Why a piano can't be perfectly in tune

Added: 2026-05-20

[00:00:00]Let's make some perfectly in tune major chords. Not approximately in tune, not good enough, but perfectly in tune using whole number frequency ratios. We'll start with C, the root of a C major chord.

[00:00:17]From there, we'll tune the major third, the interval between C and E. As the tuning changes, you can hear the beating between the notes speed up and slow down.

[00:00:38]That beating is caused by nearby frequencies interfering with each other.

[00:00:43]As the interval approaches the exact 5 to4 frequency ratio, the beating slows and finally disappears.

[00:00:57]This is when the fifth harmonic of the C and the fourth harmonic of the E are at exactly the same frequency. Those two harmonics fuse together and the interval sounds completely stable. A perfectly in tune major third. Next, the fifth from C to G.

[00:01:22]The pure fifth corresponds to the ratio 3:2. When that ratio is exact, another set of harmonics lines up perfectly and again the beating disappears.

[00:01:40]At this point, the entire major triad C is perfectly in tune.

[00:01:50]This isn't equal temperament anymore.

[00:01:52]We've adjusted two notes so that this one chord is acoustically ideal. Let's extend this to other chords. Using our perfectly tuned G as the root, we'll tune a G major chord to be perfectly in tune.

[00:02:18]Next, using the D from that chord as a root, we tune a perfectly in tune D major chord.

[00:02:41]and from the A in that chord, a perfectly in tune A major chord. Yeah.

[00:03:01]At each step, we've preserved the whole number relationships. So, all these chords are perfectly in tune. But there's a big problem. Look at what happened to the C major chord. The E that originally formed the pure major third with C has been changed. The C major chord that was perfectly in tune is no longer in tune. To repair the C major triad, we have to retune the E again. But if we do that, the A major chord breaks.

[00:03:36]So we can make C major, G major and D major perfect. But we can't also make A major perfect. And this is the central problem of tuning systems. The intervals that sound most in tune do not fit together in a system that includes all chords.

[00:03:52]The tool I've been using here contains several tunings that were popular during the last few hundred years. In them, you can see the kinds of compromises they used and follow the trend from a small number of perfectly intune intervals to equal temperament, the system most widely used today.