A bell has no fundamental

The missing fundamental is usually demonstrated in a laboratory: take a complex tone, delete its lowest partial, and observe that the pitch does not move. That is a striking result and it has the character of a trick, because somebody had to construct the stimulus.

A church bell is the same result, arrived at by founders shaving metal off castings for six hundred years, on an object weighing a ton.

The note a ringer, a listener or a score calls “the bell’s note” is the strike note. It is not any partial. It is an inference.

The partials a founder aims at #

Bell founders have names for the partials, and the names are older than the acoustics. From the bottom:

Hum — an octave below the prime, at 0.5. Prime — the nominal fundamental, at 1. Tierce — a minor third above the prime, at 1.2. Quint — a fifth above, at 1.5. Nominal — an octave above the prime, at 2, and usually the loudest partial in the sound. Above those, the deciem at 2.5, the undeciem or twelfth at 3, and the duodeciem or double octave at 4.

Those ratios are not derived from anything. A bell’s shape is a profile arrived at by iteration — cast it, listen, machine metal off the inside, listen again — and the ratios above are the targets the tradition converged on. This is the one place in this field where the numbers are a craft’s output rather than a physical model’s.

Why those three imply the prime #

Take the nominal, the twelfth and the double octave: 2, 3 and 4 times the prime. Those are consecutive members of a harmonic series whose fundamental is 1 — the second, third and fourth partials of a tone at the prime’s frequency.

Pitch is decided by the pattern of the partials rather than by the presence of the lowest one. A set of partials at 2, 3 and 4 times some frequency is a pattern with an unambiguous fundamental, and the ear reports that fundamental whether or not it is in the air. So the bell is heard at the prime.

The prime is also a partial the bell actually has, which makes this less dramatic than the laboratory demonstration and more interesting. The strike note coincides with a real partial and is not produced by it: the prime is usually a weak component and the nominal, an octave above, is much the loudest. What the listener names is neither the loudest partial nor the lowest; it is the fundamental of the series the upper partials belong to.

The evidence that it is an inference rather than the prime being heard directly is that the strike note tracks the nominal when a bell is imperfectly tuned. Retune a bell so that the nominal is slightly sharp of twice the prime and the strike note goes sharp with it, not with the prime. Founders know this and tune the nominal above all else.

What the site’s own matcher says about that #

The claim is worth running rather than repeating, because this collection has a residue matcher — it fits a fundamental to a set of partials by least squares and reports how well the fit works — and it does not agree.

Give it the three implying partials at exactly 2, 3 and 4 times the prime and it returns 2 : 3 : 4 on 293.66 hertz, fitted to nought cents. Then move the nominal alone and watch:

The strike note tracks fourteen per cent of the movement, not all of it, and the fourteen per cent is not a fitted number: a least-squares fit weights each partial by the square of its harmonic number, so the nominal’s share is 2² over 2² + 3² + 4², which is 4/29 = 0.138. The matcher has no amplitude term in it at all, so the loudest partial in the sound gets the smallest vote of the three.

That is a genuine disagreement between the model and six hundred years of practice, and the resolution is not that the founders are wrong. It is that a founder cannot move the nominal alone. The nominal, the twelfth and the double octave are all set by the same region of the profile in nearly fixed proportion, so the cut that sharpens one sharpens all three — and when all three move together the residue follows them exactly, at a hundred per cent.

So the tradition’s rule is sound and the reason usually given for it is not. The nominal is not privileged in the inference; it is privileged as a handle, being the loudest partial and therefore the one a founder can hear and measure, on a cluster that moves as a unit. What the matcher adds is the size of the error a founder would make by trusting it: a bell whose nominal alone had drifted twenty cents would still strike within three cents of its note, and would sound wrong for a different reason — the fit’s worst residual rises to seventeen cents, which is the partials ceasing to agree with each other rather than the pitch moving.

The tierce, and why a bell is minor #

The partial at 1.2 is a minor third above the prime — 316 cents, close to the pure 6:5 at 316 cents. Not a major third.

So a well-tuned Western bell has a minor third built into its sound as a matter of construction. This is why a peal of bells has the quality it has, why bell-like synthetic sounds are made with a prominent minor third, and why the “major-third bell” is a twentieth-century engineering achievement with a name and a date rather than an ordinary variant.

The kettledrum is the instructive comparison because its makers solved the same problem the other way. Its partials are tuned to near-whole ratios starting from the second, so it has a definite pitch and a missing fundamental of its own — the same trick as a bell’s, arrived at independently and on a membrane rather than a shell.

The major-third bell was developed at the Eindhoven University of Technology with the Eijsbouts foundry in the 1980s, using finite-element analysis to find a profile that puts the tierce at 1.25 instead of 1.2. The profile has a distinctive flare and the bells were cast and hung. That is a real result and it is worth noticing what it took: six centuries of ear-tuning converged on 1.2, and moving it required computing the modes of a shell.

Two ways to be heard at a pitch, and a bell uses both #

There is a subtlety worth drawing out, because it explains why the strike note is so robust.

The site’s earlier work distinguished two accounts of pitch. One is spectral: the ear finds the harmonic template that best fits the partials present and reports its fundamental. The other is temporal: the ear finds the period of the waveform, and a set of partials at 2, 3 and 4 times $f$ repeats at $1/f$ whether or not $f$ itself is present.

A bell satisfies both. Its upper partials fit a template on the prime, and they also produce a waveform whose envelope repeats at the prime’s period. The two accounts agree, which is why the strike note is heard immediately and confidently even though the partial carrying it is weak.

They come apart in the shifted-residue experiment, where a set of partials is moved bodily so that the spacing and the best-fitting fundamental disagree — and the pitch heard follows neither exactly, which is the result that decided the question. A bell is the easy case, and the fact that it is easy is what let a craft find it.

A bell has its second, third and fourth members and not its fifth, and a bar has almost nothing in common with either. “Inharmonic” is not one condition: the bar’s partials are far from whole numbers and the bell’s are near several of them and missing others, and only the second of those can imply a pitch at all.

What the founder actually does #

A bell is cast oversize and tuned by removing metal on a vertical lathe, from the inside. Each partial responds to metal removed at a different height, because each mode has its nodes and antinodes in different places — which is the same rule that governs where a hammer should strike a string and where a register vent has to sit, applied to a shell instead of a line.

Removing metal near the mouth lowers the hum; near the waist it affects the tierce; higher still, the nominal. The partials cannot be adjusted independently — every cut moves several — so tuning is an iterative search rather than a sequence of adjustments, and a bell can be flattened and never sharpened.

The order matters, and it is fixed by that asymmetry: everything is cast sharp and brought down. A bell that overshoots is scrap, or is retuned to a different note entirely.

An object tuned by ear, converging on arithmetic #

Here is what makes this a rung on the missing-fundamental ladder rather than an essay about bells.

Founders were not aiming at 2 : 3 : 4. They were aiming at a bell that sounds good and sounds at a definite pitch, and the pitch they were listening for was the strike note. Aiming at a definite strike note, iteratively, over centuries, with no theory of residue pitch and no way to measure a partial, they converged on a set of upper partials in whole-number ratios — because that is the configuration in which a residue pitch is unambiguous.

The alternative configurations do not have that property, and the matcher says in what way. Handed partials at 2, 2.4 and 2.9 it does not fail — it finds 5 : 6 : 7 on a fundamental of 119 hertz, which is a fifth and a half below the prime and nothing the bell was cast for. Two things about that answer are the difference. Its worst residual is 33 cents against the real bell’s nought, so the partials are agreeing with the fundamental only in the loosest sense; and it is unstable under the tolerance — tightened to thirty cents the same matcher names 9 : 11 : 13 on 65 hertz instead, a completely different note.

The real bell returns 2 : 3 : 4 on the prime at every tolerance from thirty cents to a hundred and twenty. A residue that survives its own tolerance parameter is what a definite pitch is, and a configuration that changes its mind about which note it is when the criterion moves by fifty cents is what a bell without one sounds like — which is a badly tuned bell, and is what a drum does by construction.

A craft with no theory converged on the arithmetic that makes a pitch inferable. That is a strong piece of evidence about pitch perception, produced by people who were not investigating pitch perception.

What a peal has to do about all this #

A ring of bells is not one bell repeated. Each is cast to its own note, and the set has to be in tune with itself — which raises a question this site has spent a whole field on: in tune according to what?

The answer for English change-ringing is the nominals, because the nominal is what the strike note tracks. A ring is tuned so that its nominals form the intended scale, and the other partials are brought into their ratios within each bell as far as the profile allows. Two constraints, one across bells and one within each.

The within-bell constraint is fixed by the profile and is therefore the same for every bell in the ring. The across-bell constraint is a scale, and the scale is a choice — most English rings are tuned to something close to equal temperament today, and older rings are frequently not, for the same reasons and with the same variety as keyboards of the same period.

There is one further complication with no counterpart elsewhere. A bell cannot be retuned upward, and a ring is often augmented over centuries — bells added to an existing set, cast by a different founder to match. So a historic ring is a record of successive decisions about a scale, made with an irreversible process, by people who could not measure a partial. That several of them are in tune to a few cents is a considerable thing.

That is the untuned baseline: a drum head with nothing done to it has no pitch, because nothing in its partials implies a fundamental. A bell and a kettledrum are both membranes or shells that have been made to lie, and the lie is the same one — a series of near-whole ratios with the first member absent.

What the picture cannot show #

These ratios are a target, not a measurement of any particular bell. Real bells depart from them by tens of cents, especially the hum and the tierce, and the departures are part of why a peal sounds like that peal. The figures draw the founder’s ideal.

A bell has far more than eight partials. The upper ones are dense and inharmonic and they decay fast, and they are a large part of the sound of the strike — the clangorous first half-second that gives way to the hum. This essay’s figures draw the tuned partials, which are what survive.

The decay is not modelled. A bell’s partials decay at very different rates, so the sound half a second after the strike is a different spectrum from the one at the strike, and the strike note is inferred from the first fraction of it. The attack is where an instrument’s identity lives and a bell is an extreme case: a listener who hears only the hum, a few seconds in, is hearing a pure tone an octave below the note.

And the sound buttons are additive synthesis at fixed ratios. They reproduce the partial structure the essay is about and nothing else — no decay differences, no dense upper spectrum, no radiation from a shell.

Why a bell fuses at all #

One more thing has to be true for any of this to work, and it is not obvious.

A bell’s partials are not a harmonic series, and what makes several simultaneous partials one note rather than several sounds is a grouping decision the ear makes. That decision is influenced by harmonicity — a partial mistuned by one per cent is heard out of a note — so a bell, whose tierce is 20 per cent away from any harmonic position, ought to fall apart into separate tones.

It does not, and the reason is the other cue. Every partial of a bell starts at the same instant, because they are all set going by one blow. Onset synchrony is a stronger cue than harmonicity — a perfectly harmonic partial with a thirty-millisecond head start leaves the note, and a wildly inharmonic one that starts with everything else stays in it.

So a bell is held together by simultaneity rather than by arithmetic, and the strike note is inferred from the subset of its partials that happens to form a series. That division of labour — fusion by onset, pitch by pattern — is visible in a bell more clearly than in almost any other instrument, because the two cues are pulling in different directions and one of them wins.

It also predicts something checkable: a bell struck twice in quick succession, or two bells struck a fraction of a second apart, should be easier to hear as separate objects than two harmonic instruments at the same interval. Ringers describe exactly that, and it is why the striking accuracy of a peal is judged in tens of milliseconds.

Whose bells, and when #

The tuning conventions here are those of the English and Dutch change-ringing and carillon traditions, and they settled between the fifteenth and seventeenth centuries. The partial names are the founders’ own.

They are not universal. A Chinese bianzhong is an almond-sectioned bell that deliberately produces two distinct strike notes depending on where it is struck, a design with no European counterpart and one that requires an entirely different profile. Japanese temple bells are tuned for a long, slow, deliberately inharmonic decay and not for a strike note at all. The claim that a bell’s partials should sit at 2 : 3 : 4 is a claim about one tradition’s goal, and the traditions that had different goals reached different shapes.

The measurement side is largely twentieth-century: Arthur Bigelow, Ernest Rossing and the Eindhoven group. The strike-note-as-residue explanation dates from the mid-twentieth century and postdates the practice by five hundred years.

Where this goes #

The missing-fundamental ladder now has three rungs — the laboratory demonstration, the distortion products that are not the explanation, and an object a craft tuned into producing one. What it does not yet have is the case where the inference fails, which is what a drum is, and that is the next essay in this field.

Sideways, a bell is the cleanest instance of the site’s oldest theme. Small whole numbers are the whole story and they do not fit: a bell’s partials are 0.5, 1, 1.2, 1.5, 2, 2.5, 3, 4, which is four whole numbers, two simple ratios and one that is neither — and the note it is heard at is a number that is not in the list at all.