The Note Is Not a Frequency

Build a sound from just three ingredients: pure tones at 600, 800 and 1,000 hertz. There is no 200-hertz tone in the recipe. Yet the three frequencies belong to the harmonic family of 200 hertz, and ordinary complexes of this kind can produce a pitch corresponding to their absent fundamental.

That is the missing-fundamental puzzle. A musical note does not have to contain a sinusoid at the frequency by which we identify its pitch. The lowest member of the family can be absent while the family resemblance survives.

I like this puzzle because the first good explanation is so satisfying that it is easy to mistake it for the last one. The arithmetic explains what information remains in the sound. A much stranger experiment asks whether that information is enough for the listener.

What went missing—and what did not

Frequency describes a physical oscillation. Pitch is a perceptual quality: the highness or lowness by which we organize melody. They are related, but they are not synonyms.

In his 2012 review, auditory researcher Andrew Oxenham describes how a harmonic complex can retain its pitch when energy at its fundamental frequency is removed or masked by noise. Changing the relative strengths of its harmonics also changes timbre—the quality that helps distinguish one kind of sound from another. Pitch can survive substantial spectral change without the sound remaining identical. Nor is pitch perfectly insulated from timbre: the same review describes difficulties comparing pitches across sounds with very different spectral qualities. [1]

For our constructed three-tone example, the remaining structure is easy to inspect. During five milliseconds, the 600-hertz tone completes three cycles, the 800-hertz tone four, and the 1,000-hertz tone five. All three arrive back at their starting phases together. Their sum therefore repeats 200 times per second.

A spectrum and a waveform answer different questions here. The spectrum lists which sinusoidal ingredients are present. The waveform shows how their sum changes through time. The spectrum can contain no line at 200 hertz while the complete waveform has a five-millisecond repeating pattern.

I checked that arithmetic with an ideal digital sum, not a listening experiment: its three component amplitudes were the intended values, its 200-hertz component was zero to numerical precision, and the waveform repeated after five milliseconds. The calculation establishes a property of that signal. It does not establish what any particular person heard.

This also avoids the wrong kind of magic. We need not imagine a missing bass sinusoid being conjured into the acoustic signal. Relationships among the remaining components can carry information about the absent fundamental.

Real ears and playback systems can produce distortion, including new frequencies. That possibility has to be controlled rather than dismissed. But the documented survival of missing-fundamental pitch with noise masking the fundamental is a problem for any explanation that depends entirely on a secretly audible low tone. [1]

The ear is not a single stopwatch

“The sound repeats, so the brain measures the repetition” is an appealing next sentence. It is not a complete account.

The cochlea separates sound by frequency: different regions respond preferentially to different frequency ranges. This frequency-to-place arrangement is called tonotopy. Its filtering matters. Low-numbered harmonics can be individually resolved, while higher-numbered harmonics often share a filter and interact within it, producing amplitude fluctuations at the fundamental's rate. The boundary is condition-dependent, not a fixed line between two harmonic numbers. [1]

The overlapping harmonics seem to offer the easy route: a ready-made envelope rhythm. Yet the review reports that low-numbered, individually resolved harmonics generally produce more robust, accurate pitch than high-numbered unresolved ones. There is a difference between making a periodic cue available and explaining how hearing uses it.

Even the signal arithmetic has a trap. Shift our three ingredients upward by 50 hertz: 650, 850 and 1,050. They are still spaced 200 hertz apart. But the full waveform no longer repeats after five milliseconds; its shortest repeating period is now 20 milliseconds. I checked this as a second ideal-signal calculation. It predicts no listener's pitch—not 200 hertz, not 50. Its narrower lesson is that component spacing and full-waveform repetition are not the same physical property.

To get beyond attractive arithmetic, we need a task in which the listener, rather than our equation, supplies the answer.

Put the timing in the wrong place

In a 2004 experiment, Oxenham, Joshua Bernstein and Hector Penagos tried an ingenious separation. Ordinary frequency changes alter both the timing information delivered by the auditory nerve and where along the cochlea the activity occurs. The researchers wanted to move the timing without giving it the usual address. [2]

They made transposed tones: low-frequency patterns shaped the amplitude of high-frequency carriers. With appropriate processing, these were intended to generate temporal patterns resembling those produced by low tones, but in cochlear regions responding to high frequencies. The resemblance was a first-order, model-supported approximation—not a direct recording showing identical activity in human nerves.

One experiment gave four young adults with normal hearing a comparison involving a single fundamental and a complex made from harmonics three, four and five. The nominal fundamental was around 100 hertz, not the 200 hertz of my illustrative example. In the transposed version, the three harmonic patterns modulated separate high-frequency carriers. Background noise masked possible low-frequency distortion products.

With the ordinary complexes, all four listeners could make meaningful fundamental-pitch comparisons. With the transposed complexes, three reached the test's maximum allowed difference without producing a measurable discrimination threshold. The fourth managed a threshold, but a much worse one. In a subsequent matching task, the three listeners available for testing matched ordinary complexes reasonably accurately to the fundamental, but produced no reliable fundamental-pitch matches for the transposed complexes. [2, Methods and Fig. 3]

That does not mean the transposed sounds were inaudible. It means their temporal ingredients failed to support the same stable complex pitch in those tasks.

The researchers also checked that the transposed stimuli could carry useful timing information for another job. A separate four-person experiment tested tiny differences in arrival time between the ears. At the lower tested frequencies, up to 150 hertz, transposed performance equaled or bettered ordinary-tone performance; above that range it worsened. So usable timing had not simply vanished. Its usefulness depended on the task. [2, Fig. 2]

Then came the particularly revealing mismatch. A tested model pooled temporal information across frequency channels and produced a clear repetition peak corresponding to the fundamental for both ordinary and transposed complexes. The listeners did not produce the matching pitch result. A model that found the right interval had still missed an important condition of hearing. [2, Fig. 4]

A clock needs an address

The experiment does not prove that pitch is encoded only by place, or that timing does not matter. The paper discusses accounts requiring both accurate timing and appropriate cochlear placement; the later review preserves that distinction. The result is a challenge to treating temporal patterns as interchangeable regardless of the frequency channels carrying them. [1, 2]

The limits matter. These were small laboratory samples of young adults. The intended neural similarity was supported by a model and a behavioral control, not established by directly measuring identical human nerve responses. And the review and experiment share an author: they are an overview and its underlying case, not two independent replications. This is a close look at a revealing experiment, not a comprehensive survey of every subsequent pitch theory.

What changed for me is the question. I began with “How can we hear a frequency that isn't there?” That wording confuses the pitch with one possible ingredient of the sound. A better question is: Which relationships among the ingredients let hearing treat them as one note?

Removing a spectral component need not remove the note. Keeping a temporal pattern need not preserve it either. The next puzzle is not just how hearing finds repetition, but how it decides which pieces belong together—and which repetition deserves to count as the pitch.

Sources and scope

[1] Andrew J. Oxenham, “Pitch Perception”, Journal of Neuroscience 32 (2012), 13335–13338. Especially “The what and why of pitch,” “Cochlear filtering,” “Relationship of auditory-nerve and brainstem responses to pitch,” and “Interactions between periodicity and other acoustic variables on pitch.” Narrative review; its historical studies are not separately reanalyzed here.

[2] Andrew J. Oxenham, Joshua G. W. Bernstein and Hector Penagos, “Correct tonotopic representation is necessary for complex pitch perception”, PNAS 101 (2004), 1421–1425. Methods, experiments 1–3; Results and Figs. 2–4; Discussion of place–time accounts. Primary behavioral experiment with model comparison.

Both complete main texts, methods where present and textual captions were consulted. The 600/800/1,000 and 650/850/1,050-hertz examples are reproducible ideal-signal calculations, not audio playback, listener measurements or the study's exact stimuli. No new listening, instrument measurement, neural recording or clinical claim is made.