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 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.

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.