A bearing set by its thirds carries five eighths of its octave

A piano tuner sets one octave in the middle of the instrument first — the bearing, here F3 to E4 — and tunes everything else from it. Inside the bearing the tool is the beat. Each tempered fourth and fifth is a little off pure, so its coinciding partials beat at a rate a printed table gives, and a tuner who counts that rate while setting a new note against an old one has laid that note at its tempered place. The major thirds give a second test: in equal temperament every third beats faster than the one a semitone below it by the twelfth root of two, so the thirds of a correct bearing climb in a smooth progression.

On a real piano those two tests disagree. The strings are stiff, so their partials run sharp of whole-number multiples, and a fourth or fifth counted to its printed rate does not land at its tempered size. A bearing counted correctly is stretched by its route laid out bearings note by note from the printed rates and found two things: the order in which the notes are taken decides how much the bearing stretches — 1.62 cents an octave for one ordinary plan, 0.51 for another — and the thirds inside the bearing zigzag, each step of their progression missing the printed step by up to seventeen per cent. The thirds, it concluded, have to set the bearing.

Its lead proposed the plan a careful tuner follows: set each new note by its fourth or fifth, then adjust it until the newest third steps from its neighbour by the printed ratio, and carry on. The question was how much stretch such a bearing carries, and whether it is the fifth of an octave’s stretch that the progression of thirds votes for the bearing found was enough for the octave’s checks to agree.

The answer has two parts. The plan as stated cannot be carried out note by note. Done properly, it carries much more stretch than a fifth.

Where the notes land #

Solved properly, the bearing is a smooth curve. F3 sits 0.19 cents flat of equal temperament, the notes rise through A3, and E4 sits 0.88 cents sharp, with the steps between neighbours widening as the curve climbs. The two counted plans are jagged by comparison: the first throws alternate notes a cent or two sharp, the second alternates either side of the line, because each note is set by a fourth or fifth from a note on the other side of the bearing and every such step lands a little off.

The least-squares stretch of the solved bearing is 1.15 cents an octave, between the two counted plans’ 1.62 and 0.51. Its shape matters as much as its slope. A bearing whose thirds progress exactly is one whose notes climb by a smoothly growing step, because stiffness lifts each note’s upper partials by an amount that grows with pitch, and keeping the thirds’ beat rates in exact geometric progression has to follow that growth.

Turn the dial and the curve scales with the stiffness, keeping its shape. On a piano half as stiff it carries 0.57 cents an octave, twice as stiff 2.27, four times as stiff — the small upright these essays use as a foil — 4.49.

Why a walk cannot follow the thirds #

The lead’s plan was a walk: place each note by its fourth or fifth, then correct it by its third. Carried out literally, it leaves most of the bearing untouched. A note can only be corrected by a third it completes, and only if a neighbouring third is already complete to step from. In the first plan’s order the first seven notes complete no such pair — each third they finish has no neighbour yet — so they keep their counted places. The last four notes each complete a third beside one already set, and each is moved by about two cents: −2.10, +2.02, −1.94, +1.93.

The progression then comes out exact at every step but one, which misses by 15.4 per cent, because the third that step leads to was finished early, by a note that had no neighbour to step from, and nothing later reaches back to correct it. And the four moved notes pull their fourths and fifths a long way off the table: one now beats at nearly twice its printed rate, another hardly at all.

That is not a failure of the arithmetic. A tuner walking a bearing can only touch the note in hand, and a correction made to the note in hand cannot undo what an earlier note committed. In practice tuners loop back — check the whole bearing’s thirds after laying it, and nudge whichever notes the progression singles out — which is exactly what a walk cannot represent. The figure’s third curve is the end point of that looping: every note moved together until the progression is the printed one everywhere, and the fourths and fifths, as far as they can be, at their printed rates.

What the fourths and fifths pay #

The thirds fix the bearing’s shape and leave its overall level free; the fourths and fifths are what fix the level. It is the same division of labour the fourth-fifth check cannot hear the note it checks found inside a single check: one test constrains differences and needs another to fix where they sit. In the solved bearing they are as close to their printed rates as the exact progression allows, and on the grand that is not very close. Every one of the twelve fourths and fifths inside the bearing beats slower than printed: the fourths by six to twelve per cent, the fifths by three to sixteen, the root-mean-square ten per cent. On the upright the same arrangement leaves them a quarter to three fifths slow, root-mean-square 38 per cent.

All of them err the same way, and that is the signature of a stretched bearing on stiff strings. A stiff string slows the fourths and quickens the fifths set out how inharmonicity moves each interval’s coinciding partials; the stretch the thirds demand widens every interval in the bearing a little beyond its tempered size, and for a fourth, already wide, that would quicken it, while for a fifth, already narrow, it slows it. The figure says the stiffness and the stretch together leave both slow. A tuner who has set the bearing by its thirds and then checks a fourth against the printed table will hear it beating slow and be tempted to correct it, and the correction would put the thirds’ zigzag back.

That is the honest form of the trade the previous essay found. Counted from the table, the fourths and fifths are right and the thirds zigzag by fifteen per cent. Set by the thirds, the thirds are right and the fourths and fifths are slow by ten. No bearing on stiff strings can have both, and the question for a tuner is which error to hear. The arithmetic does not choose; it only says that the errors are the same size, roughly a tenth of a beat rate either way on a grand, and that on an upright both grow fourfold. On a small upright a tuner following the table hears the thirds lurch by most of a step, and one following the thirds hears every fourth beat a quarter to a half slower than the table says.

The stretch keeps pace with the octave’s #

The surprise is how regular the solved bearing’s stretch is. At every stiffness from a quarter of the grand’s to four times it, the bearing set by its own thirds carries 63 per cent of the stretch the octave A3–A4 takes when it is set by the two contiguous checks that the octave that fails both checks least balanced: 0.29 against 0.46, 1.15 against 1.82, 4.49 against 7.05. Both stretches are proportional to the stiffness, so their ratio is a property of the piano’s geometry — of which partials the thirds and the octave checks compare — and not of how stiff any particular piano is.

That is consistent with what the thirds and the fourths check different octaves found when it added up chains of intervals. Each interval counted to its printed rate on a stiff string is left wide by an amount set by the squares of the partial numbers that meet in it, so a chain of intervals lands on an octave of a particular type, and a chain of major thirds lands just short of the 6:3. The bearing set by its thirds is a set of thirds forced into exact progression, and the octave that progression implies is a stretch of a fixed type too — five eighths of the way from equal temperament to the octave the contiguous checks balance — very likely for the same reason: to first order every shift on a stiff string is a sum of squared partial numbers times one coefficient, so any two stretches computed from the same strings stand in a fixed ratio. The solve has not been taken apart to show which partials contribute what, and the ratio is reported as measured — 0.63 at every stiffness tried — rather than derived.

The counted plans have no such regularity. The first plan’s stretch tracks the octave’s at about nine tenths, the second’s at about three tenths, and which plan a tuner happened to learn decides which. The thirds give the same answer from any route.

What the octave’s checks make of it #

The question the lead posed was whether a bearing set by its thirds carries the fifth of the octave’s stretch that the progression needs in order to stop voting. The earlier essay found that the progression of thirds, added as a third check on the octave, asks for an octave equal to whatever the bearing carries plus about an eighth of a cent. In an equal bearing that drags the octave back towards the pure 4:2; once the bearing carries more than about 0.32 cents an octave on the grand — a sixth of the octave’s own 1.82 — the progression’s request falls inside the other two checks’ compromise and it stops moving anything.

The bearing set by its thirds carries 1.15. It clears that threshold by more than three and a half times, and the three checks agree exactly on the two-check octave, 1.819 cents. So following the thirds does reconcile the octave’s checks — the reconciliation is something a tuner reaches by following the thirds, as the lead hoped — but not narrowly. A bearing set by its thirds is not the least stretched bearing that would do; it is a much more stretched one, and the octave checks are indifferent to the difference.

That indifference has one exception. The threshold rises faster than linearly with stiffness — 0.32 cents an octave on the grand, 2.90 on the upright — while the counted plans’ stretches rise in proportion. The second counted plan, which sets the sharps by fifths up, carries 2.05 cents an octave on the upright, under the threshold, and there the progression of thirds pulls the octave narrow. The bearing set by its thirds carries 4.49 and clears it at every stiffness.

What a tuner hears in each bearing #

Put plainly, there are three bearings on the table and each fails a different test.

Counted from the table by fourths up and fifths down, every fourth and fifth beats at its printed rate, the thirds zigzag by fifteen per cent from step to step, and the bearing carries about nine tenths of the octave’s stretch.

Counted with the sharps set by fifths up, the fourths and fifths again beat as printed, the thirds zigzag as badly, and the bearing carries three tenths of the octave’s stretch — enough on a grand, not on an upright.

Set by its thirds, the thirds climb exactly as printed, every fourth and fifth beats a tenth slow on a grand and over a third slow on an upright, and the bearing carries five eighths of the octave’s stretch on any piano.

The aural tradition settles the choice the way the arithmetic does, and the arithmetic of beats are arithmetic is why the progression is so sharp a test — a third’s beat rate scales with the gap between its coinciding partials, so a fraction of a cent at one note becomes a visible kink in a smooth climb: tuners are commonly taught that the progression of thirds is the most sensitive test of a bearing and the fourths and fifths a guide to be checked against it. The figures say why it is sensitive — a few tenths of a cent at one note turn a smooth progression into a zigzag of fifteen per cent — and what following it costs, which the teaching rarely states: the printed fourth and fifth rates will not be met.

The model under the solve #

The strings are the modelled grand’s, with each note’s inharmonicity coefficient from its string length and gauge, scaled by a single factor for stiffer or less stiff pianos; partial $n$ of a string at $f$ sits at $n f \sqrt{1 + B n^2}$.

The bearing is solved at once. The eleven notes other than A3 are moved together, in cents, by Gauss–Newton least squares, with the seven steps of the progression of eight contiguous thirds weighted a hundred times more than the twelve fourths and fifths inside F3–E4, each fourth and fifth measured as a share off its printed rate with its sign kept. The progression comes out exact to a few hundredths of a per cent.

The walk is the lead’s literal plan. Each note is set by the plan’s fourth or fifth at its printed rate and then, if it completes a third beside one already complete, moved so that the new third steps by the printed ratio from the neighbour below, or above if there is none below.

The octave’s three checks are the earlier essays’: the two contiguous checks balanced against their own beat rates, and the progression of thirds as a third vote whose request is the bearing’s stretch plus a fixed amount.

What a model bearing cannot settle #

How fine a tuner’s ear for the progression is. The solve makes the progression exact. A tuner hearing the steps of a progression with a Weber fraction of a few per cent would accept a range of bearings around the solved one, and the range on a grand might reach from the second counted plan to past the first.

Whether the fourths’ and fifths’ slowness is heard. At the rates a bearing’s fourths beat, around one a second, a tenth slow is about one beat fewer in ten seconds of counting. Whether a tuner notices that against the table depends on how long the note lets them count, and a string that decays twice is counted early found the countable stretch of a piano note short and set by its decay.

Real pianos’ stiffness profiles. Each note’s coefficient here comes from one model of string scaling; a real piano’s bearing crosses the break between plain and wound strings in some designs, and the curve’s smoothness depends on the scaling being smooth.

Still open: the bearing a tuner’s ear would accept #

The solve gives the one bearing whose thirds progress exactly. A tuner’s ear does not need exactness; it needs each step of the progression within whatever share a listener can tell, and the earlier essays used fifteen per cent for the fourth-fifth check’s tolerance. The computation is this solve with the progression as a band rather than an equality — every step within a stated share of the printed one — and the fourths and fifths pulled as close to their printed rates as the band allows, and it would say how wide a band brings the fourths and fifths back to their table, how much stretch the bearing then carries, and whether there is a band at which every check a tuner makes inside the bearing passes at once.