The octave that fails both checks least

The thirds and the fourths check different octaves found that the two contiguous tests a piano tuner leans on are exact tests of two different things. The fourth–fifth check — a fourth up from the bottom of an octave and a fifth from there to the top, compared for their beats — passes on the 4:2 octave, where the lower note’s fourth partial agrees with the upper note’s second. The minor third–major sixth check passes on the 6:3, where the lower note’s sixth partial agrees with the upper’s third. On a flexible string both are the 2:1 octave. On a stiff one they sit two cents apart at A3, and each of the two is missed on the other’s octave by about a beat a second.

That essay ended with the question it could not answer from where it stood. If neither check can pass, a tuner’s ear settles somewhere, and the place it settles is set by how the two failures are weighed against each other. The question has an answer once a rule for the weighing is stated, and the rule that hearing itself suggests turns out to make the answer almost independent of the piano.

What the two checks hear on each other’s octave #

The starting point is the conflict as it was left.

At A3 on the grand, the 4:2 octave leaves the third–sixth check failing by 1.5 beats a second and the 6:3 leaves the fourth–fifth check failing by 1.0. Read as raw differences, the third–sixth failure is the larger, and an octave chosen to split them would sit nearer the 6:3 than the 4:2.

That reading treats a difference of a beat a second as the same thing wherever it occurs, and a listener does not. The fourth A3–D4 and the fifth D4–A4 each beat about once a second in equal temperament: a check that makes one of them beat at one and the other at two has doubled a rate, and nobody misses it. The minor third A3–C4 and the major sixth C4–A4 each beat near twelve a second: a check that makes one beat at twelve and the other at thirteen and a half has changed a rate by an eighth, and that is near the edge of what a tuner sorting two fast flutters can tell. The same difference in beats a second is an obvious failure in one check and a marginal one in the other.

So the natural measure of a check’s failure is the difference between its two rates as a share of the rates themselves. That is how these essays have treated beat rates whenever a listener has to compare them: a gamelan’s beating names its goal only if each goal beats at its own rate sorts two rates as the same or different by a Weber fraction of fifteen per cent, and the same fraction is used here unless a figure says otherwise.

Weighed as shares, the balance hugs the 4:2 #

There is one more simplification, and it is exact rather than approximate. Each check’s failure — the difference between its two beat rates — does not depend on the middle note at all. The fourth–fifth check cannot hear the note it checks showed this for the fourths and fifths: both beats involve the middle note’s third partial, and in their difference it cancels, leaving the upper note’s second partial against the lower note’s fourth. The third–sixth check works the same way through the middle note’s fifth partial and leaves the upper note’s third against the lower’s sixth. The rates being compared are near the ones a bearing plan prints, and those are equal temperament’s on whole-number partials: 0.99 a second for the fourth on A3 and 11.87 for the minor third. So each failure, as a share, is the octave’s own coincidence over a number fixed by the temperament.

Both curves are V-shapes with their points on the types. The fourth–fifth check’s is steep, because its denominator is one beat a second; the third–sixth check’s is shallow, because its denominator is twelve. They cross at 1.82 cents at A3, where both checks fail by 11.3 per cent. That is 0.22 cents past the 4:2 and 1.74 short of the 6:3: 11.2 per cent of the way from one type to the other. Balanced in raw beats a second instead, the octave would be 2.78 cents, and there the fourth–fifth check fails by 61 per cent — a fourth and a fifth beating at about 0.7 and 1.3, which no tuner would accept as matching.

Turning the dial makes the strings stiffer or more flexible, and every number on the axis scales with it: at half the grand’s stiffness the balance is 0.91 cents and both checks fail by 5.7 per cent, at four times it is 7.05 cents and both fail by 43. What does not move is where the dot sits between the two types. It stays at the same fraction of the way across.

The ratio belongs to the temperament #

The reason the dot does not move is short enough to write down. Near the types, each coincidence moves linearly with the octave’s width: widening the octave by a cent raises the upper note’s second partial by two of its frequency’s cents-worth and its third partial by three. One check’s failure grows as twice the distance from the 4:2, divided by the fourth’s printed rate; the other’s as three times the distance from the 6:3, divided by the minor third’s. Setting them equal puts the balance at a share

$$\frac{3R_{45}}{2R_{36} + 3R_{45}}$$

of the way from the 4:2 to the 6:3, where $R_{45}$ and $R_{36}$ are the two printed rates. Both are proportional to the lower note’s frequency, and their ratio — 11.94 — is a property of equal temperament alone: it is how far a tempered fourth departs from 4:3 against how far a tempered minor third departs from 6:5. The share is 0.1116. Nothing about the piano’s strings appears in it.

Computed note by note with the full stiff-string law, the balance runs from 0.112 to 0.117 of the way across on the grand between A2 and A5, and from 0.113 to 0.130 on a piano four times as stiff, the small drift being the curvature the linear law leaves out. Weighed in beats a second, the balance sits at three fifths — 0.601 to 0.614 and 0.603 to 0.640 — which is the same arithmetic with the printed rates taken out, so that only the partial numbers, two and three, remain.

This is the surprising part, and it reaches past pianos. The octave a tuner’s two checks agree on is not a property of the instrument the checks are made on. It is a property of the temperament the checks are checking. The stiffness of the strings decides how far apart the 4:2 and 6:3 types are, and so how many cents the compromise is worth; where between them the compromise falls is decided by the ratio of a tempered fourth’s beat to a tempered minor third’s, which twelve fifths and seven octaves fixed long before anyone tuned a steel string. A temperament with fourths nearer pure would have a smaller $R_{45}$ and pull the balance still closer to the 4:2; one with minor thirds nearer pure would push it towards the 6:3.

Equal shares against equal beats, up the bearing #

The two rules, set against the octaves the earlier essay found, sort themselves cleanly.

At A3, from narrow to wide: the 4:2 at 1.60 cents, the share balance at 1.82, the beats balance at 2.78, the octave three thirds build at 3.20, and the 6:3 at 3.56. Up the bearing octave every curve rises with the strings’ stiffness and keeps its place in that order.

The figure separates two pictures of what a tuner is doing. If the ear weighs a check by how much its beats change in proportion, the octave that results is very nearly the 4:2 — a fifth of a cent wider at A3 — and the third–sixth check is effectively demoted to a confirmation that is allowed to be a little off. If the ear weighs the checks in raw beats a second, the octave lands most of the way to the octave the thirds build by themselves, and it is the fourth–fifth check that has to be allowed to fail, by more than half its own rate. The arithmetic does not decide which picture is right. It says that they are far apart, that the first is the one the ratio-sensitivity of hearing predicts, and that a tuner who reports favouring the 4:2 octave through the middle of the keyboard is describing what the first picture produces.

That is a different reading of the practice the thirds and the fourths described. There, a preference for the 4:2 looked like a choice among types. Here it looks like what happens anyway when two checks of very different speed are weighed by ratio: the slow check dominates, because any departure from it is a large share of its rate.

How much both fail, and where it stops being inaudible #

At the balance both checks fail by the same share, and that share has its own law. Since the balance is a fixed fraction of the way across, the common failure is proportional to how far apart the types are: about 5.8 per cent for every cent between the 4:2 and the 6:3.

On the grand the common failure is 7.9 per cent at F3, 11.3 at A3, 14.8 at C4, 16.1 at C♯4 and 22.9 at F4; by A4 it is 32 and by A5 nearly 90. The 4:2 and 6:3 types spread apart as the strings shorten and stiffen, and the failure follows them. Against a fifteen-per-cent ear, both checks pass on the balanced octave from B♭2 up to middle C, and from C♯4 upward no octave passes both. At A2, on the wound bass strings where the modelled grand’s coefficient jumps, the failure is back over the line at 15.9 per cent.

The stiffer piano never gets under it. Its common failure is 20 per cent at C3, 43 at A3 and over a hundred at A4: on such an instrument the two checks contradict each other audibly at every note a tuner would make them on, and the balance is only the least bad of a set of octaves all of which fail.

The two buttons are worth hearing in that light. On C4’s balanced octave the minor third and the major sixth beat about fifteen per cent apart; on F4’s, about twenty-three. The first is at the edge of a difference; the second is a plain one.

The octaves that pass both #

A fifteen-per-cent ear is an assumption, and the most useful thing a figure can do with an assumption is to vary it.

At fifteen per cent the band is about six-tenths of a cent wide at F3, from 0.83 to 1.41 cents, and narrows as it rises until it closes after C4: eight of the thirteen notes of the bearing octave have an octave that passes both. The band always contains the balance, which is the reason for preferring it: wherever any octave passes both, the one that fails both least is among them.

Turn the dial and the band moves the way a threshold should. At ten per cent — a tuner who can tell a rate of ten beats from eleven — only F3, F♯3 and G3 have a passing octave. At twenty per cent eleven of the thirteen do, up to E♭4; at twenty-five every note in the bearing does. On a piano four times as stiff the band does not open in the bearing at all until the listener is allowed thirty per cent, and then only below E3.

So whether the conflict between the two checks is a real problem for a tuner depends on two numbers, and the figures give the trade between them: the stiffness of the strings, which sets the gap between the types, and the listener’s rate acuity, which sets the gap the ear forgives. On a large grand in the middle of the keyboard the two checks can both pass to an ear that is good but not exceptional. On a small upright they cannot pass together for anyone.

The stretch the balance lays out #

The earlier essay asked whether a stretch curve derived from the checks would agree with the one the octave types give. Laid out octave by octave, it can be compared directly.

Laid out from an equal bearing, 4:2 octaves put A1 17.7 cents flat and A6 19.1 sharp; the share balance puts them at 20.2 flat and 21.6 sharp; the beats balance at 30.8 and 32.1; 6:3 octaves at 39.5 and 40.4. Even pure 2:1 octaves stretch a stiff piano by four or five cents at those notes, since each lower note’s own type sits a little wide.

The answer to the question, then, is that the stretch the checks produce is an octave-type curve, just one that no tuner names. Because the balance sits a fixed share of the way between the two types at every note, the curve it lays out is the 4:2 curve plus a ninth of the distance to the 6:3, everywhere. Its shape is the types’ shape; it cannot be told from a choice of octave by looking at the curve. A tuner who reaches it by weighing checks and a tuner who reaches it by aiming at an octave somewhat wider than the 4:2 produce the same piano, and the octave that is not two to one already treated the octave type as a matter of which partials are silenced rather than a matter of principle. The piano is tuned wrong on purpose derived the stretch from the stiffness of the octave’s own strings, and that derivation survives intact; what the checks add is a reason for the octave to be the slightly-wide-of-4:2 one rather than any other, and a reason that does not depend on the tuner’s taste.

That claim has an edge. Above about middle C the balanced octave stops satisfying both checks to the ear, and there the tuner is no longer compromising between two passes but between two failures. Nothing in the arithmetic says the ear keeps weighing them by share when both are audible; the treble curve drawn here is the share rule extended, not an observation.

The model under the numbers #

The stiff-string law, each note with its own coefficient. Every partial is raised by the square root of one plus $Bk^2$, with $B$ from the modelled grand’s string lengths and gauges, which rise in stiffness about 2.9 times an octave up through the bearing. A real piano’s coefficients scatter from string to string; the two checks are immune to the middle note’s scatter, but not to the outer notes’.

The printed rates are the denominators. A check’s two rates are each near the printed rate when the middle note has been set to its table, and that is what the shares are taken against. If a tuner had set the middle note badly the two rates would differ from the table in the same direction, and their difference — the failure — would not change.

Rate discrimination as a single Weber fraction. The fraction a listener needs to tell two beat rates apart surely depends on the rate, on how long each is heard, and on whether they are heard in succession or remembered. A beat once a second gives only two or three cycles in a typical comparison, which is a short sample to estimate a rate from. The band figure varies the fraction rather than settle it, and three beats at most is a reminder of how little a short sample holds.

What the balance cannot show #

That tuners balance by share. The figures show where the octave falls under each rule and how far apart the rules put it. A tuner who reports favouring the 4:2 in the middle of the keyboard is consistent with the share rule and inconsistent with the beats rule; that is evidence, not a measurement of anyone’s ear.

The other checks. A bearing also uses the progression of thirds, the tenth against the seventeenth, and the octave’s own beats, and a tuner counts beats is a reminder that the method is a web of comparisons rather than two. Each additional exact test pins a further octave type and adds a term to the weighing. The balance here is the answer for the two contiguous checks alone.

Order effects. A tuner who makes the fourth–fifth check first and then the third–sixth check is not weighing two failures at once: the first check has already set the octave. Read as a sequence rather than a sum, the order of the tests decides the octave as the thirds and the fourths argued; the balance describes a tuner who goes back and forth.

Still open: the progression of thirds as a third vote #

The progression of thirds is the one check in a bearing that asks for a direction rather than an equality, and on a stiff piano it bends: each third still beats faster than the one below on the grand, but by a smaller step towards the top, and on a stiffer piano the progression turns over at E4. A tuner who holds the octave at the share balance fixes where the thirds inside it can go, and the step from the top third of one octave to the bottom third of the next is then set by the octave rather than chosen. The same arithmetic, with the progression’s smoothness counted as a third failure in the same units — the departure of each step from the printed ratio, as a share — would say whether the octave that balances the two contiguous checks also keeps the progression smooth across the octave’s edge, or whether a tuner listening to all three is pushed towards a wider octave by the thirds alone, and by how much.