The raised tierce points the bell at its own note

The major-third bell has no mirror problem raised a bell’s tierce from a minor third above the prime to a major one and ran the masked-threshold count across ten bells from C1 to C4 at every stroke from 20 to 110 decibels. The raised tierce was the robust one: further from the prime, which is what masks a tierce, it survived harder strokes, and the major bell delivered its colour in 141 of the 190 cells.

That essay left one question it could not answer with the count. A bell is named by its strike note, a pitch no mode supplies: a bell has no fundamental traced it to the nominal, undeciem and duodeciem at two, three and four times the prime, which imply the prime as their common fundamental. The tierce is not among them, so on that arithmetic moving it should leave the strike note alone. But a tierce at 1.25 times the prime is the fifth harmonic of a fundamental two octaves below the prime, and a tierce that fits a harmonic series might lend the bell’s lower spectrum a pitch of its own — a second, lower pitch to compete with the strike.

It does lend one. The pitch it lends is the strike note’s own name.

Where the question was left #

The bells are the earlier essays’ ten. Each has eight modes at fixed ratios to the hum — 1, 2, 2.4, 3, 4, 5, 6 and 8 for a minor bell, with the tierce at 2.5 for a major one — and amplitudes falling from the hum as one over the mode’s rank. 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. The figure is the colour the count delivers; the question now is the pitch the delivered modes imply.

Four modes that make a series of their own #

The strike note comes from the top of the spectrum. The bottom of the spectrum is four modes: the hum, an octave below the prime; the prime; the tierce; and the quint, a fifth above the prime. Relative to the hum they sit at 1, 2, 2.4 and 3 in a minor bell and 1, 2, 2.5 and 3 in a major one. Any two or more of them, delivered together, are a small set of partials, and a small set of partials is something the ear can complete into a fundamental, as the missing-fundamental essays have done for every other object in this collection.

The residue fit used here is the one the missing-fundamental essays use: assign rising harmonic numbers to the partials, fit a fundamental by least squares, keep the fit if every partial lands within thirty cents of where it puts it, and prefer the fit with the fewest empty harmonic slots between its lowest and highest partial — the slots a listener completing the series would have to explain away.

With a major tierce the four lower modes are the 2nd, 4th, 5th and 6th harmonics of 65.4 hertz: half the hum, two octaves below the prime. One slot, the 3rd, is empty. That fundamental is C, the strike note’s own name, two octaves down. The prime, tierce and quint alone are 4 : 5 : 6 with no slot empty — a major triad’s own proportions, whose fundamental is two octaves below its lowest note.

With a minor tierce the same four modes are the 5th, 10th, 12th and 15th harmonics of 26.2 hertz, with seven empty slots. That fundamental is A♭: the note a major third below the strike note’s name, three octaves and more down. The prime, tierce and quint alone are 10 : 12 : 15, the proportions of a minor triad, with three slots empty.

So the lead had the direction backwards. The major tierce does make the lower spectrum more harmonic, and it does give it a lower virtual pitch — but that pitch is the strike note an octave below the hum, and it cannot compete with the strike for the bell’s name: it agrees with it. The minor tierce is the one that points elsewhere. A minor bell’s lower spectrum, read as a residue, names a note that is not the bell’s.

This is the minor triad’s ambiguity, found in a bronze object. The root an ear supplies ran the same kind of matching on triads and found the major triad fitting 4 : 5 : 6 with nothing missing, on a fundamental that is its own root, and the minor triad fitting two different series with two different answers, one of them a major third below its root. A minor bell’s prime, tierce and quint are a minor triad, and they inherit that ambiguity whole. The bell-founders’ minor third is not only a colour; it is a second candidate for the bell’s pitch, built into the spectrum’s lower half.

Read the other way, the major bell is the first bell in this collection whose two halves agree. Its upper modes, nominal, undeciem and duodeciem, are 2 : 3 : 4 on the prime; its lower modes, hum, prime, tierce and quint, are 2 : 4 : 5 : 6 on the prime’s double octave below; and all eight modes, the deciem included, lie on a single harmonic series on that low C, as its 2nd, 4th, 5th, 6th, 8th, 10th, 12th and 16th harmonics. A minor bell has never had that. Its tierce is the one mode in its lower half that lies on no series the others share, and the series it does fit, with the prime and quint, has a different fundamental. The two halves of a minor bell’s spectrum, read separately, name two notes a major third apart; the two halves of a major bell’s name one.

How often the struck bell delivers the second candidate #

What a listener could complete depends on which of the four modes are delivered at all, and the count decides that cell by cell.

With the tierce at 1.25 the figure has two colours. In 124 of the 190 cells the delivered lower modes point at the strike note’s name; in the other 66 fewer than two are delivered — the lowest bells, where the hum and prime fall below the threshold of hearing, and the hardest strokes, where the hum masks the rest. They point at no other note in any cell.

Turn the dial back to 1.2. The minor bell’s lower modes point at the strike note’s name in only 39 cells and at the note a major third below in 82. The pattern follows the tierce’s own window, which the earlier essays mapped: wherever the minor tierce is delivered alongside the prime and quint, the series they fit is the minor triad’s, and the residue moves a major third down. Where the tierce is masked — hard strokes, which the prime buries it under — the remaining lower modes are hum, prime and quint, harmonics 1, 2 and 3 of the hum, and point at the strike’s name again.

At the neutral tierce, 1.22, the lower modes already point at the strike’s name in 122 cells and at nothing else: within the thirty-cent tolerance, a tierce 344 cents above the prime is close enough to the fifth harmonic for the 2 : 4 : 5 : 6 series to absorb it. At 1.28, a little above just, the same holds. The second candidate belongs to the minor third and to nothing near the major one.

By register the shift is steady. The C1 and E1 bells deliver too little of their lower spectrum to imply anything a third away. From C2 the second candidate appears, in three strokes; by E2 it outnumbers the strike’s name eight to six; by C4 it holds seventeen of nineteen strokes, the strike’s name two. Higher bells deliver their lower modes at every stroke but the hardest, so the higher the bell, the more consistently a minor tierce is there to pull the lower spectrum’s residue down a third. The major bell’s right-hand bars are the totals of the minor bell’s left-hand ones, within a stroke: the raised tierce does not deliver more of the lower spectrum, it changes what the same delivered modes imply.

The strike note’s own support is untouched #

None of this disturbs the series the strike note is usually said to come from.

In every register the count is identical with either tierce: six strokes on C1, ten on E1, twelve on G1, fifteen on C2, sixteen on E2 and G2, eighteen on C3, nineteen on the three bells above. The tierce lies below all three of the upper modes, and its masking, which reaches upward, is too weak at their distance to change any of them. A bell can be made minor by the third nobody tunes found the deciem’s move to help the undeciem by clearing room under it; the tierce is too far away to do either.

So a bell has two places a listener could take a residue from, and they are independent. The upper three modes name the strike note whatever the tierce does. The lower four name the strike note too if the tierce is major, and a note a major third below it if the tierce is minor and delivered.

A weighted reading says the tierce is too faint to matter #

A residue fit treats every delivered mode alike, however narrowly it clears its masking. A listener presumably does not. The alternative reading weights each mode by how far it stands above its masked threshold, and sums the evidence for each candidate pitch over every mode in the bell — subharmonic summation, the method of a family of published pitch models, with each step down the subharmonics weighted by 0.84.

The two curves lie on top of each other. With either tierce the strongest candidate is the prime’s octave below — the hum, the strike note’s name — and the note a major third below the strike reaches 23 per cent of it in both. Across every cell of every register, the weighted reading names the strike note’s pitch class in all 182 cells where anything is delivered, with either tierce; the major-third-below candidate’s share of the winner is 0.36 at the median for the minor bell and 0.35 for the major.

The reason is visible in the margins. At this stroke the tierce clears its masked threshold by 4.2 decibels, while the other seven modes clear theirs by 12 to 29. The tierce is the most masked mode in the bell, which is the whole story of the earlier essays, and a reading that weights modes by their margins gives it almost nothing to say. Summed with everything else the bell sounds, the minor triad in its lower spectrum is outvoted by the hum, prime and quint’s simple 1 : 2 : 3 and by the upper modes’ 2 : 3 : 4.

The hum is the clearest case of why the two readings diverge. In the residue fit the hum is one partial among four, and its only work is to fix which octave of the series is meant. In the weighted reading it is the loudest mode in the bell, clearing its masking by nearly thirty decibels, and it pulls the strongest candidate down to its own octave: the weighted model names the bell’s note an octave below the strike, which is where the hum note is. Bell listeners do hear a hum note, as a separate low pitch, and the spectrum that will not fuse is a reminder that a bell’s modes need not be heard as one object at all. A listener who segregates the hum and prime from the rest is a listener whose residue comes from groups of modes, which is the first reading’s premise rather than the second’s.

The two readings disagree, then, and their disagreement is informative. If a listener completes small groups of partials into residues — attending, say, to the lower part of the sound as a chord — then a struck minor bell offers a pitch a major third below its strike note in most cells above C2, and a major bell offers none. If a listener integrates the whole spectrum weighted by audibility, neither tierce moves the pitch at all. Neither reading gives the major bell a competing pitch. The lead’s worry is answered on both.

What the readings rest on #

The bell’s modes are a founder’s targets, at exact ratios. A cast bell’s modes miss them by some cents each, and a tierce a few cents flat of 6 : 5 changes nothing here, since the residue fit allows thirty; a quint or hum well out of tune would weaken both series.

The amplitudes and the masking are the earlier essays’: one over the mode’s rank, at the strike, with no decay. A bell’s tierce does not arrive late, it leaves early found the tierce the first mode to fall out of hearing as a bell rings, so everything said here about the minor tierce’s second candidate is said of the strike and fades within the first second or two.

The residue fit is unweighted and prefers fewer empty slots, which is how it chooses between series an octave apart. The weighted reading’s 0.84 per subharmonic step and its use of margins as weights are a stated choice from the family of such models, not a fitted one.

What a residue cannot settle #

Whether listeners hear the second candidate. The figures say where a minor bell’s spectrum offers a pitch a major third below its name, and that a weighted model does not hear it. Bell listeners do report secondary strike notes and ambiguous pitches in some bells; whether those reports concern the lower spectrum’s minor triad is a question for a listening test, and the four lower modes the buttons play are the stimulus it would start from.

How masking and pitch interact. Both readings take the masked-threshold count as the gate: a mode below its masked threshold contributes nothing. One sound hides another set out that count, and it was built to say what is heard as a component, not what feeds a pitch. A partially masked mode may still contribute to a residue, and the weighted reading’s use of margins is one guess at how much.

What a carillon does. A carillon plays chords and tunes on bells, and a lower-spectrum pitch a major third below each bell’s name would sit inside the harmony being played. The chord is still major found that a major chord survives a great deal; a pitch that turns a bell’s lower spectrum into a different chord root is a different kind of disturbance, and none of the arithmetic here measures it.

Still open: the minor triad inside a peal #

The second candidate a minor bell offers is the note a major third below its strike note’s name. In a ring of eight bells tuned to a major scale, that note is sometimes in the scale and sometimes not: below the tonic bell it is the flattened sixth, which a major scale does not contain; below the mediant it is the tonic itself; below the dominant it is the mediant. So the lower spectrum of each minor bell in a ring would point either at another bell’s strike note or at a note the ring never rings. The computation is this one run across a ring: each bell’s delivered lower modes at the strokes ringers use, the note they imply, and whether that note is one of the ring’s own strike notes. It would say how many of a minor-tierce peal’s hidden second pitches fall on the scale it rings — reinforcing the ring’s own harmony — and how many fall off it, where they would be heard, if at all, as something foreign to the ring.