The parts list

I have started on a pair of loudspeakers. This is where they begin: with a parts list, one rule, and a good deal of physics that decides everything else before a single panel is cut.

Before these

I had Dynaudio Xeo 30s, and I liked them. They are good loudspeakers, and that is the bar to clear: deeper bass, without a separate subwoofer; dynamics where nothing audibly compresses when the music gets loud; and enough resolution that nothing in a recording hides. The measure is not a number on a data sheet. It is music good enough to move you.

The rule

No subwoofer, and no further active drivers. Passive radiators are allowed. If the bass has to go lower, it has to come from the box, not from another box.

The reason is not purity. A subwoofer is a second source in the room, in a different place from the speakers, handing over at a frequency where the ear can still sometimes tell where a sound comes from. Getting it to disappear is its own project, and it rarely does completely. A pair of speakers that goes deep enough on its own has one less thing to hide.

The parts

Three drivers, three amplifier channels. To see why a loudspeaker needs three at all, it helps to look at what they are pushing.

Sound is air being pushed

A loudspeaker cone moves the air in front of it back and forth. The air does not travel to your ear; each bit of it only swings to and fro around where it is. But it pushes its neighbour, which pushes the next, and the push travels outwards at the speed of sound, 343 metres a second. Where the air is crowded together the pressure is a little higher, where it is spread apart a little lower, and those bands of high and low pressure are the sound.

How far apart the bands are is the wavelength: the speed of sound divided by the frequency. At 40 Hz, the lowest string of a bass guitar, it is 8.6 metres. At 400 Hz, around the middle of a voice, 86 centimetres. At 4 kHz, where much of the detail of a consonant or a cymbal lives, 8.6 centimetres. Three decades of frequency, three decades of size: from larger than a room to smaller than a hand.

Why three

No single cone can do all of that well. To move enough air at 40 Hz a cone has to be large and travel far, which makes it heavy, and a heavy cone cannot follow a signal that changes four thousand times a second. It also starts to beam: once the wavelength is about as small as the cone, the sound no longer spreads evenly into the room but narrows into a beam straight ahead, and the sound off to the side, which is most of what reaches you in a room after the reflections, goes dull.

So the range is split. A woofer for the bass, large and with a long throw. A small midrange for the middle, where voices and most instruments live, which is where the ear is least forgiving. A tweeter, a dome a few centimetres across, for the top. Each only has to do what it is good at, and each is still small compared with the wavelengths it plays, so the sound leaves it evenly in all directions.

Where they hand over

The handovers are the crossover. In most loudspeakers it is a network of coils, capacitors and resistors inside the box, between one amplifier and all three drivers. Here it is not. The Hypex FA253 has three amplifier channels and a DSP, a small processor, in front of them. The signal is split into three bands digitally, each band gets its own amplifier and its own driver, and nothing passive sits between an amplifier and its cone.

That has consequences that are hard to get with a passive network: the slopes can be as steep as they need to be, each driver can be corrected on its own, a driver sitting further back can be delayed so that all three arrive together, and changing the design means loading a new file, not soldering.

One common choice of filter is the Linkwitz–Riley of fourth order: two ordinary second-order filters in a row. At the crossover frequency each side is down 6 dB, and the two are exactly in phase there, so their sum is flat, not a bump and not a dip. Away from the crossover each side falls towards 24 dB per octave: a couple of octaves past it, the driver that should be quiet is down by a factor of more than two hundred in amplitude.

The curves in the figure are computed, not drawn. A second-order Butterworth filter, as a complex function of frequency, and the Linkwitz–Riley as two of them in a row:

/** Second-order Butterworth at fc, low- or high-pass, at frequency f. */
function bw2(f: number, fc: number, high: boolean): Cx {
  const s: Cx = [0, f / fc];
  const den = add(add(mul(s, s), [Math.SQRT2 * s[0], Math.SQRT2 * s[1]]), [1, 0]);
  return div(high ? mul(s, s) : [1, 0], den);
}

/** Fourth-order Linkwitz–Riley: two Butterworths in a row. */
const lr4 = (f: number, fc: number, high: boolean): Cx => {
  const b = bw2(f, fc, high);
  return mul(b, b);
};

Why the two sides add up to a flat line takes one line of algebra. Write s for the frequency relative to the crossover, as the code does. The low side is 1 / (s² + √2 s + 1)², the high side s⁴ / (s² + √2 s + 1)², so together they are (1 + s⁴) / (s² + √2 s + 1)². And 1 + s⁴ is (s² + √2 s + 1)(s² − √2 s + 1), which leaves (s² − √2 s + 1) / (s² + √2 s + 1). Top and bottom differ only in one sign, and for every real frequency they have the same size. So the sum lets every frequency through at exactly the level it came in, and only turns its phase, slowly, around the crossover. A filter like that is called an all-pass, and it is what “sums flat” means.

Where exactly the handovers go is not decided. It depends on things that have to be measured: where each driver starts to beam, where it starts to break up, and how the three sit on the front of the box. What the figure shows is the principle: that the sum can be flat, and how quickly each driver has to get out of the way.

The lowest octave

The rule says the bass has to come from the box. A woofer in a closed box has its lowest frequency set by the box: the air inside acts as a spring against the cone, and below a certain frequency the output falls at 12 dB per octave. A port, a tube in the front, uses the air in the tube as a mass on that spring and tunes the box lower; it is the most common trick, and it can chuff and whistle when it is driven hard.

A passive radiator is the same trick without the tube: a cone with no magnet and no coil, moved only by the air in the box. Its mass does what the air in the port would do, but it has no air rushing through a narrow opening. Tuned to the octave below where the woofer alone would give up, the box plays that octave too, and below it falls at 24 dB per octave. That is the curve under the diamond on the left of the figure; the dashed one is what a closed box would do instead.

Each speaker gets two of Purifi’s 6.5-inch radiators: two, because mounted opposite each other their pushes cancel on the cabinet, and together they have the area to move the air the lowest octave needs. Whether with aluminium or fibre cones is still open.

Next

The enclosure: its volume, which sets the tuning; its walls, which must not sing along; and the front, where the midrange and the tweeter have to sit close enough to act as one source at the crossover. Then the first measurements, and the crossover fitted to them. More in the project, and here as the box takes shape.