Equal temperament and cents: why a piano’s thirds are slightly off

Play C and E together on a piano and you hear a major third. It isn’t the purest possible major third, though. In the simplest version of that interval the upper note vibrates five times for every four of the lower one, a ratio of 5:4, and a piano’s third is 14 cents wider than 5:4. Its fifths, by contrast, are almost exactly pure.

That difference comes from equal temperament, the tuning that has dominated Western music since the 20th century, and it’s a deliberate trade. This article explains the unit used to measure it, the cent; why a piano’s fifths come out nearly pure and its thirds don’t; how you can hear the difference as beats; and what it all means when you’re training your ear.

Cents: a ruler for intervals

Intervals are ratios of frequencies, which makes them awkward to compare in hertz. The octave above A3 (220 Hz) is 220 Hz higher, but the octave above A4 (440 Hz) is 440 Hz higher. Cents solve that. A cent is a hundredth of an equal-tempered semitone, the step between two neighboring piano keys, so an octave, twelve semitones, is 1200 cents wherever it starts.

Alexander Ellis introduced the unit in 1885 to compare the musical scales of many nations, and it has become the standard way to measure and compare intervals.

A single cent is too small to hear between two notes played one after the other. One author puts the smallest pitch difference people can tell apart at about 5 to 6 cents, and the threshold varies a lot from person to person and with the sound itself. Played together, though, small differences show up another way, as beating, and that’s how tuning becomes audible.

Just intervals: whole-number ratios

A piano string doesn’t vibrate at one frequency only. Above the fundamental, the pitch you hear, it adds harmonics at roughly two, three, four and five times that frequency, and so on up. Neighbors in that series make the simplest intervals: the octave, 2:1; the fifth, 3:2; the fourth, 4:3; the major third, 5:4; the minor third, 6:5.

Tuning an interval to one of these whole-number ratios is called just intonation. A just interval is pure: it doesn’t beat. Measured in cents, the just fifth is 702 and the just major third 386. Earstep draws each interval’s figure from these ratios, which is why the simple ones close into tidy shapes (here’s how).

How far a piano is from just

Of those ratios, equal temperament keeps only the octave exact. It splits the octave into twelve equal semitones, each a frequency ratio of the twelfth root of 2, and every interval is a whole number of them. Here is how far each of the piano’s intervals lands from its just ratio:

Two groups stand out. The perfect fifth and fourth are almost pure: the fifth is 2 cents narrower than 3:2 and the fourth 2 cents wider than 4:3. The thirds and sixths are well off: the major third is 14 cents wider than 5:4, the minor third 16 cents narrower than 6:5, the major sixth 16 cents wider than 5:3 and the minor sixth 14 cents narrower than 8:5. They come in mirror-image pairs, because each pair adds up to a pure octave: the fifth is exactly as narrow as the fourth is wide, and the minor sixth as narrow as the major third is wide.

Beats: the difference you can hear

Two tones at almost the same frequency don’t sound steady. They drift in and out of step, and the sound swells and fades, a beat, as many times a second as their frequencies differ.

Intervals beat through their harmonics. In a fifth, the third harmonic of the lower note and the second harmonic of the upper note nearly coincide. Tuned just, they match exactly and the fifth is steady. Tuned as a piano tunes it, they are slightly apart, and they beat. Press each tuning to compare:

In a major third the near-matching harmonics are the fifth of the lower note and the fourth of the upper one. The piano’s third is much further from just, and around middle C those two harmonics are about ten hertz apart. So the third beats about ten times a second, where the fifth beats a little less than once:

Piano tuners use exactly this. They set the middle of the keyboard by listening to beats, and Wikipedia’s article on piano tuning lists the beat rate of each interval. One of the easiest checks of equal temperament is to play major thirds a semitone apart, one after another, and listen for their beating to speed up evenly. An octave lower, every interval beats half as fast.

You can hear the same thing on a real piano, in the way a held interval fades. These are Earstep’s own clips:

Why keyboards settled on equal temperament

Why not tune a piano just? Because a keyboard gives each note one fixed pitch, and no set of twelve fixed pitches can make every interval pure in every key. The classic illustration: stack twelve pure fifths, which should bring you back to your starting note seven octaves up. You overshoot by about 23.5 cents, a gap called the Pythagorean comma.

Every tuning has to put that error somewhere. Older keyboard tunings, such as meantone, sounded in tune in some keys and more dissonant in others. Well temperaments made every key usable, but intervals still varied in size from key to key, so each key had its own character. Equal temperament spreads the error evenly. All twelve fifths are made the same size, each 2 cents narrower than 3:2, and twelve of them close the circle exactly. Every interval is then the same size in every key, so music can move to any key without changing how its notes relate. The price is the thirds and sixths you saw in the table, a little off everywhere.

Two people are usually credited with first working out the math, independently: the Ming prince and mathematician Zhu Zaiyu, who published precise figures in 1584, and the Flemish scholar Simon Stevin. When keyboards actually switched is harder to pin down, but equal temperament has been the predominant tuning of Western music since the 20th century, and pianos are generally tuned with A4 at 440 Hz, a standard agreed internationally in 1939.

What it means for ear training

Earstep plays every sound on one grand piano, the Salamander Grand Piano, sampled by Alexander Holm. Real pianos are usually tuned to a slightly stretched form of equal temperament: their strings’ overtones run a little sharp, so tuners widen the octaves to match, and the recorded piano was tuned that way too. The app measures every sampled note and retunes it to equal temperament at A4 = 440 Hz, every note within 0.8 cents.

So the intervals you practice are the piano’s own: thirds 14 cents wider than 5:4, fifths 2 cents narrower than 3:2. They’re the ones you’ll hear on pianos and most keyboards, since equal temperament is the most common temperament in use today.

A third that’s 14 cents wider than 5:4 is still a major third. A major and a minor third are a whole semitone, 100 cents, apart, and tuning moves an interval by a fraction of that, so its name doesn’t change. What changes is its color, the faster beating you heard above. Singers and string players, who can adjust every note, tend to lean toward pure intervals, so the same third can sound slightly different in a choir than on a piano. It’s the same interval either way, with the same name.

When you practice intervals played together, let each one ring. The thirds’ quick beating is one more thing to listen for next to the near-still fourth and fifth. Try it with all four:

In Earstep, Intervals together opens after level 2 of Rising intervals and starts with “Third or fifth”, the major third against the fifth.

Sources

• Cent (music), Wikipedia: the cent as a hundredth of an equal-tempered semitone, Ellis’s 1885 paper, and how small a pitch difference people can hear.
• Just intonation, Wikipedia: just intervals as whole-number ratios that don’t beat, the harmonic series, the Pythagorean comma, and singers’ and string players’ pull toward pure intervals.
• Five-limit tuning, Wikipedia: the just ratios of the thirds.
• Equal temperament, Wikipedia: twelve equal semitones, the comparison with just intervals, Zhu Zaiyu and Simon Stevin, and its place since the 20th century.
• Musical temperament, Wikipedia: why fixed-pitch instruments need a temperament, well temperament, and how equal temperament closes the circle of fifths.
• Beat (acoustics), Wikipedia: what beats are and how fast they come.
• Piano tuning, Wikipedia: harmonics that coincide and beat, tempered intervals, the table of beat rates, and the test with rising major thirds.
• A440 (pitch standard), Wikipedia: A at 440 Hz, agreed internationally in 1939.
• Inharmonicity, Wikipedia: why a piano’s overtones run sharp and its octaves are tuned wide.
• Stretched tuning, Wikipedia: stretched octaves on wire-strung instruments.
• Open Music Theory: Intervals: interval sizes, and minor intervals a half step smaller than major ones.