One walk round the bearing measures the piano and the tuner together

A piano tuner’s table of beat rates is a list of twelve instructions, and a stiff string gives the bearing a comma of its own showed that they cannot all be obeyed. The seven fourths and five fifths of the temperament octave lie end to end in one closed circle, and on a stiff string the printed rates carry the walk 2.22 cents past where it began. Share that shortfall out and every interval beats about nine per cent slower than printed. Leave it unshared and it collects in whichever interval the tuner does not count.

That essay took the shortfall as something to be calculated from the string. But it left a more useful observation lying at the end. A tuner who walks the circle does not need to know the string. The walk ends at A3, the note it started from, and whatever gap is left is sitting there in the air, as a very slow beat against the note already set. The disagreement that spoils the bearing is also a measurement of it, taken with the same instrument and in the same minute, and in principle it could be used. A tuner could walk the circle once, listen to how far home is, and correct the table.

Whether that works is a question about noise. A tuner’s count of a slow beat is not exact, and twelve counts in a row accumulate error in the same place as the thing being measured. The closing beat is also very slow and has to be heard on a decaying note. The honest expectation was that the correction might be swamped: 2.22 cents is not a large signal, and if the counting errors summed to anything like it, the tuner would be correcting the piano by a number that was mostly the tuner.

What the walk reads and what it is worth #

The arithmetic of the correction turns out to be very simple, because the walk is close to linear in the share of each rate. Slow every interval by one per cent of its printed rate and the walk comes home 0.235 cents higher. That number, 23.5 cents per whole unit of share, is nearly a property of the table alone. It changes by less than a third of a cent between an ideal string and a piano half as stiff again as the modelled grand, because it is really the sum of the printed rates themselves, referred to A3. It therefore does not have to be measured on each piano. Expressed as a beat at A3 it is 2.98 hertz. That gives a rule small enough to carry in the head: the share to take off the table is the closing beat’s rate divided by three. On the modelled grand the closing unison beats at 0.28 hertz, one beat in 3.6 seconds, and a third of that is the 9.45 per cent the earlier essay computed from the strings’ stiffness.

To test it, the model is a tuner who makes each count with a random error as a share of the rate being counted, and who reads the closing unison with an error in hertz. Two hundred such tuners are drawn from a seeded generator, so the figures are repeatable, and each one does the following: walks the circle once with the printed rates, reads the miss, divides it into a share, and then sets the bearing again with every rate corrected by that share and with a fresh count error on every interval. The twelfth interval of any setting is never counted, being what is left after the other eleven notes are placed. It is where a setting’s whole error ends up, and the worst interval is nearly always that one.

The correction takes the worst interval from about 106 per cent off to about 23, at a counting error of five per cent and a closing unison read to two hundredths of a hertz. That five per cent is a stated round figure, not a measurement of any tuner. What matters is the width of the result across any reasonable figure. At ten per cent the corrected worst is 44 per cent. At twenty per cent, a tuner who gets one beat in five wrong, it is 86 per cent, and that is still better than a perfect counter on the printed table can do, because the printed table’s error does not depend on the counting at all. The expected swamping does not happen. The reason is the arithmetic of the circle. A share common to all twelve intervals adds up across twelve of them, while independent errors add up only as the root of twelve. The signal grows by a factor of twelve and the noise by a factor of three and a half.

The stretch of the bearing comes back as well. Set from the table, the modelled grand’s bearing carries 1.34 cents an octave, whichever interval is last. Corrected, it carries 1.14, which is exactly what the shared bearing carries. The spread between tuners is about a tenth of a cent. And the thirds, which are the test a tuner actually trusts, go from a root-mean-square error of 12.4 per cent against their printed progression to 3.4 per cent, against 0.7 per cent for the exact share. The earlier essay observed that the thirds come right unasked when the shortfall is shared. It turns out that they do so even when the sharing is done by a person with a five per cent error in each count.

Which pianos it is worth doing on #

A correction that is only noise on a good string should be a mistake on a good enough string, so the sum can be asked where the crossing is.

The answer is that no real piano is on the wrong side of it. The corrected tuner’s worst interval is 23 per cent on every string, flat from an ideal string to one half as stiff again as the grand, because the correction exists to remove exactly the part of the error that grows. The table’s error rises in step with the stiffness: 54 per cent at half a grand’s, 106 at the grand and 157 at 1.5. The two cross at about a fifth of the grand’s stiffness, where the miss is 0.45 cents and the closing beat is one in twenty seconds. Nobody can read that, and they do not need to, since a string that uniform is not on the market.

This closes the question that the earlier essay ended on, whether the printed table can be kept or should be abandoned in favour of the comparison a tuner already makes. It can be kept, with one correction per instrument, and that one number is the only thing about a piano’s stiffness that matters to the bearing. The table has one dimension of wrongness, and the tuner has one reading to take of it. That is a tidier relation than anyone setting out to count twelve intervals had reason to expect, and it comes from a fact established earlier: that a stiff string slows the fourths and quickens the fifths in amounts tied together by the string’s own coefficient, so that one number scales all of them.

A second walk mostly repeats the first #

The next obvious refinement is to walk the circle again. Having set the corrected bearing, the tuner can read a new miss, correct the correction, and so on. The earlier essay asked whether that would improve on the first pass or merely add noise, and the answer depends entirely on what happens to the second reading.

Replacing the first reading with the second buys nothing that can be distinguished from the scatter of two hundred tuners: 22.5 per cent after one walk, 20.0 after two, 21.7 after eight. Each reading carries its own noise, and a later one is not cleaner than an earlier one because the noise was never in the table. A tuner who keeps adjusting to the latest closing beat is following a random walk around the right answer. The same habit will be familiar from other fields of measurement, where it is the standard way to make a controlled process worse.

Averaging the readings does what averaging does. The correction’s error falls as the root of the number of readings, from 1.72 per cent of the rate after one walk to 0.53 after eight. The worst interval does not follow it. It goes from 22.5 per cent to 17 after one more walk and then stops, near 16, however many walks are taken. Having a perfect correction would not help much either, because the setting itself is still eleven counts with a five per cent error, and all of their noise still lands in the uncounted twelfth interval. That floor belongs to the setting and not to the correction. The sharp distinction between the two kinds of error is what makes the decision simple: a tuner should take one walk, or two and average them, and then stop walking and start checking the interval that was left over. A walk costs about a minute and the second is worth about a quarter of the remaining error. The third is worth nothing.

A closing beat cannot tell the piano from the tuner #

The correction was built to measure the string. But look at what it actually reads. Every count in the walk has some error, and a share of that error may be common to all twelve: a tuner whose internal sense of a second runs fast, or whose habit is to count from the first beat rather than the first interval, or who has simply learned the table slightly wrong. A common share of every rate is exactly what stiffness looks like, by construction, because stiffness has been found to act as a common share.

A tuner whose count runs nine per cent slow is right on the printed table without knowing why. The table’s error has a minimum at a common share of about −9.45 per cent, where it falls to 6.6 per cent, and it has nothing to do with the string except that the string happens to want that share. Their slow count has cancelled the table’s fault, and every bearing they set will close, on this piano and on any piano of about its stiffness. On a stiffer upright they will have the same problem as everyone else, only smaller. And if they tune by ear and judge by the thirds, they will trust their own clock and not the table, and they will be right to.

Now apply the correction. The rms error is 8.2 per cent at every value of the bias, to the first decimal place, from fifteen per cent slow to ten per cent fast. The closing beat reads the sum of the string’s shortfall and the tuner’s bias without being able to tell them apart, and it does not need to. The same reading that repairs the table also repairs the clock. So the procedure has a property the earlier essay had no way to see: it does not assume the tuner is calibrated. A tuner who took a stopwatch to their own count would learn nothing the closing beat had not told them already, as far as the bearing is concerned. It also means a miss cannot be interpreted as a measurement of the piano, as this essay’s first section suggested, unless the tuner’s clock is known. What it measures is the shared error of that tuner on that piano. It is the right quantity to correct, and the wrong one to write down as the inharmonicity.

The surprising connection is older than the piano. The closing beat is the same move that named the Pythagorean comma: go round a circle of intervals by a stated rule, come back to the note the walk began on, and treat the failure to arrive as a quantity. The comma is a property of the integers. This one is a property of the wire and the person, and a tuner’s walk measures it the way a walk of twelve fifths measures the comma. In both cases the closure is what makes a set of local instructions answerable to something, because no count in the walk can reveal its own error and the sum of them can.

How badly the closing beat can be heard #

That leaves the weakest link. The whole method rests on hearing a beat at 0.28 hertz on a note that is dying, one cycle in 3.6 seconds, and the essay on counting in a decay has already shown that a slow beat on a piano is not simply there for the listening. A piano note decays twice over, and a beat this slow has only two or three cycles to be read in before the note is gone.

The correction pays until the error of reading the closing beat is larger than the beat. At an error of 0.1 hertz, about a third of the beat itself, the worst interval is 37 per cent. At 0.2 hertz, 63. At 0.3, which is an error bigger than the whole signal, it is 91, still under the table’s 106. It crosses at about 0.35. So a tuner who can tell a closing beat of one in three or four seconds from no beat at all, and from one in two seconds, has what the method needs. The size of the error a real tuner makes on a beat like this is what is unknown here, and it is the one measurement this calculation would most benefit from. A tuner can also shift the reading to where it is easier, since the same miss beats twice as fast on the second partials of the pair, and a miss read on a coincident partial, where a tuner normally listens, gives a number two or three times larger to hear. That has been left out of the sum, so the figure is a floor.

What this tuner is not #

The model has one tuner and no piano behaviour beyond stiffness. It has no false beats, no unison of three strings, and no drift in the pitch of the string as the pin settles while the walk is being made, which a real tuner will know is the first thing that happens. It treats each count as an independent draw, while a person’s count is probably correlated with the one before, and a correlated error is a share of the common kind that the correction would take out rather than the independent kind it would leave.

It has one plan, the walk from A3 round the circle in the order the route essay calls the ordinary one, and it sets every interval as a single measurement. The uncounted interval is left wherever it falls. A real tuner checks it, and is free to spread what is left over by ear; that step would reduce the worst interval further and has not been simulated. The bearing set by its thirds is the alternative the tuner already has, and nothing here shows the correction to be better than listening to the thirds, only that it is available and that it ends in the same place.

Nor does it ask whether a tuner wants this. Aural tuners who work from a table are a minority now, since most of them use the thirds and most of the rest use an electronic device that has measured the string directly. The electronic tuner’s approach is the one this most resembles, and a device that measures the partials would have no use for the closing beat. The method is a claim about what a table of numbers and a pair of ears can do by themselves, and the repertoire of the claim is the modern piano’s temperament octave, on a model of its strings that does not differ across the bearing’s own compass. It should not be read as advice to any working tuner.

Still open: what the leftover interval could be used for #

A tuner who has corrected the table and set eleven intervals still has one left over, off by about twenty per cent of its rate, and that interval is not a fault. It is a second measurement of the setting’s total error, at the end of the setting, for the cost of listening to one more pair of notes. If that residual were spread back across the eleven, the setting would close for the second time and the error would be divided rather than collected. The computation is the same trial with a last step in which the leftover share is distributed, once or in a second pass with only a few notes moved, and it would say how much of the 23 per cent floor is really in the setting and how much remains when a tuner gets to use what the bearing has just told them. Whether moving a few notes helps, or merely moves the noise from one interval to several, is exactly the distinction between a correction and a chase.