Perception and the listener

A bell can be made minor by the third nobody tunes

A low bell struck hard delivers its major third, the deciem, and loses its minor one, the tierce, to masking. A founder could flatten the deciem from a major to a minor third above the nominal, which puts it exactly an octave above the tierce. The masking count says that move costs the deciem only one or two decibels of stroke at the top of its window, so the flattened deciem still outlasts the tierce in every register. A bell tuned that way reads minor in every cell of register and stroke that delivers a third at all. In-between tunings read as neutral thirds, so the move only works if it goes all the way.

15 min read 7 figures The instrument is not the model The limits of the listener The design was forced

Assumes: A bell is never minor alone · A bell has no fundamental

A church bell is called minor because of one mode. Its tierce sits a minor third above the prime, and every founder who tunes a bell puts it there. A bell is never minor alone ran the masking count across ten bells from C1 to C4, each struck at every level from 20 to 110 decibels, and found the minor colour fragile. The tierce is masked from below by the prime and the hum, both louder than it, so on a low bell it survives only soft strokes, and below C2 it never survives at all. Meanwhile the bell’s other third, the deciem, a major third above the nominal, survives nearly everything: it is masked mainly by the nominal just below it, which is barely louder than itself. In fifty-nine of the 190 cells of register and stroke, the bell delivered its major third and not its minor one, and in no cell did it deliver the minor alone.

That essay ended by naming a remedy no founder seems to have tried. Founders tune five of a bell’s modes, the hum, the prime, the tierce, the quint and the nominal, and the deciem is not one of them. It falls wherever the bell’s profile puts it. Flatten it from a major third above the nominal to a minor one, and the third a hard stroke delivers would be minor.

A bell whose deciem is the tierce's octave is minor at every stroke. 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, with the deciem tuned from 2.5 times the prime to 2.4 — a minor third above the nominal and exactly an octave above the tierce: whether the masked-threshold count delivers a third at all. C1: ···mm··············; E1: ···mmmmmm··········; G1: ··mmmmmmmmm········; C2: ·mmmmmmmmmmmmm·····; E2: ·mmmmmmmmmmmmmm····; G2: ·mmmmmmmmmmmmmmm···; C3: mmmmmmmmmmmmmmmmm··; E3: mmmmmmmmmmmmmmmmmm·; G3: mmmmmmmmmmmmmmmmmm·; C4: mmmmmmmmmmmmmmmmmmm (m a minor third, a dot none; levels in steps of five from 20). 131 of 190 cells read minor and none major. Of the 59 cells in which the bell as cast delivered only its major third, 49 now deliver a minor one. — Fig. 1 For bells from C1 to C4 struck with the hum at 20 to 110 decibels, with the deciem tuned from 2.5 times the prime to 2.4 — a minor third above the nominal, exactly an octave above the tierce — whether the count delivers a third. Every cell that delivers one delivers a minor third; the dots mark cells the bell as cast delivered as major.

The move, and why it lands on the tierce’s octave

The deciem as cast is at 2.5 times the prime, the nominal at 2. Their ratio, 5 : 4, is a just major third, 386 cents. A minor third above the nominal is 6 : 5 of it, 2.4 times the prime, 316 cents above the nominal.

And 2.4 is exactly twice the tierce’s 1.2. The flattened deciem is not just a minor third above the nominal; it is the tierce’s own octave. A bell tuned this way carries one minor third in two octaves rather than a minor third in one and a major third in the other. Whatever the ear does with thirds in different octaves, which the earlier essay could not settle, it would no longer have to reconcile two that disagree.

Everything else is held where the earlier essays put it: the same eight modes at the same ratios, the same amplitudes falling from the hum, the same decay law, 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 deciem’s frequency moves. That is a simplification a founder would not have, and it is taken up below.

What it costs the deciem

The obvious objection is that moving the deciem down moves it towards the nominal, which masks it. The nominal’s spreading function falls away above it, so a mode closer to it sits higher on its skirt and needs more level to stand clear.

Flattened to a minor third, the deciem still outlasts the tierce. For each bell, the range of strokes over which the count delivers the tierce, the deciem as cast at 2.5 times the prime, and the deciem tuned to 2.4: C1: tierce never, deciem 33.5–52.5 as cast, 34–43.5 flattened; E1: tierce never, deciem 30–69 as cast, 30.5–61.5 flattened; G1: tierce never, deciem 27.5–78.5 as cast, 28–72 flattened; C2: tierce 28–43, deciem 24–90 as cast, 24.5–85.5 flattened; E2: tierce 24.5–60.5, deciem 22–97 as cast, 22–93.5 flattened; G2: tierce 22–70.5, deciem 20.5–101 as cast, 20.5–98 flattened; C3: tierce 20–83, deciem 20–106 as cast, 20–104 flattened; E3: tierce 20–90.5, deciem 20–108.5 as cast, 20–107 flattened; G3: tierce 20–95, deciem 20–110 as cast, 20–108.5 flattened; C4: tierce 20–100.5, deciem 20–110 as cast, 20–110 flattened. Moving the deciem towards the nominal costs it a few decibels of stroke at the top of its window, because the nominal below it now masks it more; it still reaches far higher strokes than the tierce in every register. — Fig. 2 For each register, the strokes at which the count delivers the deciem as cast, the deciem at 2.4, and the tierce. Flattening the deciem trims a few decibels off the top of its window in every register; the trimmed window still reaches far harder strokes than the tierce’s.

The mechanism is the one one sound hides another set out for any two tones: a masker’s influence spreads upward in frequency further than downward, and the nearer a weaker tone sits above a louder one, the more of it is taken. It does cost something, and the cost is small. On a bell whose hum is on C2, the deciem as cast is delivered up to a stroke of 90 decibels; flattened, up to 85.5. On E2, 97 against 93.5; on C3, 106 against 104. The cost is largest on the lowest bells, because there the whole window is narrowest and a few decibels is a large share of it: on C1 the deciem as cast is delivered from 33.5 to 52.5 decibels and flattened from 34 to 43.5. The bottom of every window, set by the threshold of hearing, hardly moves.

What does not change is the comparison that matters. The tierce on C2 is delivered only from 28 to 43 decibels. The flattened deciem on C2 reaches 85.5, more than forty decibels higher. On E2 the tierce stops at 60.5 and the flattened deciem at 93.5; on C3, 83 against 104. In every register where the tierce exists at all the flattened deciem outlasts it, by forty-two decibels of stroke on C2, falling to about ten on C4 where both last to nearly the hardest stroke drawn; and below C2, where the tierce never sounds, the flattened deciem still does.

The flattened deciem gives up a decibel or two; the tierce is gone by 45. A bell on C2: how far each of four modes stands above its masked threshold, against how hard it is struck. the tierce: 20 dB -7.8, 30 dB 0.9, 40 dB 0.2, 50 dB -0.5, 60 dB -1.3, 70 dB -2.2, 80 dB -3.1, 90 dB -4.2, 100 dB -5.4, 110 dB -6.6; the deciem as cast: 20 dB -3.8, 30 dB 6.2, 40 dB 9.3, 50 dB 7.9, 60 dB 6.4, 70 dB 4.7, 80 dB 2.6, 90 dB 0.1, 100 dB -2.8, 110 dB -6.2; the deciem at 2.4: 20 dB -4.2, 30 dB 5.8, 40 dB 7.3, 50 dB 6.1, 60 dB 4.7, 70 dB 3.2, 80 dB 1.3, 90 dB -1.0, 100 dB -3.7, 110 dB -6.8; the nominal: 20 dB -4.8, 30 dB 5.2, 40 dB 7.8, 50 dB 6.2, 60 dB 4.4, 70 dB 2.5, 80 dB 0.4, 90 dB -1.9, 100 dB -4.4, 110 dB -7.1. A mode is delivered while its margin is above nought. — Fig. 3 A bell on C2: how far the tierce, the nominal and the deciem as cast and flattened stand above their masked thresholds, against the stroke. The flattened deciem runs one or two decibels under the cast one; the tierce is above nought only between about 28 and 43 decibels.

On one bell the margins show the size of the trade directly. At 40 decibels the cast deciem stands 9.3 decibels above its masked threshold and the flattened one 7.3. At 60, 6.4 and 4.7; at 80, 2.6 and 1.3. The flattened deciem loses one to two decibels of margin at every stroke, which moves the top of its window by about four and a half decibels of stroke, because near the top of the window its margin falls by about one decibel for every four of stroke. The tierce is below nought at every stroke above 43 decibels. The two minor thirds of a flattened bell are not equally robust: one lasts to a moderate stroke, the other to a hard one.

Drawn as the hardest stroke at which each register still delivers its deciem, the cost becomes a single gap between two lines, and it can be watched as the deciem is moved.

Fig. 4 For each bell, the hardest stroke at which the count still delivers the deciem, at the dial’s tuning and as cast, with the tierce’s hardest stroke as dots. Drag the deciem from 2.5 down to 2.375: the line falls a few decibels in the low registers and hardly at all above C3, and stays far above the dots.

The gap is widest at the bottom of the compass and closes towards the top. On C1 it is nine decibels, on G1 six and a half, on C2 four and a half, on C3 two, and on C4 nothing: the high bells deliver their deciem to the hardest stroke drawn whatever its tuning. Flattening further, to 2.375 — a third of 298 cents, a little narrower than a just minor one — takes a few more decibels off the low bells: C1’s deciem then stops at 40.5 decibels, C2’s at 84, C3’s at 103. That still leaves the line far above the tierce’s dots, and it shows the cost growing smoothly with the distance moved rather than jumping at any particular tuning. A founder aiming at 2.4 and landing a little flat would lose a decibel or two more of window, and nothing else. The cost of the move is concentrated exactly where the tierce has already gone, in the bells that need the move most, and it is small there too.

Minor at every stroke that has a third

The hero figure puts the two together. With the deciem at 2.4, 131 of the 190 cells deliver a third, and every one of them is minor. The fifty-nine cells in which the cast bell delivered only its major third now include forty-nine that deliver a minor one; the other ten are at the top of each low bell’s window, where the few decibels the flattened deciem gave up were the whole of its margin, and now deliver no third at all.

That answers the question the earlier essay left. A low bell can be made minor at the dynamics it is actually struck at, by tuning the mode nobody listens for. On C2, struck hard enough to bury the tierce — anything from about 45 to 85 decibels — the cast bell’s only third is major and the flattened bell’s only third is minor. On C1, the lowest bell drawn, the flattened bell is minor between about 35 and 43 decibels and has no third outside that range.

A bell’s tierce does not arrive late, it leaves early found that the tierce is the first of the upper modes to go as a bell rings. Everything so far is counted at the strike, and a bell is heard for seconds afterwards. The deciem decays faster than the tierce, being higher, but starts with more margin, so whether the flattened deciem carries the minor colour into the ring is a count of its own.

Through the ring, the flattened deciem keeps the minor colour the tierce has lost. Three bells struck so that the hum is at 80 decibels, every mode decaying at forty seconds divided by its ratio to the hum: at each half-second over ten seconds, whether the count delivers the tierce, the deciem as cast and the deciem at 2.4. C2: tierce never, deciem as cast 0–1 s, flattened 0–0.5 s; E2: tierce never, deciem as cast 0–2.5 s, flattened 0–2 s; C3: tierce to 0.5 s, deciem as cast 0–6 s, flattened 0–6 s. — Fig. 5 Bells on C2, E2 and C3 struck with the hum at 80 decibels, each mode decaying at forty seconds over its ratio to the hum: at each half-second, whether the count delivers the deciem as cast, the deciem at 2.4 and the tierce. On C2 and E2 the tierce is never delivered; the flattened deciem carries a minor third for half a second and two seconds. On C3 it lasts six seconds against the tierce’s half-second.

Struck at 80 decibels, the C2 bell never delivers its tierce; the cast deciem lasts a second into the ring and the flattened one half a second. On E2 the tierce is again never delivered, and the flattened deciem lasts two seconds against the cast one’s two and a half. On C3 the tierce is delivered for half a second and the flattened deciem for six, exactly as long as the cast one. So the ring does not change the verdict: a hard stroke on a low bell delivers a third for at most a few seconds, and if the deciem has been flattened, that third is minor for as long as it lasts. It is short on C2 because a stroke of 80 decibels is already within ten of the top of that bell’s deciem window, and the ring only takes level away. A bell’s strike gives its tierce a millisecond found the onset no help to the tierce; nothing in the ring helps it either.

The way there is through the neutral third

A founder cannot move a mode by a twentieth of a ratio and nothing in between. Every bell on the way from a deciem at 2.5 to one at 2.4 has a third somewhere between major and minor, and what a bell on the way reads as depends on where it is.

Fig. 6 Of every bell from C1 to C4 struck with the hum at 60 to 100 decibels, the share that reads as a major third, as a neutral third (within twenty cents of 350 cents above the nominal), as minor, or as no third, against the deciem’s ratio to the prime. Between 2.475 and 2.425 the struck bells read neutral; from 2.4 they read minor.

The figure counts the struck bells, the cells with the hum at 60 to 100 decibels where a carillon’s bass bells and a tower’s ring sit. As cast, 73 per cent of them deliver a third, and every one of those reads major: the tierce is buried, or is outweighed by a deciem a major third above the nominal. Flatten the deciem to 2.475, 369 cents above the nominal, and on the convention used here — a third within twenty cents of 350 counts as neutral — 72 per cent read neutral. They stay neutral down to 2.425, 334 cents. At 2.4 they read minor, 68 per cent of the struck bells. Further flattening, to a deciem a little below a minor third, changes almost nothing; it costs a few more cells at the top of each window.

The neutral band is a convention, and the essays on categorical hearing are the reason for drawing one at all. The ear sorts into boxes found listeners’ categories of interval to be bounded, with a region between major and minor where identification is uncertain, and how many boxes an octave holds put a number on how fine those boxes can be. Where the boundary falls for a bell’s deciem, heard an octave and a third above its strike note, is not measured; a band of forty cents around 350 is a stand-in. What the figure shows does not depend on its exact width. There is no partial version of the move. A deciem flattened part of the way does not make a bell somewhat minor; it makes it something else.

What the move does for the strike note

A founder might worry that moving a mode disturbs the pitch the bell is named for. A bell has no fundamental set out how the strike note is implied by three upper modes, the nominal, the undeciem and the duodeciem, at 2, 3 and 4 times the prime: a harmonic series on the prime that the ear completes. The deciem is deliberately not one of them, which is why moving it changes nothing in the inference. It can change whether those three modes are heard.

Moving the deciem down clears room for the twelfth above it. For each bell, how many of the nineteen strokes from 20 to 110 decibels deliver at least two of the three modes that imply the strike note — the nominal, the undeciem and the duodeciem — with the deciem as cast and tuned to 2.4. C1 6 and 7, E1 10 and 10, G1 12 and 12, C2 15 and 15, E2 16 and 16, G2 16 and 17, C3 18 and 18, E3 19 and 19, G3 19 and 19, C4 19 and 19. The undeciem, at 3 times the prime, is masked partly by the deciem below it; moving the deciem from 2.5 to 2.4 moves it away, and at C2 struck at 80 dB the undeciem's margin goes from 3.9 to 4.8 dB. — Fig. 7 For each register, how many of the nineteen strokes from 20 to 110 decibels deliver at least two of the three modes that imply the strike note, with the deciem as cast and at 2.4. Flattening never costs a stroke, and on C1 and G2 it gains one.

It helps, slightly. The undeciem, at three times the prime, sits just above the deciem and is partly masked by it. Move the deciem down and it moves away from the undeciem, which then has more room: on C2 struck at 80 decibels the undeciem’s margin rises from 3.9 to 4.8 decibels. Counted over strokes, the strike note is supported at least as often in every register and at one more stroke on C1 and on G2. A founder who flattened the deciem would not be trading the strike note for the colour. The two modes the move concerns are on either side of the series, and the move takes one of them away from the series’s middle member.

The model the count rests on

The bell is the earlier essays’ minor-third bell with one change: eight modes at 0.5, 1, 1.2, 1.5, 2, 2.5 (or the stated deciem), 3 and 4 times the prime, amplitudes falling from the hum by 0, 6, 9.5, 12, 14, 15.6, 16.9 and 18.1 decibels, decay times of forty seconds at the hum divided by each mode’s ratio to it. The count is the masked threshold: the power sum of every other mode’s spreading function at a mode’s frequency, the upper slope flattening as the masker gets louder, with the threshold of hearing beneath it. The stroke is the hum’s level. The deciem keeps its amplitude when its frequency moves.

What the count cannot show

Whether a founder can move the deciem alone. A bell’s modes are all shaped by one profile, and the founder tunes them by turning metal from the inside of the bell at chosen heights. Each cut moves several modes by different amounts; tuning five is already the craft. A profile that put the deciem a minor third above the nominal would very probably move the nominal, the undeciem or the tierce as well, and would have to be designed rather than tuned into an existing casting. The count prices the outcome, not the route to it. It is the same distance that separates any bell’s ideal tuning from a casting, and bells whose principal third has been moved do exist: profiles designed in the Netherlands at the end of the twentieth century give a bell a major tierce in place of the minor one. That is the move in the other direction, on the other third.

Whether the deciem keeps its strength. Every figure gives the flattened deciem the cast deciem’s amplitude, 15.6 decibels under the hum. The earlier essay noted that real bells vary here and that the deciem is not always prominent. A profile redesigned to move it might also make it weaker or stronger, and its window moves by about four decibels of stroke for every decibel of amplitude. A flattened deciem three decibels weaker than modelled would lose another twelve decibels of stroke at the top of its window, which on C2 would still leave it thirty decibels above the tierce’s.

How a listener weighs a third two octaves up. The flattened deciem is a minor third above the nominal and a minor tenth above the strike note. The earlier essay’s caution applies unchanged: delivered is not the same as heard as the colour of the bell, and the chord is still major is a reminder of how broad a category a third occupies. What the flattened bell removes is the contradiction: whichever octave a listener attends to, the third it finds is minor.

What happens to the bell’s sound. A deciem a major third above the nominal is part of what a bell sounds like, and changing it would change the timbre even where no colour question arises. Whether that is better or worse is not something the count has an opinion about.

Still open: whether the major-third bell has the mirror problem

The bells whose tierce has been raised to a major third have the opposite arrangement to the one priced here: a major third low, where masking buries it, and the deciem — if it is left at 2.5 — a major third high, where it survives. On this count that bell should be the robust one: its two thirds agree, and its loud strokes deliver the agreeing one. But a raised tierce sits further from the prime and nearer the quint, and the masking it meets is different. The computation is this count with the tierce at 1.25 instead of 1.2 and the deciem left as cast, and it would say whether the major-third bell delivers its major colour across more of the register and the dynamics than the minor-third bell delivers its minor one — which is the comparison that would say whether the bell everyone casts or the bell almost no one casts is the one whose colour survives being struck.