The progression of thirds votes for the bearing, not the octave

The octave that fails both checks least asked where a tuner’s octave settles when the two contiguous checks a bearing relies on cannot both pass. The fourth–fifth check is exact on the 4:2 octave and the minor third–major sixth check on the 6:3, and on a stiff string those are different octaves. Weighed the way hearing weighs beat rates — each failure as a share of the rates being compared — the two fail equally a ninth of the way from the 4:2 to the 6:3, on every note of every piano, because the ratio between a tempered fourth’s beat and a tempered minor third’s belongs to equal temperament and not to the strings.

It ended by naming the check it had left out. A tuner setting a bearing also listens to the progression of major thirds: each tempered third should beat a little faster than the one a semitone below it, by the ratio of two neighbouring frequencies, so that a chromatic run of thirds rises in speed as evenly as it rises in pitch. That check asks for a direction rather than an equality, and on a stiff piano it bends. The question left open was whether, counted as a third failure in the same units, it would push the octave wider — towards the octave three thirds build by themselves — and by how much.

The answer reverses the prediction twice over. The progression pushes the octave narrower, not wider. And the octave it pushes towards is not a property of the strings at all, which is what makes it a different kind of check from the other two.

The third the octave decides #

The progression is a check of every third in a bearing against its neighbour, but only one of those comparisons involves the octave being set. If the bearing runs from F3 to E4 and the octave is F3–F4, then every third from F3–A3 up to C4–E4 has both its notes inside the bearing and is fixed before the octave is touched. The first third whose upper note is the new F4 is C♯4–F4. Its speed is set by where F4 goes, and the check is whether it beats faster than C4–E4 by the printed step — a twelfth root of two, about six per cent.

That is the comparison counted here: the step from the third just below the octave’s edge to the third that ends on the octave, as a share of the printed step. A step exactly six per cent faster passes; a step of twenty per cent, or of nothing, fails by the amount it departs. It is the same unit the other two checks were weighed in, which is what lets them be put on one axis.

The new curve is the olive V. Its point — the octave on which the step is exactly the printed one — sits at 0.13 cents on A3. That is narrower than the 4:2 octave at 1.60, narrower than the 6:3 at 3.56, and narrower even than the 2:1 type at 0.40 — the octave type on which the lower note’s second partial meets the upper note’s first. The progression wants an octave that is, to within an eighth of a cent, not stretched at all.

Its slope sits between the other two. Widening F4 by one cent raises its fourth partial by about one beat a second, against an edge third beating about fourteen times a second, so the step moves by about seven per cent for each cent of octave. The fourth–fifth check moves by about fifty per cent for each cent and the third–sixth by about six. So the progression is not the dominant voice and not a negligible one: it is roughly as sharp a test as the third–sixth check, and it points the other way.

Three votes put the octave back on the 4:2 #

With three V-shapes on the axis, the octave that fails all three least is the one where the largest of the three failures is as small as it can be. For two V-shapes pointing at each other that was simply their crossing. With three it is decided by whichever two bound the answer from either side, and on A3 those are the third–sixth check, which wants the octave wider, and the progression, which wants it narrower.

They cross at 1.63 cents, failing by 12.6 per cent each. The fourth–fifth check, which was one of the two failing checks before, now passes at 1.6 per cent — it is sitting almost on its own type. The octave that fails all three least is three hundredths of a cent wider than the 4:2.

This is a stranger result than it looks. The two-check balance moved the octave a ninth of the way towards the 6:3 because the third–sixth check pulled it there and nothing pulled back. The progression pulls back by almost exactly enough to cancel that, so the fast check on the wide side and the fast check on the narrow side neutralise each other, and the slow check in the middle — the fourth–fifth — is left to set the octave by itself. Adding a vote returned the octave to the type a single check chose.

The cost is in what everything else fails by. The two-check balance left both checks 11.3 per cent out; the three-check octave leaves two of the three 12.6 per cent out. That is the general rule, since a third constraint can only make the least-bad compromise worse or leave it alone, and it is the reason a tuner who listens to more things at once has a harder time, not an easier one.

The octave the thirds ask for belongs to the bearing #

Why the progression wants an unstretched octave is the centre of the whole question, and the answer is short. The other two checks were found to be blind to their middle notes: the fourth A3–D4 and the fifth D4–A4 share D4’s third partial, so in the difference between their beats it cancels and only the octave’s own two notes remain. That is what made each of them a test of an octave type.

The progression has no such cancellation. Its comparison is between two thirds that share no note at all. The third below the edge, C4–E4, is entirely inside the bearing; the third at the edge has its lower note in the bearing and its upper note on the octave. So the step between them compares the octave against the bearing’s own intervals. It is not a test of the octave against the strings. It is a test of the octave against the temperament it was laid out from.

The cleanest way to see that is to stretch the bearing and watch the progression’s preferred octave move.

Every line is the bearing’s stretch plus a constant. On A3 the constant is 0.125 cents whatever the bearing does: stretch the bearing by a cent an octave and the progression’s octave is 1.12, by two cents and it is 2.12. On F3 the constant is 0.088, on C4 0.162, on F4 0.247. On strings with whole-number partials it all but vanishes, and the progression’s octave is exactly the bearing’s.

So the constant is what the strings contribute, and it is small for a reason that can be said in a sentence. Stiffness slows every tempered major third a little, because it raises the upper note’s fourth partial by less, proportionally, than the lower note’s fifth; and it slows the higher of two neighbouring thirds by a slightly larger share, because a string’s stiffness rises faster up the keyboard than its frequency does. The step from one third to the next therefore comes out a hair short of the printed one, and widening the octave by a fraction of a cent restores it. Everything else in the progression’s preferred octave is the bearing.

That is the surprising connection, and it ties the bearing check to something nobody counts as a check at all. The progression is the place where a stretched-octave tuning and the temperament it is built on meet. The piano is tuned wrong on purpose derived the stretch from the octave’s own strings; twelve fifths and seven octaves derived the temperament from the circle; and the progression across the octave’s edge is the one comparison a tuner makes that refuses to let the two be chosen separately.

Through the bearing and above it #

In an equal bearing the result on A3 is general through the middle of the keyboard, and becomes something different above it.

Through the bearing octave the three-check octave never strays more than a twentieth of the way from the 4:2: between −0.011 and 0.034 of the distance to the 6:3, on notes where the two-check balance was 0.112. Above it the line turns down and crosses below the 4:2, reaching −0.112 at A4 and −0.157 at C5. There the edge third is beating fast enough that the progression’s narrow pull beats the third–sixth check’s wide one, and the fourth–fifth check joins them as a failing vote rather than a passing one. On the piano four times as stiff the same crossing happens before the bearing begins, and on A3 the three-check octave is a sixth of the way below the 4:2.

The range is cut at C5 deliberately. On the modelled grand the third near the octave crosses pure between E5 and F5 — its stiff partials overtake the tempered gap — and from there its beat no longer rises with the octave’s width in the way the check assumes. On the stiffer piano that happens from D4. A progression of thirds that has stopped beating on its printed side has nothing to say about an octave, and the figure stops where its vote stops meaning anything.

What the third vote costs #

The interesting question for a tuner is not where the compromise moves but what it fails by, since that decides where the ear can still be satisfied.

The two curves run close and parallel, the three-check one a little above. On F3 the worst failure goes from 7.9 per cent to 8.6; on A3 from 11.3 to 12.6; on C4 from 14.8 to 16.7; on C♯4 from 16.1 to 18.4. Against a fifteen-per-cent ear that moves the highest note on which all the checks are satisfied from C4 down to B♭3 — two semitones of the bearing’s top lost to the third vote.

The two buttons are worth hearing together. On C4’s two-check octave the edge third beats 1.26 times as fast as the third below it, nineteen per cent more than the printed step; on its three-check octave the step is 1.24, seventeen per cent more, and the third–sixth check pays the difference. Neither sounds wrong in isolation, and the difference between them is small. The trade is only audible as a trade, and on C4 it is a trade between two failures.

The passing octaves, and where they are trimmed #

A fifteen-per-cent ear is the same assumption the two-check essay varied, and varying it here shows which side of the band the progression cuts.

At fifteen per cent the progression trims the band from above and never from below. On F3 it does nothing — the band is 0.83 to 1.41 cents either way — because the progression’s own failure is still small across it. On B♭3 the band’s upper edge comes down from 2.04 cents to 1.91, and on B3 and C4, where a thin band existed for the two checks, there is none for three. Eight of the thirteen notes pass two checks and six pass three.

Turning the dial moves the cut without changing its side. At ten per cent three notes pass two checks and two pass three; at twenty eleven and nine; at twenty-five all thirteen pass two and twelve pass three; at thirty both pass everywhere and the progression only shaves the top of each band. That direction is the whole result in another form. In an equal bearing the progression wants narrow octaves, so it is the wide end of every passing range it removes, and it is the wide end that the third–sixth check was pulling towards.

How much stretch the bearing needs before the thirds agree #

If the progression’s octave rides the bearing, then the conflict it creates in an equal bearing should disappear in a stretched one. It does, and it disappears early.

With half a cent of stretch spread through the bearing, the progression’s own octave on A3 moves from 0.13 cents to 0.62, and at the two-check balance its failure is 9.8 per cent, below the 11.3 the other two share. It is no longer one of the binding votes. The octave that fails all three least is exactly the octave that fails two least: 1.82 cents, 11.3 per cent. Turning the dial further to a cent or two moves the progression’s V on past the balance without bringing it back into play until the bearing is stretched beyond the octave itself.

The threshold is small. On the grand it is 0.18 cents of bearing stretch on F3, against a balanced octave of 1.27 — fourteen per cent of it. On A3 it is 0.32 cents of 1.82, seventeen per cent; on C4 0.48 of 2.38; on F4 a cent of 3.70, a little over a quarter. A bearing that carries a fifth of the octave’s own stretch is enough for the progression of thirds and the two contiguous checks to agree. On the stiffer piano the share is larger — a third at F3, two fifths at A3, three fifths at F4 — because the gap between the 4:2 and 6:3 types grows faster than the progression’s constant does.

So the three-check story has two endings, and which one a tuner lives in is decided before the octave is set. In a strictly equal bearing the progression fights the octave stretch and wins enough to put the octave back on the 4:2. In a bearing whose thirds are each set a hair fast — enough to carry a fifth of a cent of stretch through A3’s octave — the progression agrees with the compromise the other two reached and adds nothing to it.

What this means for the bearing #

The bearing methods tuners learn set a temperament octave around middle C first and lay octaves out from it afterwards, and they ask that the thirds’ progression be smooth both inside the bearing and across its edges. The arithmetic says those two instructions are one instruction. An octave stretched by a cent or two at its edge can only continue a smooth progression from a bearing that is itself stretched by some fraction of that, and the fraction is not large.

That bears on a piece of practice that is often described but rarely explained: tuners commonly describe the temperament octave of a real piano as coming out a little wider than a theoretically equal one. The arithmetic here does not show that tuners do this. It shows that a tuner who did not would find the thirds’ progression across the bearing’s edge contradicting every stretched octave above it, and that a tuner who did would find all three checks satisfied together to the point where the two contiguous checks alone had left them. The practice claim is a claim about aural tuning of the modern piano by beat-counting bearings, where the progression of thirds is one of the standard tests; it is not a claim about electronic tuning, which sets a curve directly, or about historical temperaments, whose thirds were not meant to progress evenly at all.

It also changes what a tuner counts beats was describing when it called the method a web of comparisons. The contiguous checks are tests of octave types; the progression is a test of consistency between the octave and the bearing. They belong to different questions, and they agree only when the answer to the second has been chosen to suit the first.

The models under the numbers #

The bearing’s stretch is spread evenly. Every result with a stretched bearing assumes each of its semitones is widened by the same twelfth of the octave’s stretch. A real bearing set by beat counts on a stiff piano stretches unevenly, more at the top than the bottom, and the progression’s octave would follow whatever the top of the bearing did rather than its average. That makes the threshold figure a statement about the average stretch needed and not about any particular bearing plan.

The stiff-string law, each note with its own coefficient, from the modelled grand’s lengths and gauges, as in the two-check essay; the stiffer piano multiplies every coefficient by four. The progression’s small constant comes entirely from how fast that coefficient rises, and on a piano with an unusually flat scale design it would be smaller still.

The step’s failure is weighed as a share of the printed step. That puts it in the same units as the other two checks, which is what lets them share an axis, but a listener comparing two thirds that beat at thirteen and fourteen times a second is making a different judgement from one comparing a fourth and fifth beating once a second. The single Weber fraction for all three is a convenience, and three beats at most is a reminder of how little a short sample of a slow beat holds. The band figure varies the fraction rather than settle it.

One step of the progression. Only the step across the octave’s edge is counted, because it is the only one the octave decides. A tuner listening to a whole run of thirds also hears every step inside the bearing, and on a stiff piano those are all a little short of the printed step in an equal bearing — which is the same fact as the progression’s constant, seen from inside.

What the curves leave out #

The order of the checks. The minimax octave describes a tuner weighing all three at once. A tuner who sets F4 by the progression first, then confirms with the contiguous checks, lands on the progression’s octave and hears the fourth–fifth check fail by half its rate; one who does the reverse lands on the 4:2. The figures say which octaves exist and how much each fails, not which a particular method reaches.

Whether tuners stretch the bearing on purpose. The threshold figure says how little stretch would reconcile the checks. It cannot say whether the slight widening tuners report is a deliberate response to this conflict, a side effect of counting beats against a stiff piano’s true rates, or a habit with some other origin. All three would produce the same bearing.

The treble. Above about E5 on the grand, and above C♯4 on the stiffer piano, the third near the octave has crossed pure and the progression no longer bears on the octave in the way modelled here. What tuners listen to up there is a different set of tests, and the beat a tuner can actually use is the place where the usable rates were counted.

Still open: a bearing set by its own thirds #

The threshold figure treats the bearing’s stretch as a free number. It is not, for a tuner who sets the bearing by counting each third against the printed rate: the stiff partials make every third’s true rate differ from the printed one, so a bearing set to printed rates already carries a stretch the strings choose. The stiff string’s slowing of the fourths and the thirds’ shortfall computed here are the ingredients. The computation is a bearing built note by note from beat counts on the modelled grand — each new note set by the interval a standard plan uses to reach it — and the stretch it ends up carrying read off against the threshold here. That would say whether a tuner who simply counts beats correctly lands, without intending to, on a bearing stretched enough for the three checks to agree, or one that leaves the progression still fighting the octave.