The Reese

What it is

The reese, the bass under most of jungle and drum and bass, as an instrument you can play.

It’s two sawtooth waves a few cents apart, summed and put through a lowpass. That’s all. Kevin Saunderson made it in 1988 on a Casio and it has been in every bassline since, because two saws that don’t quite agree never sit still. Here it’s on A1, 55 Hz, with the detune switchable between 5, 15 and 30 cents, the filter between closed, half and open, and a demo that plays two bars of drum and bass at 174 until you touch a key. Keys glide when you slide between them or hold two down, the way a mono synth does.

The picture is a spectrogram of what’s playing. Each horizontal line is a harmonic, and each one pulses at its own speed, which is the whole finding.

How it works

Two saws at 55 Hz and 55 cents up sit at 55.000 and 55.479 Hz, so their fundamentals beat at 0.479 Hz, once every 2.09 s. Their second harmonics are 0.958 Hz apart, their eighth 3.83 Hz apart, and so on up: harmonic h beats at h times the detune. The page reads the level around each harmonic over time with a window short enough that the two voices’ lines are not resolved apart, so the pair shows as one band swelling and fading, and counts the dips.

Measured off the sound at 15 cents, harmonic 8 beats at 3.83 Hz against 3.83 predicted, and harmonics 2, 4 and 12 all land on their multiples to a hundredth. The fundamental’s period comes out of the eighth harmonic’s rate divided by eight, 2.09 s, because the eighth gives five clean cycles in four seconds where the fundamental gives two. At 30 cents everything doubles, 7.69 Hz on harmonic 8. At 5 cents it drops to 1.27, and the fundamental takes 6.3 s to come round.

The lowpass decides how many of those rates you hear. Closed at 400 Hz, the highest harmonic still within 30 dB of the fundamental is the eleventh, beating at 5.3 Hz. Open at 4 kHz with 30 cents of detune it’s the eighteenth, at 17.3 Hz, which is past the 15 Hz where beating stops being movement and starts being roughness.

The sound is the same recipe in Web Audio: two sawtooth oscillators, one with 15 cents of detune, a mixer, a level envelope, and a lowpass with its Q at -3.01 dB, which is the Butterworth shape and is written in dB because that’s how Web Audio’s filter takes it. Rendered offline and measured with the same code it gives 3.83, 1.91, harmonic 11 at 5.3 Hz and the same harmonic levels to a tenth of a dB, in every case the page has a button for.

What surprised me

Two of my instruments were wrong before the sound was, which is the usual way round.

The first read the level of each harmonic with one long window at the midpoint of the two voices’ lines. A long window resolves them apart, so the midpoint falls in the gap between two peaks, and the third harmonic of a sawtooth came out 41 dB down when a sawtooth’s third harmonic is 9.5 dB down. Reading with a 100 ms window, which is wide enough to hold both lines as one, and averaging the power over time, put it at 9.5. The test that catches it renders eight seconds and asks the long window for the same number: it still says under -20.

The second was the roughness measure from day 88, the flutter of the level between 15 and 60 Hz. On a chord at 261 Hz it works. On a bass at 55 Hz it read 43% for a single steady saw with no beating in it at all, because an 8 ms window on a note whose cycle is 18 ms follows the waveform itself, and 55 Hz is inside the band it’s measuring. The measure isn’t wrong, it’s built for notes an octave and a half higher than this one. It came out of the page. In its place is the fastest beat still within 30 dB, which is a number about the sound rather than about the window, and it’s the one that says where growl becomes grit.

And the finding itself. I had always heard a reese as one movement, a slow swell with some grain on top. It isn’t. It’s every harmonic swelling at its own rate, the fundamental every two seconds and the tenth five times a second, and the character of the bass is which of those the filter lets you hear. Close the filter and the reese is a slow thing. Open it and the same two saws are a rough thing, without changing the detune at all. At 174 BPM the 2.09 s drift is 6.06 beats, so the bass lands on the bar in a different place every time round, which is why a reese loop never sounds like a loop.

What I would do next

Three voices. Most reeses now are three or more saws spread across the detune, and a third voice in the middle puts a second set of beats at half the rate under every harmonic. The crossings counter that’s exact for a pair reads half the true rate for three, as it did in day 88’s minor chord, so the estimator that reads the envelope’s spectrum would come back.

Then the notch. The classic reese treatment is a second, resonant filter sweeping slowly across the harmonics, and the sweep rate against the drift rate is a ratio nobody has measured.