Piano Inharmonicity: The Hidden Challenge Every Tuner Must Understand
There’s a paradox at the heart of piano tuning: the piano can never be mathematically in tune, and attempting mathematical tuning makes it sound worse.
The reason is inharmonicity — and understanding it is what separates tuners who make pianos sound good from tuners who make pianos sound correct but wrong.
What Are Overtones?
Every musical note contains not just its fundamental frequency, but a series of partials — higher frequencies that vibrate simultaneously and give each instrument its characteristic tone color.
For most instruments, these partials are called harmonics and they fall at exact whole-number multiples of the fundamental: 2×, 3×, 4×, and so on. When these partials are exact multiples, they’re called harmonic — the octave is twice the fundamental, the fifth three times, and so forth.
This means when you tune a violin string’s A to 440 Hz, its partials will be at 880, 1320, 1760 Hz — exact multiples. Two strings tuned to the same note will share the same partial frequencies, and their octaves and fifths will mesh cleanly.
Why Piano Strings Are Different
Piano strings are not ideal strings. They have stiffness — particularly in the thick bass strings — and that stiffness causes the partials to deviate upward from their theoretical positions.
Instead of 2×, 3×, 4×… the actual partials of a piano string might be 2.001×, 3.004×, 4.010×… The deviation increases with each higher partial, and it increases with:
- String thickness — bass strings are much stiffer relative to their length
- String length — shorter strings (treble) have more inharmonicity for their pitch
- String tension — lower tension increases inharmonicity
This deviation of partials from harmonic positions is called inharmonicity, and it varies enormously across the piano’s range. Bass strings have the highest inharmonicity; long, well-designed treble strings on high-quality instruments have relatively low inharmonicity.
Why Mathematical Octaves Sound Flat
Here’s where it gets interesting.
If you tune an octave so that the fundamental of the upper note is exactly twice the fundamental of the lower note — mathematical octave purity — it will sound flat.
Why? Because when you play that octave together, the partials interact. The human ear doesn’t primarily compare fundamentals; it listens to the coincident partials — the higher-frequency overtones that overlap between the two notes.
For an octave, the relevant coincident partial is the 2nd partial of the lower note meeting the 1st partial (fundamental) of the upper note. But because of inharmonicity, the 2nd partial of the lower note is slightly sharp from its theoretical position. To match that partial — to make the octave sound pure and beatless — you have to tune the upper note slightly sharp of mathematical purity.
This means every octave in a piano is stretched slightly beyond its mathematical position. This is called octave stretching, and it’s not an error — it’s the correct tuning for a piano.
The Cascade Effect
Octave stretching isn’t just about octaves. Because temperament is built from octaves and the bass and treble are tuned by octaves from the temperament octave, inharmonicity shapes the entire tuning of the instrument.
- The bass of a piano is stretched significantly flat of where it would be on a purely mathematical grid
- The treble is stretched significantly sharp
- The overall result is a broad curve called the tuning curve or stretch curve that experienced tuners internalize
On a well-tuned grand piano, A5 (the A above middle octave) is often 10-15 cents sharp of mathematical A5 when measured from A0 at the bottom. On an upright, the stretch curves are usually even more dramatic due to shorter string lengths and higher inharmonicity.
What Good Tuners Do
Master tuners don’t think consciously about inharmonicity while tuning — they’ve internalized it through thousands of pianos, and their ear automatically adjusts.
What they’re actually listening for is beatlessness at coincident partials, not mathematical purity at fundamentals. When an octave sounds clean and full with no beating, the partials are aligned — and inharmonicity has been correctly accommodated without measurement.
This is one of the reasons aural tuning produces better results than naive ETD tuning on real pianos. An ETD that tunes mathematical octaves (or uses a poor stretch model) will produce octaves that sound slightly off even though the numbers are “correct.” A tuner with a well-trained ear naturally compensates because they’re listening to what sounds right — which, on a piano, is not what the mathematics predicts.
Modern ETDs like CyberTuner actually measure the piano’s inharmonicity profile before computing a tuning solution — they use the acoustic physics, not just math. This is what makes advanced ETD work useful. But it also means the underlying principle is identical to what a trained ear is doing: accounting for the actual vibrational character of each string.
The Practical Implications for Learners
If you’re learning to tune pianos, inharmonicity has a few important practical implications:
1. Octaves should sound pure, not measure pure. When you’re checking your octave tuning, trust your ear over a cents meter. If the octave sounds beatless and full, it’s correct — even if it measures sharp or flat.
2. Bass tuning is judgment, not formula. The lowest bass notes have extreme inharmonicity, and the “correct” tuning requires matching specific coincident partials while letting others beat. Experienced bass tuning sounds fat and resonant; novice bass tuning often sounds thin or wrong-sided.
3. Treble gets sharper. Don’t fight the stretch. If your treble sounds bright and slightly sharp to your octave expectations, that’s often correct — the inharmonicity demands it.
4. Different pianos need different stretch. A 9-foot concert Steinway has very different inharmonicity curves from a 46-inch studio upright. The same physical approach doesn’t produce the same sonic result on different instruments. Experience on many different pianos is irreplaceable.
5. Your ear will learn to hear stretch automatically. This is one of the subtle beauties of aural training — with enough practice, your ear begins to find the right stretch intuitively, without conscious calculation. The physics becomes perception.
The Deeper Point
Inharmonicity is not a flaw of pianos. It’s a defining characteristic of the instrument — part of what gives a piano its distinctive tone, its richness in the bass, its brightness in the treble.
Tuning a piano isn’t about forcing it into a mathematical ideal. It’s about listening to what this piano, with its specific string geometry and physics, wants to sound like — and helping it sound its best.
That’s a craft judgment. And it’s why no algorithm, no ETD, no formula fully replaces an experienced ear on a complex instrument.
Learn to hear inharmonicity and develop the tuning ear that compensates for it automatically — 88 Lessons covers this in depth starting in the intermediate curriculum. Start from Lesson 1 →