A bearing counted correctly is stretched by its route, not its strings

The progression of thirds votes for the bearing, not the octave found that the comparison a tuner makes between the major third below an octave’s edge and the third that ends on the octave is a test of the octave against the temperament octave it was laid out from. In an equal bearing that test pulls a stiff piano’s octave back to the 4:2 type and costs the top of the bearing its pass; in a bearing stretched by about a fifth of the octave’s own stretch — 0.32 cents across the bearing at A3 on the modelled grand — the progression agrees with the other checks and falls silent.

That essay treated the bearing’s stretch as a free number, spread evenly over its twelve semitones, and ended by naming why that was a simplification. A tuner does not choose a stretch for the bearing. A tuner sets each note by counting the beats of a fourth or a fifth against a note already set, at the rate the printed tables give, and on a stiff string those printed rates are not the rates of tempered intervals. A stiff string slows the fourths and quickens the fifths worked out the consequence for one interval at a time: counted to its printed rate, every fourth and every fifth ends up a little wider than tempered. So a bearing built note by note from beat counts already carries some stretch, chosen by the strings. The question was how much, and whether it is enough for the three checks to agree.

The arithmetic answers a different question first, and the original one turns out not to have an answer that belongs to the strings at all.

Two ordinary plans #

A bearing plan is an order of intervals. The tuner sets A3 from a fork, then sets each new note by a fourth or a fifth from one already set, until all twelve notes from F3 to E4 are in. Two plans are compared here, both of the standard kind.

The first works outward from A3 by alternating a fourth up and a fifth down: A3 up to D4, down to G3, up to C4, down to F3, then up to B♭3 and E♭4, down to A♭3, up to C♯4, down to F♯3, up to B3 and finally E4. The second sets the sharps by fifths up and fourths down — A3 to E4, down to B3, down to F♯3, up to C♯4, down to A♭3, up to E♭4 — and the flats the first plan’s way.

Each fourth and fifth is counted to the rate the printed table gives for a tempered interval with whole-number partials. On the modelled grand, each string’s partials are raised by its own stiffness, so that rate belongs to an interval slightly wider than tempered; the counted note lands slightly off its equal-tempered place, and every later note counted from it inherits the error.

Where the notes land #

Neither plan puts any note more than three cents from equal temperament, which is why the result is usually described as a tempered bearing with a little stretch in it. But the two plans put the same notes in different places. The first sends E4 to 2.78 cents sharp, B3 to 1.93, C♯4 to 1.82, F♯3 to 1.39; F3 sits at 0.43 and G3 at 0.23. The second puts E4 at 0.57, B3 at −0.28, C♯4 at −0.38, F♯3 at −0.82 and A♭3 at −1.03: its sharps lie flat, where the first plan’s lie sharp.

Fitted with a straight line, the first plan’s bearing is stretched by 1.62 cents an octave and the second’s by 0.51. Measured from F3 to E4, the first spans 2.35 cents and the second 0.14.

The route, not the strings #

The first plan’s route makes the mechanism plain. Each fourth counted up is left wide by between 0.49 and 0.85 cents, so each note set by a fourth up lands sharper than the note it was counted from. Each fifth counted down is left wide too, which, with its upper note fixed, puts the lower note flat of where it should be — by between 0.40 and 0.52 cents, about two-thirds of what the fourth before it added. A fourth up and a fifth down is a whole tone down, and each such pair leaves the tone a quarter of a cent sharp. Eleven links later, the last notes the plan sets are the ones that have accumulated most: B3 and E4, at the end of the chain, sit highest.

So the shape of the bearing is the shape of the plan. A note’s error is the sum of the errors along the path that reached it, and the path is a decision of the tuning method, not a property of the piano. The second plan reaches its sharps by fifths up, each of which, left wide, sends the next note sharp, and then by fourths down, each of which sends the next note flat by more — so its sharps drift flat. Both plans count every interval correctly by the table; they disagree about where the notes are because they took different roads.

The thirds and the fourths check different octaves found the same telescoping for chains that close on an octave: a fourth and a fifth counted to the table land on the 4:2. The bearing plans never close; each walks around the circle without coming back to where it began, and the error has nowhere to cancel.

Enough stretch, by accident or not at all #

Read as a stretch, the answer to the original question is yes for the first plan by a wide margin. Its 1.62 cents an octave is five times the 0.32 that A3 needs and more than three times C4’s 0.48. A tuner who uses it and counts correctly will, without meaning to, lay down a bearing stretched well past the point where the progression of thirds stops fighting the octave.

The second plan’s least-squares stretch of 0.51 also clears the threshold, but that number describes a fit through a bearing whose ends are only a seventh of a cent apart and whose middle wanders by a cent either way. Whether its progression across the octave’s edge agrees with the other checks depends on which notes the edge comparison uses, and for most of the bearing’s octaves those are notes the second plan has left flat.

So the honest answer is that the question was misframed. The stretch a counting tuner lands on is real, it is large enough to matter, and it is set by the route. A tuner who wants to know whether the bearing is stretched enough has to know which plan was used, and the earlier essay’s even stretch is a description of neither.

The thirds inside the bearing zigzag #

The more consequential finding is about the thirds that never touch the octave.

In an equal bearing on the grand, the eight major thirds from F3–A3 to C4–E4 beat at 6.67 up to 9.66 times a second, each a little faster than the one below, each a little slower than the printed table says — the shortfall the earlier essays traced to the stiffness. That is the smooth progression a tuner listens for.

In the first counted bearing they beat at 6.46, 6.80, 8.41, 7.63, 9.45, 8.57, 9.01 and 11.17. G3–B3 beats faster than G♯3–C4 above it; A3–C♯4 faster than B♭3–D4. The progression goes up, up, down, up, down, up, up. In the second plan’s bearing it zigzags too, in different places: 6.46, 7.99, 7.14, 8.97, 8.02, 8.57, 9.00, 9.46.

The cause is the route again, read pair by pair. A third’s width is the difference between the errors of its two notes. G3, reached early by a fifth down, sits at 0.23 cents; B3, reached near the end by a fourth up, sits at 1.93. G3–B3 is therefore 1.7 cents wide of tempered, and since a major third at this pitch beats a little over half a beat a second faster for every cent it is widened, it beats nearly a beat fast: 8.41 against 7.43 in the equal bearing. G♯3–C4 has its notes at 1.17 and 0.82 and is a third of a cent narrow. Neighbouring thirds have notes from different stages of the plan, and the stages alternate.

How far the zigzag fails #

In an equal bearing on the grand every step of the progression is within six-tenths of a per cent of the printed step. In the first counted bearing the steps are −0.6, +16.8, −14.4, +16.9, −14.4, −0.7 and +17.0 per cent. In the second they are +16.8, −15.6, +18.6, −15.6, +0.9, −0.8 and −0.8. Against a listener who tells two beat rates apart at fifteen per cent — the fraction the earlier essays used for every check — the first plan fails three steps of seven, and passes the other two large ones by six-tenths of a per cent; the second fails four. Counting every fourth and fifth exactly as the table says produces a bearing whose own progression of thirds fails audibly, by errors forty times the equal bearing’s.

The buttons make the point by ear. The two thirds from F♯3 to G3 should step up by six per cent; in the counted bearing the second beats about twenty-four per cent faster than the first, from 6.8 to 8.4. The two from A3 to B♭3 should step up by the same six per cent; the second beats nine per cent slower.

This is the familiar experience of aural tuning stated as arithmetic. A tuner who lays a bearing by fourths and fifths alone and then plays the thirds in sequence hears them stumble, and corrects the notes until the thirds progress evenly. A tuner counts beats described the method as a web of checks, and the thirds’ progression as one of them. On a stiff string it is more than a check. The fourths and fifths counted to the table cannot produce a bearing whose thirds progress; the thirds have to set it.

Which steps come out right #

Two steps of the first plan’s progression are within a per cent of printed: F3 to F♯3 and B♭3 to B3. They are not luck. A step of the progression compares two thirds, four notes, and it comes out right when the two thirds’ widths agree — when the lower notes of the two thirds sit at the same distance from their upper notes in the plan’s chain. F3–A3 and F♯3–A♯3 both join a note set in the plan’s first half to a note set by the same kind of link a few moves later; their errors are alike, the thirds are widened alike, and the step between them is the equal bearing’s step.

The failing steps are the ones where the plan crosses from one stage of its route to another between the two thirds: G3, set second, against B3, set second to last; G♯3, set in the middle, against C4, set early. Wherever a step of the progression straddles that seam, the two thirds’ errors differ by a cent or more and the step fails by fifteen per cent. That is why the two plans fail different steps. Each has its own seams, and the progression finds them.

It also says what a cleverer plan could and could not do. A plan whose route visited the notes in the order of the progression itself would put the seams between thirds that are compared less often, but it cannot remove them: twelve notes reached by eleven links from one reference must accumulate eleven links’ worth of error somewhere, and some pair of neighbouring thirds has to straddle the largest jump. The octave that fails both checks least found a balance point between two checks that could not both pass; a plan of fourths and fifths has no such balance to find, because the error it accumulates has nowhere to go but into the thirds.

The zigzag grows with the stiffness #

Every error in the counted bearing comes from the stiffness, so every error scales with it. On strings a quarter as stiff as the modelled grand’s the worst step is 4.0 per cent by the first plan and 4.4 by the second: below anything a tuner would hear. At half the stiffness, 8.1 and 8.9. At the grand’s own, 17.0 and 18.6, just past fifteen. At twice, 37 and 40; on a piano four times as stiff — the small upright the earlier essays used as a foil — 92 and 98 per cent, which is a third beating nearly twice as fast as the one above it should.

So the printed tables are good enough to lay a bearing on a very large piano, marginal on a concert grand, and useless on a small upright unless the tuner checks and corrects by the thirds. The stretch the counted bearing carries scales the same way: 0.41 cents an octave by the first plan at a quarter of the grand’s stiffness, 1.62 at the grand’s, 6.31 on the upright. On the upright the first plan’s bearing is stretched by more than many tuners would stretch its octaves.

What the earlier finding becomes #

The earlier essay concluded that a bearing carrying a fifth of the octave’s stretch is enough for the progression of thirds and the two contiguous checks to agree, and that tuners’ habit of widening the temperament octave slightly would produce exactly that. This essay does not overturn the conclusion; it changes what can be said about where such a stretch comes from.

A tuner who counts only fourths and fifths by the table produces a stretch, but an uneven one whose amount depends on the plan and whose thirds do not progress. A tuner who then corrects the bearing by the thirds — making each third beat faster than the one below by the printed step — is choosing the stretch directly, through the progression, and the progression of thirds votes for the bearing already showed that the progression, inside a bearing, wants the stiff string’s own slight shortfall corrected. So the path to the reconciled bearing runs through the thirds, not through counting the fourths and fifths more carefully.

That also separates two claims that are easy to run together. The piano is tuned wrong on purpose derived the stretch of the octaves from the strings; twelve fifths and seven octaves derived the temperament from the circle. A counted bearing is where the two meet without either being chosen, and the result is neither a stretched temperament nor a tempered one: it is a record of the plan.

The model under the count #

The modelled grand’s strings, each with its own stiffness coefficient from the lengths and gauges the beating essays used, so that a note’s k-th partial is raised by the square root of one plus the coefficient times k squared. The stiffer piano multiplies every coefficient.

Counting is exact. Every fourth and fifth is set to beat at precisely its printed rate, on the printed side of pure. A real tuner counts to a tolerance — three beats at most is a reminder of how little a short sample of a slow beat holds — and that tolerance is a few per cent of a beat a second. The errors here are systematic and would survive any amount of care in counting; a tuner’s random errors would add to them.

Two plans of one kind. Both lay the bearing out from A3 by fourths and fifths. Plans that use thirds and sixths as setting intervals rather than checks, or that start from C or F, would distribute the errors differently, and the essay’s claim is about the kind of error, not about these two distributions.

What the count cannot settle #

What tuners actually do. Aural tuners are taught to check every new note against thirds and sixths already set and to correct, so few real bearings are the pure count drawn here. The figures describe what the corrections are correcting, not the bearing a tuner ends with.

How a progression is heard. The fifteen per cent band treats every step alike. A tuner playing thirds in sequence hears a reversal — a third slower than the one below — far more readily than an uneven increase, and every counted bearing on the grand contains at least one reversal. The beat a tuner can actually use set out which rates a tuner can count at all; a reversal between two of them needs no counting.

Still open: a bearing set by its own thirds, done properly #

The obvious next plan is the one the arithmetic recommends: set each new note by a fourth or fifth, then adjust it so the newest third beats faster than its neighbour by the printed step, and carry on. That plan has to choose, at each note, between the fourth’s count and the third’s, and the choice it makes decides the stretch. The computation is a bearing laid out by that rule — the thirds’ progression as the authority, the fourths and fifths as a guide — and it would say how much stretch a bearing set by its own thirds carries on the grand and on the upright, and whether it is the fifth of an octave’s stretch the earlier essay found the progression needed, which would make the reconciliation of the three checks something a tuner reaches by following the thirds rather than something that has to be added.