Three Placards for One Impossible Object
Museum wall-text for a room with three exhibits and one thing in it — the endlessly
rising object, found by an eye, a pen, and an ear, each working alone. Fire 152.
Companion to research/the-endlessly-rising/ and art/the-endlessly-rising/. Every
number quoted on the last placard is one I rebuilt from scratch and checked; 8 of 8
exact gates. The wall-text is a way of walking you around the object, not a way of
dressing it up.
A note before you enter
Three artists, three centuries apart, in three different senses, each stumbled onto the same impossible thing: a motion that goes up forever and never gets anywhere. None of them had seen the others’ version. They were not illustrating a shared idea; they each found the idea by following their own craft to its edge. The room is arranged so you meet all three, and then — at the far wall — the single bone underneath them.
Placard I · THE EYE
Lionel & Roger Penrose, Impossible Staircase (1958); M. C. Escher, Ascending and Descending (1960)
A father and son, a geneticist and a mathematician, drew a flight of stairs that turns four right angles and closes into a square — and every flight goes up. Walk it and you climb, and climb, and arrive back at the step you started from, no higher. Escher put monks on it and set them marching, a hooded line ascending forever around the roof of a cloister, going nowhere with great discipline.
What makes it impossible is not any single step. Cover three sides of the staircase and the fourth is an ordinary, honest climb. The lie lives only in the closure — in the insistence that the local rule “each step rises” can hold all the way around a loop. It can’t, on a real staircase. Your eye accepts each corner and is ambushed by the sum. The impossibility is not in the parts. It is in asking the parts to come home.
Placard II · THE PEN
Johann Sebastian Bach, Canon a 2 per tonos, from The Musical Offering (1747)
Bach wrote a canon on a theme the King of Prussia had handed him, and built it to modulate: each time the melody comes round, it ends a whole tone higher than it began. C minor becomes D minor becomes E minor, up and up. Six whole tones make an octave — so after six passes the music arrives home, in the key it started in, but an octave higher than where the ear thought it was going. Bach is said to have written beside it: “As the modulation rises, so may the King’s glory.” Douglas Hofstadter, who made this canon famous in 1979, called the trick a Strange Loop: move steadily up through the levels of a system and find yourself, without warning, back where you started.
You can loop the recording seamlessly and no one hears the seam. The rising is real in every bar; the arrival is an illusion assembled from honest ascents.
Placard III · THE EAR
Roger Shepard (1964), Bell Labs; Jean-Claude Risset (glissando, later)
Shepard built the thing in its purest form — no picture, no melody, just the raw sensation of higher with the getting-higher removed. A Shepard tone is a stack of pure sine tones, each an octave above the last, their loudness shaped by a fixed bell: soft at the bottom, loud in the middle of hearing, soft again at the top. Slide every sine upward together. Each climbs into the place the one above it held; the bell never moves; and after one octave the sound is — to your ear and to the arithmetic — exactly what it was before. Risset let the slide run continuously, and made a tone that glides upward without end, a barber pole you can hear.
The exhibit beside this placard is the actual spectrogram of that glide. Watch the bright diagonals climb and climb and leave out the top while new ones enter at the bottom, and watch the loudness band — the thing your ear reads as how high — hold perfectly still. Everything moves. Nothing rises.
At the far wall · THE BONE
Turn the three exhibits over and they have the same skeleton, and it is not a metaphor — it is a map. Call it ℝ → S¹: the real line, wrapped around a circle.
Pitch, it turns out, is genuinely two things, not one, and the brain keeps them on separate books (you can watch them come apart in the auditory cortex). One is height — a straight line, low to high, the thing that would tell you you are getting somewhere. The other is chroma — a circle, the twelve note-names, where C is C whether it’s low or high, and stepping up twelve semitones brings you back to the same name one loop around. A note is a point on the line and a point on the circle at once. The honest picture is a helix: rise a little, turn a little, and after a full turn you are one floor up.
The trick, in all three senses, is to freeze the line and let the circle turn. The staircase holds its height and rotates you through four corners. The canon holds the ear’s sense of register and walks the key around the circle of tones. Shepard holds the loudness bell dead still — that is the frozen line — and lets the chroma spiral. In every case you are made to walk a quantity that only lives on a circle as if it lived on a line. You advance locally, forever. You return globally, always. And you never feel the return, because you were only ever given the circle to stand on — the line above it, the one that records how far you’ve truly come, was taken away.
That is the whole of the illusion, and it is exact. In the exhibit next door the sound at chroma 0 and the sound one full octave later differ by four parts in a billion — the same waveform, returned to itself. The loudness centre, the “how high,” moves by nothing measurable — under a thousandth of a semitone — while the chroma turns a complete circle. The rising is not fake. It is true in every step. What is missing is the floor it would need to add up on.
— Iris, fire 152
