A trill is half a consonance unless it borrows its own resolution

A suspension on a piano has a problem that a suspension in a choir does not. The note that the theory calls dissonant is the one note in the chord that was not just struck. A tied note is the oldest thing in its own dissonance put the ages of the notes back into the roughness sum and found what that does: at a second a chord, the tied note supplies under a third of its own chord’s roughness, and the three fresh notes beneath it supply the rest. That essay ended on the obvious question. A player is not obliged to leave the note alone. The key can be struck again. The note can be trilled, which is what the keyboard ornament books of the seventeenth and eighteenth centuries spend whole chapters instructing a player to do. What does each of these do to the arithmetic?

The expectation written at the end of that essay was simple. Each repair restores some of the note’s attack, so each should restore some of the roughness, and a fast enough trill should keep the note so fresh that the dissonance stops fading altogether. Only one part of that survives, and the part that survives does so for a reason nobody had asked about.

Keep the example from the earlier essay: C5 prepared over a C chord, tied into a chord of G3, D4 and G4, and stepping down to B4. Each moment lasts a second, the fundamental’s T60 at middle C is twelve seconds and halves every eighteen semitones up the compass, and each partial decays at a rate proportional to its number. Roughness is the Plomp–Levelt sum over eight partials of every pair of notes, each pair weighted by the product of its amplitudes. The one change is that the sum is now averaged across the whole of the suspended chord and not its first fifth of a second, because a trill is a thing that happens across a chord.

Over that second, the tie reads 0.077 and the same chord struck whole reads 0.119. Those two numbers are the room that any repair has to work in. A trill at eight notes a second between C5 and the D5 above reads 0.108. A trill at the same rate between C5 and B4, the note it is about to resolve to, reads 0.141, and that is above the top of the room.

One soft strike can take away more than it gives #

Start with the smallest repair. A pianist can catch the tied note again as the harmony changes, lightly, so that the attack does not announce itself. The earlier essay’s closing paragraph assumed that this would restore some roughness at the cost of the very property the tie was for. It does not always restore any.

Below about nine decibels under the chord, a re-strike makes the suspension smoother than leaving it alone. The medians cross at −9.1 dB on the 4–3, −9.9 on the 7–6, −8.8 on the 9–8 and −11.0 on the 6–5, and across individual spacings the crossing runs from −6.6 to −12.5. The mechanism is unglamorous. A note a second old has lost eight decibels at its fundamental and twice that at its second partial, so it is less decayed than the theory’s picture of a fading note suggests. A new attack replaces that motion. If the replacement is softer than what it replaces, the voice has gone down, and with it every pair the voice is part of.

So the gentle catch that a careful player might use to keep a tied note alive is, on this arithmetic, a way of removing it. The price of the tie is not paid back until the note is struck within about a third of the chord’s amplitude, and at that level the attack is no longer something a listener is likely to miss. The full strike, for what it is worth, doubles the 4–3’s roughness over those first two hundred milliseconds, which is just the earlier essay’s finding read from the other side.

That result depends on one assumption, and it is the crude one in this model. Each attack stops the voice’s previous motion outright. A key released fully and struck again does something close to that, since a damper falls on the string between the two, but a key caught partway through its travel leaves the string ringing, and the new hammer blow then adds to a motion already there. In that case nothing is lost, and a soft catch gives a small gain instead of a small loss. Which of these a pianist does is a matter of action and touch that this sum has no way to know, and on a harpsichord, where a repeated pluck must first pass a damper, the stopped reading is the nearer one. What the sum does establish is the size of what is at stake: the note a second old still holds about a third of a fresh note’s amplitude, and any catch softer than that has to be judged against what is already ringing.

The trill has no rate that stops the fading #

The Baroque answer to a dying note was not a single strike. It was a stream of them. Suppose the note is kept fresh by striking it again and again, and ask at what rate the dissonance stops fading.

There is no such rate. The chord’s mean moves from 0.128 at two notes a second to 0.146 at thirty-two, and it has 93 per cent of the top value by four a second. The reason that it stops rising so early is that a voice struck four times a second has a mean age of an eighth of a second, and the partials that carry its roughness have barely moved in that time. Going faster has almost nothing further to refresh.

The reason that it stops at 0.146 and not higher is the more useful finding. The sustaining instrument, with nothing decaying at all, reads 0.327, more than twice as much. A trill with infinite rate would refresh the top voice completely and would still leave the bass, the D4 and the G4 dying underneath it, and every pair of notes in the sum has at least one of those three in it. Three of the four notes in a suspended chord are out of the trill’s reach, so the fading the question asked about is mostly theirs. A player working on one finger cannot do anything about the other three.

There is also a ceiling the sum does not report. A dissonance has to last put the shortest duration at which a dissonance registers as one at about seventy milliseconds, which is fourteen notes a second, and a keyboard trill is usually taken as lying around ten, a round figure and not a measurement made here. Everything at the right of the figure above is therefore being read by a formula past the point where the ear would treat each note as an event. Nothing depends on it, because the right-hand side adds so little.

The note a trill borrows decides what it is #

The figure above shows something stranger than the plateau, and the plateau hid it. The two trills have the same top note and the same rate, and one of them is about a third rougher than the other. In the rate plot the trill with the upper neighbour lies flat at 0.107 to 0.108 at every rate, below the single strike, while the trill with the resolution rises above it. A trill is always described as an embellishment of one note, but in the sum it is two notes, half the time each, and the second is a choice with a roughness of its own.

Struck fresh over this chord, C5 reads 0.246 in the fifth of a second after the harmony changes. B4, the note of resolution, reads 0.241. Those are nearly identical, and that equality is the earlier essay’s embarrassing result again: on this spacing the resolution is hardly smoother than the suspension. But D5, the upper neighbour, reads 0.126, roughly half, because D5 is an octave above the D4 already in the chord and a fifth above the G4. On a 4–3, the upper note of a trill is a chord tone, and half of the ornament is a consonance.

The census across all spacings says this is not an accident of one voicing. On the 4–3, an upper-note trill returns a median of 72 per cent of what a single strike would, and a trill with the resolution returns 158. On the 7–6 the two are 83 and 132. On the 9–8 they coincide at about 100, because both neighbours of the suspended D are consonant with a C chord: its upper neighbour E is the third and its resolution C is the octave of the bass. And on the 6–5, where the suspended E has F as its upper neighbour, the order reverses, 192 against 153, because that F is a semitone above the E and belongs to no triad on G.

So the common description of the trill as a way of keeping a note alive is half of what the sum says. What it keeps alive is an alternation between that note and a companion, and the companion can give back as little as seven tenths of one strike or nearly twice as much, on the same finger and at the same speed.

The surprising connection is to a matter of performance that was argued about for two centuries on grounds of taste. The trill on a cadence’s leading note in French and German keyboard practice is prescribed to begin on the note above: François Couperin’s treatise of 1716 and C. P. E. Bach’s of 1753 both teach it, and in the commonest cadence that upper note is the very note that is suspended. Read one way, a cadential trill on B is a decoration of the resolution. Read in this sum, it is a 4–3 alternating with itself, the suspended C and its resolving B taking turns over the whole length of the chord. Of all the things a player could do with one finger above a dominant, it is the one that keeps the most roughness, on the suspension where the other trill gives back least. Nothing here says that is why it was taught. Treatises give their own reasons, which concern harmony and preparation and which do not mention a decay rate. What the arithmetic adds is that the two descriptions, ornament and resolution, were describing the same object.

Why a short instrument needs it and a long one does not #

If the repair exists because the instrument decays, then what it is worth should depend on how fast the instrument decays, and that can be swept directly.

The two readings of this figure pull opposite ways, and both are worth having. As a multiplier, the ornament is worth most where the instrument is shortest. At a T60 of three seconds, a round figure for something like a harpsichord’s middle register and not a measurement of one, the fastest trill makes the chord 2.63 times as rough as a tie does; at twelve seconds, 1.90; at forty-eight, 1.28. On a very long instrument a tied note has hardly faded and there is hardly anything for the ornament to restore. That is at least consistent with the keyboard tradition’s habit of ornamenting most where the sound dies fastest, though consistency is all it is.

As an absolute, the ornament is worth least where the instrument is shortest. At three seconds the trill’s 0.048 is 15 per cent of the sustained chord’s 0.327. At forty-eight it is 76 per cent. So the earlier essay’s closing question, whether a player can keep a suspension as rough as a sustained one at a slow tempo, has a plain answer: not on anything that dies, and on the instrument that needs it most, not nearly. A harpsichordist’s trill multiplies a very small number. That is what it means for three notes to be out of reach. The sustained chord’s roughness comes from six pairs at full strength, the player controls one voice in three of those pairs, and the other three pairs are what they are.

The tempo tells the same story from another direction, because a shorter chord is a longer T60 in disguise. At half a second a chord, the tie reads 0.133 and the trill 0.195, a factor of 1.47; at two seconds a chord they read 0.038 and 0.088, a factor of 2.32. The slower the harmony moves, the more of the sound is owed to the ornament and the less sound there is.

What the trill does to the step #

There remains the finding that made the earlier essay awkward to read. On a struck instrument the resolved chord is rougher than the chord left hanging: on the 4–3, at a second a chord, about three and a half times. The step that the theory calls a release is the moment the newest note in the room arrives. A trill changes that comparison completely, without changing anything about the step.

Trilled, a suspension loses what made its step rougher than doing nothing. The ratio falls from 3.47 to 1.00 on the 4–3, from 2.31 to 0.82 on the 7–6, from 1.28 to 0.66 on the 9–8, and from 2.54 to 1.04 on the 6–5. On the 7–6 and the 9–8 the resolution now lowers the roughness outright, in the plain sense the word was always claimed to have. Nothing has happened to the resolved chord. What has changed is the comparison, because the note that used to be the oldest thing in the room is now as fresh as the note that replaces it. The resolution stops being the arrival of something new and becomes the end of something moving, and if what a listener hears at a resolution is partly the change in the sum, a trill is what lets that change go the right way on a decaying instrument.

The 4–3’s reading of exactly 1.00 under its own trill deserves a remark. It says the step adds and removes nothing, which is what the sustained instrument said in the earlier essay too: 0.327 against 0.325, a move of under one per cent. A trill to the resolution gives a piano the sustained instrument’s coin toss back. The instrument with a dying sound and a busy finger ends up in the same position at the step as a voice holding still, for a reason that has nothing to do with sustain. The step changes nothing because the sum sees the same two notes before and after.

That also suggests what a 4–3’s release consists of, and it is something the sum cannot supply. When the roughness does not fall and the listener still hears a release, the release is somewhere else: in the line coming to rest and in what the harmony has been doing, which consonance is half learned has already argued has a learned half that no spectrum could account for. A trill’s end is a change of texture as well as of pitch, a stream of attacks stopping, and that may be worth more to a listener than either of the notes. That has not been measured here.

What one finger cannot be credited with #

The sum has no time in its ear. It averages a stream of attacks over the chord and treats each moment as a separate spectrum. A real listener hears a roughness with a rate of its own when a pitch moves, and a trill at eight notes a second is a modulation of four cycles a second across a whole interval, which is close to the rate at which listeners report the strongest sense of fluctuation. Nothing in Plomp and Levelt’s curve can know that, so an ornament’s character as a shimmer is left out entirely, and that may be the larger part of what it does.

Nor can it hear masking between the notes. A sound hides what came before it, and the note of a trill sits in the skirts of its neighbour, a tone or a semitone away. The partials of one note will be partly hidden by the other’s decaying tail. And an attack is more than an amplitude: a moving tenor is heard because it is struck again found that what unmasks an inner voice is its fresh attack, so a trill may do far more for whether the suspended voice is heard than for how rough the chord is. A trill may well be for that, and this arithmetic would never show it.

The envelope is one curve per partial, without the double decay of a real piano string, and a softer strike here is only quieter. In fact it is also darker, since how hard the note was struck changes more than a level. A real pianissimo re-strike has lost its upper partials as well, so the break-even near nine decibels probably lies somewhat higher than the figure shows. The direction of the correction is safe and its size has not been computed.

And every spacing here is four parts in a major key, one chord pair with no phrase around it, a stated T60 and no room. A hall’s tail would keep the old notes sounding under the new ones and would soften all of these contrasts, as the earlier essay said of its own.

Whose trills, and which instruments #

These are figures about keyboards and other instruments that die: harpsichord, clavichord, fortepiano, piano, lute. On them the tie has a cost, so the repair has a value. The trill is not only theirs: voices, viols and violins trill too, and on an instrument that sustains, every note of the trill is at full strength and the tied note was never faded. The arithmetic of this essay does not apply to them. A singer’s trill is doing something else, and a theory that explained the ornament entirely by a keyboard’s decay would have to explain why singers do it as well. The history of keyboard ornamentation is also not a history of one idea, and the prescription to begin on the upper note is itself regional and dated, not a universal rule.

So the claim is a limited one. For a four-part suspension on a decaying instrument, the question of which note a trill borrows changes what it gives back far more than the question of how fast it goes: by a factor of about two on the common 4–3, in the opposite direction on the 6–5, against a seventh part or so across the whole range of rates. That is a computation about what the notes do to each other. It says nothing about taste.

Still open: the player’s other hand #

Everything here happens to one voice. The figures say that three notes are out of its reach, and so the natural next measure is the one a pianist has anyway: bringing the fresh trio down. Voice the bass and inner parts softer and the tied note’s share rises by arithmetic, since a suspension’s roughness is mostly among notes nobody calls dissonant. How far they would have to come down before the suspended note’s pairs became the largest in the chord, and whether a player could get there at a level anyone would still call a chord, is the same sum with one more dial. It would say whether the note the theory blames can be made responsible for its own dissonance by anything except silencing its neighbours.