published account of the 1973 measurement visit.
- [3] Albera, R., Urbanelli, A., Lucisano, S., Aprigliano, A., Morando, L., Amoroso, A., Alexeev, M., & Albera, A. (2025). “Musical note recognition based on the upper adjacent harmonics without the presence of the fundamental frequency.” PMC, 2025. PMCID: PMC12022334; PMID: 40274804. https://pmc.ncbi.nlm.nih.gov/articles/PMC12022334/
- [4] Preisler, A. (1993). “The influence of spectral composition of complex tones and of musical experience on the perceptibility of virtual pitch.” Perception & Psychophysics, 54, 589–603. Also discussed in Schneider et al. (2005) and in Saus et al. (2025).
- [5] Terhardt, E. (1974). “Pitch, consonance, and harmony.” Journal of the Acoustical Society of America, 55(5), 1061–1069. https://doi.org/10.1121/1.1914648
- [6] Seebeck, A. (1841). “Beobachtungen über einige Bedingungen der Entstehung von Tönen” [Observations on some conditions of the generation of tones]. Annalen der Physik und Chemie, 2nd ser., liii, 423–436. The dispute with Ohm over periodicity pitch versus spectral pitch ran in Annalen from 1841 to 1863 and was ultimately resolved in favor of Seebeck by later experiments. For modern summary, see WTT Chapter 4-2: https://wtt.pauken.org/chapter-4-2/chapter-4
- [7] Saus, W., Seither-Preisler, A., & Schneider, P. (2025). “Harmonic vowels and neural dynamics: MEG evidence for auditory resonance integration in singing.” Frontiers in Neuroscience, 19, 1625403. https://doi.org/10.3389/fnins.2025.1625403. The individual-difference findings about fundamental-versus-overtone listening modes in this paper build on earlier work by the same Heidelberg group: Schneider et al. (2005), Seither-Preisler et al. (2007), and Schneider et al. (2023). Note: this paper’s primary topic is vowel resonance perception in singing; the spectral-versus-virtual-pitch findings appear as a secondary observation about individual differences in pitch processing mode.
- [8] Andermann, M., Reineke, A. L., Riedel, H., & Rupp, A. (2026). “The role of (missing) fundamentals, active listening, and musical expertise in cortical and subcortical correlates of consonance/dissonance.” European Journal of Neuroscience, 63(6), e70483. https://doi.org/10.1111/ejn.70483