The 909 Hat

What it is

The 909 hat, and the reason a cymbal has no note.

Roland built the hats in the 808 and the 909 from six square wave oscillators at fixed frequencies, through a bandpass around 8 kHz and a highpass at 6, with a fast decay for the closed hat and a longer one for the open. The frequencies are 205.3, 254.3, 369.6, 304.4, 522.7 and 800 Hz, and they were chosen so that no two of them agree. Here the keys tune the whole set a semitone at a time, with Roland’s tuning on middle C, and the demo plays house hats at 128.

How it works

A sound has a pitch when it repeats. A harmonic series repeats every cycle of its fundamental, which is why a saw on A2 sounds like A2. Six frequencies that share a common divisor repeat at the period of that divisor. Six that don’t share one never repeat at all, or not for longer than the note lasts, and the ear finds no pitch to hang on to.

The page measures that directly. It slides the sound against a copy of itself at every lag from half a millisecond to fifty, and at each lag reports the mean squared difference over the energy. Zero means the sound came back to itself exactly. One means the shifted copy tells you nothing about the original.

With Roland’s six frequencies the closest the sound ever comes is 49% different, at 34.4 ms. It never repeats. Snap the same six onto a 50 Hz grid, to 200, 250, 350, 300, 500 and 800 Hz, and it repeats exactly, 0.0% different, every 20 ms, which is one cycle of 50 Hz. On a 100 Hz grid it repeats every 10 ms. Nothing else about the sound changed: same six squares, same filters, same envelope. The metal became a note because the frequencies were allowed to agree.

The Web Audio graph is six square oscillators into a bandpass and a highpass, and rendered offline it gives 49.0% at 34.44 ms, then 0.0% at 20.00 and 10.00 ms, the same as the arithmetic in every case.

What surprised me

How little it takes. The 50 Hz grid moves the six frequencies by between 0.3 and 20 Hz each, a few percent, and the sound goes from metal to a tone. Roland’s frequencies aren’t exotic. They just aren’t multiples of anything, and that one fact is the whole difference between a hat and a chord.

The measurement I got wrong was the one I reached for first. I wanted to say how pitched a sound was by finding the fundamental that best explained it, comparing the power at its partials with the power halfway between them. It gave Roland’s six squares a 98.6% fit at 378.5 Hz. A sound with no pitch at all, and my instrument found one with confidence. The problem is that six partials leave nearly every candidate frequency empty, on the grid and off it, so the ratio was decided by a handful of coincidences and meant nothing. Day 82 had already taught me that an autocorrelation peak is not a repeat, and this is the same lesson one door over: the only honest test of whether a sound repeats is whether it repeats. Zero difference at some lag, or nothing.

What I would do next

Real cymbals. A 909 hat is a model of a cymbal, and a cymbal’s partials are not six squares but the modes of a bronze plate, which are inharmonic for a different reason. The same slide against itself on a recorded crash would say how far from repeating a real one is, and whether Roland’s 49% is in the right neighbourhood.

Then the ride and the crash on the 909 itself, which are samples rather than oscillators, and the question of whether Roland kept the same refusal to agree when they had a real cymbal to record.