An ending is a deceleration

The first rung of this ladder took the cadence apart and found five components with no total between them. The second found that half the cadences in tonal music are engineered to withhold some of those components so that a later one can supply them.

Both rungs are about which chords arrive and in what order. Neither of them mentions time, and time is where a closing gesture actually happens: performances slow down at the end, all of them do, and a listener who has never heard the piece knows it is ending several seconds before the last chord because of it.

The slowing has a shape, and the shape can be written down.

The runner #

Kronman and Sundberg’s observation in 1987 is that a closing ritardando does not look like a tempo dial being turned. It looks like a body stopping.

The reasoning is one substitution. Read tempo not against the clock but against position in the score — how far through the closing passage the music has got — and tempo becomes the rate at which score position is being consumed, which is a velocity. A body decelerating at a constant rate has a velocity that falls as the square root of the distance still to go, not as a straight line, because it spends more and more time covering each remaining unit.

Friberg and Sundberg’s generalisation adds one parameter. Write the tempo as

v(x) = (1 + (w^q − 1)·x)^1/q

where x runs from 0 to 1 through the closing passage and w is the final tempo as a fraction of the opening one. Then q = 1 is a straight line in score position, q = 2 is constant deceleration, and larger q holds the tempo longer and drops it later.

Where the models can be separated #

A family whose members agree at both ends is a family that has to be separated in the middle, and there are two ways to look at the middle: where the notes fall, and how long each one lasts.

Fifteen per cent of a duration is comfortably above the threshold at which two durations are told apart, so the three are audibly different performances of the same instruction. Whether the difference is describable is a separate question, and the section on the listener’s own beat below answers it in the affirmative for a reason that has nothing to do with duration.

It is worth being exact about what the fifteen per cent is a fifteen per cent of. It is the total time the twelve notes occupy, not the difference between two adjacent notes, so it is a comparison two performances apart rather than one a listener makes inside a single hearing. Nobody hears a ritardando twice. What a listener has is the shape as it unfolds, and the quantities that separate the three shapes as they unfold are in the two sections after this one.

What performance measurements say is that the curves that fit are the decelerating ones, with q between about two and three, and that the straight line in score position fits badly. That is the whole empirical content of the kinematic model, and it is worth being clear that it is a claim about the shape only.

The parameter that actually varies #

Here is the finding that makes the model useful rather than merely descriptive: across performances, the shape is fairly stable and the depth is not.

Different players and different performances of the same passage differ enormously in w — how far the tempo falls — and much less in q. So a performer’s decision is essentially one number: how much to slow down. The curve is nearly given.

That is a strong claim and it has a strong consequence for what a ritardando is. It is not a rhetorical gesture whose form is chosen; it is a single continuous parameter applied to a fixed shape, which is a much more limited kind of expressive freedom than the vocabulary around it implies. A performance instruction of molto rallentando is an instruction about a number.

What the slowing does to the beat #

A ritardando does something else that has nothing to do with its shape, and it is a second closure signal hidden inside the first.

At a hundred and nine to the minute the notated beat sits comfortably inside the window in which a rate can be counted as a beat at all, and the two-beat and four-beat levels are already at its slow edge or beyond it.

Now slow the tempo to thirty-five per cent of that. A beat of 550 milliseconds becomes one of 1,570, which is at the far edge of the window — the range inside which a rate can function as a beat runs from about a hundred milliseconds to about two thousand — and the level above it, which was already marginal, is gone entirely.

So a deep ritardando dissolves the metrical hierarchy from the top down, and it does so by arithmetic rather than by any musical decision. The beat has a preferred rate and a window either side of it, and a ritardando is a walk out of that window.

That is worth adding to the five components of the closure vector, and it is a sixth of a different kind: the others are properties of the chords and this one is a property of the clock.

The five components of the closure vector are every one of them a property of what the chords do. A ritardando supplies none of them and arrives before all of them, which is why it works on music with no cadences in it. It is a sixth component of a different kind: the others are properties of the harmony and this one is a property of the clock.

The listener is running behind #

There is a further consequence of a ritardando that has nothing to do with any of the above, and this collection has already measured it in a different field.

A listener following a beat is not reading it off a score; they are tracking it, with a mechanism that corrects its estimate a fraction of the way towards each new onset. Such a tracker following a tempo that is changing does not lose lock — it runs a constant fraction of a beat behind, and the fraction is the rate of change divided by the correction gain.

That result is for a ramp, though, and a ritardando is not one. Reading the per-note growth in the beat period straight off the kinematic curve — a slowing to thirty-five per cent over twelve notes — the rate is nothing like constant:

The last note of a ritardando grows the beat period by a quarter to nearly a half, which is not a tempo change a first-order tracker follows at all. Against the limit that tracker has — 6.3 per cent a beat at the published mid-range gains — the whole gesture crosses over partway through, and where it crosses is the one thing that separates the three shapes:

q = 1 loses the listener at note 3 of 12, q = 2 at note 7, and q = 3 at note 8. That is five notes out of twelve between the extremes of the family, and it is a difference of a completely different kind from the fifteen per cent of duration this essay has been separating them by. A straight line in score position takes the beat away in the first quarter of the gesture; a high exponent keeps it to two thirds of the way through and then removes it abruptly.

So the earlier claim that the three curves are audibly different but not describably so is too modest. They differ in when entrainment breaks, and that is precisely the kind of difference a listener can report — not as a shape but as when the ending started.

And the arithmetic puts a depth on the whole gesture. A ritardando over twelve notes that stays inside the tracker’s limit throughout can reach only 57 per cent of the opening tempo at q = 1, 62 at q = 2 and 66 at q = 3. Anything deeper than about a third loses the beat before it arrives, which is the same conclusion the metrical-window figures reach two sections above, from an unrelated mechanism and a different published constant.

So a deep ritardando does three things at once, and only the first is usually described. It lengthens the notes; it walks the metrical levels out of the window they can be counted in; and it takes the tempo out of reach of the listener’s own tracker, at a point in the gesture the curve’s shape decides. All three are consequences of one number.

That third one is the reason the effect is not merely observed but felt as an ending: an internal beat running behind the sounding one is the sensation of something slipping away, and it is arithmetic about a gain rather than a metaphor.

The test the family could fail #

The kinematic model makes one prediction that is easy to state and easy to check, and it separates it from every model that is not kinematic.

The tempo curve is a function of score position, not of time. A ritardando in which the tempo falls linearly with the clock produces a different sequence of note durations from one in which it falls linearly with position in the passage, and the two are distinguishable in a measured performance without any curve fitting: the first spends its slowing evenly in seconds and the second spends it evenly in notes.

The measurements say score position, and they say it consistently enough that the substitution is the model’s main content. Everything else — the exponent, the depth — is fitting.

The tracker gives that test a second edge the literature does not use. A ritardando linear in the clock has a per-beat rate that grows more slowly than a kinematic one, because the kinematic curve’s rate accelerates toward the end; so the two predict entrainment breaking at different points in the gesture even when their total durations match. That is a prediction about a listener rather than about a stopwatch, and it needs a tapping task rather than a score alignment — which makes it available to a kind of experiment the curve-fitting work has not used.

And the last note has a predicted length. If the final tempo is w times the opening one, the final note is exactly 1/w times the opening note, whatever the exponent is. That is a direct measurement with no fitting at all: measure the first note of the closing passage and the last, take the ratio, and w is its reciprocal. For the curves drawn here, 0.35 and 2.86.

The failure condition is clean and worth stating. If measured ritardandos fitted the time-linear family, or if the exponent varied as much between performances as the depth does, the model would be describing a convention rather than a shape. Neither is what the measurements find, and both were checkable before the model existed.

The relation to the other timing result in this collection #

There is a second measured acceleration in these essays and it moves the opposite way, which is worth putting side by side because the two are often confused.

There is a second measured acceleration in these essays and it moves the opposite way. The sentence and the period differ in that one of them halves its constituent unit halfway through — an acceleration in unit length at a constant tempo, which is the exact opposite kind of quantity from a ritardando’s slowing of the tempo at a constant unit. The two are often confused because both are described as the music speeding up or slowing down, and only one of them is a change in the clock.

Fragmentation accelerates the structure while the clock stays put. A ritardando slows the clock while the structure stays put. Both are closing gestures, both are measurable, and they routinely occur in the same eight bars — the units shortening as the tempo falls, which is why a cadence can feel simultaneously hurried and heavy.

The relation to the phrase window is the same kind of thing. A phrase is a number of seconds rather than of bars, so a passage that slows to a third of its tempo has trebled the real duration of its written units, and a four-bar group that was inside the psychological present at the start of the ritardando is outside it by the end.

An ending with no chords in it #

The strongest reason to treat the ritardando as a closure component in its own right rather than as decoration on a cadence is that it works where there is no cadence.

A cyclic piece cannot cadence — a form that returns to its beginning has no harmonic arrival available, because every point in it is followed by the same point it was always followed by. What such a repertoire ends with is a signal from outside the harmony, and a coordinated slowing is the commonest one on earth.

The same argument covers a great deal else. A piece that never establishes a key has no tonic to arrive at; a texture that is a single line has no voice-leading to converge; a passage in a mode with no usable dominant has no authentic cadence to make. In every one of those cases the clock is still available, and the clock is what gets used.

Which suggests a reading of the whole closure ladder. The five harmonic components measured at the first rung are the specialised devices of one repertoire; the timing devices are the general ones, they need no theory of harmony at all, and they are what a listener with no knowledge of the style still has access to.

Whose performances, and when #

The kinematic measurements are of twentieth-century recordings of Western art music, mostly of the common-practice repertoire, and the convention they describe has a history.

A marked final ritardando is a Romantic-era norm. Baroque and early-Classical performance practice used a tempo that was steadier and closing gestures that were more often written into the notes — a cadential trill, a longer final note, a fermata — than performed as a change of rate. The point d’orgue and the written-out slowing are the older devices; the continuously graded ritardando is later.

Outside that repertoire the picture divides sharply.

A Javanese gamelan slows to end, and does it as a marked, coordinated gesture — the suwuk — cued by the drum, which is a ritardando by any description and one that is a signal rather than an expressive freedom.

A great deal of dance music does not slow at all, for the obvious reason: a ritardando is unusable if people are moving to it, and the ending is a stop rather than a slowing.

And North Indian classical music does the opposite at its climaxes, accelerating through a jhala and closing with a tihai whose ending is arithmetical — a phrase repeated three times so that its last stroke lands on the sam. That is a closure device with the precision of a cadence and nothing kinematic in it at all.

What the picture cannot show #

A real ritardando is not smooth. The curves here are two-parameter idealisations. Measured tempo tracks are noisy, they contain local lengthenings at phrase boundaries that have nothing to do with the closing gesture, and the fit is to a smoothed trace.

Where the ritardando starts is not in the model. w says how far the tempo falls and q says how; nothing says at which bar the slowing begins, and that is a musical decision with a large effect. A model that describes the shape of a gesture and not its onset is describing half of it.

And the runner is an analogy that should not be pressed. Nothing here claims that music imitates locomotion or that listeners are simulating a body. The claim is much smaller: the same one-parameter family of curves fits both, because both are what happens when a rate falls smoothly to a stop, and that is a fact about differential equations rather than about embodiment.

The window arithmetic assumes the beat is where the notation says. A ritardando deep enough to push the notated beat towards the slow edge of the window is also a ritardando in which a listener may simply switch to counting a faster level — the subdivision becomes the beat, and the metre survives at half the value. That relocation is exactly what the preference rules do when a rate leaves the window, and nothing here says which listeners do it.

The fifteen per cent is a spread across an artificial family. q = 1 fits performances badly, so the honest range across fitted models is narrower, and the figure’s spread is the spread of the candidate space rather than of the practice.

The ladder from here #

If a ritardando announces an ending several seconds before it arrives, the question is how much earlier than that an ending is predictable at all. The next rung finds that the statistics and the cues disagree — the closing bars are the least surprising material in a piece, and its closure signals arrive in the last few seconds.