Why Does One String Contain So Many Sounds?

Hearing Mathematics · How Sound Enters the Body and Mind · Article Three

When a piano hammer strikes a string, we usually say that it produces “one note”. A spectrum analyser tells a more crowded story. Alongside the fundamental used to name the pitch, higher components appear. They decay at different rates and are altered by the string, soundboard, hammer and resonant state of the instrument. What is heard as a single object is physically a population of simultaneous vibrations.

Ravel’s Boléro turns this fact into something close to a public experiment. The same melodic material returns repeatedly, supported by an insistently stable rhythmic pattern, yet the music does not stand still. Flute, clarinet, bassoon, saxophones, brass and increasingly composite orchestrations take over. A listener recognises the melody each time and simultaneously encounters a new material. Much of the change lies not in which notes are played, but in what exists inside each note, how it begins and decays, and how its spectrum combines with those of other instruments.

An ideal string can vibrate in many modes at once

A string fixed at both ends does not only swing back and forth as a whole. It can form a node at the centre and vibrate in two opposing halves. It can form nodes at thirds and vibrate in three sections. In general, its permitted modes depend on length, tension and linear density. If the lowest mode has frequency f, the modes of an ideal flexible string occur near 2f, 3f, 4f….

These are harmonics. The first harmonic is the fundamental. The second is an octave above it. The third stands an octave plus a perfect fifth above the fundamental, and the fourth returns to a higher octave. When a string is plucked, struck or bowed, the initial force and position distribute different amounts of energy among the modes. Plucking near the centre suppresses some modes according to their nodes and symmetry. Plucking near the bridge usually excites more high-frequency energy, producing a brighter, more pointed sound.

The statement that one string contains many sounds is therefore not mystical. A linear system permits normal modes to be superposed, and the string’s motion is their sum at each moment. Fourier analysis lets us estimate frequency components from a complex waveform. A real piano string, however, has stiffness, and its partials do not fall at exact integer multiples. Article Seven will show that this deviation is large enough to influence how octaves are tuned.

Why do we not hear every partial as an independent melody?

If several frequencies are physically present, why does experience usually contain a single “piano tone”? Because auditory systems do not only separate; they also group. Common onset, correlated changes in level, harmonic relations and a common apparent source encourage components to be assigned to one auditory object. The partials have not vanished. They have changed status from candidate pitches into the timbral structure of an object.

This grouping solves a practical problem. Natural sources commonly generate families of related components. If every spectral line were heard as an independent thing, voices and instruments would be nearly impossible to recognise. Hearing must balance the analysis of components against inference about sources. A spectrum tells us what frequencies are present; perception estimates how many objects those components are likely to belong to.

The missing fundamental makes the distinction vivid. Suppose a complex contains 400, 600, 800 and 1,000 Hz but no energy at 200 Hz. Because those components remain near integer multiples of 200 and share a five-millisecond periodicity, listeners often hear a pitch near 200 Hz. A review of pitch perception treats the missing fundamental as a major test for pitch theories: pitch cannot be obtained simply by selecting the lowest spectral line. The system estimates a periodic relation that need not be represented by energy at the fundamental itself.

This is not the ear inventing an arbitrary error. Telephone systems and small loudspeakers may reproduce upper harmonics more effectively than very low fundamentals, yet listeners can still identify a low voice. Perception produces a stable estimate of source structure rather than reading the spectrum aloud component by component.

Timbre is multidimensional, not a count of “how many overtones”

Musicians often call a tone “rich in overtones”, but the phrase is too coarse. What matters is not only how many partials are present. Their relative amplitudes, the rate at which energy falls towards high frequencies, noise at the onset, evolution through time and room reflections all contribute.

A higher spectral centroid is often associated with greater brightness. A short, steep attack can make a sound more impulsive. The way the spectrum changes through time helps distinguish a sustained bow, a breath-driven tone and a strike. A review of the neurocognition of timbre shows why researchers use multidimensional “timbre spaces” to represent similarity judgements. No single line running from dark to bright can stand for timbre as a whole.

Timbre also affects judgements of melody, voice and consonance. A pair of fundamentals presented as sine waves does not interact in the same way as the same fundamentals produced by instruments with many partials. When Article Four asks why 3:2 matters, the answer cannot be separated from the spectra of the sounds that form the interval.

Boléro: the melody remains, while the auditory object changes

Ravel composed Boléro in 1928 around a persistent snare-drum pattern, two repeatedly stated melodies and a long orchestral crescendo. The manuscript and full scores make the strategy visible: much of the process comes not from conventional melodic development but from changing the instruments and combinations that carry the material.

On one hearing, set aside the harmonic direction and follow only the melody’s material. The opening flute gives the line a light, defined edge. Clarinet and bassoon shift the spectral balance and breath character. Saxophones introduce different attacks and resonances. Later combinations do more than place two instruments on a single pitch. Different registers and harmonic relations can produce a compound colour that belongs to no individual instrument alone.

The mathematics here is not a hidden “magic number” used by Ravel. It lies in superposition, spectral distribution, periodic relation and accumulating intensity. The sequence of pitches preserves melodic identity; orchestration alters the internal structure and boundary of every sound. The listener repeatedly makes two judgements: this is still that melody, and this is no longer the sound heard before.

The work also shows why timbre is not merely decoration applied to musical structure. Replace every statement with an unchanging electronic tone, retaining pitch and rhythm, and the principal process of the piece disappears. In this case colour is not painted onto the outside of form. Changing colour is how the form occurs.

An instrument’s identity exists through variation

Even on one piano, playing the same key softly and forcefully changes more than loudness. Hammer speed and contact affect the excited spectrum; the sustaining pedal permits other strings and the soundboard to join the resonance. A violin can produce the same pitch with markedly different balances of partials through bow speed, pressure, contact point and articulation. We nevertheless recognise “piano” and “violin” because an instrument is not a fixed spectral photograph. It is a family of temporal and spectral behaviours that remains relatively stable across permitted variation.

This is one reason sampling and synthesis are difficult. A static recording for each piano key may preserve fundamentals and some colour, yet fail to preserve continuous transitions among dynamic levels, sympathetic interactions and onset details. To sound instrument-like, a model must reproduce the way relations change with action rather than copying one spectrum.

A note is not a smallest physical unit; it is an organised result

Notation represents a note with a notehead because composition, performance and reading need a manageable unit. Acoustic analysis expands that unit into a fundamental, partials, noise and an amplitude envelope. Hearing organises the components again into a trackable object. All three representations are valid because they answer different questions.

Mathematics exposes the modes concealed within a string and helps explain how one melody can be carried by changing spectra. It cannot determine from the number of frequencies alone how many sounds will be heard. That question also depends on common onset, harmonic relation, space, attention and musical context.

While listening to Boléro, treat each recurrence as a test. What does pitch identity preserve, and what does timbre transform? As the orchestra fills the space, we hear more than an addition of partials. The boundary, weight and inferred source of an auditory object are repeatedly reorganised. One string sustains many vibrations. Music can make them one note, a colour, or a new voice emerging from the background.

Primary sources and further listening

• Oxenham, “Pitch Perception”, including the missing fundamental
• A review of research on the neurocognition of timbre perception
• Ravel, Boléro: manuscript, scores and recordings

Continue reading: Explore the Hearing Mathematics series.

If you would like to bring these ideas about listening, understanding, and practice to the keyboard, you might try ScoreFlow, an app I developed to make score reading and daily practice flow more naturally together.