Why Is a Piano Slightly Out of Tune Everywhere? A Compromise Nobody Can Avoid

Intervals sound pure when the frequencies of the two notes stand in a simple ratio, with an octave at two to one and a fifth at three to two being the clearest. Building a scale by stacking pure fifths and reducing by octaves should eventually return to the starting note, and it does not, since twelve pure fifths overshoot seven octaves by a small but audible amount. The same mismatch appears with other intervals. That discrepancy is a fact of arithmetic rather than a defect of instruments, and it means no tuning can make every interval pure in every key. Every system therefore decides which intervals to keep pure and where to put the unavoidable error.

Each distributes the error differently and each suits different music:

• •Pythagorean tuning, stacking pure fifths, which leaves one fifth badly wrong
• •Just intonation, keeping thirds and fifths pure in one key at the cost of others
• •Meantone temperaments, narrowing fifths slightly to make thirds pure, which was standard for centuries
• •Well temperaments, distributing the error unevenly so every key is usable and each sounds different
• •Equal temperament, dividing the octave into twelve identical steps, so every key is equally slightly wrong
• •Systems dividing the octave differently, used in several musical traditions

The universal modern solution is a compromise and knowing what it gives up is worth doing. Octaves remain pure, since they are not compromised by the division. Fifths are very slightly narrow, by an amount most listeners never notice. Thirds are noticeably wide, by considerably more than in the systems it replaced, which means the most consonant interval in ordinary harmony is the one that suffers most, and singers and string players playing without fixed pitch instinctively correct it. What is bought is that every key is equally usable and modulation anywhere is possible, which is precisely what nineteenth and twentieth century music requires and what the older systems could not supply.

The whole problem applies only to instruments whose pitches are set in advance, and the ones that are not behave quite differently. Singers, string players without frets, trombonists and wind players with sufficient control adjust every note as they play, tuning intervals by ear towards purity and correcting continuously against what they hear around them, which is why an unaccompanied choir does not sound like a piano. That adjustment is largely unconscious and is what training develops. Playing with a keyboard forces those musicians towards its fixed compromise, which is one reason accompanied and unaccompanied performance sound different. Fretted instruments sit in between, with fixed frets imposing equal temperament and with players bending strings to correct particular notes.

The uneven temperaments that preceded the modern one gave each key a distinct character, since the errors fell differently in each, so one key had pure thirds and a smooth sound while another was tense and harsh. Composers of that period wrote about key character and chose keys accordingly, and a great deal of what they say about it makes no sense under equal temperament, where every key is identical apart from pitch. Performances on instruments tuned historically restore that variation and are one of the more audible results of historically informed practice. The transition was gradual, contested and later than commonly stated, with equal temperament becoming standard for keyboards through the nineteenth century rather than in Bach's lifetime.

Twelve pure fifths overshoot seven octaves, so no tuning can keep every interval pure in every key and each system places the error somewhere. Equal temperament keeps octaves pure, narrows fifths imperceptibly and leaves thirds noticeably wide, buying the ability to play in any key. Earlier uneven systems gave each key a distinct character that composers wrote about deliberately.