The arch belongs to hearing, not to the series

A low chord stops being a chord counted, register by register, how many of a close triad’s partials stand above the threshold the rest of the chord puts over them, and found an arch: in the deep bass almost nothing survives, in the middle of the compass most of it does, and in the treble the share falls again. A clarinet keeps what a string loses ran the same sweep for six spectra and found the arch in all of them, higher or lower depending on how far apart each spectrum’s strong components stand.

Every one of those spectra had its components at whole multiples of a fundamental. The clarinet leaves out the even ones and the organ weakens the third, but none of them puts a component anywhere a harmonic series does not. So the essay ended on the question it could not answer from its own figures: whether the arch is a property of harmonic series — of components that get closer together in log frequency as they rise — or of hearing, whose critical bands are wide in the bass and narrower in hertz terms above it.

The two spectra that would decide it are already computed in this collection: a bell, whose modes sit at a founder’s ratios, and a stiff piano string, whose partials are stretched sharper as they rise.

Three spectra that differ only in where their components stand #

To make the comparison about spacing, the amplitudes have to be taken out of it. Each spectrum here has eight components, and in each the k-th lowest component has an amplitude of 1/k — the law the string spectrum in the masking essays has always used, applied by rank rather than by harmonic number.

The harmonic series has its components at one, two, three and so on up to eight times the fundamental. The bell has its components where a founder tunes them, relative to the hum: 1, 2, 2.4, 3, 4, 5, 6 and 8 — the hum, the prime an octave above it, the tierce a minor third above the prime, the quint, the nominal, and three modes above. The spectrum called “bell” in the earlier essay was a different object, a harmonic series with its even partials removed; this one is the tuned instrument that a bell has no fundamental is about. The stiff string has its $n$-th partial at $n\sqrt{1 + Bn^2}$, and at $B = 0.01$ its eighth partial is 28 per cent sharp of eight times the fundamental — far more than any piano string of normal proportions, chosen to make the effect unmistakable, and swept below.

The arch survives both #

The answer to the essay’s question is in the first figure, and it is plain.

A close major triad in the harmonic series delivers 4 per cent of its components with its root on C1, climbs to 79 per cent on E3 and stays near there through the middle of the keyboard, and falls to 42 per cent on C7. In the bell spectrum it delivers 4 per cent on C1, 75 per cent at its peak on C4, and 38 per cent on C7. On the stiff string it delivers 4 per cent, 71 per cent at its peak on E3, and 46 on C7.

Three sets of frequencies with nothing in common but their amplitude law produce one shape. The peaks fall within a major third of one another and within eight percentage points. The arch belongs to hearing. Its bass side is the critical bands being several of a spectrum’s components wide at low frequencies, whatever the components’ ratios; its treble side is the upper components climbing into a region where the triad’s three notes interleave their partials within single bands. Neither needs the components to be whole multiples of anything.

That leaves what the spacing does change, and it is more interesting than the shape.

A bell crowds its middle #

The arch’s detail comes from which pairs of components are closest together in the ear’s own units.

On C2, every gap in the harmonic series is between 0.57 and 0.64 Bark: the partials are nearly evenly spaced on the ear’s axis, each a little over half a critical band from the next. The stiff string’s gaps widen as they rise, from 0.67 to 0.91, because stretching pushes each partial further from the one below. The bell’s gaps are wildly uneven. Its hum and prime are 0.64 Bark apart, like a harmonic series, but its prime and tierce are 0.26 apart, and its tierce and quint 0.38 — three components packed into about two thirds of a critical band. Above the nominal its gaps open out again, to 1.16 Bark for the top pair.

On C5 the picture reverses. The harmonic series’ upper partials are now the closest thing in any spectrum, 0.76 Bark apart at the top, while the bell’s closest gap is 1.08 and the stiff string’s 1.05. A bell in the treble has every component more than a critical band from its neighbours.

What one note loses by itself #

Those gaps predict which components a single note hides from itself, and the prediction can be checked one note at a time.

The harmonic series, at the bottom of the keyboard, loses its middle partials: on C1 it keeps only two of eight, on E1 six, and from F♯1 upward all eight. The stiff string does better at every root — five on C1, all eight from E1 — because its stretched partials sit further apart in the bass.

The bell does worse, and exactly where its gaps said it would. From A♭1 to C2 it keeps its hum, prime and upper modes and loses its tierce and its quint — on F♯1 its nominal goes too — the two modes crowded into less than half a Bark above the prime, each masked by its neighbours. On E2 it still loses its tierce. Only from A♭2 does a bell keep all eight modes by itself — more than an octave higher than the harmonic series does.

The earlier essay’s closing section had predicted a bell losing its two lowest components in the bass. The two it loses are not the lowest. The hum and the prime survive, because they are an octave apart like any fundamental and second harmonic; what goes is the pair the founder has tuned into the gap just above the prime. That matters to anyone listening to a low bell. The tierce is the component that makes a bell sound minor, and a large bell’s tierce is exactly the mode its own spectrum hides.

Where a founder would have to put the tierce #

If the tierce is lost because it is crowded against the prime and the quint, moving it should rescue it, and the question has a practical form: how far above its usual place would a founder have to tune a low bell’s tierce for the bell to keep it?

The bell was recomputed with its tierce alone moved, from 2.10 to 2.95 times the hum, everything else fixed and the amplitudes reassigned by rank. On C2 the tierce is masked at every position up to 2.75 times the hum and only just reaches threshold between 2.80 and 2.90, with a margin of a tenth of a decibel: there is effectively nowhere between the prime and the quint where a C2 bell can put a tierce and have it heard. On E2 the tierce survives from 2.55 upward, by up to 1.6 decibels. On A♭2 it survives from 2.40 — its usual place — and on C3 from 2.30.

The positions have names. A minor-third tierce sits at 2.40 times the hum, and a major-third tierce, which some modern founders tune for bells meant to play major harmony, sits at 2.50. On E2 the minor tierce is masked by 0.9 decibels and the major tierce sits exactly on its threshold; on C3 the minor tierce clears its threshold by 1.9 decibels and the major one by 3.3. Moving the tierce up a semitone does not move the register below which a bell loses it — both tierces first clear their thresholds on A♭2, the minor one by 0.4 decibels and the major one by 1.5 — but it raises the margin by which the tierce is heard from there upward, from 1.9 to 3.3 decibels by C3. So a major-third bell’s tierce is more secure through the tenor and alto range, and a founder casting a large bourdon cannot keep either kind audible by tuning alone: below about E2 the bell’s own prime and quint take the tierce whatever interval it is tuned to, and 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.

The arch at other dynamics #

A count of masked components depends on the level, since masking skirts widen as a sound gets louder. The census was run again at 50 and at 90 dB.

At 50 dB every spectrum keeps more: the harmonic series peaks at 83 per cent on E3, the bell at 79 on A♭3, the stiff string at 83 on C3, and the treble end falls only to 46, 58 and 46. At 90 dB every spectrum keeps less: the harmonic series peaks at 54 per cent on E3, the bell at 58 on A♭4, the stiff string at 63 on C4, and all three fall to 33 on C7. The arch is present at both levels for all three spectra.

Two things move. The loud arch is lower and flatter, because at 90 dB the upper skirts of the lowest components reach across several bands. And the bell’s peak climbs with level — A♭3 at 50 dB, C4 at 70, A♭4 at 90 — while the harmonic series’ stays on E3: as the level rises the bell’s crowded middle modes need a higher register to stand clear of each other, and a harmonic series has no crowded middle to protect.

A stiff string buys its bass back #

The stiff string’s stretch was set very large so that its effect would be visible. Swept, it shows where the stretch matters and where it does not.

On C2 the triad delivers 29 per cent of its components on a string with no stiffness, the same at B = 0.001, 33 per cent at 0.003 and 0.01, and 54 per cent at 0.03. The bass is where a harmonic series’ partials are crowded on the ear’s axis, and stretching them apart is the one change that helps.

On C4 and C6 the effect has no direction. The share on C4 is 79 per cent with no stiffness, 75 at 0.001, 79 at 0.003, 71 at 0.01 and 88 at 0.03; on C6 it falls a little and then rises. In a triad, a stretched partial of one note can land nearer to a partial of another note as easily as further from it, so in the middle of the compass stiffness shuffles which components collide rather than separating them.

Real piano strings, whose stiffness the piano is tuned wrong on purpose to accommodate, have B around 0.0002 to 0.001 in the bass and middle — the region where this sweep shows almost nothing — and large values only at the very top. So a piano’s stiffness, though audible in its tuning, is not what decides how much of a bass chord it delivers.

Which computation produced the numbers #

Each note’s components are placed at the stated ratios above its fundamental, and each is given a level of 70 dB plus 20 log of its amplitude. A component’s threshold is the power sum of the masking spreading functions — the lopsided skirts one sound hides another drew — of every other component sounding — the other components of its own note and those of the chord’s other notes — floored at the threshold of hearing, exactly as the masking essays have always computed it; a component is delivered when its level exceeds that threshold. A triad’s share is delivered components over twenty-four. Bark gaps use the collection’s frequency-to-Bark conversion. The bell’s ratios are the founder’s targets the bell essays use, relative to the hum rather than the prime so that its lowest component plays the part a fundamental plays in the other spectra.

Where the model stops #

A bell’s modes are not at 1/k. The amplitudes were made equal across spectra to isolate spacing, and a real bell’s nominal and hum are typically strong, its prime and tierce weaker, and its upper modes strong at the strike and quick to die. With a bell’s own amplitudes the tierce could be more or less masked than drawn.

The level is one level. Every count is at 70 dB, and a loud chord is a smaller chord found the masking skirts widening with level, so a bell struck hard would crowd its middle modes further.

Nothing decays. The count is of steady components. A bell’s upper modes die quickly and its hum rings on, so in its tail a bell is a much simpler spectrum, and what survives masking then is different from what survives at the strike.

The masking is simultaneous. Components that start together and decay at different rates would unmask one another over time, which a sound hides what came before it measured in the other direction.

What a count of components cannot say #

Whether a masked tierce changes what a bell sounds like. A component below its masked threshold is inaudible as a separate component and still contributes to the sound’s loudness and colour, and a listener’s sense of a bell’s minor quality may come from the tierce’s effect on the strike note’s pitch rather than from hearing it separately.

Whether the arch is heard as a register effect at all. The share of components delivered is an account of what is available to a listener, and the listener is given the top voice found that what a listener attends to within a chord is not proportional to it.

Whose bells #

Change-ringing towers and carillons put bells into the bass range on this figure: a tenor bell of a large ring sounds in the second octave, where the bell spectrum here loses its tierce to its own prime and quint. It is commonly said of carillons that their low bells are hard to make sound in tune in chords, and that the minor third in every bell is what a major-key harmony has to fight. On this arithmetic the low bell’s tierce is not only in the wrong place for a major chord but also partly inaudible, which is a second reason a large carillon’s bass sounds less minor than its treble — though nothing here has measured a real bell’s amplitudes, and a founder’s tuning of a bell’s profile can change the balance.

Still open: the bell’s own amplitudes, and the tierce in time #

The equal amplitude law was the right choice for asking whether the arch needed harmonicity and the wrong one for saying what a real bell loses. Founders’ measurements of their bells give each mode a level and a decay time, and with those the same count can be run at the strike and a second later.

The prediction worth testing is sharp. 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, and the minor third a carillon’s bass is known for would be a property of its tail rather than of its attack.