Tuner and Reference Tones

A tuner shows a needle, and the needle looks like a fact. It is closer to a comparison. Before any meter can say a string is flat, somebody has to decide what frequency the note ought to have, and that decision was made in meeting rooms, not in a laboratory. The measuring part is real physics. The target is a treaty.

That distinction is worth holding on to, because most arguments about tuning are arguments about the target, and most frustration with tuners is about the measurement. They are different problems with different cures.

A number chosen by committee

A musical note is a vibration that repeats. Count the repeats in one second and you have its frequency in hertz. Pitch is what a listener makes of that: higher or lower, this note or that one. For a steady tone the two travel together closely enough that musicians name notes by frequency, and by convention the anchor is the A above middle C.

The anchor has moved. Johann Heinrich Scheibler recommended 440 Hz in 1834, after inventing a device to measure pitch. In the 1860s France fixed its own standard at 435 Hz, and many countries followed. British piano tuners settled on 439 in 1899. An international conference in London in 1939 agreed on 440, a recommendation followed in 1955, and in 1975 it was formalized as the standard ISO 16.

The United States had already been broadcasting the number. In August 1936, at the request of several music organizations, the national standards laboratory added a 440 Hz tone to its shortwave station, and it still goes out in most hours of the day. An orchestra with a radio could tune to the government.

None of this made 440 a law of nature. Early-music groups play at lower pitches, a modern baroque convention sits near 415 Hz, and at least one major orchestra was tuned to 442 for years. A reference tone that cannot move its A is only useful to people who already agree with it.

Cents, and the compromise inside the scale

Once one note is pinned down, the rest follow from ratios. An octave is a doubling of frequency. Equal temperament, the system fretted instruments and modern pianos are built around, divides that doubling into twelve identical steps, each a ratio of the twelfth root of two, about 1.059. To get from the A at 440 Hz to any other note, multiply or divide by that ratio once per semitone.

Ratios are awkward to compare by eye, so tuning uses a logarithmic unit. An octave is 1200 cents and each equal semitone is 100. The number of cents between two frequencies is 1200 times the base-two logarithm of their ratio. One cent near the A at 440 is about a quarter of a hertz; one cent at the bottom of the piano is under two hundredths. That is why a tuner speaks in cents: the same reading means the same musical error anywhere on the instrument.

Equal temperament is a compromise, and cents show exactly how much of one. A pure fifth, two notes at a ratio of 3 to 2, is 702 cents wide, and the equal-tempered fifth is 700. Almost nobody hears that. A pure major third, a ratio of 5 to 4, is 386 cents, and the equal-tempered third is 400. That gap of 14 cents is audible as a shimmer in any sustained major chord on a piano.

Older keyboard tunings spent the error differently. Quarter-comma meantone kept most major thirds pure at 386 cents and paid for it with a handful of unusable keys. Well temperaments such as Werckmeister III, from 1691, made every key playable and let each sound a little different. Pythagorean tuning stacked pure fifths and accepted thirds of 408 cents. Each is a coherent answer to a problem with no perfect one, since three pure major thirds add up to 1159 cents, not 1200.

There is one more layer of convention. Some instruments read a different note from the one they sound. A clarinet in B flat that plays a written C produces a B flat, a major second lower, and an alto saxophone in E flat sounds a major sixth below what is on the page. A tuner hears concert pitch. The player has to know which name to expect.

What beats tell the ear

Long before meters, tuners had a measuring instrument built into the physics of sound. Play two steady tones that are close in frequency and the result swells and fades. Those are beats, and their rate is simply the difference between the two frequencies. Two tones one hertz apart beat once a second. As the pitches converge the beating slows, and when they match it stops.

Here is an example to reproduce with the second tone on this page. Take the A at 440 Hz and add a tone 5 cents above it. Five cents is a ratio of two to the power of 5 over 1200, so the second tone is 441.27 Hz. The difference is 1.27 Hz, and you hear 1.27 beats per second, a slow, obvious pulse. Halve the offset and the pulse halves. An error far too small to hear as a change of pitch is easy to hear as a rhythm.

Beats have limits. Above about 20 a second the ear stops following them as pulses and hears roughness instead. They also measure sameness, not correctness: two strings can be beatless together and both wrong. And a harmonic on a string is a pure interval while frets are tempered: matching the harmonics of two neighboring guitar strings sets a pure fourth of 498 cents where the frets expect 500. Real strings are also slightly stiff, which pulls their overtones sharp. So a guitar tuned entirely by matching harmonics is only approximate.

That is the case for the drone. Hold a reference tone, play against it, and listen for the beat to slow. It trains the thing a meter cannot: the habit of hearing an interval settle.

How a meter finds a note, and where it struggles

A tuner has no ear, so it has to find pitch in a list of numbers. The useful fact is that a steady note repeats. Add any set of harmonics to a fundamental and the combined wave still repeats with the period of the fundamental, however different it looks. Pitch follows that period. It follows it so faithfully that a tone built only from harmonics two, three and up is still heard at the pitch of the absent fundamental, an effect known as the missing fundamental.

So a good detector does not look for the strongest frequency. It looks for the repeat. The method used on this page, published at the University of Otago in 2005, slides a short stretch of the signal against itself and scores how well it matches at each delay, normalised so that a perfect repeat scores one. The first strong peak marks the period, and a parabola through the peak and its neighbours places it between samples. The height of that peak doubles as a measure of clarity: noise never repeats, so it scores low and can be ignored.

The weak points follow from the same idea. To see a repeat, the detector needs at least two full cycles inside its window. The lowest note of a piano, A0, is 27.5 Hz, and one cycle lasts 36 ms. A window of 4096 samples at a common audio rate of 48,000 per second is 85 ms long, barely more than two cycles. Low notes therefore leave less to average over, respond more slowly, and suffer most from background noise. High notes have the opposite problem: a period only about ten samples long must be located to a small fraction of one sample.

Choosing the wrong peak produces the classic failure, a reading one octave off. A sound whose second harmonic is far stronger than its fundamental really does repeat almost perfectly at half the period, and a machine may reasonably pick that. The authors of the method say plainly that pitch is a subjective quantity and that no detector gets it right every time. A reading that jumps an octave is a hint to pluck more gently or move the microphone, not a verdict on the instrument.

A reference tone has a simpler job, and a browser does it well. The standard that defines web audio describes an oscillator whose phase advances at exactly the stated frequency, counted in samples of the audio clock. The tone is as true as that clock. What a page cannot know is how true the clock itself is, and an honest tool says so.

Three habits follow from the physics. Let a note ring for a second before trusting the number, because a detector needs several steady cycles and the first instant of a pluck offers none. Tune low strings with patience, or tune their octave harmonic, which gives the detector twice as many cycles. And when a note must sit inside a chord, use the meter to get close and your ears to finish, listening for the beating to slow.

The page’s table adds one reminder that sound is a physical thing: each note has a length. At 20 degrees Celsius sound travels about 343 metres a second, so the A at 440 Hz is a wave 0.78 metres long, and the lowest A on a piano stretches to roughly 12.5 metres. A tuner fits all of that into a needle. It helps to remember what the needle is comparing, and who chose the mark at its centre.