A stiff string slows the fourths and quickens the fifths

A piano tuner laying a temperament is told what to listen for in beats per second. The fifth from A3 to E4 should pulse about three times every four seconds; the fourth from A3 to D4 about once a second; the major third from A3 to C♯4 between eight and nine times. A tuner counts beats, and that is the whole method derived those numbers from two things only, the ratio of the interval and the register it sits in, and they are exact for a string whose partials are whole-number multiples of its fundamental.

A piano string is not that string. It is steel under tension, and steel resists bending, so each partial is pushed sharp of its whole-number place by an amount that grows with the square of its number. The same essay said so, and drew a conclusion from it that the essays since have repeated in passing: every true beat rate is faster than the printed one, on every interval, on every piano, and a tuner on a small instrument should “beat everything a little faster than the table says”.

Half of that is right. The other half is the wrong way round, and the error is visible in that essay’s own table, where the major third’s true rate on the grand is 6.67 beats a second against the printed 6.93 — slower, not faster.

Which partial is on top #

The beat a tuner counts in a fifth is between the third partial of the lower note and the second partial of the upper, and like every beat its rate is the difference of the two in hertz. In a fourth it is the fourth partial of the lower against the third of the upper; in a major third, the fifth against the fourth; in a minor third, the sixth against the fifth; in a major sixth, the fifth against the third. In every one of them the lower note supplies the higher-numbered partial, because the lower note has to climb further up its own series to meet the upper one.

Stiffness raises the k-th partial by a factor of the square root of one plus Bk², where B is the string’s inharmonicity coefficient. A higher-numbered partial is raised more. So in every coincidence the lower note’s partial is lifted more than the upper’s, and the question of what that does to the beat is a question about which of the two was on top to begin with.

That is decided by the temperament. Because twelve fifths overshoot seven octaves, equal temperament makes the fifth 1.955 cents narrow — the upper note sits a little low — so the third partial of A3 sits a little above the second partial of E4. Lifting the higher-numbered partial further widens a gap that already pointed upward, and the beat quickens. The fourth is tempered by the same 1.955 cents in the other direction. It is wide: D4 sits a little high, so the third partial of D4 is above the fourth partial of A3. Lifting A3’s fourth partial now closes the gap, and the beat slows.

The rule is general. Equal temperament makes the fifth, the minor third and the minor sixth narrow and the fourth, the major third and the major sixth wide. Stiffness quickens the beats of the first group and slows those of the second. The essay on the bearing plan saw only fifths, which happen to be narrow, and generalised.

On the grand modelled here — the same string model, scaled up the compass by its own lengths and gauges, that the piano is tuned wrong on purpose used to derive the Railsback curve — the numbers at A3 are these. The fifth’s printed rate is 0.74 beats a second and its true rate 0.96, 29 per cent fast. The fourth’s printed rate is 0.99 and its true rate 0.63, 36 per cent slow. The minor third beats 8 per cent fast, the major third 5 per cent slow, the major sixth 11 per cent slow. The departures are largest for the fourth and fifth because their beats are slow to begin with — the same partial-by-partial arithmetic that makes a fast beat insensitive to a small shift makes a slow one very sensitive to it: the stiffness shift is a fraction of a hertz in every case, and a fraction of a hertz is a third of a slow beat and a twentieth of a fast one.

Stiff enough, and the fourth is pure #

The grand’s strings are long and thin for their pitch, which is what a concert grand is for. A small upright’s are short and thick, and its inharmonicity coefficient is several times larger at the same note. The essay on the bearing plan priced the upright at four times the grand’s coefficient, and the same factor serves here.

Drawn against the stiffness, the two groups separate cleanly. The fifth’s share rises from 1.20 on the grand at F3 to 1.41 at twice its stiffness and 1.81 at four times. The minor third’s rises more slowly. The three wide intervals fall, and the fourth falls fastest: 0.75 on the grand, 0.49 at twice, and at 3.96 times the grand’s stiffness the tempered fourth F3–B♭3 does not beat at all. The lifting of F3’s fourth partial has exactly cancelled the 1.955 cents by which the fourth was tempered. The interval is equal-tempered in its fundamentals and pure in the partials an ear compares.

Past that point the fourth beats again, but from the other side: its lower partial is now on top, which is the arrangement of a narrow interval. A tuner on such an instrument listening for the fourth’s beat hears one, slow, and has no way to know from the beat alone that it is the wrong side of silence. The essays on the bearing plan said that a pure interval is the strongest instruction a plan can give, because silence is easier to hear than a rate. On a stiff enough piano the tempered fourth gives that instruction falsely.

The major sixth crosses too, much later, at about thirteen times the grand’s stiffness at F3; the major third does not cross at that note in any range a piano reaches. The fifth and the minor third never cross, since their gap only grows.

Up the keyboard #

Inharmonicity rises steeply up the compass. The strings shorten faster than their diameters shrink, and the coefficient of the model’s grand goes from 0.00029 at F3 to 0.00083 at F4 and 0.0015 at C5. So the departure is not a constant correction to the table; it grows note by note through the octave the temperament is laid in.

On the grand, the fifth on F3 beats at 1.20 times its printed rate, the fifth on B3 at 1.35, and the fifth on F4 at 1.60. The fourth goes the other way, from 0.75 on F3 through 0.56 on B3 to 0.25 on F4, and somewhere around A4 it reaches zero. The bearing octave is placed in the middle of the keyboard partly to keep its beats countable, and that placement is also what keeps its fourths beating at all on a grand. On the upright, at four times the stiffness, the fifth on F3 already beats at 1.81 times the printed rate, the fourth on F3 at none, and every fourth above it beats on the wrong side.

The first essay put the fifth’s departure on the grand at 20 per cent at the bottom of the bearing octave and 60 per cent at the top, and on an upright at 81 and 237 per cent. The figure reproduces those numbers exactly. What that essay did not compute was the fourth beside it, and the fourth is where its rule of thumb fails. “Beat everything a little faster” is right for the fifths and wrong for the fourths. On a small upright, a tuner obeying it would widen fourths that are already too wide, by the same mechanism that makes them beat too slowly.

Where a tuner who counts the table leaves each interval #

The rule of thumb does have a correct half, and it is the part the first essay put in a single sentence: a tuner who hits the printed rate exactly has left the interval slightly wide. That turns out to be true of every interval, narrow or wide, and for opposite reasons.

Take the fifth. Its true beat is too fast, 0.96 instead of 0.74. To slow it, the tuner must close the gap between A3’s third partial and E4’s second, and since A3’s is on top that means raising E4. The fifth gets less narrow — that is, wider — by 0.57 cents before it beats at the printed rate.

Take the fourth. Its true beat is too slow, 0.63 instead of 0.99. To speed it, the tuner must open the gap between D4’s third partial and A3’s fourth, and since D4’s is on top that again means raising D4. The fourth gets wider still, by 0.71 cents. Two opposite errors in the beat rate correct into the same direction in the interval.

On the grand at A3 the thirds and sixths come out 0.72 cents wide (the major third), 1.31 (the minor third) and 1.73 (the major sixth). On an upright four times as stiff: 2.24 for the fifth, 2.74 for the fourth, 2.73 and 4.89 for the thirds and 6.61 for the sixth. The sixth moves most because its coincidence is between the fifth and third partials, the widest gap in partial number of any interval here, so the two partials are lifted by the most different amounts.

The direction is the same as before, and the reason is simple once it is found. Whatever the interval, the stiffness has lifted the lower note’s partial relative to the upper’s. A tuner who restores the whole-number beat rate is restoring the whole-number relationship between the two partials, and the only way to do that is to raise the upper note by the amount the lower partial was lifted, referred back to the fundamental. An interval tuned that way is pure-relative, not tempered-relative, to stiff partials, and every interval comes out wider than its equal-tempered size for the same reason a stiff octave tuned beatless comes out wide.

The table stretches its own octave #

That last sentence has a consequence the bearing plan never states. A bearing octave is laid as a chain of fifths and fourths, and a fifth and a fourth stacked make an octave. If both are left wide, the octave they make is wide too.

The arithmetic is additive to within a hundredth of a cent: the octave’s stretch is the fifth’s widening plus the fourth’s. Tuning up a fifth from A3 and then up a fourth from wherever E4 landed, each to its printed rate, reaches an A4 1.89 cents wider than 2:1. Laying the fourth first and the fifth second reaches one 1.60 cents wide, because each link’s widening depends on the note it starts from and the two orders start the second link on different notes. The table does not close its own octave the same way twice.

Against the octave types, both lie in a definite place. A beat is never one beat found that a mistuned octave on a stiff string beats in a whole family, and that zeroing any one member of the family defines an octave of its own width — the “octave types” every tuning manual tabulates as 2:1, 4:2, 6:3 and so on. Each type compares a partial of the lower note with a partial of the upper, and each partial is raised by its own string’s stiffness; the upper note’s string is nearly three times as stiff as the lower’s on the grand modelled here, so the types are narrower than a formula giving both notes one coefficient would say. At A3 the 2:1 octave is 0.40 cents wide, the 4:2 octave 1.60 and the 6:3 octave 3.56. Laid fourth first, the table’s octave is the 4:2 type to within a hundredth of a cent, at every note from C3 to C5; laid fifth first it lies a little past it — 0.20 cents past at F3, 0.29 at A3, 0.99 at C5 — and well short of the 6:3. (Corrected: this essay first quoted the types with the lower note’s coefficient for both notes, 1.07 and 4.26 cents at A3, and placed the table’s octave between the 2:1 and the 4:2. The fourth–fifth check cannot hear the note it checks found why the fourth-first chain lands on the 4:2 exactly.)

That is a stretch nobody chose. The essay on the stretched piano attributed the stretch to the tuner’s decision about which partials of the octave to make agree, and the octave types are that decision written down. But a tuner who never listens to an octave at all, who only counts the fifths and fourths of the bearing octave against a printed table, has already stretched it — by an amount fixed by the string’s stiffness, the ratios involved and the order in which the links are laid — to the 4:2 type itself when the fourth is laid first. When that tuner then checks the octave by ear, they find it beating at the 2:1 type’s member, and silent at the 4:2 only if the fourth was laid first. Whatever they do next is a correction to a stretch the table built.

Struck once, the fourth gives little to count #

A tuner does not hear a rate; they hear a beat inside a struck note that is dying. Counted in the decay, or not at all and the essays after it followed the countable beats of a mistuned octave through a real decay — the auditory filter broadened by the level, the partials falling below a listener’s margin, the window in which a beat can be separated from its neighbours opening and shutting. That count takes any ratio, and run on the tempered intervals with each note’s own stiffness it says how much of a struck note each one gives a tuner to work with.

The temperament is set with a strip of felt woven through the middle section so that every note is a single string, and on a single string the window is short: a few tenths of a second, as a string that decays twice is counted early found for the octave. Inside it the five intervals fare very differently, and the reason is the rate window three beats at most put round a countable beat. The thirds and the sixth beat at eight to thirteen a second and deliver three or so countable beats each. The fifth beats at 0.96 a second and delivers 1.8. The fourth beats at 0.63 a second, and its window on A3 runs from 0.27 to 0.77 seconds after the strike: half a second, holding a third of a cycle. Set to its printed rate, it beats faster and delivers half a beat.

Over the unisons that replace the mute afterwards, left two and a half cents wide as the octave is counted best over a unison left two and a half cents wide recommended, every count roughly doubles: the fifth to 3.9, the thirds and sixth to between 7.4 and 9.3, the fourth to 0.8. The fourth is still the poorest interval on the piano to count, and the stiffness is the reason. It takes a beat printed at one a second and turns it into one at less than two-thirds of that, in a window that does not lengthen to compensate.

This sits oddly beside the table, which says the fourth is the livelier of the two. On whole numbers the fourth on any note beats a third faster than the fifth on the same note: 0.99 against 0.74 at A3, 0.79 against 0.59 at F3. On the grand’s strings the order is reversed at every note of the bearing octave. At F3 the fourth beats at 0.59 and the fifth at 0.71; at A3, 0.63 and 0.96; at F4, 0.40 and 1.89. The tempered fourth is the slower interval throughout the octave in which a temperament is laid, and by the top of it the fifth is beating almost five times as fast.

A tuner who counts fourths at their printed rate does not hear this, because they have not left the fourths tempered. They have widened them by 0.71 cents at A3 until they beat at the printed rate, and a fourth tuned that way beats as clearly as the table promised because the tuner made it do so. What the strings take away, the tuner’s hand puts back, and the price is the widening that the octave inherits.

What the first essay’s rule should have said #

At an upright’s stiffness the same picture is sharper.

At four times the grand’s stiffness, at A3, the fifth beats at 1.61 a second against 0.74 printed — more than twice. The fourth has crossed pure, at 2.76 times the grand’s stiffness on this note, and beats at 0.44 a second with its partials the other way round. The major sixth beats at 5.57 against 9.98, the major third at 6.90 against 8.73; the minor third at 15.82 against 11.87. Of the five numbers a tuner on this instrument would hear, not one is within twenty per cent of its printed value, and two of the five differ from it by a factor of two or more.

So the rule should have been stated as two rules. To reach equal temperament on a stiff string, beat the narrow intervals faster than printed and the wide intervals slower. And to reach whatever a tuner reaches by counting the table, expect every interval to come out wide, by a widening that adds up through the octave into a stretch to the 4:2 type or a little past it. The first is a recipe for equal temperament in the fundamentals; the second is a description of what the recipe printed in every tuning manual actually produces. They are different instruments, and a working tuner’s ear is the arbiter of which one is wanted — most would say the second, stretched, which is the one the table delivers without being asked.

What the model assumes #

The partials follow the stiff-string law exactly, each raised by the square root of one plus Bk², with B computed from the model grand’s lengths and gauges at each note. Real strings depart from it near their terminations and in the wound bass, and a real piano’s coefficients scatter from note to note by tens of per cent. The scatter would make the corrections ragged from note to note; it would not reverse their signs, which depend only on which side of pure the temperament puts each interval.

The upright is the grand scaled by four. A real small piano is stiffer in some registers than others and its scaling breaks are in different places. The factor was chosen to match the bearing-plan essay’s own comparison and is a stand-in for a family of instruments.

The tuner’s target is the whole-number rate. Printed bearing plans differ slightly in their assumed pitch and in rounding; none known here corrects for inharmonicity, which is the premise of the comparison. Electronic tuning devices that measure a piano’s partials before setting it do correct for it, and on them none of this applies.

The count through the decay is the one the octave essays used, with the stiffness of each note now its own rather than the lower note’s for both. The octave counts those essays published are unchanged by that: they used the lower note’s coefficient for the upper note because an octave’s upper note was the quantity being varied.

What the picture cannot show #

What equal temperament is on a piano. The essays on the bearing plan took it as twelve equal divisions of a 2:1 octave in the fundamentals. A piano’s octaves are not 2:1, and a tuner aiming at “equal temperament” on a real instrument is aiming at twelve equal divisions of whatever octave they set. Against that target the table’s widening of fourths and fifths is not an error but a contribution. The figures here measure everything from the fundamentals’ equal temperament because that is the only version with a number attached; which version a listener prefers is a separate question that the piano is tuned wrong on purpose took up.

How tuners actually use the table. No working tuner counts a printed rate and stops; they check with thirds, tenths, fourth–fifth comparisons and octaves, and correct. The figures say what a tuner who obeyed the table to the letter would produce, which is the right baseline for saying what the checks have to correct and the wrong description of any particular tuning.

Still open: whether the checks cancel the table’s stretch #

The tuner’s standard check on a fourth and a fifth is to compare them directly: a fourth and a fifth that together fill an octave are sounded one after the other, and the tuner listens for whether the two beat alike, as the whole-number table says they should when they share their middle note. On a stiff string the fourth and fifth on one lower note change their order — on the grand, left exactly tempered, the fourth is the slower of the two at every note of the bearing octave — and the contiguous pair may move too. A tuner who adjusts until the pair agrees is adjusting something the figures here can price: the pair of settings that satisfies the check at every note, and the octave it builds. Whether the check pulls the table’s stretch back towards the 2:1 type or pushes it further out is arithmetic with the same count, and it would say whether the most common check in the bearing is correcting the stiffness or compounding it.