What Is Equal Temperament? Making Everything Slightly Wrong on Purpose

Pure intervals come from simple frequency ratios, with an octave at two to one and a perfect fifth at three to two, and those ratios are what the ear recognises as consonant. The difficulty is arithmetic. Going up by twelve pure fifths and down by seven octaves should return to the same note if the two are compatible, and instead the fifths arrive slightly higher, by an interval called the Pythagorean comma, roughly a quarter of a semitone and easily audible. The same incompatibility appears between pure thirds and fifths. Any instrument with fixed pitches, meaning keyboards, fretted strings, harps and most wind instruments, must therefore decide which intervals to make pure and which to sacrifice, because it cannot adjust in the moment. Instruments without fixed pitch, principally voices, unfretted strings and trombones, do adjust continuously and play closer to pure intervals in context, which is why a good string quartet does not sound like a piano.

Several systems preceded equal temperament and each failed differently:

• •Pythagorean tuning stacked pure fifths, giving excellent fifths and noticeably wide thirds, which suited medieval music where thirds were treated as dissonant
• •Just intonation made the intervals of one key exactly pure, producing extraordinarily sweet harmony in that key and intervals in others that are unusable, including a badly out-of-tune fifth known as the wolf
• •Quarter-comma meantone narrowed each fifth slightly to obtain pure major thirds, which suited Renaissance and early baroque music and confined usable keys to a limited set
• •Well temperaments, of which there were many competing recipes in the seventeenth and eighteenth centuries, distributed the error unevenly so that all keys were usable while each retained a distinct character, which is what gave keys their traditional associations
• •Bach's collection of preludes and fugues in every key is frequently cited as advocating equal temperament, and the title refers to a well-tempered rather than an equal-tempered instrument, which are not the same thing
• •Equal temperament divides the octave into twelve identical steps, each a ratio of the twelfth root of two, making every fifth slightly narrow and every major third noticeably wide

The gain is complete freedom of key. Any piece can be transposed to any key and sounds identical apart from pitch, modulation to distant keys becomes available, and a fixed-pitch instrument can accompany anything. That freedom is what nineteenth-century harmony is built on, since composers from Beethoven onward moved through remote keys in ways that earlier tunings could not accommodate. The loss is that no interval except the octave is pure. Equal-tempered major thirds are audibly wide, by about fourteen cents, which is why a sustained major chord on a piano has a faint beating that a barbershop quartet singing the same chord does not. Key character disappears, since every key is a transposition of every other, which means the traditional descriptions of keys as bright or mournful no longer have a physical basis on a modern piano and survive as convention and as the effect of different string tensions and instrumental ranges.

Adoption was gradual and contested, with theoretical descriptions appearing in China and Europe in the sixteenth century, growing use through the eighteenth, and full dominance in Western practice by the later nineteenth as pianos standardised and tuning became a trade with instruments to measure it. Piano tuning is further complicated by inharmonicity, since real strings are stiff and their upper harmonics fall slightly above exact multiples, which means a piano tuned mathematically sounds wrong and tuners stretch the octaves, spreading the extremes apart to match what the ear expects. Outside Western music, twelve equal divisions is one choice among many: Arabic maqam and Turkish traditions use intervals that fall between the twelve, Indian classical music works with a system of finer divisions and expects instruments to be tuned to a piece rather than universally, and gamelan ensembles are tuned as a set with their own scales and are deliberately slightly mismatched between paired instruments to produce a shimmering beating. A modern movement in microtonal composition works explicitly outside the twelve.

Pure intervals cannot be fitted into a fixed set of pitches, since twelve stacked fifths overshoot seven octaves by an audible amount, so every fixed-pitch instrument compromises. Earlier systems made some keys pure and others unusable, while equal temperament divides the octave into twelve identical steps so every key is equally usable and every interval except the octave is slightly wrong, with thirds noticeably wide. That trade enabled free modulation and removed key character.