A bell's strike gives its tierce a millisecond

A bell’s tierce does not arrive late, it leaves early settled where a low bell’s minor third goes. The tierce — the mode a minor third above the prime that gives a bell its minor colour — is masked on a large bell, and the masking comes from below: the prime a minor third under it and the hum an octave and a third under that. Taking the quint away moved the tierce’s threshold by a fifth of a decibel; taking the prime away moved it by three to five and a half. Since the lowest modes of a bell are the ones that ring longest, the tierce’s shadow never lifts. A bell that has a tierce at the strike loses it within a few seconds, and a bell that has none at the strike never gains one.

That makes the strike the whole question, and the essay ended by pointing out that the strike was the one moment its count had treated as a steady state. It gave every mode its full level at time zero and let each one decay. A real bell’s first hundred milliseconds are not like that. The modes are set ringing by an impact and build up, and a listener’s ear builds them up too, since every auditory filter takes a few milliseconds to respond to a component that has just begun. If the modes that mask the tierce build more slowly than the tierce itself, there might be a window in the attack where the tierce is ahead of its shadow. The prediction written into that essay was specific: the tierce’s margin would have a maximum inside the first two hundred milliseconds, and for the bells the steady count calls hopeless it might be above zero there.

The first part is wrong in a way that decides the second. The tierce’s margin is largest at the very first instant of the strike and falls from there, under every law of rise tried. On the bells the steady count calls hopeless it is above zero for less than a millisecond by the ear’s own rise, and for at most twenty-five under the most generous law of rise anybody could propose — far too briefly to be heard.

Giving the strike a shape #

The bell is the one the whole bell sequence has used: eight modes, hum, prime, tierce, quint, nominal and three above, at the ratios of a well-tuned minor-third bell, with amplitudes falling from the hum and every mode decaying at a rate proportional to its frequency, the hum’s sixty-decibel time forty seconds. The masking is the count one sound hides another set up: each mode’s masked threshold is the power sum of the spreading patterns every other mode casts on it, a pattern steep below the masker and shallower above it, and a mode is delivered when it stands above both that threshold and the threshold of hearing. The hum is struck at 70 dB.

To that the onset adds one multiplication. Each mode’s amplitude is scaled by $1 - e^{-t/\tau}$, rising from nothing towards its full level with a time constant $\tau$ of its own. Three laws for $\tau$ are tried, because nobody has measured the onset of a bell’s modes separately and the laws bracket what they could be.

The ear’s own. An impact sets every mode going at once and at full amplitude, and what rises is the listener’s response: each auditory filter builds up a component at a rate set by its bandwidth, $\tau = 1/(2\pi \cdot 1.019 \cdot \mathrm{ERB})$, three to four milliseconds at the frequencies of a low bell’s prime and tierce. This is the physically honest law, and the one the hero uses.

All together. Every mode rising with the same twenty-millisecond constant — a control, to separate the rise’s shape from its differences.

Higher modes faster. The lead’s own proposal: the modes rising with a time constant proportional to $1/f$ or $1/f^2$, from twenty or fifty milliseconds at the hum. The steeper law gives the tierce the largest head start over the prime it could have — at $1/f^2$ it rises 1.44 times faster than the prime — and so it is the most generous to the prediction.

The first instant is the best #

The hero’s curves share a shape. Each starts at its highest and falls, steeply through the first ten or twenty milliseconds while the modes are building and then slowly as they decay. Not one has a peak inside the window. The C2 bell’s tierce is 0.3 decibels above its threshold half a millisecond after the strike and 2.5 below it by 400; the E2 bell’s goes from 1.6 above to 1.2 below; the G1 bell’s never reaches zero.

Changing the law of rise changes how high the curve starts and how long it takes to fall, and nothing else. Under every law the maximum is the first moment computed. The laws that let the tierce rise faster than the prime give it more to start with — three decibels at the head of the most generous law against a third of one under the ear’s own — and keep it above zero longer, but the shape is the same. There is no stage of the strike at which the tierce catches up with its shadow, and then loses it again. It starts ahead and is caught.

The shadow grows faster than the tierce #

The reason is not in the onset at all. It is in the masking.

A masker’s spreading pattern is steep on its low side and shallow on its high side, and its high side becomes shallower as the masker becomes louder. That is one of the oldest facts about masking, and one sound hides another wrote it into the count this whole sequence uses: the upper slope falls by a fifth of a decibel per Bark for every decibel the masker gains. So a masker that grows by ten decibels raises its shadow on a component above it by rather more than ten.

Drawn through the strike, the consequence is plain. Between half a millisecond and 400, the prime rises 17.4 decibels and the tierce 16.6, since they build at nearly the same rate. The threshold over the tierce rises 19.4. The tierce’s margin is the gap between its own curve and its threshold’s, and that gap closes by almost three decibels while both are climbing. The early instant is the best not because the tierce has raced ahead but because everything is quieter, and at a lower level the prime’s shadow does not reach as far.

So the attack’s advantage is a level advantage, not a timing one. It would be there if every mode rose together, and the second figure’s control shows it is: with every mode rising at the same rate, the tierce still starts 0.4 decibels above zero at C2. A faster rise for the higher modes adds to it, and cannot change its sign.

Why a peak was expected, and where the reasoning slipped #

The prediction was not careless, and seeing exactly where it went wrong is the most useful thing the figures say. It reasoned from rates. The prime masks the tierce; if the prime builds more slowly than the tierce, then early in the strike the tierce is relatively louder than it will be once both have settled, and its margin should rise above the steady value. Later in the note the decay takes over, the higher tierce falling faster than the prime below it, and the margin falls again. Between the two a maximum, somewhere in the first two hundred milliseconds.

Both halves of that are right about the levels. What they leave out is that a masked threshold is not a fixed number of decibels below its masker. A moving tenor is heard because it is struck again found the same thing from the other side: a note re-struck into a decaying chord gains partials because the chord’s shadow has shrunk faster than the chord, and the shadow shrinks faster because its reach shrinks with its level. A bell’s strike is that process run forwards. As the modes build, the prime’s reach grows along with its level, and a tierce rising a shade faster than the prime cannot outrun a shadow that is lengthening as well as deepening. The rates matter to how much the tierce starts with. The shape of the curve belongs to the masking.

The same asymmetry runs under a clarinet keeps what a string loses, where what a spectrum’s upper partials keep is decided by how strong its lower ones are, and under every low chord that hides its tenor: a louder low component takes more of what lies above it than its extra level alone would take. The bell’s tierce is one more instance, stretched out in time.

A millisecond is not enough #

Standing above a steady-state threshold for a few milliseconds is not the same as being heard. The ear integrates a sound’s energy over roughly the first two hundred milliseconds, and a short sound needs more level than a long one to reach the same detectability: about ten decibels more for every factor of ten it is shortened. A tierce above its threshold for a twentieth of that window has to be thirteen decibels above it to be heard, not a fraction of a decibel.

Against that requirement the attack’s windows are hopeless for every bell the steady count calls masked. The C2 bell’s tierce is above threshold for two tenths of a millisecond, by a tenth of a decibel, where it needs twenty-nine. Under the most generous rise it is above for 24 milliseconds by just over a decibel, where it needs nine. The E2 bell’s does better under that law, 48 milliseconds at 1.3 decibels, and still needs 6.2. The G1 bell’s tierce gets 11.7 milliseconds at 0.6 decibels under the generous law and nothing under the ear’s own.

Only bells whose tierce is already audible at the steady strike — G2 and above — have an attack window long enough to count, and they do not need it. The onset rescues no bell the steady count calls hopeless. The minor third of a large bell is not a property of its attack’s shape; if the count is right, a large bell’s tierce is not heard through the masking at any moment.

One bell at a time #

The dial walks the curve up the register. Each step up the scale lifts the whole curve by about a decibel, because a higher bell’s modes sit where the masking slopes are steeper and the threshold of hearing matters less, and the point at which the curve crosses zero slides later. It crosses zero after a fraction of a millisecond at C2, after four at E2, after a tenth of a second at G2, and not at all at C3. The register boundary the earlier essays found — E2 to G2, below which the tierce is masked — is the same boundary seen through the strike, and the strike does not move it.

What the onset visits, a soft stroke keeps #

The mechanism that makes the first instant the best has a consequence that outlasts the instant. If the tierce’s advantage early in the strike is that everything is quieter, then a bell struck more softly should have that advantage for the whole of its note. It does. A C2 bell struck with its hum at 40 dB keeps its tierce above threshold at the strike, by 0.2 decibels, where the same bell at 70 dB is 2.2 below. An E2 bell struck at 60 dB or below keeps its tierce, and one at 70 dB loses it. Every ten decibels of stroke costs the tierce about a decibel of margin, in every register.

That is the one way this count lets a low bell’s tierce be heard, and it points somewhere the lead did not. A carillon’s bass sounds more minor played softly than played loudly. A carillonneur controls exactly that, since a carillon’s clappers are driven by the player’s hands and feet through a mechanical action and the stroke’s force is theirs to choose. The earlier essays’ statement that a low bell has no tierce needs a dynamic attached: it is a statement about bells struck at the level the count assumed. Whether the difference, a decibel or two of margin, is large enough to hear is a question for the ear rather than the arithmetic — but it is a larger and longer effect than anything the attack offers.

The model under the curves #

Eight modes with the amplitudes and ratios of the earlier bell essays; each decaying at a sixty-decibel time of forty seconds divided by its frequency ratio to the hum; each built up by $1 - e^{-t/\tau}$ with $\tau$ from one of the three laws. At each of 160 moments spaced logarithmically from half a millisecond to 400, each mode’s level is computed and the count run: every other mode’s spreading function evaluated at the tierce and summed in power, the larger of that and the threshold of hearing taken as the tierce’s threshold. The longest stretch of positive margin is found, its time-weighted mean excess, and the integration penalty $10 \log_{10}(200\,\mathrm{ms}/d)$ for a stretch of duration $d$ under 200 milliseconds. The steady-state margin quoted is the earlier essay’s own count at time zero.

What the picture cannot show #

The clang. A struck bell’s first milliseconds carry a broadband impact noise — the clapper’s thud, spread across every band — that decays in tens of milliseconds. The count has no noise in it. Noise can only add masking, so its absence makes every curve here an upper bound, and the attack’s windows shorter still in reality.

Forward masking by the attack. A sound hides what came before it and what comes after it too, for a couple of hundred milliseconds, and the attack is loud. That would weigh against the tierce in the early window as well.

Onset cues. An onset can only hand over what the chord leaves found that a component starting tens of milliseconds after its neighbours is heard out of a mixture more easily. A bell’s tierce starts with every other mode, at the same impact, and gets no such help. A bell whose tierce took longer to build than its prime would have an onset cue; this count has none, and none of the laws tried gives the tierce a later start.

Measured onsets. None of the rise laws is measured. The ear’s own is set by the listener, not the bell; the other two bracket what a bell’s modes could do. A founder’s recording of a single bell, filtered mode by mode through its first hundred milliseconds, would replace all three.

Which bells #

The bells are well-tuned minor-third bells of the kind cast for carillons and chimes in the last century, from G1 to C3, and the claim is about their tierces as the count of masking sees them: that the attack gives a masked tierce an instant and no more, that the instant is too short to be heard, and that the one way to keep a low bell’s tierce is to strike it softly. A bell has no fundamental and the arch belongs to hearing, not to the series are the two essays whose bell this is. Whether listeners hear a large bell as minor, and whether they hear it as more minor when it is struck gently, are the measurements that would test the count, and none is reported here.

Still open: whether a soft stroke is heard as minor #

The count gives a low bell’s tierce a decibel or two of margin for every twenty decibels the stroke is softened, and the earlier essays’ boundary between bells that have a tierce and bells that do not moves up and down the register with the dynamic. The question that follows is whether that margin is heard as the bell’s colour. A bell has no fundamental found the pitch listeners name for a bell comes from its upper partials, and the minor quality is the tierce’s contribution to the same judgement. The computation would join the two: the pitch and the colour a listener assigns a bell as a function of which modes are delivered, run across dynamics, so that the prediction becomes a specific statement — a C2 bell struck at forty decibels is heard as minor, and at seventy as neither — which a single session with a carillon and a sound-level meter could check.