A tuner counts beats, and that is the whole method

A piano tuner arrives with a small kit: a lever, a felt strip, a couple of rubber wedges, and — in most cases — a tuning fork. Two hours later the instrument is in equal temperament to within a couple of cents across seven octaves. Nothing in that kit measures pitch.

What the tuner uses instead is a wristwatch, or a foot, or nothing at all except a sense of how fast a thing is pulsing. Every interval in the first octave laid is supposed to beat at a particular rate, the rates are written down, and hitting them produces the temperament as a by-product.

The rates are small numbers — a little over half a beat a second at the bottom of the octave, a little over one at the top. That is a pulse slow enough to count against a watch and fast enough that being wrong by ten per cent is obvious.

The interval that is supposed to be wrong #

The starting point is the discrepancy that makes tuning a problem at all. Twelve pure fifths overshoot seven octaves by 23.46 cents, so a keyboard cannot have twelve pure fifths and also close. Every temperament is a decision about where to put that error, and equal temperament puts a twelfth of it into each of the twelve links.

A twelfth of 23.46 cents is 1.955 cents. That is the amount by which every fifth on a modern piano is narrow, and it is far too small to hear as a pitch difference. Nobody can tell a 700-cent fifth from a 702-cent one by comparing them to a memory.

They can hear it instantly as a beat.

This is the arithmetic the ear does without being asked: two frequencies close together produce a sum whose amplitude rises and falls at their difference, and the difference is a number in hertz that a listener can count.

Which partials are doing it #

The two notes of a fifth are not close together, so they do not beat with each other. Their partials do.

An unequal temperament gives an unequal set of beat rates, and that is what makes it layable by ear at all: the tuner is counting different numbers on different links rather than the same number twelve times, so a mistake shows up as a rate that does not match the instruction.

Every note from a string is a stack of partials at whole-number multiples of its fundamental. For two notes a fifth apart the arithmetic of the stacks is simple: if the ratio were exactly 3:2, the third partial of the lower note and the second partial of the upper would be the same frequency.

A pure fifth is silent, which is the strongest instruction a bearing plan can give: the tuner is told to remove the beat entirely rather than to count it, and a beat that should not be there is far easier to hear than a rate that is slightly wrong.

Tempering the fifth moves those two partials apart. By how much is a question with an exact answer: the beat rate is the difference in hertz between the third partial of the lower note and the second of the upper.

For the F–C fourth at the bottom of the bearing octave, with F at 174.61 Hz and C at 261.63 Hz, the coincidence is between the fourth partial of the F and the third of the C — 698.5 Hz against 784.9 Hz for a pure fourth, and the tempered pair beats at 0.59 per second. At the top of the octave the same interval type beats at 1.12. The interval is identically tempered in both cases; the rate differs because a beat rate is a difference in hertz and hertz scale with register.

That last point is the one that makes the printed tables necessary. A tuner cannot learn one number and apply it everywhere.

The bearing octave, and why it is where it is #

The whole temperament is laid inside a single octave, usually F below middle C up to the F above it, and the rest of the instrument is then tuned in pure octaves outward from it. That octave is a compromise between two limits.

Lower, and the beats become too slow to count reliably — half a beat a second is already a four-second cycle at the fifth. Higher, and they become too fast: the same fifth two octaves up beats at four or five per second, which is a rattle rather than a pulse.

Doubling is exactly what the arithmetic predicts. The tempering is a ratio, so it scales with frequency, and a beat rate is a difference, so it scales with frequency too — which is why the plan an octave up is the same plan with every number multiplied by two.

How accurate a counted beat actually is #

The method looks crude. Counting a pulse against a watch is not obviously a precision technique, and the quantity being aimed at — two cents of narrowing — is a fiftieth of the gap between two adjacent keys. It is worth asking what a tuner’s accuracy actually comes to, because the answer is not what the method looks like.

The relationship is direct. A beat rate is the difference between two partials, so an error of one beat per second in the rate is an error of one hertz in that difference. If the coincidence is between the third partial of one note and the second of another, an error of a tenth of a beat per second corresponds to an error of a twentieth of a hertz in the upper note — and a twentieth of a hertz at 262 Hz is 0.22 cents.

That is a fifth of the precision of a good electronic tuner, obtained by counting to ten. It is also better than the instrument holds: a piano moves by more than that as the room warms.

The leverage comes from the partial number. The ear is judging a coincidence between the third and second partials, so it is working at three times the frequency of the note being adjusted, and an error at the partial is divided by three when it is referred back to the fundamental. The higher the coinciding pair, the finer the discrimination — which is the reason a major third, whose coincidence is between the fifth partial of the lower note and the fourth of the upper, is the interval tuners use to check a temperament rather than to lay it. It is too sensitive to set by, and exactly sensitive enough to test with.

Being wrong by half a beat, which is a gross error nobody would leave, is 1.24 cents. Being wrong by the entire rate — tuning a fifth pure when it should have been tempered — is the 1.955 cents that the twelfth of a comma amounts to, which then has to go somewhere else.

An irregular temperament has an irregular plan #

Equal temperament is the easy case for a tuner, because every fifth is tempered by the same amount and the rates form a smooth curve. It is also historically recent as a keyboard practice. The temperaments in use before it were mostly irregular: they narrowed some fifths and left others pure, which is what gave each key a character of its own.

The contrast is worth dwelling on. In equal temperament every rate lies between 0.59 and 1.12, a spread of half a beat, and none of them is zero. In Werckmeister III eight of the twelve links are exactly pure, and the four that are not beat at around two and a half to three per second. The instructions for laying it are correspondingly different in kind: “make this one perfectly still” rather than “make this one pulse a little under once a second”.

A pure interval is the easiest target there is, because the absence of beating has no rate to misjudge. Which means the irregular temperaments were, if anything, easier to lay by ear than the regular one that replaced them, and the reason equal temperament won had nothing to do with the difficulty of hearing it.

Where the computation stops describing the instrument #

Everything above assumes the partials of a string sit at exact whole-number multiples of its fundamental. They do not.

A real piano string is stiff, so its partials are sharp of whole multiples and the coincidence a tuner listens for is not exact. The third partial of the lower note and the second of the upper are each displaced, and by different amounts — which means the beat rate a tuner actually counts is not quite the one the arithmetic above gives, and the discrepancy grows up the compass.

A real steel string has bending stiffness, which raises the nth partial above n times the fundamental by an amount that grows with n. This is the reason a piano’s octaves are stretched on purpose, and it puts a systematic error into every number in the bearing plan.

The size of the error depends on the instrument. On a concert grand with long bass strings the inharmonicity coefficient is small and the printed rates are close enough to be usable. On a small upright, whose strings are short and thick, the third partial of a low note can sit several cents above where the arithmetic says, and a tuner who insisted on the printed rates would produce something that is not equal temperament and does not sound like it either.

The direction of the error is worth having, because it is the same on every instrument and a tuner can use it. Stiffness raises the n-th partial by a factor growing as n², so of the two partials in any coincidence the higher-numbered one — always the lower note’s — is raised more. Every coincidence is therefore pushed apart in the same direction, and every true beat rate is faster than the printed one, on every interval, on every piano. A tuner who hits the printed rate exactly has left the interval slightly wide, and the amount grows with the register and with how small the instrument is. Which is a rule of thumb the arithmetic gives for nothing: on a small piano, beat everything a little faster than the table says.

What a good tuner does about this is not to abandon the numbers but to treat them as a starting point and then check the result differently — by listening to intervals the plan does not contain, particularly major thirds and tenths, whose beat rates should increase smoothly as they move up the octave. A sequence of thirds that suddenly slows or reverses is the audible signature of an error, and it does not require knowing what the correct absolute rate is.

That is a real methodological point rather than a piece of craft folklore: the check is on the monotonicity of a sequence rather than on the value of any single rate. Whether it survives inharmonicity is a question with an answer, and the answer is on some instruments.

Computing the major-third rate at every semitone of the bearing octave, with each note’s partials raised by its own inharmonicity coefficient:

On a grand the sequence stays monotone and on an upright it does not. At four times the site’s own coefficient the thirds climb to 7.98 near the top of the octave and then turn back down, so a tuner following the rule would find a reversal and conclude they had made an error where they had not. At sixteen times the sequence is not remotely monotone and the check reports errors everywhere.

That is exactly the wrong way round for a fallback. The printed rates fail on small uprights, and the check offered in their place fails on the same instruments — because both are defeated by the same quantity, and the check’s independence from it was assumed rather than shown. What the monotonicity rule really survives is moderate inharmonicity, which is where the printed rates were nearly right anyway.

The single-rate half of the claim is worth quantifying too, because “close enough to be usable” turns out to mean something looser than it sounds. On a grand the F–C fifth’s true rate is 20 per cent above the printed one at the bottom of the bearing octave and 60 per cent above at the top; on an upright it is 81 and 237 per cent. Sixty per cent of a 1.18-beat rate is 0.7 beats a second, which referred back to the fundamental is about a cent and a half — comparable with the 1.955 cents of tempering being aimed at. So even on a good instrument the printed table is not a small correction away from correct; what makes it usable is that a tuner treats it as a starting point, which is what the paragraph above says and is the load-bearing part of it.

The same arithmetic, everywhere #

None of this is specific to pianos or to fifths. Any two notes whose partials nearly coincide beat at the difference between the coinciding pair, so the rate can be computed for any interval and any register from the ratio alone.

It is how a string quartet checks a chord, how an organ builder voices ranks against each other, and how a guitarist tunes by harmonics — the fifth-fret and seventh-fret harmonics of adjacent strings are the fourth and third partials, and the beat between them is the same quantity this whole essay is about. The guitar case has a well-known trap attached: tuning by those harmonics produces pure fifths, and six pure fifths do not close against the frets, which are equal-tempered. An instrument tuned that way is progressively out with itself, by exactly the comma this site keeps returning to.

The most exacting use of the arithmetic is the wolf. A fifth carrying an entire comma is 23.46 cents wide, and at the bottom of the bearing octave that is a beat rate of about seven per second — no longer a pulse at all, but the audible roughness that gives the interval its name.

Whose practice, and when #

The written record of tuning by counted beats is more recent than the practice. Instructions in the seventeenth and eighteenth centuries are qualitative: a fifth is to be tuned “a little flat”, a third “as sharp as the ear will bear”. Werckmeister’s own directions are of that kind, and they work because the intervals they describe are mostly pure or mostly a quarter-comma out, which are two states a trained ear distinguishes without counting anything.

Beat-rate tables in the modern sense belong to the nineteenth and twentieth centuries, and to equal temperament in particular — a scheme in which nothing is pure, every interval is slightly wrong, and there is therefore nothing qualitative left to aim at. The tables are what replaces the pure interval as a target. Owen Jorgensen’s Tuning (1991) collects both the historical instructions and the arithmetic for reproducing them, and it is the standard reference for the claim that the two traditions are aiming at different kinds of thing.

Electronic tuning devices, which arrived in the 1970s and measure frequency directly, did not make the arithmetic obsolete so much as move it. A device that measures cents has to be told how much stretch to apply, and the good ones compute it from the instrument’s own measured inharmonicity — which is the same calculation, run the other way round.

What the picture cannot show #

A bearing plan is a set of target rates, and it says nothing about the order in which the links are laid, which is most of the skill. Nor does it show the unisons: nearly every note on a piano has two or three strings, they must be tuned to each other to well within a beat, and that work is perhaps half the job and has no rate at all — the target is exactly zero.

It also flattens a real dependence on time. A wrestplank moves, a string settles after being pulled, and a tuner is aiming at where the instrument will be in an hour rather than where it is at the moment of the blow. None of that is in the arithmetic, and all of it is why tuning remains a trade.

The ladder continues from here into the places where beating stops being a tool and becomes the subject: why roughness can be computed from the same partial coincidences that a tuner is counting, and why the same interval is rougher in the bass for a reason that is the exact opposite of the one that makes beats easier to count down there.