The bass a small loudspeaker does not make

The first rung of this anchor opens with an observation and does not compute it: a laptop speaker reproduces essentially nothing below two hundred hertz, a bass guitar’s bottom string is at forty-one, and the bass line comes through at the right pitch with nobody noticing.

The observation is correct. What it leaves out is how far from nothing “essentially nothing” is, and the answer is a long way — far enough that the residue stops being a curiosity about perception and becomes the load-bearing assumption in a piece of consumer engineering that ships in billions of units a year.

The argument #

Two independent curves conspire against a low note and a third finishes the job, and all three are computable.

The source falls as one over the harmonic number for any ordinary plucked or bowed spectrum. The radiator rises as the square of frequency at a fixed cone displacement, which more than cancels that fall. And the ear is roughly fifty decibels less sensitive at forty hertz than at two hundred and fifty. Multiply the three and the loudest component of a forty-hertz note is nowhere near forty hertz.

The surprising part is not that this is true through a three-inch driver. It is that it is true through anything at all. The residue is not a repair for bad loudspeakers; it is how low bass is heard on any system, and the small speaker only widens a gap that was already there.

Which computation produced the number #

On axis in a baffle the far-field pressure from a piston of area $S$ moving with peak displacement $x$ is

$$|p| = \frac{2\pi \rho f^2 S x}{r},$$

which is the whole of the first curve. It falls as $f^2$, so every octave downward costs twelve decibels before anything about the amplifier or the enclosure has been mentioned. Below the system’s resonance the displacement is already at its limit and the level goes with it; above resonance the displacement needed for a given output falls as fast as the pressure would rise, the limit stops binding, and the maximum flattens.

For a radiator of effective radius 3.2 cm at a 1.5 mm limit, one metre away, that gives 66 decibels at forty hertz. None of those three numbers describes a product: they are the dimensions of a cone of a stated size at a stated limit, and every figure here prints all three.

The second computation is ISO 226, the published equal-loudness contours, evaluated at each harmonic’s own frequency rather than only at the standard’s tabulated third-octaves. Sixty-six decibels at forty hertz is seventeen phons. The same sixty-six decibels at a kilohertz would be sixty-six phons.

Why the conversion is so brutal down there is the equal-loudness contours: every point on one of them is a level that sounds equally loud, and they crowd together at the bottom of the spectrum, so a decibel lost at 40 hertz costs several times what a decibel lost at a kilohertz does.

What actually arrives, harmonic by harmonic #

The fourteen-phon figure on the unrestricted system is the result this rung was not slated to find, and it is the more interesting one. A forty-one-hertz note played through a system that reproduces forty-one hertz perfectly still has its loudness carried by its fourth to sixth harmonics. The fundamental is present, is audible, contributes body — and is not what the ear is measuring the note by.

That reframes the whole anchor. The missing fundamental is normally introduced as what happens when something goes wrong: a channel is band-limited, a partial is deleted, a bell has no prime. The arithmetic says that in the bottom octave and a half of the musical range the fundamental was never carrying the note in the first place, and the residue is the ordinary mechanism rather than the exceptional one.

How much of that depends on the source falling as one over n #

The caveat below records that a real bass spectrum is not exactly one over n, and the two halves of the finding turn out to depend on it completely differently.

Recomputing with the source falling as n to the power −α, for α from 0 to 2, through the small radiator:

Through a small driver nothing is at risk. The loudest component is the sixth to eighth harmonic across the whole plausible range of source spectra, and the fundamental is between forty and eighty phons below it. A margin of forty phons does not care what the source was doing.

The full-range case is where the assumption is load-bearing. Recomputed on the eighteen-centimetre driver of the second figure:

At a source falling as one over n squared, the fundamental is one phon below the loudest thing in its own note — which is to say, tied.

Where the claim flips, and it is close #

Solving for the exponent at which the fundamental takes the lead: it happens at 1/n^2.07.

That is uncomfortably near the edge of what a real string does. A plucked string’s spectrum falls roughly as one over n at low harmonic numbers and steepens above the plucking point’s first null, and a bass with a soft attack, a heavy string or a tone control turned down can present something close to one over n squared into the low band. So the claim that the fundamental is never the loudest part of a low note holds for every source spectrum in the plausible range and stops holding just outside it, on a full-range system.

That is worth separating from the essay’s main argument rather than blurred into it, because the two claims now have very different standing. On a small device the residue is the mechanism, by a margin nothing could close. On a full-range system the fundamental is not the loudest component for ordinary sources and is within a phon or two of being so for dull ones, which is a much weaker statement than “never, on any system at all” — and it is the statement the arithmetic supports.

The reason the two behave so differently is the resonance. Below it the driver’s displacement limit binds and radiation rises as frequency squared, which multiplies the harmonics up by n² against the source’s fall; above it the limit stops binding and the source’s own slope takes over unopposed. A small driver resonating at 250 hertz has the whole first six harmonics of a low E inside the rising region. A large one resonating at 45 has none of them.

The three curves also explain why the loudest harmonic lands where it does. Radiation rises as $f^2$ and the source falls as $1/n$, so the radiated level of harmonic $n$ rises as $n$ — six decibels per doubling — until the system resonance, above which radiation is flat and the source’s fall takes over. The turning point is therefore at the resonance, which for this driver is the sixth harmonic of a forty-one-hertz note. On a driver resonating at 150 hertz it would be the fourth. In every case it lands close to the dominance region, and it lands there for reasons that have nothing to do with hearing.

What the listener has to work with #

So what reaches a listener from a bass line on a small device is a set of unresolved partials whose sum repeats forty-one times a second. That is precisely the object the previous rung built in a laboratory: no template to match, and a repeat rate for something to read.

So what reaches a listener from a bass line on a small device is a set of unresolved partials whose sum repeats forty-one times a second — the autocorrelation of harmonics two to five alone returns to one at 24.3 milliseconds, the period of 41.2 hertz, and that repeat is the entire evidence for the note. The price is determinacy rather than accuracy. Four consecutive harmonics from the fifth up are consistent with two fundamentals inside thirty cents: 41.2 hertz with nothing unaccounted for, and 20.6 which fits every partial exactly and predicts three slots that nothing occupies. With eight harmonics the octave candidate carries seven empty slots and is easy to reject; with two it carries one. A narrower passband does not shift the pitch, it makes the pitch less determined.

The trick, and it is not a repair #

A device that cannot radiate a fundamental can synthesise its harmonics instead. Generate the second to fifth harmonics of whatever is in the low band, put them where the driver works, and let the listener’s own machinery supply the note. The category is called psychoacoustic bass enhancement, it is in phones, laptops, televisions and car doors, and it is a deliberate application of everything on this anchor.

It is worth being precise about what it substitutes. For an ordinary plucked bass the harmonics are already there, and the enhancement is adding little. For a synthesised bass or a kick drum whose low band is close to a pure tone, they are not there at all: a forty-hertz sine through the driver of the hero figure is seventeen phons and inaudible in any real listening situation, and there is nothing above it to imply anything. The enhancement manufactures the evidence.

Two consequences follow from the arithmetic and one of them is the opposite of what is usually claimed.

The substitution is not paid for in masking, which is where a cost would be expected. To be as loud as its own harmonics, a forty-hertz fundamental needs 104 decibels where the harmonics need 82 — because the ear’s contour is so steep down there. Masking spreads upward much further than downward and its upward reach grows with level, so the very loud low tone buries slightly more of the lower midrange than the substitute does: at 300 hertz the real fundamental raises the threshold to 82 decibels and the substitute to 78. Four decibels is not much either way, and the direction is the one nobody would guess.

What it does cost is determinacy and headroom. The octave ambiguity above is real and it grows as the passband narrows. And the substituted harmonics have to be loud enough to be heard against the rest of the mix, in a band where the driver is also reproducing everything else, so a device that enhances aggressively is spending its excursion and its amplifier on synthetic content.

And the trick is worth more the quieter the listening is.

And the substituted harmonics are not resolved from each other either. Harmonics two to five of a forty-one-hertz note sit at 82, 124, 165 and 206 hertz, forty-one apart, against an analysis bandwidth that runs from 34 to 47 hertz across that span. Only the lowest two of them have a filter to themselves. The rest arrive together, and what a filter delivers from a pair of partials inside it is the fluctuation that this site computes as roughness — here at forty-one a second, which is fast enough to be a buzz rather than a beat. That is the honest form of the complaint that a small speaker’s bass sounds synthetic. The added partials are not heard as separate tones; they are heard as one rough thing whose roughness rate is the note.

The claim that the restored bass has a timbre the real note does not is the one the computation does not support in the form it is usually made. Through the same driver, an ordinary bass note’s harmonics arrive at very nearly the levels the enhancement supplies, because the driver has already thrown the fundamental away. The timbre difference is between the small device and a full-range system, and it belongs to the driver rather than to the enhancement. What the enhancement changes is which sources get the treatment: a note that had harmonics keeps its own, and a note that had none is given a set it never had.

What the picture cannot show #

The room, the enclosure and the placement, all of which move the curve by more than the differences argued about here. A driver in a corner gains several decibels at the bottom for free; the same driver in free air on a small box loses several more than the figures show.

Distortion, which at the excursion limit is severe. The hero figure draws the maximum a cone can reach and says nothing about what it sounds like there, and a driver run to its limit at forty hertz generates harmonics of its own — which, by an irony this essay cannot resolve, are the same harmonics the enhancement would have added deliberately.

Directivity is not the problem, and it is worth showing that it is not.

The spectrum is drawn as one over n and a real bass is not. A plucked string’s spectrum depends on where it was plucked, an electric bass adds a pickup’s own filtering on top, and a recorded one has usually been compressed and equalised before it reaches any loudspeaker. The section above sweeps the exponent and finds the small-driver result untouched across the whole range and the full-range result marginal at the steep end — so the one-over-n assumption is doing no work in half the essay and a great deal in the other half.

And loudness is not pitch. Every number here is about level and audibility. A note whose fundamental is seventeen phons and whose sixth harmonic is seventy-eight has the right pitch and the wrong weight, and the difference between a bass line that is audible and one that is felt is not in any of these figures.

The spectrum sweep varies one exponent and holds the rest of the model still. A real source departs from a power law in ways an exponent cannot express — a plucking point puts a null in the series, a pickup adds a resonance, a compressor moves the balance with the envelope — and the sweep says only that the conclusion is insensitive to the slope. What it establishes is where the sensitivity lives: not in the small-driver case, which is decided by the radiator, and squarely in the full-range case, which is decided by the source.

Whose recordings, and when #

The engineering decision is old and the physics did not change.

The telephone channel of the 1920s excluded the fundamental of every adult voice on cost grounds, and the engineers had the psychoacoustic literature. Small transistor radios of the 1950s and 60s were voiced with a deliberate lower-midrange lift for exactly the reason the loudness figure gives. Recording practice in popular music since roughly the 1960s has routinely added harmonic distortion to bass parts — described in the trade as making the bass “translate” — which is the enhancement above, applied at the mix rather than at the device, and applied because most listeners were on small speakers.

The orchestral parallel runs the other way and was arrived at by ear. Doubling a bass line at the octave with cellos or bassoons, standard from the eighteenth century onward, puts a partial in the resolved range that belongs to the same series as the double bass’s note. It strengthens the fit without adding a pitch, and it is the same operation as an organ’s quint rank and the same operation as a bass-enhancement stage, arrived at three times independently by people who could not have agreed about the mechanism.

Where the ladder goes next #

This anchor now has eight rungs and they divide in a way the table did not predict. The first four establish that pitch is inferred from a pattern — in a laboratory, in a bell, in a kettledrum. The last four price the inference, and the price turns out to be paid in the same currency every time: how many partials arrive separately, and what is left when none of them do.

Three things the last four rungs found that the first four did not have. The residue’s ceiling is not a resolvability ceiling, because resolvability improves with frequency and never fails at the top. The shifted-residue experiment does not separate the two surviving accounts, because a correlation predicts the shift as accurately as a template does. And the fundamental of a low note is not the loudest part of it on any system, so the residue is the normal mechanism in the bass rather than an artefact of bad equipment.

What the ladder has not done is decide between a template and a correlation, and it now looks as though the question is badly posed: the two agree wherever partials are resolved and only one of them applies where they are not. The rungs still open are the ones that would test the division rather than the models — pitch strength as a measured quantity across the resolved-to-unresolved boundary, what happens to a residue when a room adds a reflection at a fraction of the period, and whether a listener’s octave errors track the empty-slot count these figures keep printing.