(And why that’s actually a good thing.)
If you’ve ever played a freshly tuned piano and thought, “It still sounds a little off,” you’re not imagining it.
Even after a professional tuning, a piano is never perfectly in tune. Not in the mathematical sense, anyway.
And yet, that slight imperfection is exactly what makes it sound warm, balanced, and alive.
This is the story of how physics, perception, and musical compromise work together to create the illusion of a perfectly tuned piano.
Equal Temperament: A Brilliant Compromise
Modern music depends on a system called 12-tone equal temperament. It’s the reason you can play in every key on the same piano without it sounding completely wrong.
Before equal temperament, instruments were tuned using older systems like just intonation and meantone temperament. Those tunings made some intervals sound beautifully pure, but others horribly dissonant. You could tune your instrument to sound perfect in one key, but shift to another and the harmony would fall apart.
Equal temperament changed that. It divides the octave into 12 equal logarithmic steps, each with the same ratio of 2^(1/12) between them. That means every semitone is spaced identically on a frequency scale.
Mathematically, this works beautifully. For example:
- If A4 = 440 Hz,
- Then A5 = 880 Hz,
- A6 = 1760 Hz,
- And each half-step in between multiplies by 1.059463.
In a perfect world, this would give us an instrument that’s both logical and consistent across all keys.
But the piano doesn’t live in a perfect world.
Real Strings, Real Problems
A theoretical string vibrates in perfect whole-number multiples: 2x, 3x, 4x, and so on.
A real piano string doesn’t.
It’s thick, short, and under extreme tension, which means it behaves more like a rigid metal rod than a perfectly flexible line. This stiffness causes each overtone (also called a partial) to vibrate slightly sharp compared to its theoretical frequency.
This phenomenon is called inharmonicity, and it’s built into the very design of the piano.
Every string has its own unique pattern of inharmonicity. Thicker, shorter strings—like those in the bass—have much more of it. Long, thin strings—like those in the treble—have less.
That’s why a small upright often sounds “bright” or “tight,” while a 9-foot concert grand sounds open and resonant: its long strings produce overtones that align much more closely with pure harmonic relationships.
The Problem with “Perfect” Octaves
Let’s take a simple example: tuning an octave.
In theory, if you tune A4 at 440 Hz, you’d expect A5 to be 880 Hz. That’s a perfect 2:1 frequency ratio.
But because of inharmonicity, the second or fourth partials of the lower note (A4) won’t align perfectly with the fundamentals or partials of A5. When that happens, you hear slow “beats” — a kind of wobbling interference between nearby frequencies.
The octave will sound restless, even sour, even though the math says it’s correct.
So to make the octave sound pure, tuners raise the upper note ever so slightly—just enough for the overtones of both notes to align to the ear. The beats disappear, and the octave suddenly feels fused and stable.
That’s an aurally perfect octave — it sounds pure even though it isn’t mathematically pure.
What “Stretch Tuning” Really Means
If every octave needs to be widened a little bit to sound perfect, what happens when you keep doing that all the way up and down the piano?
You get stretch tuning.
Each octave is stretched slightly wider than a 2:1 ratio, just enough to make the octaves sound right locally. But when you plot the entire instrument, the top notes end up sharper and the bass notes flatter than the mathematical model would predict.
It’s not that the tuner is “stretching too much.” It’s that each octave is aurally perfect in its own register, and those small deviations accumulate as you move outward from the center.
The result is a tuning curve that looks like a shallow “S”:
- The low bass falls below the theoretical line.
- The treble rises above it.
- The middle of the keyboard sits closest to perfect 12-TET.
That’s how every well-tuned piano behaves—from a small spinet to a full concert grand.
Equal Temperament Still Holds Within Each Octave
This is where many players get confused: “If the octaves are stretched, doesn’t that mean the tuning isn’t equal temperament anymore?”
Not quite.
Equal temperament is about the spacing of the 12 notes within each octave, not the overall size of the octave itself.
Within each register, tuners still divide the octave evenly on a logarithmic scale.
In other words, all the semitone steps are consistent. The whole system just “rides” on top of a gently curved tuning line caused by inharmonicity.
So equal temperament is preserved—it’s just applied to a piano that bends the rules of physics a little.
Why Tuners Don’t Stretch Past What Sounds Right
A common misconception is that tuners stretch octaves past what sounds pure to make other intervals (like fifths and thirds) sound more balanced.
That’s not true.
In modern equal temperament, the goal is always aural perfection for each octave—no more, no less.
If a tuner pushed the upper note beyond that point, the octave would start to beat again, this time in the opposite direction. It would sound restless and thin.
Instead, the tuner stops exactly where the octave locks in and the partials line up cleanly. That “sweet spot” varies slightly depending on the piano’s string scale, but it’s always found by ear (or, in modern tools like Pianoscope, by analyzing inharmonicity curves).
When tuners move up or down the keyboard, they use those aurally perfect octaves as the reference for building new ones. Each one sounds correct in its own neighborhood, and the resulting curve is a natural consequence of that process—not an intentional exaggeration.
The Accumulated Stretch
Let’s quantify that a bit. On a well-scaled concert grand:
- Each octave might be stretched by only 1–2 cents near the middle.
- By the top, that stretch accumulates to 30–40 cents sharp.
- By the bottom, the lowest strings might be 20–30 cents flat.
If that sounds like a lot, don’t worry. The human ear perceives pitch logarithmically, and the octave relationships remain aurally clean. You don’t hear the stretch—you hear balance.
That cumulative stretch is also why recordings of acoustic pianos rarely match pitch-perfect synthesizers. Digital pianos often use exact equal temperament without any stretch, which can make them sound sterile when played alongside an acoustic instrument.
Why This Imperfection Sounds Beautiful
A “perfectly” tuned piano—one where every octave is an exact 2:1 ratio—would actually sound terrible.
Its octaves would beat against themselves.
Its chords would shimmer unevenly.
Its bass would feel muddy, and the treble would feel tense.
Stretch tuning corrects that by letting each register breathe according to its own physics. It aligns the overtones so that the piano feels cohesive from bottom to top.
And our ears prefer it.
The human brain doesn’t judge pitch by pure frequency ratios—it judges by harmonic alignment. When the partials line up, we hear smoothness, warmth, and depth. When they don’t, we hear tension and noise.
That’s why a well-tuned acoustic piano always sounds more “alive” than a digital one. The tiny imperfections in its tuning are what make it human.
Why Different Pianos Stretch Differently
Every piano has its own fingerprint of inharmonicity.
A 9-foot Steinway D has much longer strings than a small studio upright, which means its overtones are closer to harmonic. As a result, a concert grand needs less stretch to sound balanced.
Small uprights, on the other hand, require more. Their shorter strings exaggerate inharmonicity, so tuners must stretch the octaves more aggressively to make them sound pure.
That’s also why comparing the tuning of two very different pianos can be misleading. You might measure them both with an app and see slightly different curves—but both can still be perfectly in tune for their scale.
Equal Temperament vs. Real-World Temperament
It’s worth noting that equal temperament itself is an abstract concept.
It’s a distribution system—a way to space notes evenly so that all keys are usable. It doesn’t prescribe exactly how wide the octave must be in absolute terms.
That’s why two equally well-tuned pianos might both be “equal tempered” but still differ slightly in their overall stretch. Each one follows the same internal relationships, just mapped onto its own physical limitations.
So when we talk about a piano being “in tune,” what we really mean is that it’s balanced within itself. Every interval fits naturally with every other, even if the frequencies don’t add up perfectly on paper.
A Piano That’s Never in Tune, Yet Always Sounds Right
Here’s the paradox:
A piano that’s “perfectly” in tune would sound wrong.
A piano that’s slightly “out” in every octave sounds right.
The imperfection is the point.
Every piano tuner’s job is to find that delicate line between physics and perception—to make an inherently inharmonic instrument feel harmonious. It’s part science, part art, and part illusion.
The next time you hear a pianist play and think, “That piano sounds incredible,” remember: it’s not really in tune. It just sounds that way because every little imperfection has been perfectly placed.
The Takeaway
- Equal temperament divides the space within each octave evenly.
- Piano strings are stiff, causing inharmonicity and sharpened overtones.
- Tuners stretch octaves slightly to make them sound aurally pure.
- Each octave is tuned by ear, not by math.
- The result: no piano is ever truly “in tune,” but the best ones sound perfectly balanced.
That’s the beauty of the piano. It’s not a machine that obeys physics perfectly—it’s an instrument that lives somewhere between math and emotion.
