A louder final chord is a deeper silence and a brighter sound

A final chord stands out for a twentieth of a second followed a closing chord through the two smoothers that turn sound into a listener’s sense of loudness — one quick, for the note, one slow, for the passage, the pair loud is relative and it comes down slowly introduced — and found that the chord stands above the running impression only while the slow one is catching up. After a bar of silence it stands highest, 5.14 decibels, fifty-four milliseconds in, and by half a second the two are level again. Every chord in that essay was played at the tutti’s own level.

It ended by proposing the obvious next case, a final chord marked louder than the passage, and by stating what it expected: that a louder chord “adds a stand that the impression cannot close, since the impression can rise only to the chord’s own level”. That sentence has a mistake in it, and the mistake is the useful thing. The impression can rise only to the chord’s own level — and it does rise to it. A stand is a difference between the note and the impression, and nothing in the smoother lets the impression stop short of a steady note. A louder final chord’s stand closes exactly as a quiet one’s does.

So the question the proposal was after has to be asked differently. If a louder mark does not buy a lasting stand, what does it buy? It buys two things, and they turn out to be different in kind: a depth for the chord to climb out of, which behaves in every respect like a silence, and a change of spectrum, which does not close at all.

What a step louder is, as a number #

A dynamic mark has no units, and the mark that is not a level made that point at length: somebody has to turn f and ff into a number, and on a struck string the number is a hammer velocity rather than a decibel. The same stated scale is used here, a factor of two in velocity for each step. The chord is a six-note tonic from C3 to C5, each note a struck string whose spectrum comes from the hammer’s contact time, and the passage chord is set to the loudness the closing essays have always used for a full tutti.

Struck at twice the velocity, that chord is 7.12 phons louder. The number is computed from the whole spectrum put through the ear’s excitation pattern — every partial of every note casting its skirt along the critical-band axis — rather than from the pressure, so it is the ear’s sum.

Straight out of the tutti, then, the louder chord arrives with the running impression 7.12 decibels below its own steady loudness, in the units the closing essays use. It rises out of that exactly as a chord rises out of a silence. It stands highest, 4.51 decibels, thirty-six milliseconds in, and it is level with the impression by half a second. The whole stand is transient.

A mark and a silence are one kind of preparation #

The earlier essay found that a chord takes back a nearly fixed share of whatever fall preceded it — between 59 and 68 per cent — because the share is set by the ratio of the two smoothers’ attack constants and not by the size of the fall. A louder mark is a fall of a different origin, so the natural test is whether it lands on the same line.

It lands on the same line, and so does everything else. Six silences from three tenths of a second to five seconds, three marks from one step louder to three steps, and all three marks with a bar of silence before them: twelve preparations, and the chord takes back between 60 and 68 per cent of the depth in every one. A step louder is 7.12 decibels of depth and 4.51 of stand, 63 per cent. Three steps louder, from mp to ff, is 21.23 decibels and 13.00 of stand, 61 per cent. A step louder after a bar of silence is 15.21 decibels — within two hundredths of the silence’s 8.09 plus the mark’s 7.12 — and 9.46 of stand.

The smoother cannot tell a depth written as a rest from a depth written as a dynamic. Both are a gap between where the listener’s sense of the passage is and where the next sound is, and the chord spends its first few tens of milliseconds closing it at the same rate whatever made it. The depths also add: a silence and a mark together are simply a deeper preparation.

That makes the exchange rate between the two worth stating. The depth a step louder buys — 7.1 to 7.5 phons, depending on which step — is the depth of a silence of 1.06 to 1.11 seconds, which is a bar of common time at about 220 beats a minute or half a bar at 110 — shorter than the three and a half seconds a silence long enough to be an ending needs before a gap is heard as a close in its own right. A composer who writes a final ff after a f tutti has, as far as the stand is concerned, written a general pause of about a second; one who writes both has written two.

The mark and the silence differ in the one place the stand does not look, which is what happens after it. A general pause is spent by the note after it because the impression falls through a silence — at the fixed rate a rest is a diminuendo measured — and then has to climb back; a louder mark never lets the impression fall at all. That difference is the next figure.

What a louder stamp leaves behind #

The earlier essay’s central trade was between two readings of a final chord. A short stamp stands out for most of its life but leaves the impression short of the chord’s level, so the passage ends up heard as quieter than it was played; a held chord leaves the impression at full level but stands out for only a fifth of a second of a much longer life.

A louder mark changes the terms of that trade completely. The stamp after a silence starts from an impression already fallen 8.09 decibels, climbs for a tenth of a second, and leaves the passage 3.90 phons below the tutti. The louder stamp starts from the tutti’s own level, climbs toward a chord seven phons higher, and leaves the passage 4.27 phons above the tutti after the same tenth of a second — above it after fifty milliseconds, at 2.21, and at 6.14 after two hundred.

Both stamps stand out by about the same amount while they sound — 5.14 and 4.51 decibels at their peaks — and they leave the passage eight phons apart. The short ending that stands above the music and the ending that leaves the passage at full weight are not a trade when the preparation is a dynamic. A silence buys its stand by lowering the impression and a louder mark buys the same stand by raising the chord, and only the second leaves the listener’s sense of the passage higher at the end than it was at the start.

That is a fact a composer has used whether or not anybody wrote it down. A final sforzando stamp after a forte tutti gets a stand the size of a bar’s rest and gives up nothing to get it; the same stamp after a written rest gets a larger stand and pays for it in a passage that finishes softer than it was played. Which ending a movement wants is its own business, and the arithmetic says the two are different objects rather than one gesture with a longer or shorter rest in front of it.

Brighter for the whole chord #

The other thing a louder mark buys is a spectrum, and here the earlier essay’s instinct was right in a way its sentence was not: there is something about a louder chord that does not close, and it is not loudness.

The split is made by computing the chord three times. First, the passage’s own spectrum with every partial raised by the doubling of velocity, which is what a louder chord would be if a harder blow only moved the string further: 5.69 phons. Second, that same spectrum raised further to the louder chord’s true total power — because the shorter hammer contact excites partials that the softer blow left almost silent, and they carry energy of their own: 0.70 phons more. Third, the louder chord’s actual spectrum at that same power, so that the only difference is where the energy sits: 0.73 phons more, because energy spread into more critical bands is summed by the ear as more loudness, which is the mechanism a chord is not as loud as its notes is about.

So a fifth of a louder chord’s loudness, 1.43 of its 7.12 phons, is the spectrum. The other four fifths are the blow. On the stand, that fifth is worth about nine tenths of a decibel of peak. The answer to “is a louder final stamp heard as a louder ending or only as a brighter one” is therefore that it is mostly heard as a louder one, for the fraction of a second the stand lasts, with a fifth of that loudness itself made out of brightness.

But the brightness is not only converted into loudness. The louder chord’s spectral centroid — the balance point of its power along the frequency axis — sits at 410 hertz against the passage chord’s 372: 1.66 semitones higher. That difference is a property of the chord’s spectrum, not of any smoother. It does not wait for either smoother — the note’s own loudness needs about fifty milliseconds to arrive, as a staccato is a dynamic mark measured, and the spectrum needs none. It is present from the strike and it is still present when the chord is released, whether that is a tenth of a second later or three seconds. The stand closes; the colour does not.

Where in the compass the spectrum matters #

The split depends on register, because the hammer’s contact pulse removes partials above a frequency that is fixed in hertz, and a note’s partials sit at multiples of its own fundamental.

The blow is worth between 5.64 and 5.72 phons in every register — a doubling of velocity is a doubling of velocity. What the spectrum adds rises from 1.00 phon with the chord at C1–C3 to 2.71 at C5–C7, so its share of the gain doubles from 15 to 32 per cent. The centroid moves 0.88 semitones in the bass and 1.78 in the treble.

The top register has an oddity that is worth a sentence because it shows the ear rather than the string. There the extra partials carry a great deal of energy — 3.50 phons’ worth — and spreading that energy is worth minus 0.79 phons. The louder chord’s upper partials land above four or five kilohertz, where the ear’s sensitivity falls away, so moving energy up into them adds pressure the ear does not sum as loudness. A fortissimo in the top two octaves of a piano is louder mostly because it is more powerful, and not at all because it is wider.

For the closure question the register result says something simple. A louder final chord in the bass is almost entirely a louder chord: its stand is the blow’s and its colour barely changes. The same mark in the treble is a third spectrum, and a listener’s impression of it being brighter has twice as much to work with.

Which computation produced the numbers #

The chord is C3, G3, C4, E4, G4 and C5, each a struck string with twenty-four partials whose amplitudes come from the hammer-contact model the notation essay used: a contact time that shortens as a quarter power of velocity, a pulse whose spectrum nulls above about 1.5 over the contact time, and a strike point at an eighth of the string. The velocity doubles per mark, from 1 for mp to 8 for ff; it is stated rather than measured. The passage chord’s overall level is set, by six rounds of adjustment, so that its loudness on Zwicker’s excitation pattern equals the tutti loudness every closing essay has used.

Loudness gains are phons of that excitation pattern. The stand, depth and impression come from the closing essays’ two smoothers — the published attack constants of 22 and 99 milliseconds and a two-second release — driven by a final chord whose level is the passage’s plus the gain. The silence with the same depth as a mark is found by bisection on that prelude.

Where the model stops #

A struck chord decays and this one does not. The spectrum is fixed for the chord’s whole length, which is right for its first tenth of a second and wrong after: a piano’s upper partials die faster than its lower ones, so a held louder chord gets duller as it sounds and the centroid gap narrows. The note that gets duller as it dies computes exactly that and it has not been joined to this.

Not every instrument brightens. The spectrum model is a hammer’s. A bowed or blown instrument also brightens when played louder, by different mechanisms; an organ does not brighten at all, since its pipes speak at one pressure and a louder chord is more stops. On an organ a louder final chord is purely a deeper preparation plus a different registration, which is a decision about colour rather than a consequence of loudness.

The loudness model grows too fast with level. Zwicker’s excitation integral grows by a factor of about 2.3 per ten decibels where the sone scale says 2.0, so every gain here is a little large in absolute terms. The split between blow and spectrum is a comparison at nearly equal levels and suffers much less.

And a mark is not a velocity. A player asked for ff after f supplies a velocity and a touch, and the factor of two is a convenience that sets every number above together.

What a centroid cannot say about a listener #

Whether 1.66 semitones of centroid is heard as brighter. The centroid is the conventional single number for spectral balance and it is a proxy. Whether a listener distinguishes two chords whose centroids differ by 1.66 semitones at equal loudness, in the context of an ending, is a discrimination measurement this essay does not have.

Whether the stand or the colour is what an ending uses. The arithmetic prices both and weighs neither. An ending is five components with no total, and the dynamic one has now split again, into a transient loudness and a lasting colour.

Whether a louder mark and a silence feel alike. They are identical to the smoother. A listener hears a silence as a silence and a louder chord as a louder chord, and what the smoother’s indifference says is only that their effect on how far the chord stands out is interchangeable.

Whose endings #

The louder final stamp is everywhere in classical and romantic orchestral writing: the ff or sf tonic after a f close, frequently with no rest before it. The two-stamp ending with a rest between, which the earlier essay priced, is the same repertoire’s other habit. On this arithmetic they are two different preparations for the same stand, and the louder mark is the one that does not cost the passage its weight. That is consistent with how often the louder mark is used at the ends of fast movements, where a general pause would halt the momentum, and how often the rest is used in slower ones — but nothing here has counted endings, and the consistency is an observation about what the numbers allow, not about what composers chose.

Still open: the stamp that is both louder and later #

The earlier essays priced a silence and this one prices a mark, and they turn out to add as depths. They do not add in their consequences. A silence leaves the passage quieter and a mark leaves it louder, so a final chord that comes after a rest and is marked louder leaves the passage at a level that depends on how the two are balanced — and there is a specific balance at which the passage ends exactly where the tutti was, while the chord still stands out by the sum of both preparations.

Finding it is a search over two numbers the closing essays already have: the length of the rest and the size of the mark, with the impression left by a stamp of a stated length read against the tutti. A first reading of the same smoother already locates it for one case: a stamp a tenth of a second long and a step louder leaves the passage 1.37 phons above the tutti after a bar’s rest, 0.42 after two seconds and 0.03 below it after two and a half, where it stands 14.69 decibels above the impression — nearly three times what either the rest or the mark buys alone. The reason to want the whole surface, over every stamp length and every mark, is that it would say what the most common written ending — a rest, then a ff chord — is doing to the passage it closes: whether composers write the rest long enough to end the passage at its own weight, or leave it short and end louder than they began.