The Equal Temperament Compromise: Why Every Piano Is Slightly Wrong
Every interval in Western classical music, with the exception of the octave, is slightly out of tune. This is not a defect in instrument construction or in the ear of the listener. It is the intended result of a deliberate mathematical decision made, formalized, and adopted across the Western musical tradition over several centuries.
The decision is called equal temperament. Its consequences are present in every piano ever built, every fretted guitar, and every harmony notated in the Western tradition.
The Problem of Pure Intervals
A string vibrating at 100 Hz produces a tone. A string vibrating at 200 Hz produces a tone exactly one octave higher, and the relationship between the two frequencies is acoustically pure. The ratio is 2:1, and this ratio produces the smooth, beating-free consonance that the ear perceives as resolution.
Other interval ratios also produce pure harmonies. A perfect fifth, the interval between C and G, has an ideal frequency ratio of 3:2. A perfect fourth has a ratio of 4:3. A major third has a ratio of 5:4. These ratios, collectively called just intonation, produce intervals free of acoustic beating and, in isolation, more consonant than any tempered equivalent.
The problem arises when attempting to build a keyboard instrument that plays in multiple keys using pure intervals throughout.
The Comma Problem
Twelve perfect fifths stacked on top of each other should, in theory, equal seven octaves. A pure fifth has a ratio of 3:2, so twelve fifths equal (3/2)¹² = approximately 129.75 times the starting frequency. Seven octaves equal 2⁷ = 128 times the starting frequency. The two values do not match. The discrepancy, roughly 23.5 cents or nearly a quarter of a semitone, is called the Pythagorean comma.
A keyboard instrument tuned to pure fifths accumulates this comma across its range. Instruments tuned this way produce beautiful music in their home key and in closely related keys, but generate harsh, beating harmonies in distant keys. A harpsichord tuned for the key of C becomes effectively unplayable in F-sharp.
The Compromise
Equal temperament distributes the comma across all twelve semitones of the chromatic scale. Each of the twelve fifths is made slightly smaller than pure, shrunk by exactly one twelfth of the Pythagorean comma. The result is that no interval except the octave is acoustically pure, but equally, no key is more out of tune than any other.
The tradeoff is a general slight dissonance throughout, in exchange for complete harmonic flexibility. Johann Sebastian Bach's collection of preludes and fugues in all twenty-four major and minor keys, published in 1722 and known as The Well-Tempered Clavier, demonstrated that a tempered keyboard instrument could produce musically satisfying results in any key. The collection was as much a technical argument as a musical one.
What the Ear Accepts
Equal temperament does not go unnoticed by trained ears. Professional string players, who are not bound by frets or fixed pitches, routinely play intervals slightly wider or narrower than their equal-tempered equivalents when performing with other strings. The major third, which is 14 cents sharp in equal temperament compared to its just equivalent, is often narrowed by ensemble string players for a smoother sound. The equal-tempered major triad is measurably more dissonant than the just major triad.
The ear has been trained over centuries to accept the equal-tempered piano as the reference pitch. Western listeners perceive its intervals as normal because they are surrounded by instruments that produce them. The compromise is so thoroughly normalized that pure intervals of just intonation can, to a Western ear trained from childhood on the piano, sound subtly wrong. Familiarity has recalibrated the perceptual standard, and the slight wrongness of every note has become invisible.
The equal temperament compromise is the harmonic foundation of Western civilization's musical output for the last three centuries. Every symphony, every sonata, every chord is built on a mathematical approximation. The beauty turns out to require the error.