Henkjan Honing - Specialized Research AI at MIT

Henkjan Honing
Music, Mind, Machine Group
Music Department,
ILLC, University of Amsterdam
NICI, University of Nijmegen
The Netherlands
honing@uva.nl

The Final Ritard:
On Music, Motion,
and Kinematic Models

Motion plays an important role in music, a fact ev-
idenced not only by the wealth of terminology used
by musicians and music theorists that refer to mu-
sic in ‘‘motional’’ terms. Consider, for example,
how we speak of music as ‘‘slowing down,’’ ‘‘speed-
ing up,’’ ‘‘moving from F-sharp to G,’’ etc. A con-
siderable amount of theoretical and empirical work
tries to illustrate apparent relation between physi-
cal motion and music (see Shove and Repp 1995 for
an overview). However, it is very difﬁcult to spec-
ify—let alone validate—the nature of this long-
assumed relationship. Is there a true perceptual
experience of movement when listening to music,
or is it merely a metaphorical one owing to associa-
tions with physical or human motion?

Some scientists have looked at music and mo-

tion in a very direct way, for instance, relating
walking speed to preferred tempi (e.g., Van Noor-
den and Moelants 1999) or body size to timing pat-
terns found in music (Todd 1999). However, these
direct relationships between the human body and
music seem too simplistic to generally hold. Others
have approached the relation more as a metaphori-
cal one, arguing that musicians allude to physical
motion in their performances, imitating it in a mu-
sical way (cf. Shove and Repp 1995). These theories
tend to be difﬁcult to express in computational
terms.

This article reviews a family of computational
models (e.g., Sundberg and Verillo 1980; Feldman,
Epstein, and Richards 1992; Todd 1992; Friberg and
Sundberg 1999) that do make the relation between
motion and music explicit and therefore can be
tested and validated on real performance data.
These kinematic models attempt to predict the
timing patterns found in musical performances
(generally referred to as expressive timing). Most of
these studies focus on modeling the ﬁnal ritard:
the typical slowing down at the end of a music per-
formance, especially in music from the Western Ba-

roque and Romantic periods. But this characteristic
slowing down can also be found in, for instance, Ja-
vanese gamelan music or some pop and jazz genres.
In this kinematic approach, one looks for an expla-
nation in terms of the rules of mechanics: that is,
how expressive timing might relate to, or can be
explained by, models of physical motion that deal
with force, mass, and movement.

A discussion of these kinematic models is pre-
sented below in the form of a story (see Figure 1),
with three ﬁctitious characters who represent the
different disciplines involved in this research (psy-
chology, mathematics, and musicology). The story
is a continuation of Desain and Honing (1993; see
also http://www.nici.kun.nl/mmm/tc for additional
sound examples), an article that dealt with the
state of the art in expressive timing research some
ten years ago. In addition, it brought forward a cri-
tique on the usefulness of the tempo curve (a con-
tinuous function of time or score position) as the
underlying representation of several computational
models (including most computer music software
at that period). The main point of critique was that
the predictions made by models using this repre-
sentation are insensitive to the actual rhythmic
structure of the musical material: they make the
same predictions for different rhythms. All this
suggested the existence of a richer representation of
timing in music perception and performance than
is captured by an unstructured tempo curve.

The present article attempts to offer an informa-
tive but informal discussion of models of the ﬁnal
ritard, including some of the problems that these
kinematic models do not address. Experimental
support for an alternative view, as brieﬂy presented
in the discussion, will be the topic of a forthcoming
article.

The Final Ritard: A Tale on Music and Motion

Computer Music Journal, 27:3, pp. 66–72, Fall 2003
(cid:1) 2003 Massachusetts Institute of Technology.

In the following text, P, M, and their musical friend
MF continue their enthusiastic search in trying to

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Figure 1. The ﬁnal ritard, a
tale on music and motion.

Part 1: In Which MF Had an Important Insight
and P Found the Appropriate Literature

Not so long ago, MF remembered those Christmas
holidays while he was reading a book on the his-
tory of tempo rubato. He was still convinced his
friends were on the wrong track with their silly
computer models. But the more he read about
tempo rubato, the more he was convinced that they
might have overlooked an obvious link between
music and biological motion. Blatantly obvious
(once he realized it) was the explicit reference of
much music terminology—words like andante or
accelerando—to qualities of human movement.
And therefore, he reasoned, a successful model of
expressive timing—unlike the unsuccessful models
made by his friends—should be based on the rules
of movement and the human body.

MF couldn’t help making a phone call to P, the
amateur psychologist, to tell him about his new in-
sight. ‘‘My dear friend P,’’ he said, ‘‘for expressive
timing to sound natural in a performance, it must
conform to the principles of human movement.
Isn’t knowledge about the body—the way it feels,
moves, reacts—what musicians share with their
listeners?’’ P almost immediately became enthusi-
astic. He saw a new opportunity to continue the in-
vestigations that had ended so brusquely before.
P decided to go to the library, and there he found a
lot of interesting psychological literature on the re-
lation between motion and music. Much of it,
however, involved some formidable mathematics.
MF then proposed to have a new gathering with
the ‘‘old team,’’ including their mathematical
friend, this time at MF’s home, safe from modern
technology!

Part 2: In Which the Friends Met Again
and Explored Elementary Mechanics

A few days later, P and M found themselves at
MF’s kitchen table, which was well stocked with a
pot of tea and a tin full of cookies. They returned
to a lively discussion of expressive timing in mu-
sic. After browsing through the books that P
brought, M (the amateur mathematician) stated

Honing

unravel the mystery of timing in music perfor-
mance. This time they will ﬁnd out about the ki-
nematic approach to expressive timing and
computational models that are also based on the
notion of a tempo curve; as such they are likely to
continue their argument.

Prologue: What Happened Before

Quite some time ago, P, who is interested in psy-
chology, and M, an amateur mathematician, got to-
gether during the Christmas holidays with their
musical friend MF. Those were the days before cel-
lular telephones, a time of herbal tea and the just-
arrived technology of MIDI. MF, while duly
impressed by P’s and M’s well-equipped music stu-
dio and expertise in computer modeling, remained
unimpressed by their musical results and, sadly,
left, rather irritated, to spend his Christmas else-
where.

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Figure 2. Prediction of the
ﬁnal ritard by the kine-
matic models described in
Equations 1, 2, and 3 (with
w (cid:2) 0.3). Tempo and score
position are normalized.

)
v
(
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i
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N

1.0

0.8

0.6

0.4

0.2

Eq. 1

Eq. 2 (q=2)

Eq. 3

Longuet-Higgins and Lisle (1989) and Todd (1992)
propose an identical model, but express it rather as
tempo (v) linear in time (t):

v (t) (cid:2) u (cid:3) at

(1b)

Friberg and Sundberg (1999) generalize this model
by adding a variable q for curvature (varying from
linear to convex shapes; see Figure 2), w (a non-zero
ﬁnal tempo), and normalize it:

v (x) (cid:2) [1 (cid:3) (w (cid:4) 1)x]

1/q

(2)

0.2

0.4

0.6

0.8

1.0

Normalized score position (x)

with some authority, ‘‘These models borrow from
elementary mechanics and kinematics. They talk
about mass, force, and speed of an object in terms
of velocity, time, and place. And, interestingly,
tempo variations in music performance are com-
pared with the behavior of physical objects in the
real world.’’ P was all ears; MF just took another
sip of his tea.

M wrote most of the formulas, one below the
other, on a piece of paper, patiently explaining their
formal differences. A tidier version of M’s jottings
is given next.

Interlude: Formalizations of the Final Ritard

Now, some of the existing formalizations of the ﬁ-
nal ritard are brieﬂy summarized. Kronman and
Sundberg (1987) deﬁne the ﬁnal ritard as a square
root of score position, a model of constant braking
force (a convex function; see Figure 2):

v (x) (cid:2) (u (cid:3) 2ax)

1/ 2

(1a)

Todd (1985) and Repp (1992) suggest quadratic

Inter-Onset Interval (IOI, or beat duration) as a
function of score position:

IOI(x) (cid:2) c (cid:3) kx (cid:3) lx

(3)

where c is a constant reﬂecting vertical displace-
ment, and k and l are coefﬁcients reﬂecting the de-
gree of curvature. This results in a concave
function when expressed as tempo as a function of
score position (see Figure 2). In addition, Feldman,
Epstein, and Richards (1992) and Epstein (1994) dis-
cuss a model of force dynamics. However, they
tested it with a model of beat duration that is in
fact unrelated to a model of force, just like Equa-
tion 3 (cf. Friberg and Sundberg 1999). Figure 2 il-
lustrates the equations above.

Part 3: In Which the Friends Built a ‘‘True’’
Physical Model

After seeing so many formulas and equations, MF
protested ‘‘But M, please! We are investigating mu-
sic here, not mechanics!’’ ‘‘Look,’’ P swiftly inter-
rupted, ‘‘I found the studies of these music
researchers. They explain ritardandi in music per-
formance as alluding to human motion, like the
way runners come to a standstill. Let me read a
passage for you: ‘Performers aim at this allusion,
and listeners, with some education, ﬁnd it aestheti-
cally pleasing’ (Repp 1992). Isn’t this exactly what
you described to me on the phone?’’

where v is velocity (or tempo), x is distance (or
score position), u is initial tempo, and a is accelera-
tion.

P and M seemed conﬁdent that they had now

found what they had been searching for all the
time. MF too was quite pleased with the fact that

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Figure 3. A mechanical
implementation of a con-
stant braking force model,
consisting of a music box
(1), a piece of piano roll
(2), solid-metal ﬂywheel

(3), belt (4), and a handle
(5). For a short movie
showing the machine at
work, see www.hum.uva
.nl/mmm/fr/.

these respected researchers had found evidence for
his intuitive ideas about bodily motion. But he still
had reservations. ‘‘How does the math of elemen-
tary mechanics compare to a ﬁnal ritard in music?
Can’t we listen to these formulas?’’ M replied with
a frown on his face, ‘‘Well, if we would have met in
our studio, we could have programmed them for
you. Now, we must think of something else.’’ But
after a small pause he began to smile. ‘‘Let’s see
how far we can get with the material in your ga-
rage.’’

That morning, MF’s kitchen turned into a real
workshop. ‘‘Can we use one of your music boxes?’’
P asked sheepishly. With some hesitation, MF col-
lected one of his beloved machines from the living
room. And after some hours of triﬂing and ham-
mering, they had built it—a ‘‘true’’ physical model
of constant braking force! (See Figure 3.)

The machine they built contained a music box

with the crank replaced by a ﬂywheel. This ﬂy-
wheel was connected to the music box with a belt,
as shown in Figure 3. When turning the new han-
dle, the music box would start playing, and when
released—owing to the inertia of the ﬂywheel—it

would continue playing, slowly coming to a halt
from the friction of the machinery.

MF inserted his favorite piano roll, a Bach fugue,

into their newly made contraption. He turned the
ﬂywheel, and the music started playing. A few bars
before the end, he released the handle, and the mu-
sic came slowly to a standstill over the last few
notes. ‘‘Wonderful, wonderful!’’ They all jumped
with joy. MF thought his antique music box had ﬁ-
nally become truly musical.

Part 4: In Which Some Disappointment
Was Unavoidable and They Decided to Look
at Real Performances

When they had calmed down a bit, M had a second
look at his paper full of formulas, and said with a
tone not atypical of a young mathematician, ‘‘But I
have to say that these models are actually under-
speciﬁed. They make no claims about how to de-
rive the ‘metaphorical’ mass or speed from the
music. In our contraption, we just arbitrarily de-
cided on the mass of ﬂywheel, and we can freely
decide the speed at which the handle is released.’’
M also realized that their contraption had some
shortcomings. ‘‘Our ﬂywheel has a ﬁxed braking
force, caused by the friction of the contraption. But
it should actually be dependent on when and at
what speed you release the handle and stop when
the right ﬁnal tempo is reached, like the equations
show. That’s difﬁcult to make mechanically.’’

But P responded ‘‘Oh come on M, don’t be so
strict. Let’s just try another one, a slightly more
modern piece. What do you think?’’ After some
searching, MF returned with a piano roll of Beetho-
ven’s Paisiello Variations. ‘‘Remember this?’’ he
teased, alluding to their previous Christmastime
investigations using the same piece. MF inserted
the piano roll, and they listened again for the last
measures of each variation. But whatever they
tried, releasing the handle early or late, at higher or
lower speeds, it never sounded quite right. ‘‘It
doesn’t do the rhythmic ﬁgures right,’’ MF com-
plained. ‘‘Apparently, it only works with the re-
peated eighth notes of the fugue.’’

‘‘We could be here forever trying to change this
or that factor,’’ P warned. He was convinced they

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Figure 4. Final ritards in
performances of the last
three measures of Schu-
mann’s Tra¨ umerei from
Kinderszenen, Op.15.
(Tempo 1 is M.M. (cid:2) 60;
after Repp 1992.)

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had to return to the empirical approach. ‘‘Why
don’t we look at how MF performs ﬁnal ritards?’’

Part 5: In Which They Looked at Graphs from
Famous Pianists, But Couldn’t Please Their
Musical Friend

P opened his briefcase and removed a folder with
the performance data they had collected during
that ﬁrst Christmas gathering. ‘‘These are the
graphs of MF performing the ﬁnal measures of
Tra¨ umerei by Schumann.’’ And enthusiastically

holding up an article, P added, ‘‘And here are some
interesting measurements made from recordings by
some of your colleagues. Look, you played it just
like Alfred Brendel!’’ (See Figure 4.)

There was quite some diversity among these fa-

mous pianists; they all seemed to play the ﬁnal
measures differently. MF said questioningly, ‘‘I do
not see how one single curve could describe all
these performances.’’ P responded, ‘‘But the point
here is to model the average, normative perfor-
mance,’’ to which M added, while pointing at
Equation 3, ‘‘This research showed that the last six
notes of these averaged performances can be ﬁtted

Computer Music Journal

closely by a quadratic function. That is an impor-
tant ﬁnding, isn’t it?’’ ‘‘Indeed, M,’’ P conﬁrmed,
‘‘but we must be aware that an average curve is a
statistical abstraction, not a musical reality.’’

Their musical friend smiled and took another
close look at the diagrams. ‘‘So if I understood your
explanations,’’ he asked M, ‘‘this function should
have a hollow, concave shape. But doesn’t our con-
traption generate a convex-shaped deceleration?’’ M
conﬁrmed this. ‘‘A convex shape indeed is what the
other research found. Apparently, there is evidence
for a variety of shapes. However, what worries me
is the complete freedom in deciding on the mass
and amount of force applied; ﬁtting these curves to
the data is too ﬂexible.’’ ‘‘Maybe all these pianists
have their own speciﬁc force and mass,’’ MF inter-
jected optimistically. They looked at each other
with some disappointment. It seemed that once
again they had failed to ﬁnd a model of expressive
timing that could please their musical friend. MF,
who this time wanted to end their endeavors in a
more optimistic manner, proposed ‘‘Let’s go to the
living room. I will play my favorite fugue for you.’’

Discussion

This tale addresses kinematical models of expres-
sive timing, and it questions how well expressive
timing can be explained by models of physical mo-
tion. One point of critique is that the predictions
made by these models are insensitive to the actual
rhythmic structure of the musical material. This
was stated more generally with respect to tempo
curves in the original article (Desain and Honing
1993) and elaborated upon subsequently (Desain
and Honing 1994; Honing 2001). However, more
central is the concern that these descriptions do
not, in principle, teach us anything about the na-
ture (whether ‘‘motional’’ or not) of the underlying
perceptual or cognitive mechanisms. Even if we as-
sume that these tempo curves do give a good ap-
proximation of the empirical data (despite the
contrasting results in the research discussed above),
the mere fact that the overall shape (e.g., a square-
root function) can be predicted by the rules that
come with human motion is not enough evidence

for an underlying physical model of expressive tim-
ing, however attractive such a model might be.

An alternative explanation could be based on the
relation between rhythmic structure and expressive
timing (Desain and Honing 1996). For example, a
ritard of many eighth notes can have a deep rubato,
while one of only a few notes and possibly a more
elaborated rhythmical structure (i.e., with differen-
tiated durations), might be less deep (i.e., exhibit
less ‘‘slowing down’’ and/or ‘‘speeding up’’). Along
these lines, it is not a class of functions (originating
from mechanics) that best describes the timing pat-
terns observed but a set of constraints that describe
the boundaries of possible ﬁnal ritards: the con-
straints on expressive timing are a consequence of
the need not to break the perceptual rhythmic cate-
gories while decelerating quickly. (For example,
slowing down more would be perceived as a differ-
ent rhythm altogether.)

Models of tempo tracking and rhythmic categori-

zation (e.g., Longuet-Higgins 1987; Desain and
Honing 2001) predict the boundaries for which the
rhythmical structure can still be perceived. Apart
from explaining the dependency of a ritard on the
performed rhythmic material, this yields con-
straints on the shape of the ritard. Such restrictions
are not made by a physical motion model, because
any metaphorical mass, force, and amount of decel-
eration are equally likely. As such, a ﬁnal ritard
might coarsely resemble a square-root function,
with the added characteristic that the detail de-
pends on the rhythmical material in question.

Finally, this does not mean that all timing pat-
terns in music performance can be solely explained
in terms of musical structure alone; therefore, the
role of the body (Clarke 1993), its physical proper-
ties (Todd 1999), and the way it interacts with a
musical instrument (Baily 1985) is too evident. The
challenge is to construct a theory of music cogni-
tion that incorporates both the cognitive and em-
bodied aspects of music perception and
performance.

Acknowledgments

Special thanks to Robert Gjerdingen and Doug
Keislar for valuable suggestions on an earlier ver-

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sion of this article, and to Bruno Repp for his con-
structive criticisms and for kindly providing the
original data for Figure 4. We also thank the De-
partment of Mechanics, University of Amsterdam,
for actually making the contraption shown in Fig-
ure 3. And last but not least, thanks to Peter De-
sain with whom the characters of P, M, and MF
were invented.

This article is based on a text ﬁrst published in
2003 in Music Theory Online 9(1), (available on-
line at societymusictheory.org/mto/issues/mto
.03.9.1/toc.9.1.html). It was written during a sab-
batical at New York University by kind invitation
of Robert Rowe. The research was funded by the
Netherlands Organization for Scientiﬁc Research
(NWO) in the context of the Music, Mind, Machine
project.

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