Why Can’t a Piano Be Tuned Exactly to Its Theoretical Frequencies?

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

Given A4 at 440 Hz and f(n)=440×2^(n/12), tuning a perfect piano might seem to require nothing more than setting eighty-eight keys to the frequencies in a table. Electronic measurement can display deviations to decimal places. A professional tuner, however, does not finish by calibrating every isolated string to its theoretical fundamental. A well-tuned piano is commonly sharper than ideal equal temperament in the treble and flatter in the bass. Its octaves are deliberately “stretched”.

This is not hearing obstructing science, nor a tuner damaging an exact equation with subjective taste. The equation assumes ideal periodic sources. Piano strings have stiffness. Hammer contact affects excitation. Bridges and the soundboard couple energy. Unison strings must be coordinated, and the entire instrument must sound continuous across registers. Piano tuning optimises relations among complex sounds rather than copying a list of fundamentals detached from materials.

The difference between an ideal flexible string and real steel wire

The modes of an ideal flexible string occur at f, 2f, 3f, 4f…. If a bass fundamental is 110 Hz, its second partial is exactly 220 and its fourth exactly 440. The fundamental of the A an octave above can align exactly at 220. Tuning an octave by eliminating beats among corresponding partials appears straightforward.

A piano string is not a geometrical line without bending resistance. Steel stiffness adds restoring force to higher modes, making partials slightly sharper than ideal integer multiples. The deviation generally becomes greater for higher partial numbers. This is inharmonicity. A string’s “second partial” may lie a little above twice its fundamental, and its fourth a little above four times.

Tune the upper-octave fundamental to exactly twice the lower fundamental and some of the upper tone’s components will fail to align with the sharpened partials of the lower string, producing beats. Raising the upper tone slightly can calm the partial relationship actually being compared. Moving downwards can stretch bass fundamentals in the opposite direction. The resulting deviations from theoretical equal temperament open out towards the ends of the keyboard, a pattern commonly associated with the Railsback stretch or Railsback curve.

An open study in the Journal of the Acoustical Society of America combined measured piano spectra, inharmonicity and a model of sensory dissonance, producing results close to the empirical Railsback curve. A separate subjective tuning experiment involving piano tuners and orchestral musicians found that synthetic tones with piano-like inharmonicity produced a stretched tuning curve resembling Railsback’s and significantly different from harmonic comparison tones.

The Railsback curve is not a template every piano must copy

“Pianos need stretch” can become a new misconception if it is taken to mean “apply one universal curve”. Inharmonicity depends on string length, diameter, tension, material and scale design. A small upright may need shorter, thicker or wound strings to make its bass; a concert grand has a different scale. Individual notes can also have local differences.

The Railsback curve is a classic summary of measured tunings, not a law detached from instruments. A tuner must hear the beats, register transitions, unisons and resonance of the specific piano. Electronic tuning systems likewise need to estimate the instrument’s inharmonicity. Results can vary with use, tonal goal and tuning tradition.

An average graph cannot therefore decree that every upper note must be raised by one fixed number of cents. The recurring structure of the problem is more important: theoretical equal temperament supplies reference relations; material shifts the partials; tuning distributes deviations across the instrument so that important comparisons remain coherent.

One piano key often controls more than one string

In the middle and upper ranges, one key commonly strikes two or three unison strings. Exact agreement can reinforce a sustained tone; slight deviations produce beats and change how the sound develops and decays. A tuner establishes a reference string and brings the others into a suitable unison. Here, too, “one note” is a dynamic organisation. A listener should hear one tone rather than competing sources, while the sound retains the life associated with the instrument.

With the sustaining pedal depressed, strings not struck by the hammer can resonate sympathetically when they share components with played tones. The bridge transmits energy to the soundboard, which moves a much larger area of air. Room reflections further alter the spectrum reaching a listener. A fundamental measured at one point cannot represent the complete object experienced in performance.

Measurement remains useful. Modern tools can estimate fundamentals, partials and inharmonicity with great accuracy, preserve a target and detect drift. The error begins when “accurate measurement” is mistaken for “a complete decision about where to tune”. Data describe the relations. A tuning standard must still select which partials to coordinate, how registers should join and how the piano will operate in a room and repertoire.

The beats heard by a tuner are not vague intuition

Aural tuning is not a matter of feeling that something is “close enough”. Tuners select intervals and partial pairings, using beat rates to establish a temperament region and extend it across the keyboard. Equal-tempered fifths, fourths, thirds and sixths have expected beat-rate patterns. Octaves can be checked through 2:1, 4:2, 6:3 and other partial pairings. Decisions in the bass and treble are cross-checked.

This expertise translates mathematical relations into temporal experience. A frequency difference written as a number is heard as a number of fluctuations per second. The way beat rates should progress across semitones and registers supplies redundant tests. A poor relation appears in several checking intervals rather than depending on one mysterious judgement.

Electronic and aural tuning are not opposing camps. A good device can calculate a stretch target from the measured inharmonicity of an individual instrument; a tuner can then verify it through intervals, colour and continuity. The machine expands what can be measured. A person defines the goal and accepts the integrated result.

Debussy’s La cathédrale engloutie needs the resonance of a whole piano

Debussy’s Tenth Prelude from Book I, La cathédrale engloutie, constructs the image of a cathedral rising from the sea through distant open intervals, chordal planes and bass resonance. The original piano score specifies changing dynamics, register and pedalling relations. The work plainly does more than emit theoretical frequencies in sequence.

Its opening intervals need silence and decay to define distance. Later, massive chords extend across a broad range. Bass fundamentals, upper partials, soundboard resonance and the sympathetic field released by the pedal combine into an imagined bell-like sonority. Replace every note with an unrelated pure sine wave and much of the weight and depth disappears.

Stretched tuning does not make the work more “impressionistic”. It helps the complex tones of an actual piano coordinate across register. High partials of bass notes continually encounter fundamentals and partials in the middle and treble. Tuning makes the whole instrument available as a resonant space. Touch and pedal then decide which of those potential relations become audible.

Theoretical frequencies remain essential, but they are a first-level standard

Without the structure of equal temperament, tuning lacks a transposable framework. Without a reference pitch, instruments cannot readily coordinate. Without spectral measurement, inharmonicity is difficult to explain. Practice does not refute theory; it makes theory specific.

The levels are better stated this way. Equal temperament specifies ideal logarithmic relations among pitch classes. Piano scaling gives each string material conditions. Inharmonicity alters partial locations. The tuner chooses stretch and beat rates across the instrument. Performance and the room turn those relations into actual sound. Each layer adds constraints and requires judgement.

A piano cannot be tuned “exactly to theoretical frequencies” because those frequencies define only a model of fundamentals. The physical piano demands a richer precision: not that every isolated number be correct, but that many non-ideal partials form an audible, comparable and usable order across one instrument.

While listening to La cathédrale engloutie, notice what seems to emerge above a bass after it has sounded. The effect belongs to no single key. It is maintained by strings, soundboard, pedal, room and hearing. The task of tuning is not to conquer material, but to make the material capable of answering itself.

Primary sources and further listening

• Giordano, “Explaining the Railsback Stretch in Terms of the Inharmonicity of Piano Tones and Sensory Dissonance”
• Rigaudon et al., “Effect of Inharmonicity on Pitch Perception and Subjective Tuning of Piano Tones”
• Debussy, Préludes, Book I, including La cathédrale engloutie

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.