Faster Forever
A click track that speeds up without limit, built out of a nine-second loop that repeats exactly.
What it is
Headphones or speakers, and give it twenty seconds. Five click tracks at once, an octave of tempo apart, every one of them continuously accelerating. Each fades in when it’s too slow to notice and fades out when it’s too fast to follow.
The top strip shows the voices drifting rightward along a tempo axis. The lower panel is every click of the last six seconds.
It never stops speeding up. It also never actually gets faster.
How it works
Voice L has tempo 0.9 · 2^(L + t/9) clicks per second, so every voice doubles every
nine seconds. Its gain is a raised cosine over the five octaves of the stack, silent at
both ends, and crucially it’s periodic in log tempo with a period of one octave spacing.
That periodicity is what makes it close. After nine seconds, voice L sits at exactly the tempo and exactly the gain voice L−1 had. The whole ensemble has slid down one notch and become itself.
What surprised me
I expected the loop to be approximate. It’s exact, and exact in a way I can print:
| t (seconds) | tempos sounding (clicks/sec) | total gain |
|---|---|---|
| 0 | 1.80 3.60 7.20 14.40 | 2.5000 |
| 4.5 | 1.27 2.55 5.09 10.18 20.36 | 2.5000 |
| 9 | 1.80 3.60 7.20 14.40 | 2.5000 |
| 90 | 1.80 3.60 7.20 14.40 | 2.5000 |
| 900 | 1.80 3.60 7.20 14.40 | 2.5000 |
Not close to identical. Identical. Meanwhile any individual voice, over that ninety-second span, has gone from 0.9 clicks a second to 921.6. That’s 1024 times faster.
The total gain is exactly 2.5 at every instant, which turns out to be forced. Five raised cosines sampled an octave apart are five fifth-roots of unity and their cosines sum to zero. The ensemble’s centre of mass wobbles by 0.013 octaves, a 1% tempo variation, and comes back. Forever.
So there are two completely defensible descriptions of this signal. Every component is accelerating without bound. The signal is a nine-second loop. Both are true, and choosing between them is choosing what counts as a component.
What actually unsettled me is the listening. Knowing all of the above, having written the test that proves the loop, I still hear it accelerate. The knowledge doesn’t touch the percept at all.
Hearing evidently tracks individual voices and doesn’t compute the ensemble. Once a voice has your attention it takes it up and out and hands you to the next one without ever admitting the handover.
What I would do next
Find where the illusion breaks. Widen the octave spacing until you can hear the handover, then see whether that spacing matches anything known about auditory stream segregation.