FIELD GUIDE

RUNGS: WHERE THE WATER ANSWERS

The cell can only ring in its own shapes, its modes, each at its own pitch. Play a note close to one of those pitches and the water swings hard; play between them and it barely stirs. We call each mode's pitch a rung of the cell's ladder.

A rung is narrow: in this water about a quarter of a semitone either side. That is why brightness means resonance: on a rung the surface tilts steeply enough for the lamps to catch it, and the cell lights up.

Each mode is labelled (m, n): m petals around, n rings out. The Atlas lists the rungs your needles can reach; the Sweep plays a fork through all of them and draws the peaks.

THE CENTRE DRIVE

Drive the whole wall at once, through the meniscus all the way round, and the push is the same in every direction. Only the round modes can answer: (0, n), concentric rings and nothing else.

So a plate of rings is read by counting them: n rings is the (0, n) rung. The round rungs climb by about a minor third each, a little less as they go up, one ring more each time.

LOUD AND SOFT: WHAT A LAMP SEES

A camera looking straight down sees a lamp reflected wherever the water tilts toward it by half the lamp's angle off the axis: a lamp at 22° lights slopes of tan 11°, about 1 in 5; a lamp at 44° needs 1 in 2.5.

Over a cycle the slope at each point swings up to its peak and back. A lamp shows the points whose peak just reaches its slope, so its bright lines are contours of how steeply the water moves. Play louder and the water tilts further: a steep lamp, dark for a quiet note, lights up for a loud one, and every contour moves outward from the peaks.

THE SIZE OF THE CELL

Ripples this small are held by surface tension, and their pitch climbs steeply as they shorten: a capillary wave's frequency grows as the wavelength to the power −3/2. Shrink the cell and every mode shortens with it, so the whole ladder moves up, roughly as the cell's radius to the power −3/2.

A 24 mm cell rings in the cello's and the low flute's range; a 14 mm cell in the flute's.

STROBES: SAMPLING MADE VISIBLE

A strobe lamp flashes instead of shining, f_s times a second, and the camera sees the water only at its flashes. A pattern that repeats a whole number of times between flashes looks frozen; one that repeats a little more or less creeps forward or back. A partial at f appears to move at f − f_s · round(f / f_s): the strobe tuner's trick.

So a strobe is a comb on the water's spectrum: it freezes every harmonic that is a multiple of its rate and lets the rest blur. Two strobes in two colours freeze two different sets. And a turning pattern, too fast to see, shown by a strobe a hair off its pitch, turns slowly: one way if the strobe is a little slow, the other way if it is a little fast.

The screen never flashes: a strobe only chooses which instants the picture is made of.

RINGS OF N NEEDLES

Needles spaced evenly round the rim, all moving together, push the water in a pattern with N-fold symmetry. A mode with m petals is pushed only if its petals line up with every needle at once, which happens when m is a multiple of N: a ring of 4 wakes the 0-, 4-, 8-, 12-fold modes and no others.

More needles, fewer modes in reach: a ring of 8 skips the 4-fold rungs a ring of 4 also wakes, so its mandalas come out clean. Read a plate's petals, and you know how many needles it wants.

TURNING THE RING

A standing mode's petals sit where its needles push. Turn the ring of needles round the rim and the mandala turns with it, slot by slot. Most plates don't care which way up they are, but a plate judged as it stands must be matched turn for turn.

WINDING: PATTERNS THAT TURN

Wind the ring: let each needle push a little later than the one before, q whole cycles all the way round. The push now travels round the rim, and the water's pattern travels with it, turning at the note's pitch divided by its number of petals.

A pattern turning hundreds of times a second is a blur to the eye and to any lamp: on its own its petals would smear into rings. But a wound ring drives waves turning both ways at once (orders q and N − q), and their mix leaves a twist: petals that all lean the same way, like a spiral. A plate with a handedness is a winding.

ORDERS MODULO N

A ring of N needles wound q steps drives exactly the modes whose order m satisfies m ≡ ±q (mod N): q, N − q, N + q, 2N − q … So four needles wound 2 steps (each needle half a cycle behind the last) push a 10-fold pattern, because 10 = 2 + 2 × 4. Few needles can wake many petals, if they are wound right.

BEATS AND BREATHING

Two flutes a hair apart don't sound like two notes but one that swells and fades: a beat, as often per second as their pitches differ. The water does the same thing. Their two patterns pass in and out of step, so the mandala breathes, swelling and dissolving in time with what you hear.

Bring them to exactly the same pitch, a just unison, and the beat stops: the mandala holds still. Equal temperament splits the difference between intervals, so a tempered note next to a just one always beats a little.

TIMBRE: PARTIALS YOU CAN SEE

A flute, a clarinet and a pipe playing the same note sound different because each is a different stack of harmonics, f, 2f, 3f… at different strengths. The water hears each harmonic separately: each one lands on the cell's ladder on its own and draws its own pattern where it finds a rung. So timbre is visible.

A clarinet's low notes have almost no even harmonics (its bore is a cylinder closed at the reed); overblown a twelfth higher, into its clarion register, they come back. An open organ pipe sounds every harmonic, a stopped pipe only the odd ones, a tuning fork none. A singing bowl's partials are not harmonics at all, and each comes as two close pitches, because no bowl is perfectly round: the bowl beats with itself, and under a strobe its partials never stand still.

SUPERPOSITION AND CANCELLING

Waves add. Two rings on two rungs draw both their patterns at once, one over the other. Two rings on the same note add where they push together and cancel where they push against each other.

Set two rings of four between each other, 45° apart: their four-fold pushes are exactly opposite and cancel, while their eight-fold pushes agree and double. Two rings of four, interleaved, are a ring of eight.

COINCIDENT PARTIALS

A note is not one pitch but a stack of harmonics: f, 2f, 3f… In a just fifth (3 : 2) the lower note's third harmonic and the upper note's second land on exactly the same pitch. Temper the fifth and they miss by a whisker, and that pair beats: rate = |3 f_low − 2 f_high|.

A tempered fifth is the same fraction narrow wherever it is played, so its beat is proportional to its pitch. A3 and E4 beat 0.74 times a second, G♯3 and D♯4 0.70, F3 and C4 0.59. Piano tuners count these beats to tune.

THE THIRD TONE

Play two loud notes and listen below them: a third, quieter pitch hums at the difference of their frequencies, f_high − f_low, and another at 2 f_low − f_high. The violinist Giuseppe Tartini heard it in 1714 and tuned his double stops by it. Organ builders use it too: two pipes a fifth apart sound a bass an octave below the lower, with no pipe that long.

Water is not perfectly linear either. Where two steep waves cross, they push the surface at the difference of their pitches. Where that lands on one of the cell's rungs, a slow, broad swell rises that neither note could draw alone. It grows with the square of the loudness, so it shows only at forte. In a just chord the third tones fall on one pitch and hold still. Temper the chord and they scatter and beat.