Every concert grand you have ever heard is tuned out of equal temperament. Not slightly. The highest notes are stretched sharp by up to 30 cents, and the lowest notes pulled flat by a similar margin. A piano tuned to exact mathematical equal temperament sounds wrong to every trained ear in the room. This is not a tuning style preference or a historical artifact. It is a direct consequence of physics, and understanding it matters for anyone working with piano in a virtual instrument context.
The Problem: Stiff Strings Do Not Vibrate Like Theory Predicts
Ideal string theory assumes a perfectly flexible string under tension. Real piano strings are steel wire under enormous tension, but steel is stiff. That stiffness acts as a restoring force in addition to string tension, and the extra restoring force affects different vibrational modes unequally.
The result: upper harmonics are stretched sharp relative to the fundamental. The frequency of the nth partial of a real string is not n times the fundamental. It is:
f_n = n * f_0 * sqrt(1 + B*n²)where B is the inharmonicity constant. For small values of B, this simplifies to:
f_n ≈ n * f_0 * (1 + 0.5*B*n²)The sharpening grows with the square of the partial number. The 8th partial is not just slightly sharp. It is 64 times more affected than the 1st.
The formula for B is derived from the physics of a stiff beam under tension:
B = (π³ * E * d⁴) / (64 * T * L²)Where E is Young's modulus of the string material (roughly 200 GPa for steel), d is the string diameter, T is the string tension, and L is the vibrating length.
For middle C on a typical grand piano (d = 1 mm, T = 667 N, L approximately 620 mm), B comes out to approximately 0.00038. That sounds small. The 2nd partial of this string sits at about 523.66 Hz instead of the expected 523.25 Hz (2 x 261.63 Hz), a stretch of roughly 1.4 cents for just one partial. Barely audible in isolation. But the effect compounds across the keyboard, and this is where the real problem emerges.
The Railsback Curve: What Happens When You Chain Octave Tuning
In 1938, O.L. Railsback published measurements of 16 grand pianos and 12 upright pianos tuned by professional technicians, published in the Journal of the Acoustical Society of America. He measured the pitch of every note immediately after tuning using a chromatic stroboscope and found something consistent: all well-tuned pianos follow approximately the same deviation pattern from equal temperament. This pattern is now called the Railsback curve.
The shape is a stretched S-curve:
- The middle octaves (roughly C3 to C5) are close to equal temperament, within about 2 cents.
- The highest treble notes are stretched sharp by 20 to 30 cents.
- The lowest bass notes are pulled flat by a similar margin.
This happens because professional tuners tune octaves by ear, specifically by listening for the beat rate between the 2nd harmonic of the lower note and the fundamental of the upper note. If a piano's A3 string has its 2nd partial at 440.33 Hz instead of 440.00 Hz, a tuner will tune A4 to 440.33 Hz to eliminate beating, not to 440.00 Hz. A4 is now about 1.3 cents sharp relative to equal temperament.
Repeat this process upward across the keyboard and the deviation accumulates. By A6, the note might sit 15 cents sharp. By the top of the keyboard, 25 to 30 cents sharp. Moving downward from the temperament octave produces the opposite: bass notes get pulled progressively flat.
This is not a mistake. A piano tuned to exact equal temperament, with all notes at mathematically precise frequencies, sounds consistently wrong because the harmonics of low notes beat against the fundamentals of high notes. The beating is audible, distracting, and the result is described universally as "dull" or "flat-sounding."
Why B Varies Across the Keyboard
Look at the formula again: B = (π³Ed⁴) / (64TL²).
The L² term in the denominator means shorter strings have higher inharmonicity. Treble strings on a grand piano are only a few centimeters long, compared to well over a meter for the lowest bass strings. This alone would suggest treble inharmonicity is enormous and bass inharmonicity is small. But the d⁴ term tells a different story for bass strings.
Bass strings are wound with copper wrap to add mass without making the string prohibitively long. But they are still thicker than treble strings, and d⁴ scales aggressively with diameter. A string with twice the diameter has 16 times the contribution to B from the diameter term. So both ends of the keyboard have high inharmonicity for different reasons: the treble because of extreme string shortness, the bass because of extreme string thickness.
The middle of the keyboard has the best balance of length and diameter, producing the lowest B values. This is why the middle register is tuned closest to equal temperament on the Railsback curve.
The Implications for Virtual Instruments
This is where the physics becomes directly relevant to audio production.
Early digital pianos and many budget sample libraries store a recording of each note played at a calibrated equal-temperament pitch, then transpose that sample across the keyboard range. A sample library that captures A4 at 440.00 Hz and uses that sample untransposed for A4 is presenting inharmonic audio with inharmonic partials at specific frequencies. Play a two-octave chord and the 2nd partial of A2 (if the sample was recorded from a stretched piano) may clash with the fundamental of A4 (tuned to 440.00 Hz on the host).
Synthogy's Ivory implements a stretch tuning option precisely because of this. Roland's digital pianos include stretch tuning as a configurable setting. When stretch tuning is off, the piano sounds "lifeless" compared to an acoustic instrument, and the culprit is the mismatch between where the harmonics actually land and where the fundamentals of adjacent notes are tuned.
Pianoteq takes a different approach: as a physical modeling instrument, it computes string inharmonicity from first principles. The B values for each string are parameters in the model. The result is that chords spread across the keyboard sound coherent because the harmonics of each note are generated to match the expected inharmonicity pattern for that string length and diameter, not sampled from a recording that may have been made under different conditions.
The practical implication for layering: if you combine a piano sample library with a synthesizer patch tuned to exact equal temperament, wide-register chords will produce subtle beating in the mid and upper range. The synth's perfect harmonics do not line up with the piano's stretched harmonics. In dense arrangements this disappears into the texture. In sparse, high-register passages with long release tails, it becomes audible as a slight roughness.
What This Means in the Studio
Piano tuning is not a solved problem handed off to instrument makers. It is an active engineering decision in every virtual instrument that includes piano. Stretch tuning is not an option that makes the piano sound slightly warmer or more "vintage." It is the physically correct behavior that every real piano already has. Disabling it is what creates an artifact.
The Railsback curve is also a reminder that equal temperament is a mathematical idealization that real acoustic instruments only approximate. A piano's pitch deviates from equal temperament by as much as 30 cents at the extremes. Yet a well-tuned piano sounds more in tune than a piano mathematically locked to 440 Hz equal temperament. The reason is that "in tune" is defined perceptually, based on the absence of beats between coincident harmonics, not on adherence to a frequency table. The harmonics define what in-tune means for that instrument, and for the piano, the harmonics are always sharp.
Railsback measured this 88 years ago with a stroboscope and graph paper. The physics has not changed. The only question is whether your software accounts for it.