A bell's tierce does not arrive late, it leaves early

The arch belongs to hearing, not to the series put a founder’s bell through the masking census and found that a large bell loses its own tierce — the minor third above the prime, the mode that makes a bell sound minor — and that no interval a founder could tune it to rescues it below about A♭2. Every level in that computation is the level at the instant of the strike, and it ended by naming the one thing that could change the answer:

At the strike a low bell’s tierce is masked by its prime and quint, as here; but the quint and the upper modes decay faster than the prime and tierce, so a second into the note the tierce may emerge above its masked threshold. If so, a low bell would sound less minor at the strike than it does as it rings.

Given decay times, the prediction fails in both halves. The tierce does not emerge. And a bell that does have a tierce at the strike loses it — so a low bell sounds more minor at the strike than as it rings, which is the opposite conclusion from the same premise.

The quint was never masking it #

The prediction rested on a diagnosis, and the diagnosis is the part that was wrong.

The arch belongs to hearing described the tierce as one of “the two modes crowded into less than half a Bark above the prime, each masked by its neighbours”. The tierce has two neighbours — the prime a minor third below it and the quint a minor third above — and it is not masked by both. Taking each of the bell’s other seven modes away one at a time settles it.

On a C2 bell the tierce sits at 60.5 decibels against a threshold of 62.6. Remove the quint and the threshold falls to 62.5 — a relief of 0.14 decibels. Remove the prime and it falls to 58.2, a relief of 4.38, and the tierce is audible. The hum is worth 1.75 and every mode above the quint is worth less than a hundredth of a decibel.

So the tierce has one masker with an accomplice, and both are below it. That is the rule these essays have been running on from the first: one sound hides another, and it hides upward. A masking skirt is steep on its low side and shallow on its high side, so a mode at 131 hertz reaches up to one at 157 and a mode at 196 hertz barely reaches down to it.

The tierce’s position is not crowded between two neighbours. It is a shadow cast from underneath. And that changes what a decay law can do to it, because the modes that cast the shadow are the lowest in the bell and the lowest in the bell are the ones that ring longest.

What the register boundary was, and what it is now #

It is worth restating the earlier result in the terms this one supplies, because the two are not the same statement and the difference is the essay.

That essay asked a static question — at which register does a bell keep all eight of its modes — and got a static answer: from about A♭2 upward. A low chord stops being a chord had found the same shape for a triad and a clarinet keeps what a string loses for six harmonic spectra, and in every one of those the answer is a boundary on a pitch axis.

A bell is the one object in this account for which that is the wrong axis. A triad on a piano is struck and decays too, but its partials all belong to the same three strings and fall together; a bell’s eight modes are eight separate resonances of one casting with decay times that differ by an order of magnitude, and nothing in the masking arithmetic was ever asked to hold two components at levels that drift apart. Adding time to the census turns a boundary into a window, and a window has a width as well as a position.

The width is the new quantity and it is small. C3’s tierce is delivered for three and a half seconds out of a note that rings for the better part of a minute — six per cent of it — and G2’s for a sixth of a second out of the same. Stated as a proportion rather than as a boundary, a bell’s tierce is present for almost none of its note at any register this model can reach.

Which is why a steeper law hurries it #

A bell’s decay times are its most conspicuous property — a bourdon’s hum rings for the better part of a minute and its upper modes for a few seconds — and founders’ measurements of them are not in this collection. What is available is the shape: a mode’s decay time falls with its frequency. That can be written as one exponent and swept.

The prediction assumed a steeper law would help, and it does the reverse, for the reason the masker figure gives. A steeper law is one in which low modes outlive high ones by more; the tierce’s maskers are its low modes; so a steeper law preserves the shadow after the thing in it has gone. On C3 the tierce is heard for 3.55 seconds under an exponent of one, 1.40 under 1.5, 0.70 under 2 and 0.25 under 3.

The one exception is instructive and it is not a bell. At an exponent of nought, every mode decays at the same rate, the whole spectrum quietens together, and the tierce of a C2 bell becomes audible after 17.8 seconds and an E2’s after 6.35 — bells whose tierces were masked at the strike. That is not the decay uncovering anything; it is masking’s own level dependence. A loud masker casts a wider shadow than a quiet one, so a spectrum that falls without changing shape emerges from under itself.

So there is a route by which a tierce arrives late, and taking it requires a bell whose modes all decay together, which is not a property any bell has. The earlier route — the quint getting out of the way — does not exist, because the quint was never in the way.

How long each bell has one #

Read across the compass, the result is a register boundary with a clock attached.

C1, E1, G1, C2 and E2 never have a tierce, at any moment of the note. G2 has one for a sixth of a second — about the length of the strike itself. C3 has one for three and a half seconds, which is a musical duration.

The earlier boundary was a register: below A♭2 a bell’s tierce is masked. This one is a register and a duration, and the duration is the part that bears on what a listener hears. A bell rings for tens of seconds and its tierce is present for the first few or not at all, so on any bell large enough for the question to be interesting, the minor third is an attack phenomenon.

That is a claim about what a carillon’s bass sounds like, and it goes the opposite way to the received description. A large bell is usually described as having a minor character that emerges as the strike note settles — the clangour goes and what is left is the hum and the tierce. On this arithmetic what is left is the hum.

A G1 bell delivers five of its eight modes at the strike, three after a second, two after two seconds and one after four. A C2 bell delivers six, six, four and one. A C3 bell delivers eight at the strike and is down to two by eight seconds.

And the last mode standing is always the hum, which is the one mode a listener does not hear as the bell’s pitch: a bell has no fundamental is the essay about that — the strike note is inferred from the nominal, the twelfth and the double octave, and all three of them are gone within a second or two. A bell four seconds after its strike is delivering exactly one component, and it is an octave below the note anybody would name.

One note, before any of this #

The earlier figure of a single bell’s modes at the strike is the right thing to read this against, because it is the picture with the time taken out.

That figure is a set of verdicts at one instant, and the two modes it shows a bell losing are the two this essay has been about. What it cannot show is that they are lost for different reasons. The quint is masked by the prime and the nominal, which sit a fifth below and a fourth above it; the tierce is masked by the prime and the hum, which are both below it. Two adjacent modes, two different shadows, and only one of them has a neighbour above it doing any work.

That asymmetry is entirely a consequence of the shape of a masking skirt, and it is why the same figure supports two opposite predictions depending on what is assumed about decay. That essay read the pair as a crowd and predicted the crowd would thin. Read as two shadows cast upward, the pair does not thin: the casters are the slowest things in the bell.

What the model states and what it assumes #

Two of the numbers above are assumptions rather than measurements and it is worth separating them from the result.

The hum’s sixty-decibel time is set at forty seconds, which is a plausible figure for a large bell and is not read off anything. It scales the durations and it does not affect any comparison: doubling it doubles every span in the window figure and changes no ordering, because every mode’s decay time scales with it.

The exponent is the real assumption, and the sweep exists because of it. What the sweep shows is that the conclusion does not depend on its value: at every exponent that falls with frequency at all, a masked tierce stays masked and an audible one is lost, and the only thing the exponent decides is how quickly. The single value at which the result reverses is zero, and a bell with a flat decay law would be a bell whose hum and nominal ring for the same length of time, which is not what anyone has ever reported about a bell.

The amplitudes are the other assumption and they are the earlier essay’s. Each mode has an amplitude of one over its rank, which is a law imported from strings so that the three spectra it compared would differ only in their frequencies. A real bell’s modes have their own levels at the strike, which depend on where the clapper lands, and a tierce excited harder would survive longer. That is the one change that could rescue the result and it is exactly what that essay said a remedy would have to be: any remedy would have to change the modes’ amplitudes rather than their ratios, which in a bell means its profile or where it is struck.

Which computation produced the numbers #

The bell’s eight modes sit at the founder’s ratios relative to the hum — 1, 2, 2.4, 3, 4, 5, 6 and 8 — with the k-th lowest given an amplitude of one over k. The hum’s level at the strike is 70 decibels and each other mode’s is that plus twenty times the log of its amplitude.

Mode k’s sixty-decibel decay time is the hum’s, divided by its frequency ratio to the hum raised to the stated exponent, and at time t its level has fallen by sixty times t over that. A mode is delivered when its level exceeds both the threshold of hearing and the power sum of the masking spreading functions of every other mode still sounding, which is the criterion the whole masking anchor uses and is unchanged here except that the levels move.

The relief attributable to a single mode is the difference between the tierce’s masked threshold with the full spectrum and with that one mode removed, everything else held. The arrival and departure times are found by stepping the note in fiftieths of a second to a horizon of twenty.

Where the model stops #

The thresholds are steady-state thresholds. A masking threshold measured on a tone that has been sounding for a while is not the threshold that applies in the first tenth of a second, and how long a note has to be measures how long a percept needs to exist at all.

A bell’s modes are not pure tones with exponential envelopes. They beat against their own doublets, since a bell is nearly axisymmetric and each mode is really a close pair, and a beating mode’s level oscillates rather than falls. What that does to a threshold is not in this arithmetic.

Only the bell’s own modes mask it. A tower holds several bells and a carillon plays chords of them, which would put every other bell’s modes into the same sum — and the essays before this one say what a chord of a spectrum does to itself.

The strike is treated as instantaneous. The first tenth of a second of a bell is a clang with energy well above the tuned modes, and G2’s whole tierce window is a sixth of a second, so the one bell whose result is marginal is the one the model is least able to speak about.

What a count of modes cannot say about a bell #

Whether a listener hears a minor third. A mode’s being above its masked threshold is a statement about a filter bank, and hearing a chord’s quality is a statement about what a listener does with what gets through. The tierce could be delivered and not attended to, or masked in this sense and still colour the sound.

Whether the hum is heard at all. Four seconds into a large bell the model delivers the hum and nothing else, and the hum is a very low tone at a modest level. Whether that is a note or a residue of one is the question a bell has no fundamental is about from the other end.

Whether any of it is what a recording would show. The count here is of modes above a masked threshold, and a spectrogram of a bell shows every mode that is present whether a listener could hear it or not — which is the same gap the top that falls while the note lasts records between a string’s partials being there and being delivered.

What a founder would hear. Founders tune bells by ear against their own modes, listening to each in turn while damping the others — which is a procedure designed to defeat exactly the masking measured here, and which means a founder’s judgement of a tierce is not a prediction of what a listener in the square will get.

Still open: the strike, where the whole question is decided #

Every bell in this essay either has its tierce at the instant of the strike or never has it, and the instant of the strike is the one moment the model treats as a steady state.

A real bell’s first hundred milliseconds are not a set of modes at their steady levels. They are an impact: a broadband clang whose energy is spread across every band, falling away as the modes sort themselves out. During that window the tierce is masked by something much larger than the prime, and after it the modes are already decaying — so the level the earlier essay and this one both call “the strike” is a level the bell never actually has.

The computation that would settle it needs an onset rather than a new mechanism: give each mode a rise as well as a fall, with the higher modes rising faster, and run the same count across the first two hundred milliseconds. The prediction the two figures above make between them is that the tierce’s margin has a maximum somewhere inside that window rather than at its start — and if the maximum is above zero for bells the steady-state count says are hopeless, then the minor third of a large bell is not merely an attack phenomenon but a phenomenon of the attack’s own shape.