Noise and the Hat
Noise has a colour, and the colour is a number: white is flat, pink falls 3 dB an octave, brown falls 6, measured. A hi-hat is what noise is for, and the 808's is six square waves chosen to share no harmonic. How little pitch that leaves is a number too, 45% for the six and 16% for the finished hat.
What it is
Three noises and a hi-hat.
White, pink and brown noise, which differ by how fast their spectrum falls with frequency, and the page measures that slope off the sound and reports it in dB per octave. Then the TR-808 hi-hat, which isn’t noise, or not only. It’s six square waves at frequencies Roland picked so that no two of them share a harmonic anywhere useful, and the reason that sounds metallic rather than like a chord is that nothing in it repeats. No period, no pitch.
How little pitch is also a number. The strip in the middle of the page is the sound’s autocorrelation, how much it looks like itself a little later, and its tallest peak is the pitch strength. A chord of squares reads 92%. The 808’s six squares read 45%. Through a 7 kHz highpass with noise mixed in and an 80 ms decay, which is the hat, 16%. White noise, 3%.
How it works
The colours are measured from an averaged spectrum: two seconds of the noise, cut into 2048 sample windows, the power at forty log spaced frequencies from 100 Hz to 12 kHz summed over all of them, and a straight line fitted to the result in dB against octaves. White comes out at -0.01 dB an octave, pink at -3.01, brown at -5.97, against 0, -3 and -6.
Pitch strength is the tallest normalised autocorrelation peak at lags between 50 Hz and 2 kHz, over 8192 samples for a steady sound and 2048 for the hat. A single tone gives 1.000. Six squares at multiples of 200 Hz give 1.002, since they share a period. A three note major chord gives 0.95 and the six note voicing on the page 0.92, because the intervals nearly share one.
The sound on the page is the real thing: noise from rendered buffers, the tone sets as six oscillators with a band limited square wave each, the hat as those six plus a noise burst through a highpass into a fast decay. Rendered offline and measured the same way, the real chord reads 0.915 against the maths’ 0.915, the 808 set 0.447 against 0.447, a random set 0.395 against 0.395, and the hat 0.161 against 0.161.
What surprised me
That Roland’s six numbers are good and not magic. The 808 hat’s oscillators sit at 205.3, 369.6, 304.4, 522.7, 540 and 800 Hz, and the folklore is that they were chosen to be as inharmonic as possible. Measured, they leave 45% of a pitch, and the autocorrelation says where: a ghost fundamental at 268 Hz, which three of the six sit near, at 1.95, 2.01 and 2.98 times it. The set is nearly a chord on a note nobody played.
Then I tried twelve random sets of six between 200 and 800 Hz. They average 60% and range from 39% to 75%, so Roland’s 45% beats the average and only one of the twelve beat Roland. Good, then. Not the least pitched set there is, and eleven of twelve random ones are worse.
I also tried the set the folklore would design, six frequencies each 1.618 times the last, on the theory that the golden ratio is as far from any simple ratio as a number gets. It reads 64%, worse than the 808 and worse than the random average. Part seven found the same thing from the other side: consecutive powers of the golden ratio have rational near misses, 1.618 squared is 2.618 and cubed is 4.236, close enough to 5 over 2 and 17 over 4 to line harmonics up. The golden ratio is a bad way to make things not repeat, twice now.
What actually takes the pitch out of a hat is the two things that come after the oscillators. The highpass at 7 kHz brings the six squares from 58% to 16% with the noise in, and the noise does most of it: without noise the highpassed set still reads 58%, with the noise mixed in at the level the page uses it reads 16%, with twice that 10%. The metallic set is the character, the noise is what makes it a hat.
The instrument failure was where I measured. Pitch strength takes a window from the middle of its input, which is right for a steady tone and wrong for a hat: the middle of a half second render of an 80 ms decay is the tail, where the sound is 4% of what it was, and the number I first reported, 19%, was measured on what was left. The real oscillators disagreed with it, 0.150 against 0.190, and a disagreement between the sound and the maths has meant a mistake in the instrument every time in this series. Measured at the onset, where the hat is, both say 0.161.
One more, about the colours. One window of white noise reads -0.63 dB an octave, which would pass for pink if you weren’t paying attention. Four windows read -0.39, sixteen read -0.10, sixty four read 0.02. A spectrum of noise is itself noise until it has been averaged, and the page says how many windows it averaged next to every slope it reports.
What I would do next
Part twelve is distortion, and what separates warm from harsh is which harmonics the shape of the curve adds: odd, even or both.
The 808 set could be improved, and the page nearly says how. A search over six frequencies for the lowest pitch strength would find a set with no ghost, and whether that sounds better or just different is a listening question the numbers can’t settle.
The other measurement is the 909 hat against the 808. The 909’s is a sample of a real cymbal, and a real cymbal’s partials are inharmonic by physics rather than by choice. Its pitch strength against the 808’s would say whether Roland’s six oscillators got close to the thing they were standing in for.