The fourth–fifth check cannot hear the note it checks

Every piano tuner who lays a temperament by ear checks it as they go, and the check taught first and used most is a comparison of two slow beats. With an octave in place — A3 to A4, say — the tuner sounds the fourth from the bottom note to a note inside the octave, A3 to D4, and then the fifth from that note to the top, D4 to A4. The fourth is tempered wide and the fifth narrow, both beat about once a second, and the instruction is to listen to whether they beat alike.

A stiff string slows the fourths and quickens the fifths ended by asking what this check does on a real piano. On whole-number partials, a fourth and a fifth on the same lower note beat at different rates — the fourth about a third faster — and that essay found the order reversed on a grand’s stiff strings, so it framed the check as one in which the fourth should lead the fifth and asked whether a stiff string moved the lead. That framing was the wrong check. Nobody compares a fourth and a fifth that share their lower note. The check that is actually made shares the middle note, and in that arrangement the whole-number table says the two beat at exactly the same rate. On A3 it prints 0.99 a second for each.

Put that way, the question changes shape, and the answer is more surprising than either a correction or a compounding. The check cannot see the middle note at all.

Three partials near 880 hertz #

The beat a tuner hears in the fourth A3–D4 is between the fourth partial of A3 and the third partial of D4, both near 880 hertz. The beat in the fifth D4–A4 is between the third partial of D4 and the second partial of A4, also near 880. Three partials of three notes meet at one frequency, and the middle note’s partial is in both pairs. Like every beat, each rate is a difference of two frequencies in hertz, and every partial beats at its own rate, so these three are the pairs a tuner attends to rather than the only ones sounding.

Written as arithmetic, the fourth’s rate is D4’s third partial minus A3’s fourth, and the fifth’s rate is D4’s third partial minus A4’s second. Their difference is A4’s second partial minus A3’s fourth. D4 is not in it. Raise D4 by a cent and both beats quicken by the same amount — about half a beat a second for every cent at this register — and the difference does not move by a hundredth.

The three rows of the figure make the point visually. In each, the middle dot is the same tempered D4. In the bottom row the octave is a pure 2:1 in its fundamentals, and A4’s second partial sits 0.82 hertz below A3’s fourth: the fifth beats at 1.45 a second and the fourth at 0.63. In the middle row the octave has been widened until A4’s second partial lands exactly on A3’s fourth, and the two beats are both 0.63. In the top row the octave is a little wider still, A4’s partial has passed A3’s, and the fourth now beats 0.15 a second faster than the fifth. In none of the three rows did anything happen to D4.

A check of the octave, and of one octave in particular #

The hero figure draws the same thing as two lines. The horizontal axis moves the middle note three cents either side of its tempered place; the vertical axis is the beat rate of each interval, signed so that a wide fourth and a narrow fifth both count as positive.

The two lines are parallel. A tuner who hears the fourth and fifth beating unequally and tries to fix it by moving D4 will chase the fault up and down without closing it, because the fault is not in D4. The only way to make the two lines meet is to move one of the outer notes, and the dial shows where they meet: at an octave 1.60 cents wider than 2:1, and nowhere else.

That octave has a name. A beat is never one beat showed that a mistuned octave on a stiff string beats in a whole family at once — the lower note’s second partial against the upper’s first, the lower’s fourth against the upper’s second, and so on — and that a tuner choosing which member to silence is choosing an octave type. The type that silences the lower note’s fourth partial against the upper’s second is the 4:2 octave. The fourth–fifth check is a 4:2 test, carried out through a third note that happens to sit at the meeting point. A tuner using it is not checking the temperament; they are checking which octave they have.

The middle note does contribute one thing: the common speed of the two beats. Set D4 three cents sharp and the two beats run at 2.2 and 2.0 a second rather than 0.6 and 0.5, and a difference of 0.15 between beats that fast is harder to hear than the same difference in beats near one. So D4 decides how easy the check is to read, and nothing about what it reads.

What the 4:2 octave is on a real piano #

Here the arithmetic of the earlier essay has to be corrected, because it drew the octave types too wide.

An octave type is defined by two partials, one on each note, and each partial is raised by its own string’s stiffness. That essay computed the types the way the octave-type literature usually tabulates them, with one inharmonicity coefficient for both notes — the lower note’s. But the upper note of an octave is on a shorter, relatively thicker string, and on the grand modelled here its coefficient is 2.87 times the lower note’s. Its second partial is raised nearly three times as much as a string with the lower note’s stiffness would raise it, so it reaches A3’s fourth partial sooner, and every type is narrower than the shared-coefficient formula says.

At A3, with each note its own stiffness, the 2:1 octave is 0.40 cents wide, the 4:2 octave 1.60 and the 6:3 octave 3.56. The earlier essay printed 1.07 for the first and 4.26 for the second, and the same formula gives 9.53 for the third — the three octaves on the assumption that A4 was as flexible as A3. Its own octaves built by counting fifths and fourths to the table were computed correctly, with each note’s own stiffness, and came out at 1.89 cents laid fifth first and 1.60 laid fourth first. Against the inflated types those looked like an octave “between the 2:1 and 4:2”. Against the right ones they are something more definite: laid fourth first, the table builds the 4:2 octave exactly, to within a hundredth of a cent, at every note from C3 to C5; laid fifth first, it builds an octave 0.29 cents past the 4:2 at A3, 0.20 past at F3 and 0.57 past at F4, and well short of the 6:3 everywhere.

The first of those is not a coincidence, and it is the same arithmetic as the check. Counting a fourth up from A3 to the printed rate fixes D4’s third partial at a set distance above A3’s fourth. Counting the fifth from D4 to A4 at its printed rate fixes A4’s second partial at the same distance below D4’s third — because the table prints the same 0.99 for both, the contiguous pair beating alike being what equal temperament means on whole numbers, since twelve fifths and seven octaves are reconciled by tempering the fifth and the fourth by the same 1.955 cents in opposite directions. The two distances cancel, and A4’s second partial lands on A3’s fourth. That is the 4:2 octave, reached without either partial ever being listened to. A tuner who lays the bearing fourth first has already passed the fourth–fifth check before making it.

Why the other order overshoots #

Laid fifth first, the chain goes up a fifth to E4, comparing A3’s third partial with E4’s second near 660 hertz, and then up a fourth from E4 to A4, comparing E4’s fourth partial with A4’s third near 1320 hertz. The two coincidences are an octave apart, and nothing cancels exactly. Whether the result overshoots depends on how the strings’ stiffness changes from one note to the next.

Two rates of change give no overshoot, and both are worth understanding.

With one stiffness everywhere, a link of ratio m:n counted to its printed rate is left wide by an amount proportional to $m^2 - n^2$, the difference of the squared partial numbers that meet in it. The fifth contributes $3^2 - 2^2 = 5$, the fourth $4^2 - 3^2 = 7$, and the octave they build is proportional to 12 in either order. The 4:2 octave type is proportional to $4^2 - 2^2 = 12$ as well. The sums telescope, and both orders land on the same octave.

At four times stiffer an octave up, the coefficient grows with the square of the frequency, and a partial’s rise then depends only on its own frequency. Two partials that meet are raised by exactly the same amount, so every coincidence is as it would be on a flexible string, every link beats at its printed rate when tempered, and the table builds an octave that is 2:1 and 4:2 at once. It is a stiffness the table cannot see.

Between those two rates neither cancellation works, and the overshoot rises to 0.34 cents at about two and a quarter times and falls away. The grand modelled here, at 2.87, sits just past the peak at 0.29. Measured pianos’ coefficients rise through the middle of the keyboard at a rate of the same order — well above one and well below four — which is the condition the overshoot needs, though they scatter from note to note around the trend.

The margin the check hears #

On whole-number partials the fourth and fifth beat alike, and the check has one correct answer. On the octave the table builds fifth first, the fourth beats faster.

On the grand the lead is 1.12 at F3, 1.18 at A3, 1.27 at C♯4 and 1.42 at F4 — rising through the octave because the coefficient rises and the overshoot with it. On a piano four times as stiff it is 1.63 at F3 and 2.13 at A3, and towards the top of the octave the table’s octave has moved so far past the 4:2 that the fifth from the middle note to the top is nearly pure: the ratio reaches 15 at F4 because the fifth is beating at about a tenth of a second.

So the instruction some teachers give — that in a well-set octave the fourth should beat a shade faster than the fifth — has an exact reading. It describes the octave a fifth-first table builds on a stiff string. On the grand at A3, “a shade” is 18 per cent; on a small upright it is more than double. A tuner who has learnt to expect that margin has learnt the overshoot, and will leave it in.

A difference of a sixth of a beat a second #

The comparison is made by ear between two slow beats, and the gap it has to detect is small.

With D4 counted up from A3 the fourth beats at its printed 0.99 and the fifth at 0.85. In three seconds the fourth’s swells gain 0.44 of a cycle on the fifth’s. Three beats at most and counted in the decay, or not at all found that a struck note offers only a short stretch in which a slow beat can be followed, and on single strings — the strip mute in, as when a temperament is set — the earlier essay found the tempered fifth delivering fewer than two countable beats from one strike. Inside so short a window a tuner is comparing about two swells of one beat with about two swells of the other and judging whether they arrive together. At 0.15 a second they drift apart by a quarter of a swell in the time available. It can be done — tuners do compare rates this close, by striking each interval several times — but it sits at the edge of what the comparison resolves, and nowhere near its middle.

The gap grows in proportion to the stiffness: 0.15 a second on the grand at A3, 0.29 at twice its stiffness, 0.53 at four times, where the octave is a full cent past its 4:2 type. So the check is easiest to hear on exactly the pianos where the table’s overshoot is largest. On a concert grand it is a fine judgement. On a small upright, where the fourth beats more than twice as fast as the fifth for the upper half of the bearing octave, a tuner using the check cannot miss it.

What the check corrects, and what it cannot #

The question the earlier essay left was whether the commonest check in the bearing corrects the table’s stretch or compounds it. With the octave types measured properly, the answer comes in three parts.

Laid fourth first, the check has nothing to correct. The table has already built the 4:2 octave, the fourth and fifth beat alike, and the tuner hears agreement. The check confirms an octave chosen by the order of the chain.

Laid fifth first, the check corrects the overshoot, and only the overshoot. A tuner who narrows A4 until the fourth and fifth beat alike takes 0.29 cents out of the octave at A3, and lands on the 4:2. That is a correction towards narrower, but it stops at the 4:2 and never approaches the 2:1. The stretch the table builds is mostly the 4:2 type’s own, and the check endorses it.

Either way, the octave a tuner checks this way is the 4:2. Not because the 4:2 was chosen for its sound, but because the fourth–fifth check is a 4:2 test by the arithmetic of which partials meet. Every tuning manual discusses the choice between octave types as though it were made when the octave is tuned. On this evidence it is made when the check is chosen. A tuner who wanted a 2:1 octave on a stiff piano would have to stop trusting the most familiar test in the bearing, since a 2:1 octave fails it — the fifth beating 0.82 a second faster than the fourth at A3 — and would fail it however the temperament inside were adjusted.

What the arithmetic rests on #

Every partial follows the stiff-string law, the k-th raised by the square root of one plus $Bk^2$, with B from the modelled grand’s lengths and gauges at each note. Real coefficients scatter by tens of per cent from note to note, so a real octave’s overshoot will be ragged rather than the smooth curve drawn here. The blindness of the check to its middle note does not depend on any of this: it holds for any partials whatever, so long as the middle note’s partial is the same one in both intervals.

The upright is the grand’s coefficients times four, kept from the earlier essay so the numbers can be compared. A real small piano’s coefficients do not grow up the keyboard at the grand’s rate, and the overshoot depends on that rate more than on the overall stiffness.

The tuner hits the printed rate exactly. Real tuning is iterative, with every interval adjusted against several checks, and no working tuner follows a chain once and stops. The figures say what the chain and the check would do if obeyed literally, which is the baseline a real tuning departs from.

What none of these figures can show #

Which octave sounds best. The figures show that the fourth–fifth check selects the 4:2 octave on any piano; they say nothing about whether a listener prefers it to the 2:1 or the 6:3 in music. The piano is tuned wrong on purpose and the octave that is not two to one took up that question, and the answer depends on the register and the music, which no beat count contains.

How well a tuner compares two rates. The figures put the gap at 0.15 a second on the grand at A3 and argue that it sits at the edge of what a short comparison resolves. The discrimination threshold for the rate of a slow amplitude modulation inside a decaying note has not been measured in this setting, and the figures cannot supply it.

Still open: the checks that test other octaves #

The fourth–fifth check tests the 4:2 octave because three partials meet at the fourth, third and second. Tuners use other checks built the same way, and each pairs different partials. A major third and a major tenth from the same lower note share the lower note’s fifth partial and compare the upper notes’ fourth and second; a minor third and a major sixth inside one octave meet at the sixth, fifth and third partials. By the arithmetic above, each is a test of some octave type, and the telescoping sum of squared partial numbers says which: for a chain of three major thirds, $3 \times (5^2 - 4^2) = 27$, which is the 6:3 type’s $6^2 - 3^2$. If the thirds counted to the table build a 6:3 octave while the fourths and fifths build a 4:2, the two most common families of checks in the bearing disagree by a whole octave type, and no temperament on a stiff piano can satisfy both — which would make the tuner’s final octave a compromise between checks, not a choice of type. Whether they disagree by that much, and how the progression of thirds that every tuner listens for moves on a stiff string, is the same arithmetic with three links instead of two.