The major-third bell has no mirror problem

Nearly every church bell is tuned with a minor third, its tierce, a minor third above the prime. A bell is never minor alone ran a masking count across ten bells from C1 to C4 at every stroke from 20 to 110 decibels and found that colour fragile: the tierce is masked from below by the louder prime and hum, so it survives only soft strokes on a low bell, while the bell’s other third, the deciem, a major third above the nominal, survives nearly everything. A bell can be made minor by the third nobody tunes then flattened the deciem to the tierce’s octave, and the bell read minor in every cell that delivered a third — 131 of the 190.

That essay ended by turning the question round. A small number of bells have been cast with a major third: the tierce raised to 1.25 times the prime, a just major third, while everything else is left as a founder would have it. Those bells have the opposite arrangement — a major third low and the deciem, left at 2.5, a major third high. Their two thirds already agree. But a raised tierce sits further from the prime and nearer the quint, at 1.5, and the masking it meets is different. If the quint buried a raised tierce the way the prime buried a minor one, the major bell would have the minor bell’s problem in mirror image, with its lower third lost to the mode above it rather than below.

The count says it does not. The major-third bell is the robust one, and by a margin that follows from which mode actually does the masking.

The major-third bell, cell by cell #

The bells are the earlier essays’ ten: the same eight modes at the same ratios to the hum, the same amplitudes falling from the hum, the same masked-threshold count, in which a mode is delivered when it stands above both its threshold of hearing and the power sum of every other mode’s spreading function at its frequency. Only the tierce moves, from 1.2 times the prime to 1.25. A founder who raised the tierce would in practice change the bell’s profile and move other modes with it, and that is set aside below.

With the tierce at a major third, 141 of the 190 cells deliver a major third and none delivers thirds that disagree. The other 49 deliver no third: the lowest bells struck very softly, below the threshold of hearing, and every bell at the hardest strokes, where the nominal and the modes around it bury both thirds. Dragging the tierce down to 1.2 brings back the minor bell as cast, with 82 cells in which the minor tierce and major deciem are both delivered and disagree, and 59 in which only the deciem survives. At 1.22, a neutral third, the disagreement disappears for a different reason: the tierce is then neither minor nor major on the twenty-cent convention the earlier essay used, and the cells read as a neutral third where both are present.

The dots mark ten cells, one at the top of each register’s window, in which the major bell delivers a third and the flattened minor bell delivers none. Those are the strokes at which the deciem as cast still stands clear and the flattened deciem has been pulled down into the nominal’s skirt.

The raised tierce lasts longer #

The fear was for the tierce, so that is where to look first.

Raising the tierce does not shorten its window. It lengthens it, in every register where a tierce sounds at all. On a bell whose hum is on C2 the minor tierce is delivered from 28 to 43 decibels and the major one from 27.5 to 54.5 — eleven and a half decibels more stroke at the top. On E2, 60.5 against 69.5; on G2, 70.5 against 78; on C3, 83 against 89; on C4, 100.5 against 103.5. The gain is largest on the low bells, where the window is narrowest, and shrinks to about three decibels at the top of the range, where both tierces last to nearly the hardest stroke drawn. Below C2 neither tierce is ever delivered: the raised tierce does not appear anywhere the minor one was absent, because there the threshold of hearing and the prime’s masking both outweigh it.

The bottom of each window hardly moves — it is set by the threshold of hearing, and a tierce 70 cents higher is not noticeably easier to hear in quiet.

What actually masks a tierce #

Taken mode by mode, the masking at a C2 bell’s tierce, struck at 50 decibels, is a story about one mode. The tierce stands at 40.5 decibels. The prime puts a threshold of 39.6 decibels on the minor tierce — within a decibel of burying it — and the hum 34.2. The quint, a fifth above the prime, puts 27.7 on it; the nominal 8.7. Masking is a power sum, so the prime decides: the tierce is masked because it sits just above a louder neighbour, as one sound hides another, and it hides upward found for any pair of tones, and the quint above it is weaker and casts its masking downward, which spreading functions do much less.

Raise the tierce to 1.25 and the prime’s threshold at it falls to 38.6, a decibel lower, and the hum’s to 33.1. The quint’s rises to 29.4, nearly two decibels higher. The trade looks unfavourable — the quint gains more than the prime loses — but the quint’s threshold is still 9.2 decibels below the prime’s, and in a power sum a contribution nine decibels down counts for about an eighth. The total threshold falls. Near the top of the window the tierce’s margin shrinks only slowly as the stroke rises, because every mode grows louder together, so a small fall in the threshold moves the top of the window a long way: on C2, eleven and a half decibels of stroke.

The asymmetry is the whole of the answer to the mirror question. The minor bell’s problem was a strong mode below its tierce, casting the long upward skirt that masking has. Masking is lopsided in time as well as in frequency — a loud sound raises the threshold for a fifth of a second after it ends and only for milliseconds before it begins — and in both dimensions the long side is the side the minor tierce happens to be on. The major bell’s feared problem would have been a weaker mode above the tierce, casting the short downward one. The two are not mirror images, because masking itself is not symmetric.

A decibel, and eleven decibels of stroke #

The size of that effect looks out of proportion to its cause, and one bell shows why it is not.

On the C2 bell neither tierce is ever far from being masked. The minor tierce stands 0.9 decibels above its threshold at a stroke of 30, 0.2 at 40 and −0.5 at 50; the major tierce stands 2.0, 1.2 and 0.4 at the same strokes. Both then fall away together, crossing nought about a decibel apart in margin. But the margin falls very slowly with the stroke — every mode grows louder together as the bell is struck harder, and only the spreading functions’ growing breadth at high levels turns the balance against the tierce — so the stroke at which a margin crosses nought is extremely sensitive to the margin itself. Here a decibel of margin is about eleven decibels of stroke. The same sensitivity is why a loud chord is a smaller chord: the losses masking imposes grow with level, and they grow slowly enough that small differences between components decide which of them a hard stroke takes first.

The deciem lives in a different regime. As cast it stands 9.2 decibels clear at 40 and still 2.5 at 80; it is never close to nought until the hardest strokes, so moving it costs a few decibels of stroke and no more, which is what the earlier essay found when it flattened it. A tierce is a mode on the edge, and it is the mode a small retuning can most easily push off the edge or pull back from it.

Wherever the tierce is put #

The two tunings are points on a line, and the count can be run along all of it.

Moving the tierce from a narrow minor third up to almost a perfect fourth, the hardest stroke that still delivers it rises at every step on every bell. On C2: at 287 cents, 37 decibels; at the minor third, 316 cents, 43; at the major third, 386 cents, 54.5; at 494 cents, 65.5; at 583 cents, 70. On C3 the same curve runs from 80 to 97.5. It flattens as the tierce rises — each step gains less than the last — but it never turns over. No tuning of the tierce between the prime and the quint meets the quint’s masking before it has escaped the prime’s. The minor third is, of all the places a tierce could be put, among the most exposed, because it is among the nearest to the prime.

That gives the earlier essays’ finding a sharper form. The minor bell’s colour is fragile not because thirds are fragile in a bell but because the minor third is a small interval above a loud mode. A founder choosing where to put the bell’s third, if masking were the only consideration, would put it as high as the bell’s character allowed.

The colour, counted #

The question the earlier essay left was whether the major-third bell delivers its major colour across more of the register and the dynamics than the minor bell delivers its minor one. It does, but the reason is not the tierce.

The minor bell as cast never delivers a minor third alone, delivers both thirds in 82 cells and the major deciem alone in 59. Made consistent by flattening its deciem, it delivers a minor third in 131. The major bell delivers a major third in 141, and in 88 of those both its thirds are delivered together, agreeing. At the struck levels, 60 to 100 decibels, the counts are 61 for the consistent minor bell and 66 for the major one.

The ten-cell difference is all at the top of the windows, in the cells the hero’s dots marked, and it belongs to the deciem rather than the tierce. In every one of the 141 cells the major bell’s deciem is delivered, so its longer-lasting tierce adds agreement, not coverage. The minor bell, to be made consistent, had to move its deciem towards the nominal, which the earlier essay found costs it a few decibels of stroke at the top of every window. The major bell, made consistent, moves its tierce away from the prime, which costs nothing and gains. Consistency is cheaper for the major bell because its consistent arrangement leaves the robust third where it was.

The bell everyone casts #

The comparison the earlier essay asked for has a clear answer on these terms: the bell almost no one casts is the one whose colour survives being struck. Its lower third outlasts the minor tierce at every register, its upper third is the deciem exactly as founders already leave it, and the two never disagree.

This does not say major-third bells sound better. The minor-third bell’s sound is the tradition — the “minor” quality of a carillon or a ring of change-ringing bells is part of what listeners expect a bell to be, and the founder’s profile is a set of constraints, not a free choice of mode frequencies. What the figures do say is that the minor colour is the one that needs protecting from the listener’s own masking, and the major colour largely protects itself.

They also say something about the bells’ history that would otherwise be puzzling. Major-third bells have been designed and cast, and a bell has no fundamental noted how far a bell’s character depends on which modes a listener hears. If a major-third bell’s colour is heard more consistently than a minor bell’s, then the minor bell’s peculiar fragility — sounding minor when soft and major when loud, as a bell’s tierce does not arrive late, it leaves early traced through the decay — is part of what the tradition is, and a major-third bell would sound not only major but steadier.

The model the count rests on #

The bell’s spectrum is the earlier essays’ eight modes, with amplitudes falling from the hum by the stated law and every mode decaying faster the higher it lies. Only the tierce is moved. A real major-third bell is cast to a different profile, and raising its tierce changes the frequencies and amplitudes of other modes, the quint and the nominal especially. The count here isolates the tierce’s own effect; a real major-third bell’s numbers would differ by whatever its profile changes.

The masked-threshold count is the one every bell essay here has used: each mode against the power sum of the other modes’ spreading functions and the threshold of hearing, at the moment of the strike. It does not model the strike’s own transient, which a bell’s strike gives its tierce a millisecond took up, nor the decay after it, nor what one stroke of a ring does to the next. In change ringing a bell is struck again within a second or two, while its own previous stroke and its neighbours’ are still sounding, and a sound hides what came before it found a note masking itself harder than its predecessor can; a tierce that sits on the edge of its threshold at the strike is the mode most exposed to that kind of crowding, and the raised tierce’s extra decibel would matter there too, in a way a single stroke cannot show.

A third’s quality is named by a twenty-cent band round 350 cents, so that 330 to 370 is neutral. The convention decides only how the dial’s in-between tunings are labelled.

What the count cannot settle #

Whether a listener hears a bell’s colour from its lowest delivered third, its highest, or both. The cells count which thirds stand above their thresholds. With both thirds agreeing, as on the major bell, that question disappears; on the minor bell as cast it is the question, and nothing here answers it.

And whether the extra stroke on the tierce matters when the deciem is there anyway. In every cell where the major tierce is delivered, the major deciem is too. The tierce’s eleven extra decibels on C2 make a bell whose two thirds agree for longer, not a bell whose colour is heard for longer, and whether agreement itself is audible — whether two major thirds in two octaves make a stronger major colour than one — is a question about listeners.

Still open: whether the raised tierce changes the strike note #

The strike note of a bell is a pitch a listener hears that no mode supplies: a bell has no fundamental traced it to the nominal and the modes above it, which imply a common fundamental, and the flattened deciem was checked for what it did to that implication. A raised tierce sits in the lower half of the spectrum, below the modes that imply the strike, and so on the earlier arithmetic it should leave the strike note alone. But the tierce at 1.25 is now the fifth harmonic of a fundamental two octaves below the prime, and a tierce that fits a harmonic series may lend the hum note or a lower virtual pitch a support the minor tierce never gave. The computation is the strike-note count run with the tierce at each tuning on the dial, and it would say whether a major-third bell is heard at the same pitch as a minor one — or whether raising the tierce, by making the lower spectrum more harmonic, gives the bell a second, lower pitch to compete with its strike.