Why You Can Hear a Note That Is Not in the Sound
The Note Survives the Missing Frequency
Imagine a sound assembled from just three pure tones: 600, 800 and 1000 vibrations per second. There is no 200-hertz tone in the recipe. Nothing is quietly playing that frequency underneath the others.
Yet a sound like this can carry a pitch near 200 hertz. The lowest component in the sound and the note a listener perceives need not be the same thing.
This is the missing fundamental. I like the name less now that I understand it better: it makes the phenomenon sound as though hearing has conjured something from nothing. A frequency component is absent. But the sound has not lost every clue to the note.
The pattern is still there
Consider those three frequencies as the third, fourth and fifth harmonics of 200 hertz. In five milliseconds, the 600-hertz component completes three cycles, the 800-hertz component four, and the 1000-hertz component five. All three return to their starting phases together. The combined waveform repeats 200 times a second.
That is true even though its spectrum—the list of sinusoidal ingredients—has no component at 200 hertz. The waveform's common repeat rate survives the missing ingredient. For this ideal sum, five milliseconds is the shortest common period, not merely one convenient reunion.
This is arithmetic, not a listening report. It shows a physical clue available to hearing; it does not predict exactly what every listener will perceive. In his 2012 review of pitch perception, Andrew Oxenham describes the established result: harmonic sounds can retain their fundamental pitch when energy at the fundamental is removed or masked by noise.
The distinction matters for music. A note is not simply whichever frequency happens to sit at the bottom of a spectrum. Changing the strengths of its harmonics can alter its timbre while leaving its pitch broadly intact—though the review also warns that timbre changes can interfere with pitch judgments. The categories are useful, not hermetically sealed.
The arithmetic is the doorway. It is not the whole hearing mechanism.
A repeating signal does not explain its own listener
The cochlea separates incoming sound into frequency-selective channels. Low-numbered harmonics can be sufficiently separated that a channel responds mainly to one of them. These are called resolved harmonics. Higher-numbered harmonics can crowd together within a channel and interact, producing fluctuations in its output. These are unresolved harmonics; the boundary between the two is not a fixed harmonic number under all conditions.
It is tempting to make those fluctuations the entire answer: adjacent components beat together, hearing reads the repetition, mystery solved.
But Oxenham's review describes a complication. Low, resolved harmonics generally provide a stronger and more accurate pitch than high, unresolved ones, even though the latter can offer an obvious repeating envelope. A good account has to explain both kinds of evidence, not just point to a convenient beat.
There are timing-based accounts, accounts using the pattern of frequencies represented across the auditory system, and accounts combining them. Showing that a sound repeats does not tell us which information a listener used, where it was combined, or how accurately it can support a musical decision.
That last question is where the missing fundamental becomes more interesting than a party trick.
Can you find the third, not just notice a change?
In an independent 2021 experiment, Hedwig Gockel and Robert Carlyon asked musically trained listeners to adjust intervals between synthetic tones. The task was concrete: make the second note a perfect fifth, a major third or a major second below the first. They also tested matching to unison.
This demands something different from noticing that a melody has changed. You can detect a changed step without knowing precisely how large the step is. Accurately adjusting a third tests interval size, not just detection of a changed step.
The reference sounds contained only harmonics six through ten. For a fundamental of 280 hertz, that meant actual components from 1680 to 2800 hertz. For a fundamental of 1400 hertz, it meant components from 8400 to 14,000 hertz. Neither reference contained its fundamental. In the high condition, the very high frequencies belonged to the ingredients, not to a supposed 14,000-hertz fundamental.
The listeners were unusually well qualified for the challenge. Twenty-nine young trained musicians were screened; nine passed the hearing and pitch-discrimination requirements and entered the main experiment. These were not nine people picked at random, and their results are not a forecast for every musician's ears.
The experiment also worked to obstruct shortcuts. Continuous noise was used to mask possible distortion products. Harmonic levels and starting phases varied. The button increments varied so listeners could not reliably calculate an interval by counting clicks. Some references split odd and even harmonics between the ears. Listeners could replay the pair until satisfied, without being told how accurate their answer was.
The high-frequency sounds supported interval adjustments that were not random. A separate control strengthens the missing-fundamental interpretation: three relatively good performers matched complexes containing different, non-overlapping sets of harmonics above chance. That test credited octave-equivalent answers, so it supports a pitch related to the fundamental, not necessarily an exact register judgment. The missing-fundamental pitch had not simply disappeared. [Results III.B; footnote 3.]
But it was a much less dependable musical tool. Adjustments had larger systematic errors, varied more from attempt to attempt, and required more presentations before listeners were satisfied. Across the musical-interval conditions, the geometric mean of the high-to-low ratios of within-listener standard deviations was about five. That is a statement about repeated-adjustment variability—not a claim that every error or every listener became five times worse. Individual differences were large. [Gockel and Carlyon, Methods II and Results III.A.]
There is a small but important boundary here: the high reference started with an 8400-hertz component, but the adjustable note moved downward. At the target fifth, its lowest component was about 5600 hertz. Not every sound in a high-condition trial stayed above 8400 hertz.
A pitch can survive without surviving equally well
The useful result is neither “hearing restores the note perfectly” nor “the missing fundamental stops working.” A residue pitch can remain available while the precision needed to place a musical interval deteriorates.
This also puts earlier reports of successful high-frequency melody discrimination in perspective. The 2021 authors discuss several differences: their reference components were higher, component levels varied, edge components were quieter, and the task demanded interval adjustment. Their study does not isolate task difficulty as the sole cause, and it is not an exact repeat that cancels the earlier findings. It asks more of the listener under different conditions.
Nor does it settle the neural argument. Auditory-nerve firing can synchronize with a waveform's cycles, a behavior called phase locking. How far that timing information remains usable at high frequencies in humans is disputed. The poorer interval judgments are consistent with reduced timing information, but unfamiliarity with extreme high harmonics remains another possible contributor. These behavioral measurements do not directly measure nerve firing or uniquely choose a pitch algorithm. [General Discussion IV.]
I started with a missing ingredient. I ended with three separate questions: what relationships remain in a sound, what pitch a listener can extract, and how precisely that pitch can be used.
The first has a neat five-millisecond answer in our toy example. The others require listeners, tasks and limits. The note is not a little object concealed inside the waveform, waiting to be found intact. It is a percept supported by evidence—and some evidence supports a steadier note than other evidence does.
The next question I want to follow is what happens when the pitch stays nominally the same but the timbre changes radically. How much can a sound change its character before comparing its note to another becomes difficult?
Sources and limits. Oxenham, “Pitch Perception,” Journal of Neuroscience 32 (2012), 13335–13338, DOI 10.1523/JNEUROSCI.3815-12.2012, is a review, not a second new experiment. Gockel and Carlyon, “On musical interval perception for complex tones at very high frequencies,” Journal of the Acoustical Society of America 149 (2021), 2644, DOI 10.1121/10.0004222, supplies the independent primary experiment. I read the review's full main text and caption, and the primary paper's full accessible main text, methods, main figure captions and footnotes. I did not inspect its supplementary Appendix, figure images or raw data, or conduct a listening experiment. The illustrative waveform and frequency ranges were checked mathematically; the behavioral findings are the researchers' reported results.