Why Tuning a Piano is Mathematically Impossible

If you've ever noticed that a piano sounds slightly different from a guitar playing the "same" note, you've brushed up against one of the most elegant impossibilities in mathematics. Tuning a piano perfectly is not a practical problem waiting for a clever engineer — it is provably, irrevocably impossible.

The reason sits at the intersection of number theory, acoustics, and thousands of years of musical frustration.

What Does "In Tune" Even Mean?

Two notes are considered consonant — pleasant-sounding together — when their frequencies form simple integer ratios. The reason is physical: simple ratios mean the pressure waves produced by the two strings reinforce each other regularly, creating a stable combined waveform.

The most important intervals, and their ideal frequency ratios:

These are pure or just intervals. When two strings are in a pure interval, you hear no beating — no wobble. Play slightly out of tune and you hear a pulsing amplitude modulation called beats.

Beat Frequencies: The Sound of Imperfection

When two notes with frequencies $f_1$ and $f_2$ are played simultaneously, the combined waveform beats at a rate equal to their difference:

$$f_{\text{beat}} = |f_1 - f_2|$$

This follows from the sum-to-product identity:

$$\sin(2\pi f_1 t) + \sin(2\pi f_2 t) = 2\cos\!\left(2\pi \frac{f_1 - f_2}{2}\, t\right)\sin\!\left(2\pi \frac{f_1 + f_2}{2}\, t\right)$$

The sine term is the "average" pitch you hear. The cosine term is a slow amplitude envelope that pulsates at half the difference frequency — so you perceive one full beat per $\tfrac{1}{f_1 - f_2}$ seconds.

Piano tuners exploit this directly. They tune fifths to beat at a specific slow rate (usually 0–2 Hz), rather than aiming for zero beats. This is not sloppiness — it's the whole point, as we'll see.

Notice how at 1 Hz you hear a slow, clear throb. At 6–8 Hz it sounds rough and dissonant. Piano tuners listen for these beats constantly.

The Pythagorean Comma: Where the Math Breaks

Here is the core problem. Suppose we want to tune a piano so that:

1. Every octave is a perfect 2:1 ratio, and
2. Every fifth is a perfect 3:2 ratio.

Start on C. Go up by 12 perfect fifths and you should arrive at C again — seven octaves higher. Let's check:

$$\left(\frac{3}{2}\right)^{12} = \frac{531{,}441}{524{,}288} \approx 1.01364$$ $$2^7 = 128$$

After 12 fifths: $\left(\frac{3}{2}\right)^{12} = \frac{531{,}441}{524{,}288}$

After 7 octaves: $2^7 = 128$

Divide them:

$$\frac{(3/2)^{12}}{2^7} = \frac{531{,}441}{524{,}288} \approx 1.01364$$

They don't match. The ratio $\frac{531{,}441}{524{,}288}$ is called the Pythagorean comma, and in musical units (cents, where 100 cents = one semitone):

$$\text{Pythagorean comma} = 1200 \log_2\!\left(\frac{531{,}441}{524{,}288}\right) \approx 23.46 \text{ cents}$$

That's nearly a quarter of a semitone — easily audible.

Step through the circle one fifth at a time. Notice how the orange dots drift clockwise relative to the grey equal-tempered positions — about 2 cents per fifth. After 12 fifths you've overshot C by 23.46 cents. The circle of fifths doesn't close.

Why This Is Mathematically Inevitable

The deeper reason is this: no positive integer power of 2 can ever equal a positive integer power of 3.

Formally: $2^a \neq 3^b$ for any positive integers $a, b$.

Proof: $2^a$ is divisible only by 2. $3^b$ is divisible only by 3. By the Fundamental Theorem of Arithmetic (unique prime factorisation), they can never be equal. $\blacksquare$

More generally: an octave (ratio $2/1$) and a fifth (ratio $3/2$) are incommensurable — their logarithms have an irrational ratio:

$$\frac{\log(3/2)}{\log(2)} = \log_2(3/2) = \log_2 3 - 1 \approx 0.58496\ldots$$

This is irrational (since $\log_2 3$ is irrational — $3 = 2^x$ has no rational solution). So no finite number of perfect fifths will ever stack up to a whole number of octaves. The circle of fifths never closes.

The same argument applies to thirds: $\frac{\log(5/4)}{\log 2} = \log_2(5/4)$ is irrational. Every pure interval except the octave is incommensurable with the octave.

The Three Historical Approaches

Since perfection is impossible, every tuning system is a different philosophical answer to the question: which imperfections are most tolerable?

1. Pythagorean Tuning

Stack 11 perfect fifths, then close the circle by making the 12th fifth absorb the entire Pythagorean comma. That gap lands on one specific fifth — called the wolf fifth — which is reportedly unlistenable (roughly 678 cents instead of 702 — a deviation of ~24 cents, nearly a quarter semitone).

In Pythagorean tuning, all fifths and fourths are pure. The major thirds (81/64 ≈ 407.8 ¢) are noticeably sharp compared to the pure 5:4 (386.3 ¢). Works well for medieval music that stays in a few keys; catastrophic if you modulate.

2. Just Intonation

Choose simple integer ratios for all notes: $\frac{9}{8}, \frac{5}{4}, \frac{4}{3}, \frac{3}{2}, \frac{5}{3}, \frac{15}{8}$. Within a single key, I'm told this sounds great — every consonant interval is mathematically pure, with no beating at all.

The problem: modulate to a new key and you need entirely different frequencies. On a fixed-pitch instrument like a piano, this limits you to one key. Worse, even within a key, the "D" you get from the major second of C differs from the "D" you get as the minor third of B♭ by a syntonic comma (81/80 ≈ 21.5 ¢).

3. Equal Temperament

Divide the octave into 12 equal semitones, each with ratio $2^{1/12}$. Every fifth is then:

$$2^{7/12} \approx 1.4983 \quad \text{vs pure } \frac{3}{2} = 1.5000$$

A difference of about 1.955 cents — barely perceptible. Every key sounds identical, and the comma is distributed evenly across all 12 fifths. This is the system used on all modern pianos.

The tradeoff: major thirds in equal temperament are 400 ¢, but the pure 5:4 major third is 386.3 ¢ — a difference of 13.7 cents, which is quite audible on sustained chords. Orchestral players and singers instinctively sharpen their fifths and flatten their thirds toward just intonation; the piano cannot.

Toggle between systems above. Notice that just intonation has the purest thirds and fifths (zero cents deviation for the ones it targets) but introduces large errors elsewhere. Equal temperament distributes the error evenly but no interval except the octave is pure.

4. Meantone Temperament (Bonus)

Quarter-comma meantone, used extensively from the 15th to 18th centuries, splits the difference differently: it tunes the major third pure (5:4) and shrinks each fifth by a quarter of the syntonic comma (~5.4 ¢). The result: pure thirds in the "home" keys (I'm told these sound noticeably better than equal temperament), but an even worse wolf fifth than Pythagorean (about 737.6 ¢, nearly 40 ¢ flat).

Bach's Well-Tempered Clavier was written in a well temperament — not equal temperament, but a specific unequal temperament where each key has a different character. The famous "key colours" come from the varying amounts of comma distributed to each interval.

Stretch Tuning: Even Equal Temperament Isn't Enough

Even if we accept equal temperament, there's another wrinkle: real piano strings don't vibrate in perfect harmonics.

An ideal vibrating string produces harmonics at exactly $f, 2f, 3f, 4f, \ldots$. A real string has nonzero stiffness, which causes inharmonicity — the overtones are stretched slightly sharp:

$$f_n = n \cdot f_1 \cdot \sqrt{1 + B(n^2 - 1)}$$

where $B$ is the inharmonicity coefficient (larger for short, thick bass strings). The 2nd partial is slightly sharper than $2f_1$, the 3rd is sharper still.

This means that if you tune the octave using the 2nd partial of the lower note (as a tuner naturally would, listening for the beat to vanish), you'll end up with the higher octave sharp of equal temperament. This compound effect across 7 octaves requires stretch tuning — treble notes are tuned progressively sharp, bass notes progressively flat.

The resulting curve (octave width vs. note number) is called the Railsback curve, named after O.L. Railsback who measured it in the 1930s. A piano tuned with a flat Railsback curve apparently sounds dull and unclear; one tuned to strict equal temperament sounds thin at the extremes. The "right" amount of stretch is an aesthetic judgment the tuner makes for each individual piano.

Summary: Three Layers of Impossibility

Each layer would individually make perfect tuning impossible. Together they guarantee that every piano is a carefully engineered compromise — a machine for making music despite what the mathematics demands.

The next time you hear a piano, know that the tuner has made hundreds of tiny peace treaties with the irrationality of $\log_2 3$. Every note is slightly wrong in a carefully chosen way — and apparently that's good enough.

Further Reading

• Ross W. Duffin, How Equal Temperament Ruined Harmony (and Why You Should Care)
• Harry Partch, Genesis of a Music — the case for just intonation taken to its logical extreme
• Owen Jorgensen, Tuning: Containing the Perfection of Eighteenth-Century Temperament — exhaustive historical survey
• The Railsback curve: O. L. Railsback, "Scale Temperament as Applied to Piano Tuning," Journal of the Acoustical Society of America (1938)