Perception and the listener

A bell is never minor alone

A tuned bell carries two thirds: the tierce, a minor third above the prime, which is why a bell is called minor, and the deciem, a major third above the nominal. Run the masking count across every register and every stroke and the minor one is never delivered without the major one. Struck softly a bell delivers both; struck harder, only the major; and below C2 the tierce is never delivered at all. The deciem survives because it is masked by one mode barely louder than itself, the tierce by two that outweigh it.

14 min read 6 figures The limits of the listener The instrument is not the model What the listener supplies

Assumes: A bell's strike gives its tierce a millisecond · A bell's tierce does not arrive late, it leaves early

The minor colour of a church bell comes from one mode. A well-tuned bell’s eight principal modes stand at fixed ratios — the hum an octave below the prime, the tierce a minor third above it, the quint a fifth above, the nominal an octave above, and three more above that — and a bell has no fundamental set out how the pitch a listener names, the strike note, is implied by three of the upper ones. The tierce is the reason a bell is called a minor-third bell.

The three essays before this one followed the tierce on low bells and found it in trouble. It is masked from below, by the prime and the hum, so on a bell lower than about E2 it is not delivered at the strike and is not delivered later either, since the modes that mask it outlast it. A bell’s strike gives its tierce a millisecond found the onset no help, and found one thing that does help: a softer stroke. Each ten decibels of stroke costs the tierce about one decibel of margin, so a C2 bell struck with its hum at 40 dB keeps its tierce where the same bell at 70 dB loses it.

That essay closed by asking what a listener would make of that. Join the tierce’s fate across dynamics to the pitch the bell is heard at, and the prediction becomes specific: a C2 bell struck softly is heard as minor, and struck at seventy as neither minor nor anything else.

The first half of that prediction overlooked a mode. A bell has a second third, and the count delivers it more readily than the first.

A bell is never minor alone. For bells whose hum is on C1, E1, G1, C2, E2, G2, C3, E3, G3, C4, struck so that the hum is at 20 to 110 decibels: whether the masked-threshold count delivers the tierce, the minor third above the prime, and the deciem, the major third above the nominal. C1: ···MMMM············; E1: ··MMMMMMMM·········; G1: ··MMMMMMMMMM·······; C2: ·MBBBMMMMMMMMMM····; E2: ·BBBBBBBBMMMMMMM···; G2: ·BBBBBBBBBBMMMMMM··; C3: BBBBBBBBBBBBBMMMMM·; E3: BBBBBBBBBBBBBBBMMM·; G3: BBBBBBBBBBBBBBBBMMM; C4: BBBBBBBBBBBBBBBBBMM (B both thirds, M the major only, m the minor only, a dot neither; levels in steps of five from 20). Of 190 cells, 82 deliver both thirds, 59 the major third only, 49 neither, and 0 the minor third alone. A dot in a cell marks a stroke at which fewer than two of the three modes that imply the strike note survive. — Fig. 1 Bells whose hum stands from C1 to C4, struck so that the hum is at 20 to 110 decibels: whether the masked-threshold count delivers the tierce, the minor third above the prime, and the deciem, the major third above the nominal. Of 190 cells, 82 deliver both thirds, 59 only the major one, 49 neither, and none the minor one alone.

The other third

Above the nominal, at two and a half times the prime, sits the deciem. It is a major third above the nominal and so a major tenth above the prime and the strike note. Founders tune it, it is in every list of a bell’s partials, and it was in the count these essays have run all along; no essay on the bell’s masking had asked what happens to it.

The count is the one 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, steep below each masker and shallower above it, the upper slope growing shallower as the masker grows louder; a mode is delivered when it stands above both that threshold and the threshold of hearing. The bell’s modes take the amplitudes the bell sequence has always used, falling from the hum, so the prime is six decibels below it, the tierce 9.5, the nominal 14 and the deciem 15.6. Struck, they have those levels relative to the hum’s, which is the stroke’s level.

The first figure is the result, bell by bell and stroke by stroke. Every cell is one of three kinds. The softest strokes deliver neither third, because both are under the threshold of hearing. A band of strokes above that delivers both. Above it lies a band in which the deciem is delivered and the tierce is not. At the loudest strokes on the lowest bells, neither.

There is no fourth kind. In no register and at no stroke does the count deliver the tierce without the deciem. A bell is minor, in the sense that its minor third reaches the ear, only when its major third reaches the ear as well.

Each third’s window of strokes

The major third's window of dynamics contains the minor third's. For each bell, the range of stroke levels, as the hum's level at the strike, over which the count delivers each third: C1: tierce never, deciem 33.5 to 52.5; E1: tierce never, deciem 30 to 69; G1: tierce never, deciem 27.5 to 78.5; C2: tierce 28 to 43, deciem 24 to 90; E2: tierce 24.5 to 60.5, deciem 22 to 97; G2: tierce 22 to 70.5, deciem 20.5 to 101; C3: tierce 20 to 83, deciem 20 to 106; E3: tierce 20 to 90.5, deciem 20 to 108.5; G3: tierce 20 to 95, deciem 20 to 110; C4: tierce 20 to 100.5, deciem 20 to 110. The lower end of each window is the threshold of hearing; the upper end is where the modes below mask it. The deciem's window begins lower and ends higher in every register, so between the top of the tierce's window and the top of the deciem's, a bell delivers its major third and not its minor one. — Fig. 2 For each bell, the stroke levels over which the count delivers each third. The tierce’s window sits inside the deciem’s in every register; below C2 the tierce has no window at all.

Read register by register, each third is delivered over a window of stroke levels. The bottom of each window is the threshold of hearing: a soft enough stroke leaves the mode inaudible on its own. The top is masking: a hard enough stroke lets the modes below cast their shadows over it.

On a C2 bell the tierce’s window runs from 28 to 43 decibels; the deciem’s from 24 to 90. On E2 the tierce has 24.5 to 60.5 and the deciem 22 to 97; on G2, 22 to 70.5 against 20.5 to 101; on C3, 20 to 83 against 20 to 106. The deciem’s window starts at or below the tierce’s and ends well above it every time. On G1, E1 and C1 the tierce has no window at all, and the deciem still has one: 27.5 to 78.5 on G1, 33.5 to 52.5 even on C1.

So on a C2 bell, every stroke between 43 and 90 decibels delivers a bell with its major tenth and without its minor third. That is forty-seven decibels of dynamic range — most of what a bell can be struck at. On E2 the corresponding band runs from 60.5 to 97, on C3 from 83 to 106.

The prediction for a soft stroke needs correcting accordingly. A C2 bell struck at 40 dB does deliver its tierce, by 0.2 decibels. It also delivers its deciem, by 9.3. It is not a bell whose only third is minor; it is a bell carrying both.

One mode barely louder against two that outweigh it

The tierce is masked from far below, the deciem from just below. A bell whose hum is on C2 struck with the hum at 70 dB: how far each third's masked threshold falls when each other mode is taken away. The tierce, at 60.5 dB against a threshold of 62.6: hum 1.8, prime 4.4, quint 0.1, nominal 0.0, deciem 0.0, undeciem 0.0, duodeciem 0.0. The deciem, at 54.4 dB against a threshold of 49.8: hum 0.2, prime 0.4, tierce 0.4, quint 0.6, nominal 4.0, undeciem 0.2, duodeciem 0.0. The tierce's shadow is cast by the prime, a minor third below and 3.5 decibels louder, and the hum, nearly two octaves below and 9.5 louder; the deciem's by the nominal, a major third below and 1.6 decibels louder. A masker's reach upward grows with its level, so the louder the stroke the more the tierce loses to modes that outweigh it, and the less the deciem loses to one that barely does. — Fig. 3 A C2 bell struck with the hum at 70 dB: how far each third’s masked threshold falls when each other mode is taken away. The tierce is held down by the prime and the hum; the deciem by the nominal alone.

Why the two thirds fare so differently is in what masks each. Take each other mode away in turn and see how far the threshold over each third falls. Over the tierce, removing the prime lowers it by 4.4 decibels and removing the hum by 1.8; nothing else matters. Over the deciem, removing the nominal lowers it by 4.0, and every other mode by less than a decibel.

The two maskers differ in the one way the masking count is most sensitive to: how much louder they are than what they mask. The prime is 3.5 decibels louder than the tierce, and the hum 9.5. The nominal is 1.6 louder than the deciem. A masker’s shadow above it grows faster than the masker itself, because its upper slope flattens as it gets louder — the effect a bell’s strike gives its tierce a millisecond traced through the onset — and so every decibel added to a stroke adds more to the tierce’s threshold than to the tierce. The deciem’s masker has almost no head start on it, so it gains on the deciem far more slowly.

A clarinet keeps what a string loses found the same arithmetic in a different instrument: what survives in a spectrum’s upper reaches is decided by how much louder the components below it are. The founder’s amplitude law puts the tierce under two strong modes and the deciem under one weak one. That decides which third outlives the other at every stroke and in every register.

One bell, swept

Fig. 4 A bell whose hum is on C2: the tierce’s, the deciem’s, the nominal’s and the prime’s margins above their thresholds against the stroke. The tierce is above zero only from 28 to 43 decibels; the deciem from 24 to 90. The dial moves the bell from C1 to C4.

Drawn for one bell, the margins show the window’s two edges as two different slopes. Every mode rises steeply out of the threshold of hearing at the soft end, peaks where the stroke is just loud enough to clear it, and then falls slowly as the masking grows. On the C2 bell the tierce peaks at a margin of 0.9 decibels at a stroke of 30 and is below zero again by 45. The deciem peaks at 9.3 at 40 and stays above zero until 90. The nominal follows the deciem’s shape a few decibels lower and leaves at 81.5; the prime leaves at 87.

The dial carries the picture up and down the register. Each step down lowers every margin’s peak and shortens every window, the tierce’s fastest. At G1 the tierce’s peak never reaches zero. At C3 and above the tierce clears its threshold for most of the range a bell is ever struck at, and the difference between the thirds matters only at the hardest strokes.

A carillon at one stroke

A carillon is played across its whole compass at once, so the windows are worth reading the other way: fix the stroke and walk down the bells. At a stroke that puts each bell’s hum at 70 decibels, every bell drawn from G2 upward delivers both thirds. From E2 down to G1 the tierce has gone and the deciem remains, so those bells deliver only the major third. E1 and C1 deliver neither, because by 70 decibels their deciem is masked too.

So a single passage played at a single dynamic crosses a boundary in colour somewhere in the instrument’s lowest octave and a half. The boundary moves with the touch. Played at 50 decibels, it falls between C2 and E2: the C2 bell’s tierce is gone by 43, but the E2 bell keeps both thirds to 60. Played at 90, it rises to between C3 and E3, E3 being the lowest bell drawn whose tierce survives a stroke that hard.

That is the same shape the arch belongs to hearing, not to the series found in a bell’s whole spectrum: the modes a bell delivers thin out at the bottom of the compass, and the thinning is a property of the ear’s masking rather than of the bell’s tuning. A low chord stops being a chord found its chord tones going the same way, in the same order of loudness, as a chord was moved down. The bell adds one thing to that picture. Its two thirds sit in different places on the arch, and the lower one, which gives the instrument its name, goes first.

A carillonneur therefore has two ways to make the bass of a passage sound more like the treble. Playing the bass softly keeps more of the tierce. Playing the passage higher, if the arrangement allows it, moves the bass above the boundary. The arithmetic cannot say which a listener would notice more.

What happens to the strike note

The prediction’s second half was that a hard stroke on a low bell leaves it “neither”: not minor, and perhaps not clearly pitched. The count bears on the pitch too, through the three modes that imply it.

A hard stroke takes the strike note's support from the bottom. How many of the nominal, the twelfth and the double octave — the modes at two, three and four times the prime, which imply the strike note — the count delivers, against the hum's level at the strike, for bells whose hum is on E1, G1, C2, E2, C3: E1 0023333322221111000; G1 0133333333222211100; C2 0333333333333222110; E2 0333333333333332211; C3 2333333333333333321 (levels in steps of five from 20). The nominal goes first, at 81.5 dB on the C2 bell, then the twelfth at 95.5, while the hum is delivered at every level above its threshold of hearing. At the hardest strokes on the lowest bells, the series that names the strike note has gone and the hum, an octave below it, is what is left. — Fig. 5 How many of the nominal, the twelfth and the double octave — the modes at two, three and four times the prime that imply the strike note — the count delivers, against the stroke, for five bells. A hard stroke takes them from the bottom; the hum outlasts them all.

The strike note is implied by modes at two, three and four times the prime, a harmonic series on the prime that the prime itself need not be present to name. Struck hard, a low bell loses those three modes from the bottom: on C2 the nominal goes at 81.5 decibels, the twelfth at 95.5, and only the double octave survives to 108. Below two of the three, the series that names the strike note is no longer there to name it. The hum, an octave below the strike note and the bell’s loudest mode, is delivered at every stroke above its threshold of hearing.

So the loudest strokes on the lowest bells do produce a bell that is “neither” — no third of either kind, and no strike note — and what is left is the hum. On C2 that takes a stroke above 95 decibels; on G1, 90; on E1, 80. Below that, a hard stroke on a low bell is not neither. It is a bell with a strike note and a major tenth.

As the bell rings

As a bell rings, the minor third leaves first. Which of five modes the count delivers at moments from the strike to eight seconds, for a C2 bell struck softly (hum at 40 dB) and harder (70), and an E2 bell at 50: C2 at 40 dB: tierce to 0.25 s, deciem to 2 s, nominal to 2 s, undeciem to 1.5 s, duodeciem to 1 s; C2 at 70 dB: tierce never, deciem to 2 s, nominal to 1 s, undeciem to 3 s, duodeciem to 3 s; E2 at 50 dB: tierce to 1.5 s, deciem to 3 s, nominal to 4 s, undeciem to 3 s, duodeciem to 2 s. Every mode decays faster the higher it is, and the masking from below lasts. The tierce is gone first and the deciem after it. The strike note's series goes from the top at a soft stroke, where the threshold of hearing takes the fastest-decaying modes, and from the bottom at a harder one, where the nominal is masked first. — Fig. 6 Which of five modes the count delivers from the strike to eight seconds, for a C2 bell struck at 40 and at 70 decibels and an E2 bell at 50. The tierce leaves first, the deciem after it.

Everything above is at the strike. A bell then rings for tens of seconds, each mode decaying faster the higher it is, and the masking from below lasting longest. The order of departure follows from that. On the C2 bell struck at 40 the tierce is gone within a quarter of a second, the deciem at two seconds, the series modes between one and two. Struck at 70 the tierce was never there, and the deciem lasts two seconds. On E2 at 50 the tierce lasts a second and a half and the deciem three.

So the brief band of strokes in which a low bell delivers both thirds delivers both only for a moment; for the rest of the note’s useful life the deciem is the one third left. And the strike note’s series fades in a different order at different strokes: from the top at a soft one, where the threshold of hearing takes the fastest-decaying modes first, and from the bottom at a hard one, where masking takes the nominal first.

The model the cells rest on

Eight modes at the ratios of a minor-third bell, from the hum: 0.5, 1, 1.2, 1.5, 2, 2.5, 3 and 4 times the prime, with amplitudes falling from the hum by 0, 6, 9.5, 12, 14, 15.6, 16.9 and 18.1 decibels, and decay times of forty seconds at the hum divided by each mode’s ratio to it. A mode’s masked threshold is the power sum, at its frequency, of every other mode’s spreading function, with the upper slope flattening as the masker’s level rises; it is delivered when it stands above both that threshold and the threshold of hearing at its frequency. The stroke’s level is the hum’s at the strike. Bells are named by the hum, following the earlier essays.

What the count cannot say

How a listener weighs the two thirds. The tierce is a minor third above the prime, close to where the strike note is heard. The deciem is a major third above the nominal — a major tenth above the strike note, an octave and a third higher. Delivered, both are present in the spectrum the ear receives. Whether a listener hears a bell carrying its deciem and not its tierce as major, as neutral, or as a minor bell with a bright top is a question about how thirds in different octaves are weighed against a pitch, and nothing here measures it. The chord is still major is a reminder of how wide a category a third occupies when it is a third at all. What the count says is narrower: at most strokes on low bells, the only third the ear receives is a major one.

The amplitude law. Every amplitude here is the bell sequence’s single law. A founder who cast the deciem weaker, or the tierce stronger relative to the prime, would move both windows. Real bells vary, and the deciem in particular is not always prominent; the figures describe a bell with these amplitudes, not every bell.

The clapper and the room. A clapper’s impact adds broadband noise in the first tens of milliseconds, and a tower or a church adds reverberation. Both add masking and can only narrow the windows. Neither changes which third is masked by the louder modes.

Who listens to bells at what level

A carillonneur controls the stroke with hands and feet through a mechanical action, and plays the bass bells loud because they carry least. A change-ringing tower strikes its bells at whatever level the ropes and the wheel give it. A clock bell is struck hard by a hammer. On this arithmetic all of those sit in the band where a low bell’s only delivered third is major, and the minor colour that names the bell is the property of its soft strokes and its higher bells. Whether listeners in a carillon’s square hear the bass bells as less minor than the treble, and whether that difference changes with the player’s touch, is a question a recording session and a few listeners could answer.

Still open: a founder who tunes the deciem

Founders tune five of a bell’s modes and the deciem is usually not one of them: it falls where the profile puts it. The count says the deciem is the third a low bell actually delivers across most of its dynamic range. A founder who moved it — flattening it from a major to a minor third above the nominal — would give a low bell a minor third that survives a hard stroke. The computation is this one with the deciem’s ratio moved from 2.5 towards 2.4, the minor tenth, and its masking recomputed; it would say whether a low bell can be made minor at the dynamics it is actually struck at, by tuning the mode nobody listens for.