Why Is a Piano Slightly Out of Tune on Purpose? Equal Temperament Explained

Musical intervals are ratios of frequencies. An octave is 2 to 1, a perfect fifth 3 to 2, a major third 5 to 4, and the pure intervals sound smooth because the harmonics of the two notes line up. The trouble is that no whole number of fifths equals any whole number of octaves: twelve fifths is 3 over 2 to the twelfth power, about 129.7, and seven octaves is 128, and the difference is the comma. The same happens with thirds, since four fifths up and two octaves down give a third of 81 to 64, sharper than the pure 5 to 4 by a smaller comma. An instrument with fixed pitches cannot have all its fifths pure and all its octaves pure, and it cannot have pure thirds and pure fifths at once; something has to give.

A temperament is a decision about where to put the error:

• •Pythagorean: pure fifths, with the whole comma dumped on one fifth, the wolf, which howls and cannot be used; fine for medieval music that avoided thirds
• •Meantone, from the 1500s: pure or nearly pure major thirds, with each fifth narrowed by a quarter of the smaller comma; sweet in the common keys, unusable in the remote ones, and the standard of the Renaissance and early Baroque
• •Well temperaments, from the late 1600s: the error spread unequally so that every key was usable but each had its own colour, C major pure and F sharp major edgy; the tuning Bach probably meant by the Well-Tempered Clavier
• •Equal temperament: the octave divided into twelve identical semitones, each fifth narrowed by a twelfth of the comma, every key identical and every interval except the octave slightly impure

In equal temperament each semitone is the twelfth root of two, about 1.0595, an irrational number that no ratio of whole numbers equals, and the intervals it produces are all a little off. The fifth is two cents flat, two hundredths of a semitone, which nobody hears; the major third is fourteen cents sharp, which a trained ear does, and which gives the tempered major chord a faint shimmer of beating that a pure chord does not have. Musicians who play instruments without fixed pitch, singers, string players, wind players, drift toward pure intervals when they play together and toward the piano's when they play with it. The gain is that every key is available, that modulation to any other key costs nothing, and that an instrument can be built with twelve notes to the octave rather than the nineteen or thirty-one that some Renaissance keyboards had in the attempt to have it both ways.

The mathematics was known in China in 1584, when the prince Zhu Zaiyu calculated the twelfth root of two to nine decimal places, and in Europe within a few years, and it was resisted for two centuries because it sounded worse in the keys people used than meantone did and because it made every key sound alike, which composers who wrote for the colours of the well temperaments regarded as a loss. It won as music moved further from home, through the chromatic harmony of the nineteenth century, when Chopin and Wagner modulated to keys that a well temperament made unpleasant, and as the piano became the instrument of the middle-class home and had to be tuned by a trade rather than a musician; by about 1850 it was the standard, and it has been the tuning of nearly all Western music since, so thoroughly that the shimmer in a major chord is what most people now think a major chord sounds like.

The piano departs from equal temperament for a second reason. A real string is stiff, and its harmonics are slightly sharp of the exact multiples of the fundamental, so an octave tuned pure by the numbers sounds flat against the lower note's harmonics; tuners therefore stretch the octaves, tuning the treble progressively sharp and the bass progressively flat, by up to a third of a semitone at the extremes, so that the instrument sounds in tune with itself. A concert piano is thus out of tune on purpose twice over, by temperament and by stretch, and the tuner's skill is in hearing the beats between harmonics and setting them to the rates that make the compromise inaudible, which is why a piano tuned by ear and one tuned by an electronic tuner to the theoretical frequencies do not sound the same.

You can calculate what a tuner hears. When two notes sound together, any harmonics that almost coincide interfere and produce a pulsing called beats, at a rate equal to the difference between their frequencies. Take the A below middle C at 220 hertz and the E above it. In a pure fifth the E would be 330 hertz, so the A's third harmonic and the E's second harmonic both sit at 660 hertz and nothing beats. In equal temperament the E is about 329.63 hertz, its second harmonic falls at about 659.26, and the pair beats roughly three times every four seconds, a slow, gentle wave.

Now try the major third from the same A to the C sharp above it. Pure, the C sharp would be 275 hertz, and the A's fifth harmonic and the C sharp's fourth harmonic would both lie at 1,100 hertz. In equal temperament the C sharp is about 277.18 hertz, putting its fourth harmonic near 1,108.7, so the pair beats nearly nine times a second, a clear flutter you can hear if you hold the chord and listen. Tuners use exactly these rates, counting beats against a watch or by feel, to set each interval, which is why the fifths sound almost still and the thirds shimmer.

Temperament decides how the notes relate to each other; it says nothing about where the whole set sits, and that has been a separate argument. For centuries the A that musicians tuned to varied from city to city and decade to decade, sometimes by a semitone or more. The tuning fork, invented in England in 1711 by the trumpeter John Shore, let a pitch be carried from place to place, and forks surviving from the eighteenth and nineteenth centuries record that variety. As orchestras pushed pitch upward for brilliance, singers complained of the strain, and in 1859 France fixed its standard A at 435 hertz by decree.

An international conference in London in 1939 recommended 440 hertz, the standard most of the world now uses, and the International Organization for Standardization confirmed it after the war. It has never been universal. Some orchestras in continental Europe tune a little higher, around 442 or 443, for a brighter sound, and ensembles playing Baroque music on period instruments commonly tune to 415, almost exactly a semitone lower. A piano tuner therefore sets two separate things: the reference pitch, usually 440 hertz for the A above middle C, and the temperament, which spreads the comma across the twelve notes built on it.

Even a perfectly tuned piano does not stay that way, and the main culprit is the weather. The soundboard, a large thin sheet of wood that amplifies the strings, absorbs moisture from the air and swells slightly in humid conditions, pushing up against the strings and raising their pitch. In dry conditions, such as a heated room in winter, it shrinks and the pitch falls.

Because the effect is uneven across the instrument, the piano drifts out of tune with itself, not just up or down as a whole. Tuners therefore recommend tuning at least once or twice a year, keeping pianos away from radiators, fireplaces and draughty windows, and keeping room humidity reasonably steady. A new piano also settles noticeably in its first year as the strings stretch.

A piano is tuned slightly out of tune because the arithmetic of pure intervals does not close: twelve pure fifths overshoot seven octaves by the Pythagorean comma, so a fixed-pitch instrument must distribute an error somewhere, and equal temperament distributes it evenly by dividing the octave into twelve identical semitones, leaving every fifth two cents flat and every major third fourteen cents sharp. It replaced meantone and the well temperaments as music modulated more freely, and piano tuners stretch its octaves on top to match the stiff strings' sharp harmonics.