Equal Temperament: What It Costs and What It Buys

A piano cannot retune between two chords, so its tuning has to work for every key in advance. Equal temperament is the arrangement that does that, and it earns the ability by making every fifth slightly narrow and every major third noticeably wide. The compromise is easy to state and worth understanding in numbers, because the numbers explain what a keyboard sounds like and why ensembles that can adjust pitch do something else.

#Twelve equal steps

Equal temperament divides the octave into twelve steps of identical size. A step is the frequency ratio 2 to the twelfth root of two, about 1.0595, and the system is usually described in cents, where an octave is 1200 and a semitone is therefore exactly 100.

The arithmetic is compact enough to do on the back of an envelope. If A4 is 440 Hz, the note a semitone above it sits at 440 × 1.0595, about 466.2 Hz, and three semitones above A4, which is C5, sits at 440 × 2^(3/12), about 523.3 Hz. An octave above A4 is 880 Hz, which is exactly twice the frequency and exactly 1200 cents, since cents are defined by 1200 × log₂ of the frequency ratio.

Two properties follow from the equality of the steps. Transposition is exact: the same fingering, fret or valve combination produces the same interval in every key. And enharmonic pairs collapse, so F sharp and G flat are the same key on a keyboard and the same fret on a guitar, which means a piece can move through remote keys without the instrument changing shape underneath the player.

#Two commas, one problem

The reason a compromise is needed at all is that pure intervals do not add up.

Take the fifth, the ratio 3:2. Stack twelve of them and the result is 129.746 times the starting frequency. Seven octaves is 128 times. Twelve pure fifths overshoot seven octaves by the ratio 531441 to 524288, roughly 1.0136, which is about 23.5 cents. That discrepancy is the Pythagorean comma, and it means a keyboard tuned in pure fifths cannot close its own circle: one fifth has to absorb the error, and that fifth is the one players call the wolf.

The second problem is inside the third. Four stacked pure fifths produce a major third of 81:64, which measures about 407.8 cents. The pure major third, the one the ear hears as a 5:4, measures about 386.3 cents. The gap between them is the syntonic comma, roughly 21.5 cents. Pure fifths and pure thirds cannot both be had, because the fifths generate thirds that are too wide by exactly that amount. Tuning a chain of pure fifths therefore produces screaming thirds, and tuning pure thirds requires fifths that are narrowed. The temperaments built on that trade-off before equal temperament are the subject of meantone and well temperament.

Equal temperament resolves both problems by refusing purity everywhere. Each fifth is narrowed by about two cents, which is enough to make twelve of them fit seven octaves exactly, and the accumulated slack is distributed across all twelve fifths instead of being dumped into one.

#What the tuning distorts

The errors are not equally audible, and the table shows where the cost is concentrated. The just column gives the pure five-limit value, the size the interval has when it is built from small whole-number ratios.

The fifth’s error is small, which is why a string quartet can tune its open strings in fifths and still play with a piano without much trouble. The third’s error is seven times larger and in the opposite direction, and it has an audible consequence: the partials that a pure third would align are left slightly apart, and they beat. Near middle C the numbers are easy to check. C4 at 261.6 Hz has its fifth partial at 1308.1 Hz, and E4 at 329.6 Hz has its fourth partial at 1318.5 Hz, so the two partials sit about 10 Hz apart and produce roughly ten beats per second. That is too fast to hear as a wobble and slow enough to be heard as a roughness, which is part of why an equal-tempered piano chord has a shimmer that a just-tuned chord does not.

Semitones lose their two sizes as well. In just intonation a diatonic semitone such as C sharp to D is 16:15, about 111.7 cents, and a chromatic semitone such as C to C sharp is 25:24, about 70.7 cents. Equal temperament gives both 100, so the small step that makes a leading tone lean and the step that merely colours a note become the same distance.

The octave, the one interval equal temperament gets exactly right, and the only one where the just and tempered sizes are identical.

An equal-tempered major third and an equal-tempered fifth. The fifth is two cents away from pure, which is why it sounds settled; the third is about fourteen cents wide, which is why it carries the beating.

Two equal steps of 100 cents. In just intonation the first of these steps, the chromatic semitone, is about 71 cents and the second, the diatonic semitone, about 112, so the two moves are audibly different distances rather than the same one repeated.

#What it buys

Four things, and they are the reasons the compromise won.

The first is that fixed-pitch instruments work in every key. A keyboard, a fretted instrument, a xylophone and a set of tuned percussion have no way to adjust individual notes while playing, so any tuning that privileges some keys makes the others unusable. Equal temperament privileges none and therefore permits all of them.

The second is exact transposition. Every interval has the same size in every key, so a shape learned in C major transfers unchanged to F sharp major. For players of instruments with a fixed layout this is not a convenience but the basis of technique.

The third is enharmonic equivalence. Treating F sharp and G flat as one pitch makes chromatic harmony navigable and makes remote modulations cheap, since the new key does not need new notes, only different ones. Tonal music after roughly 1800 exploits this constantly, and much of the harmonic freedom of the nineteenth century depends on an instrument on which every key is equally available.

The fourth is that the compromise is uniform. A meantone tuning with pure thirds leaves one fifth so wide that the keys using it are avoided or retuned, while equal temperament spreads the same total error thinly enough that no interval is unusable. What the system gives up is the differentiation between keys: when every key is tuned identically, no key has a distinct character, and the historical claims about the sound of particular keys lose their acoustic footing.

#Why ensembles still play unequal

The piano is equal-tempered, and most of the music played by flexible-pitch instruments is not. String players tune their open strings in fifths and then adjust continuously, playing leading tones high and thirds narrow when the texture allows. Wind players lip notes into place, and brass players work with an instrument whose natural resonance is the harmonic series, which is a just system rather than a tempered one. Singers in an unaccompanied ensemble drift towards pure intervals, and one consequence is measurable in performance: an ensemble that tunes pure thirds and pure fifths does not return to the same pitch after a long passage, because the intervals do not add up to the arithmetic of the octave.

Where a piano is present it becomes the reference, and the flexible instruments move to it, which is why a string quartet with piano sounds subtly different from the same quartet alone. The tuning a listener hears in a concert hall is therefore a negotiation, and the equal-tempered grid is one participant in it rather than the definition of being in tune. The practical use of the cents figures above is diagnostic: a tuner reading of two cents sharp is inside the noise of an ensemble, and a reading of fifteen cents sharp on a third is the difference between a tempered interval and a pure one, which players can hear even when they cannot name it.

Questions

Is equal temperament out of tune?

Every interval except the octave is slightly wrong, and the major third is the worst at about fourteen cents wide. The point of the system is that the error is small, uniform and identical in every key, rather than concentrated in one unusable interval.

Why do orchestras sound in tune if they use equal temperament?

They do not use it. String, wind, brass and voice players adjust intonation continuously while playing, so an ensemble moves towards pure intervals and away from the fixed grid of the piano.