The chirp a staircase sings back

The gap a repeat opens found that a structure repeating at a fixed spacing reflects strongly the one wavelength whose half fits the spacing, because the small reflections from every repeat come back in step. The sandbars that reflect the sea found the same thing for ocean waves meeting a row of sandbars, and the mirror that works from every direction asked what happens when the wave arrives at an angle: the spacing that matters is then the spacing measured along the wave’s direction, and the reflected colour shifts. In all of these the structure is seen from far away, so every repeat is seen at the same angle.

A staircase seen from its foot is different, and the difference can be heard. At the base of El Castillo, the stepped pyramid at the centre of Chichén Itzá in Yucatán, a hand clap returns from the great stair not as a clap but as a chirp — a falling whistle lasting a fraction of a second. The acoustical consultant David Lubman drew attention to it in 1998, and suggested that it resembled the call of the quetzal, a bird sacred to the Maya, and that the builders might have intended it. Whether they did is a question for archaeologists. Why the stair chirps at all is a question for the physics of periodic structures, and the answer is that each step is seen at a different angle.

Ninety-one copies of one clap #

El Castillo has four stairways, one up each face, of ninety-one steps each; with the platform at the top they make 365, which is generally taken as deliberate. The steps are roughly twenty-six centimetres deep and twenty-six high, so the stair climbs at about forty-five degrees through some twenty-four metres. A listener standing ten metres from its foot claps once.

Sound spreads from the clap and reaches each step in turn, and each step’s vertical face, its riser, sends some of it back. The first reflection comes from the bottom step, a little over ten metres away; the last from the top step, about forty metres away. What the listener hears is the sum of ninety-one copies of the clap, each delayed by the round trip to its own step and weakened by the distance.

What matters for the echo’s character is not the distance to each step but how much farther each step is than the one below. Near the bottom, the steps are below the listener’s ear and are seen looking slightly down, nearly horizontally; the next step is about one tread farther away — slightly less, in fact, because looking down the next step’s rise brings it a little closer. Near the top, the line of sight runs almost up the slope of the stair, and the next step is nearly a whole step’s diagonal farther away, $\sqrt{t^2 + h^2}$ for tread $t$ and rise $h$. The magnified inset shows the rule that connects the two: the extra path from one step to the next is the step’s projection onto the line of sight, $t\cos\alpha + h\sin\alpha$, for a step seen at elevation $\alpha$. For a listener ten metres away it grows from about 22 centimetres at the bottom to 36 at the top.

From clicks to a pitch #

Every extra centimetre of one-way path is two centimetres of round trip, so successive reflections arrive separated by twice the extra path divided by the speed of sound.

The echo is a train of ninety-one clicks lasting from about 60 to 230 milliseconds after the clap. At the start they are 1.3 milliseconds apart and at the end 2.1, and they fade as they come, since the farther steps are farther away and seen at a slant. A regular train of clicks faster than about thirty a second is not heard as clicks. The ear fuses it into a tone at the repetition rate — a “repetition pitch” — and that is what makes the echo musical: a click every 1.3 milliseconds is a tone of about 770 hertz, and a click every 2.1 milliseconds is about 480 hertz.

The idea is old. In 1693 Christiaan Huygens, visiting the gardens of the château of Chantilly, noticed that the splashing of a fountain, reflected from a long flight of stone steps, came back as a musical note, and he explained it in exactly these terms: each step returns the sound a little later than the one before it, and the regular succession of returns is a tone whose pitch is set by the depth of the steps. What Huygens’s fountain lacked was a single sharp sound to start the echo; a hand clap supplies one, and with it the change in pitch through the echo becomes audible. The clap is not the only sound that works: a single footstep, a shout or the crack of a stick produces the same chirp, since any short sound is copied by the steps, and the echo’s pitch depends on the steps rather than on the sound. A long sound, by contrast, overlaps its own copies and the chirp smears into a coloured roar.

The figure checks that the ear’s interpretation is the physicist’s. The dots are the strongest frequency in successive slices of the echo, synthesised by adding ninety-one copies of a short click at their computed times and strengths; the lines are the arrival rate and twice it. The loudest frequency follows the arrival rate down from about 630 hertz in the first measurable slice to 480 at the end, and its second harmonic follows twice the rate an octave above. The steepest fall is at the start, where the steps below and near eye level are passing through a wide range of elevations in a short time; later the steps are all seen at nearly the same steep angle and the pitch levels off. To a listener the sound is a quick downward sweep, settling — which is the shape of a bird’s chirp, and of the recordings made at the pyramid.

How sharp the note can be #

A tone made of clicks is only as sharp as the clicks are regular, and the stair’s clicks are regular only over a stretch. That makes the echo’s pitch a question about resolving power, which what a thousand slits buy that two cannot answers for a grating: $N$ equally spaced lines define a wavelength to about one part in $N$, because a wavelength differing by more than that puts the first and last line’s contributions out of step. Ninety-one evenly spaced reflections would define a pitch to about one per cent, a sixth of a semitone.

The stair’s reflections are not evenly spaced over all ninety-one. The interval changes by three or four per cent from one step to the next at the bottom, by a tenth of a per cent two-thirds of the way up and by less near the top, and the ear can only add up, coherently, the run of steps over which the interval stays within its own resolution. Near the bottom that run is short — a handful of steps — and the pitch is correspondingly vague, a rushing sweep rather than a note. Higher up dozens of steps share nearly the same interval and the tone sharpens, which is why the echo seems to settle on a pitch at its end. The sharpening is the same accumulation that the pile of glass that stops every colour shows for weak reflections from many surfaces: each step returns only a little of the clap, and only in combination, in step, do they make something the ear can name.

The direct clap matters too. A listener hears the clap itself and then the echo, and when the two overlap — as they do for a sustained sound like Huygens’s splashing fountain or a stream of footsteps — the sum of a sound and its delayed copies reinforces the frequencies at multiples of the repetition rate and cancels those between, the comb filtering that gives the sound of a voice beside a wall its faint metallic colour. With a single sharp clap the echo is separated from the direct sound by sixty milliseconds of silence at ten metres, and the chirp is heard on its own.

A grating seen at a changing angle #

The same arithmetic can be read as Bragg reflection. A wave reflected from a periodic structure is strong at the wavelength for which the reflections from successive repeats arrive back in step, which means a round trip of exactly one wavelength, or two, or more, between neighbouring repeats. For a stair seen at elevation $\alpha$, the condition for the $m$-th order is $2(t\cos\alpha + h\sin\alpha) = m\lambda$. A click train’s repetition rate and the frequency satisfying Bragg’s condition are the same number, because a regular train of reflections is exactly what a wave reflected in step from a periodic structure is.

Plotted against elevation, the reflected pitch has a minimum where the line of sight runs along the slope — for a forty-five-degree stair, at forty-five degrees, where the projection of each step on the line of sight is its whole diagonal. Either side of that the projection is shorter and the pitch higher. A step seen dead level returns $c/2t$, about 660 hertz for these steps; one seen looking down returns more. The coloured bars show the span of elevations the whole stair covers from three distances. From five metres the bottom step is seen sixteen degrees below the horizontal and the top one at thirty-nine above, and the echo sweeps across most of the curve.

This is what separates the staircase from the layered mirrors and the sandbars of the essays it follows. They were seen from far enough away that every repeat was at the same angle, so they reflected one wavelength and its multiples. A staircase seen from its foot is a grating each of whose lines is seen at a different angle, and it reflects a different wavelength from each part, arriving at a different time: a frequency sweep encoded in space and played out in time. The closest analogue elsewhere is the rainbow a passing electron leaves on a grating, where each direction of emission sees the grating’s period differently and so receives a different wavelength; there the spread is in angle, here in time.

Where to stand #

The span of elevations, and with it the width of the sweep, depends on how far away the listener stands.

Close to the stair, the bottom steps are below the ear and the top ones high above it, and the echo sweeps from above the level-sight pitch to almost the along-the-slope one: 923 down to 470 hertz from five metres, 767 to 476 from ten. Far from the stair, every step is seen at nearly the same small elevation, the clicks come back almost evenly spaced, and the chirp flattens into a steady tone near $c/2t$ — 669 to 569 hertz from a hundred metres, and, more importantly, much fainter, since the reflections weaken with the square of the distance. The chirp is a near-field effect: it exists because the listener is close enough to the stair for the angles to differ. The best place to hear it, by this geometry and by the accounts of visitors, is a few tens of metres out from the foot of the stair, where the sweep is wide and the reflections are still loud.

The echo’s details depend on more than the geometry here. The steps of the real pyramid are worn and irregular, their faces are not perfect mirrors, and a riser 26 centimetres high is comparable to the wavelength it returns, so it does not reflect sound as a mirror reflects light but diffracts it. Nico Declercq and colleagues at Ghent analysed the stair in 2004 with the full theory of diffraction by a periodic surface and found the chirp’s frequencies close to those of the simple repetition picture, with the strength of each frequency depending on how the diffracted orders of a periodic surface enter and leave. They also accounted for a second effect reported at the pyramid: footsteps of people climbing the stair, heard from its foot, sound like raindrops falling into water, an effect produced by the same periodic structure filtering the sound of a footstep high on the stair.

Chirps of different kinds #

A falling whistle can arise in several ways, and comparing them sharpens what is particular to this one. The whistle that arrives sorted describes the whistlers of the ionosphere, falling tones produced when the radio crackle of a lightning stroke travels along the Earth’s magnetic field through a plasma whose wave speed depends on frequency: the high frequencies arrive first, the low ones later, and a click is stretched into a sweep by dispersion. A staircase’s medium, air, is not dispersive at all. Every frequency travels at the same speed; the sweep comes entirely from geometry, from the spacing of the reflectors as the listener sees them. The two chirps sound alike and their causes have nothing in common.

The opposite case is a clap in a long corridor between parallel walls. The reflections arrive at intervals that grow steadily as the paths become more oblique — the same kind of slowing train — and the result is a “flutter echo” whose pitch also falls. There the reflectors are the two walls repeatedly and the periodicity is in the bounces; on the stair it is in the structure. Both are periodic arrangements seen at changing angles, and both turn a click into a descending tone. The picket fence that seems to sing when a car drives past it is the same physics with the angle changing in time rather than along the structure: the car’s own noise reflects from each post, and as the car moves the posts it hears are at a different angle, so the pitch of the reflection changes as it passes.

Gratings that reflect different wavelengths at different points along their length have been deliberately engineered for light, where they are called chirped gratings: the period of the layers, or of the ruled lines, changes along the structure, so each wavelength is reflected from a different depth and returns at a different delay. Such gratings compress and stretch laser pulses, and correct the spreading of pulses in optical fibre — the reverse of the effect the wavelength a fibre does not smear describes. The staircase is a chirped grating made by perspective rather than by construction: its period is fixed, and the chirp is in how it is seen.

A riser is not a point #

The model treats each riser as a point reflector at its centre, returning a copy of the clap whose strength falls as the inverse square of its distance and with the slant at which it is seen. That captures the timing of the reflections, which sets the pitch, and roughly their relative strength, but not the frequency dependence of each reflection: a real riser 26 centimetres high reflects wavelengths longer than itself poorly and diffracts shorter ones in several directions, so the actual echo’s loudness at each frequency differs from the synthesised one. The treads, the stone balustrades either side of the stair, the ground between the listener and the pyramid and the other faces of the building also return sound, and none of them is included. The step dimensions are approximate and uniform; the real steps vary by a few centimetres, which smears the higher harmonics while leaving the fundamental’s sweep intact. The synthesised echo uses an idealised click of a fraction of a millisecond, whereas a hand clap has a spectrum peaked around one to two kilohertz, which affects which harmonic is loudest but not where any of them lies.

Still open: what the builders heard #

The physics of the chirp is settled; its history is not. The steps of El Castillo were built to a religious and calendrical design — the 365 steps, the play of shadow on the northern balustrade at the equinoxes that makes a serpent of light descend the stair — and an acoustic effect that any regular stair produces could be a by-product of that design rather than a goal. Arguments that it was intended have pointed to other Mesoamerican sites with stepped structures and to the importance of the quetzal in Maya iconography; arguments against point out that the chirp’s resemblance to the bird’s call is a matter of opinion and that the effect is generic. Acoustic archaeology, which asks how ancient sites sounded and whether their builders cared, has measured the stair repeatedly since 1998 without settling the question, and probably cannot without evidence from the builders themselves.

The habit worth carrying away is to ask at what angle each repeat of a structure is seen. A periodic structure reflects the wavelength whose half equals its period projected on the line of sight, and a long one seen from close by is seen at a different angle along its length — so a stair of 26-centimetre steps, clapped at from ten metres, returns its ninety-one copies at intervals stretching from 1.3 to 2.1 milliseconds, and the ear hears a chirp falling from about 770 to 480 hertz. Nothing was tuned; the pitch is perspective, made audible.