a computational model of rubato (todd).pdf
TRANSCRIPT
-
8/11/2019 A computational model of rubato (Todd).pdf
1/21
PLEASE SCROLL DOWN FOR ARTICLE
This article was downloaded by: [Ingenta Content Distribution Psy Press Titles]
On: 5 December 2009
Access details: Access Details: [subscription number 911796916]
Publisher Routledge
Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-
41 Mortimer Street, London W1T 3JH, UK
Contemporary Music Review
Publication details, including instructions for authors and subscription information:
http://www.informaworld.com/smpp/title~content=t713455393
A computational model of rubato
Neil Todd a
aDepartment of Psychology, University of Exeter, Exeter, UK
To cite this ArticleTodd, Neil'A computational model of rubato', Contemporary Music Review, 3: 1, 69 88
To link to this Article DOI
10.1080/07494468900640061
URL http://dx.doi.org/10.1080/07494468900640061
Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf
This article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.
The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.
http://www.informaworld.com/smpp/title~content=t713455393http://dx.doi.org/10.1080/07494468900640061http://www.informaworld.com/terms-and-conditions-of-access.pdfhttp://www.informaworld.com/terms-and-conditions-of-access.pdfhttp://dx.doi.org/10.1080/07494468900640061http://www.informaworld.com/smpp/title~content=t713455393 -
8/11/2019 A computational model of rubato (Todd).pdf
2/21
Contemporary Music Review
1989, Vol. 3 pp. 69-88
Photocopying p ermitted b y license on ly
9 1989 Harw ood Academic Publishers Gm bH
Printed in the United Kingdom
c o m p u t a t io n a l m o d e l o f r ub a to
e i l T o d d
Department of Psychology Un iversity of Exeter Exeter UK
Presented is a model o f rubato, im plem ente d in Lisp, in which expression is view ed as the
m ap pin g of mu sical structure into the variables of expression. T he basic idea is that the
per form er use s "ph rase finallengthening" as a device to reflect some internal representation
of the p hra se structure. The representation is bas ed on Lardahl and Jackendoff's time-span
reduction. The basic heuristic in the m odel is recursive involving look-ahead a nd planning
at a nu m ber of levels. The planned phrasings are superp osed beat by beat and the ou tput
from the program is a l is t of durations which could easily be adapted to be sent to a
synthesiser given a suitable system.
KEYWORDS computational modelling, music cognition, musical performance, rubato,
mental representation, m enta l process.
n t r o du c t i o n
O n e o f t h e m o s t u b i q u i t o u s e x p r e s s i v e d e v ic e s in m u s i c a l p e r f o r m a n c e i s
r u b a t o . M o s t n o t a b l y i t i s u s e d i n m u s i c o f t h e r o m a n t i c e r a , b u t i s a l s o
e v i d e n t i n a v a r i e t y of o t h e r s ty le s . R e s e a r c h o n m u s i c p e r f o r m a n c e
( S e a s h o r e , 1 9 3 8 ; S h a f f e r , 1 9 81 ; C l a r k e , 1 9 84 ; T o d d , 1 98 5 ; B e n g t s s o n &
G a b r i e l s s o n , 1 98 0; S u n d b e r g & V e r iU o , 19 80 ) i n v o l v i n g t h e p r e c i s e
m e a s u r e m e n t o f d u r a t i o n h a s s h o w n t h a t th e r e a r e a n u m b e r o f b a s i c
o b s e r v a t i o n s w h i c h c a n b e m a d e . T h e f i rs t i s t h a t s k i ll ed p e r f o r m e r s c a n
s h o w a r e m a r k a b l e d e g r e e o f r e p ro d u c i b i l i ty f r o m o n e p e r f o r m a n c e t o t h e
n e x t ( S h a f f e r , 1 9 8 4 ; G a b r i e l s s o n , 1 9 8 7 ) . T h i s p r e c i s i o n i n t i m i n g s h o w s
t h a t th e p e r f o r m a n c e m u s t i n v o l v e th e u s e o f g e n e r a t i v e p r o c e d u r e s a n d a
p r e c is e i n te r n a l r e p r e s e n t a t i o n o f u n d e r l y i n g e x p r e s s i v e fo r m . A s e c o n d
o b s e r v a t i o n is t h e u s e o f s l o w i n g t o m a r k a p h r a s e b o u n d a r y ( T o d d , 1 98 5) ,
w h i c h h a s b e e n s h o w n t o a p p l y r e c u r s i v e ly a t a n u m b e r o f l e v el s ( S h a ff e r
& T o d d , 1 9 87 ).
I n T o d d (1 98 5) a m o d e l o f r u b a t o w a s e s t a b l i s h e d w h i c h g e n e r a t e d a
d u r a t i o n s t r u c t u r e f r o m a s t r u c t u r a l d e s c r i p t i o n o f a p i e c e o f m u s i c . T h e
i d e a o f t h e m o d e l w a s t h a t t h e p e r f o r m e r u s e s " p h r a s e f in a l l e n g t h e n i n g "
t o s ig n a l a b o u n d a r y - - t h e d e g r e e o f s l o w i n g d e t e r m i n e d b y th e
69
Downl
oad
ed
By:
[Ingenta
Content
Di
strib
uti
on
Psy
Press
Ti
tl
es]
At:08
:065
Decemb
er2009
Contemporary
usic Review
1989, Vol. 3 pp. 69-88
Photocopying permitted by license only
1989 Harwood Academic Publishers GmbH
Printed in the United Kingdom
computational model
o
rubato
Neil
Todd
Department of Psychology University of Exeter Exeter UK
Presented is a model of rubato, implemented in Lisp, in which expression is viewed as
the
mapping of musical structure into the variables of expression. The basic idea is that the
performer uses phrase final lengthening as a device to reflect some internal representat ion
of the phrase structure. The representation is based on Lardahl and Jackendoff's time-span
reduction. The basic heuristic in the model is recursive involving look-ahead and planning
at a number of levels. The planned phrasings are superposed beat by beat and the output
from
the
program is a list of durations which could easily be adapted to be
sent
to a
synthesiser given a suitable system.
KEYWORDS computational modelling, music cognition, musical performance, rubato,
mental representation, mental process.
Introduction
One
of
the
most ubiquitous expressive devices in musical performance is
rubato. Most notably it is used in music of the romantic era, but is also
evident in a variety of
other
styles. Research on music performance
(Seashore, 1938; Shaffer, 1981; Clarke, 1984; Todd, 1985; Bengtsson
Gabrielsson, 1980; Sundberg Verillo, 1980) involving the precise
measurement of duration has shown that there are a number of basic
observations which can be made. The first is
that
skilled performers can
show
a remarkable degree of reproducibility from one performance to the
next (Shaffer, 1984; Gabrielsson, 1987). This precision in timing shows
that
the performance must involve the use of generative procedures and a
precise internal representation of underlying expressive form. A second
observation is the use of slowing to mark a phrase boundary (Todd, 1985),
which has
been shown
to apply recursively at a
number
of levels (Shaffer
Todd, 1987).
In Todd
1985)
a model of rubato
was
established which generated a
duration structure from a structural description of a piece of music. The
idea of the model
was
that the
performer uses
phrase
final lengthening
to signal a
boundary
- the degree of slowing determined by the
69
Contemporary
usic Review
1989, Vol. 3 pp. 69-88
Photocopying permitted by license only
1989 Harwood Academic Publishers GmbH
Printed in the United Kingdom
computational model
o
rubato
Neil
Todd
Department of Psychology University of Exeter Exeter UK
Presented is a model of rubato, implemented in Lisp, in which expression is viewed as
the
mapping of musical structure into the variables of expression. The basic idea is that the
performer uses phrase final lengthening as a device to reflect some internal representat ion
of the phrase structure. The representation is based on Lardahl and Jackendoff's time-span
reduction. The basic heuristic in the model is recursive involving look-ahead and planning
at a number of levels. The planned phrasings are superposed beat by beat and the output
from
the
program is a list of durations which could easily be adapted to be
sent
to a
synthesiser given a suitable system.
KEYWORDS computational modelling, music cognition, musical performance, rubato,
mental representation, mental process.
Introduction
One
of
the
most ubiquitous expressive devices in musical performance is
rubato. Most notably it is used in music of the romantic era, but is also
evident in a variety of
other
styles. Research on music performance
(Seashore, 1938; Shaffer, 1981; Clarke, 1984; Todd, 1985; Bengtsson
Gabrielsson, 1980; Sundberg Verillo, 1980) involving the precise
measurement of duration has shown that there are a number of basic
observations which can be made. The first is
that
skilled performers can
show
a remarkable degree of reproducibility from one performance to the
next (Shaffer, 1984; Gabrielsson, 1987). This precision in timing shows
that
the performance must involve the use of generative procedures and a
precise internal representation of underlying expressive form. A second
observation is the use of slowing to mark a phrase boundary (Todd, 1985),
which has
been shown
to apply recursively at a
number
of levels (Shaffer
Todd, 1987).
In Todd
1985)
a model of rubato
was
established which generated a
duration structure from a structural description of a piece of music. The
idea of the model
was
that the
performer uses
phrase
final lengthening
to signal a
boundary
- the degree of slowing determined by the
69
-
8/11/2019 A computational model of rubato (Todd).pdf
3/21
70 Neil Todd
importance of the boundary. The input to the model was the time-span
reduction of Lerdahl and Jackendoff's theory (1983). Whilst the model
gave a reasonable description of the data from actual performances of
some pieces there were, however, a numbe r of objections to the model as
iL stood. This has led to the formulation of a new model. In this paper I will
describe the new model and the reasoning which led to its formulation.
he reduction hyp othesis and k no w ledg e representat ion
The first problem with the Todd (1985) model stems from the fact that it
inherits the reduction hyp othes is of Lerdahl & Jackendoff's theory.
That is, the listener, and therefore the performer, sees each event in a
single coherent s truc ture.
This hypothesis places too high a demand on
working memory to be psychologically plausible. In terms of the model it
means that wh en computing a boundary strength, every event in time ~
span reduction is taken into account, irrespective of how close, or how far
apart, the events are in time. This leads to the prediction of more degrees
of boundary strength, and therefore degrees of relative slowing, than can
be discerned from the data. On the other hand , it is both psychologically
plausible and musically necessary that the performer should have some
kind of global overview of the piece as well as being able to look ahe ad
to some degree in order to plan a phrase.
A solution to this problem, which is the first premise of the updated
model, is to suppose that the internal rep resent atio n-- rather than being
a single, simply connected tree - - is composed of a set (or forest) of trees
organised on a number of hierarchic levels with each subset of trees at one
level being bound by a tree at a higher level. This accords with
Anderson's ACT* theory of cognition (1983). In the theory knowledge
comes in chunks or cognitive units which can be such things as
propositions, spatial images or temporal relations. A cognitive unit
encodes a set of no more than about five elements. Larger structures are
created by the hierarchical embedding of cognitive units. Of particular
interest to us here are cognitive units encoding temporal information
which Anderson refers to as temporal strings . The notion of temporal
strings accords well with the idea of musical groups.
A model of performance constructed on this basis predicts a duration
structure determined by the superposition of a number of hierarchic
timing components, from a global component, span ning the whole piece,
to a local component spanning a few beats with each component
corresponding to structural level. This overcomes the objections
discussed above because for any event at one level the number of other
events directly connected is limited. At the same time it allows for look
ahead and gives the performer global overview.
he process o f performance
A second object ion to the Todd (1985) model is that it is off line . In other
Downl
oad
ed
By:
[Ingenta
Content
Di
strib
uti
on
Psy
Press
Ti
tl
es]
At:08
:065
Decemb
er2009
70
Neil Todd
importance of the
boundary.
The fnput to the model was the time-span
reduction of Lerdahl
and
Jackendoff's theory (1983). Whilst the model
gave a reasonable description of the data from actual performances of
some
pieces there were, however, a
number
of objections to
the
model
as
it stood. This has led to the formulation of a new model.
In
this paper I will
describe the
new model and
the reasoning which led to its formulation.
The reduction
hypothesis
and knowledge representation
The first problem with the Todd 1985) model stems from the fact that it
inherits the reduction hypothesis of Lerdahl Jackendoff's theory.
That is, the listener, and therefore the performer, sees each event in a
single
coherent structure
This hypothesis places too high a
demand
on
working
memory
to
be
psychologically plausible. In terms of the model it
means that when computing a boundary strength, every
event
in time
span reduction is taken into account, irrespective of how close, or how far
apart,
the
events are in time. This leads to
the
prediction of more degrees
of boundary strength, and therefore degrees of relative slowing, than can
be discerned from
the
data. On the
other hand,
it is
both
psychologically
plausible and musically necessi'1ry that the performer should have some
kind
of global overview of the piece as well as being able to look ahead
to some degree in
order
to
plan
a phrase.
A solution to this problem, which is
the
first premise of the
updated
model, is to suppose that the internal representation -
rather
than being
a single, simply connected tree - is composed of a
set
(or forest) of trees
organised on a number of hierarchic levels with each
subset
of trees at one
level being bound by a tree at a higher level. This accords with
Anderson's ACT* theory of cognition (1983). In the theory knowledge
comes in
chunks
or cognitive units which can be
such
things as
propositions, spatial images or temporal relations. A cognitive
unit
encodes a
set
of
no
more than about five elements. Larger structures are
created by the hierarchical embedding of cognitive units. Of particular
interest to
us
here are cognitive units encoding temporal information
which
Anderson
refers to as temporal strings . The notion of temporal
strings accords well with the idea of musical groups.
A model of performance constructed on this basis predicts a duration
structure
determined
by
the
superposition of a number of hierarchic
timing components, from a global
component, spanning
the whole piece,
to a local component spanning a few beats with each component
corresponding to structural level. This overcomes the objections
discussed above because for any event at one level the number of other
events directly connected is limited.
At
the same time it allows for look
ahead
and
gives the performer global overview.
The process of performance
A second objection to the Todd 1985) model is that it is off line . In other
70
Neil Todd
importance of the
boundary.
The fnput to the model was the time-span
reduction of Lerdahl
and
Jackendoff's theory (1983). Whilst the model
gave a reasonable description of the data from actual performances of
some
pieces there were, however, a
number
of objections to
the
model
as
it stood. This has led to the formulation of a new model.
In
this paper I will
describe the
new model and
the reasoning which led to its formulation.
The reduction
hypothesis
and knowledge representation
The first problem with the Todd 1985) model stems from the fact that it
inherits the reduction hypothesis of Lerdahl Jackendoff's theory.
That is, the listener, and therefore the performer, sees each event in a
single
coherent structure
This hypothesis places too high a
demand
on
working
memory
to
be
psychologically plausible. In terms of the model it
means that when computing a boundary strength, every
event
in time
span reduction is taken into account, irrespective of how close, or how far
apart,
the
events are in time. This leads to
the
prediction of more degrees
of boundary strength, and therefore degrees of relative slowing, than can
be discerned from
the
data. On the
other hand,
it is
both
psychologically
plausible and musically necessi'1ry that the performer should have some
kind
of global overview of the piece as well as being able to look ahead
to some degree in
order
to
plan
a phrase.
A solution to this problem, which is
the
first premise of the
updated
model, is to suppose that the internal representation -
rather
than being
a single, simply connected tree - is composed of a
set
(or forest) of trees
organised on a number of hierarchic levels with each
subset
of trees at one
level being bound by a tree at a higher level. This accords with
Anderson's ACT* theory of cognition (1983). In the theory knowledge
comes in
chunks
or cognitive units which can be
such
things as
propositions, spatial images or temporal relations. A cognitive
unit
encodes a
set
of
no
more than about five elements. Larger structures are
created by the hierarchical embedding of cognitive units. Of particular
interest to
us
here are cognitive units encoding temporal information
which
Anderson
refers to as temporal strings . The notion of temporal
strings accords well with the idea of musical groups.
A model of performance constructed on this basis predicts a duration
structure
determined
by
the
superposition of a number of hierarchic
timing components, from a global
component, spanning
the whole piece,
to a local component spanning a few beats with each component
corresponding to structural level. This overcomes the objections
discussed above because for any event at one level the number of other
events directly connected is limited.
At
the same time it allows for look
ahead
and
gives the performer global overview.
The process of performance
A second objection to the Todd 1985) model is that it is off line . In other
-
8/11/2019 A computational model of rubato (Todd).pdf
4/21
A c o mp u ta tio n a l mo d e l o f ru b a to 71
w o r d s i t d o e s n o t d e s c r i b e th e p r o c e s s o f p e r f o r m a n c e . W h i ls t it is
r e a s o n a b l e t o s u p p o s e t h a t t h e p e r f o r m e r c a n h o l d t h e w h o l e s t r u c t u re i n
l o n g - t e r m m e m o r y , i n d e e d a m u s i c i a n s ' s a b il it y t o m e m o r i s e i s q u i t e
r e m a r k a b l e , i t s e e m s i m p l a u s i b le t h a t t h e p e r f o r m e r c o u l d a c c e s s t h e
w h o l e s t ru c t u r e a t a n y o n e t im e . I n t h e e a rl y m o d e l t h e c o m p u t a t i o n s
w e r e d o n e f o r e a c h c o m p o n e n t a n d t h e n a d d e d t o g et h e r. I n a n a ct u al
p e r f o r m a n c e t h e c o m p u t a t i o n s a r e d o n e a s e a c h p h r a s e i s a c c e s s e d i n tu r n
a n d t h e c o m p o n e n t s s u p e r p o s e d n o t e b y n o te .
T h e o b v i o u s a n s w e r , a n d t h is is t h e s e c o n d p r e m i s e o f t h e n e w m o d e l ,
i s t h a t in o r d e r t o d e sc r i b e th e p r o c e s s o f p e r f o r m a n c e t h e m o d e l n e e d s t o
b e f o r m u l a t e d i n c o m p u t a t i o n a l t e r m s a n d i m p l e m e n t e d i n a s u i t a b l e
h i g h - l e v e l la n g u a g e s u c h a s L i sp . In p a r ti c u l a r, w h a t i s i m p o r t a n t h e r e i s
t h e i d e a t h a t a p r o c e s s s h o u l d b e c a st i n te r m s o f a n e f f ec t iv e p r o c e d u r e
( L o n g u e t - H i g g i n s , 1 97 8, 1 98 1; J o h n s o n - L a i r d , 1 98 3), t h u s e n a b l i n g t h e
t h e o r y t o b e p r e c i s e a n d t e s ta b l e .
he indeterminism of individu al perform ances
W h i l s t s u c h a t h e o r y d o e s m a k e p r e d i c t i o n s , g i v e n a c e r t a i n i n p u t , t h e
g o a l o f t h e t h e o r y i s n o t t h e p r e d i c t io n o f i n d i v i d u a l p e r f o r m a n c e s a s
s u c h , b u t t h e p r i n c i p l e d e x p l a n a t i o n o f p e r f o r m a n c e d a t a . T h i s i s s o i n
p s y c h o l o g y in g e n e r a l, a n d m u s i c p s y c h o l o g y in p a rt ic u la r , b e c a u s e if t h e
t h e o r y w e r e c o m p l e t e l y d e te r m i n i s ti c i t w o u l d n e g a t e t h e c r e at iv e a s p e c t
o f p e r f o r m a n c e . J o h n s o n - L a i r d (198 3) h a s e x p r e s s e d t h is i n d e t e r m i n i s m
o f i n d i v i d u a l p e r f o r m a n c e s i n t h e l a n g u a g e o f c o m p u t e r s c i en c e :
I f h u m a n b e in g s a r e a t l e a st a s co m p l i c a te d a s T u r i n g m a c h i n e s a n d t h e ir
i n d i v i d u a l p r o c e s se s o f t h o u g h t d i f f e r a s a r e s u l t o f t h e i r g e n e s a n d e x p e r ie n c e ,
t h e n t h e i r b e h a v i o u r i s m o s t u n l i k e l y t o b e c o m e w h o l l y p r e d i c t a b l e , b e c a u s e
t h e r e is n o e f f e c ti v e p r o c e d u r e t h a t c a n p r e d i c t t h e b e h a v i o u r o f a n a r b i t r a r y
T u r i n g m a c h i n e . T h e r e i s t h u s l i t t l e d a n g e r o f c r e a t i n g a p s y c h o l o g y c a p ab le o f
m o d e l l in g a n i n d i v i d u a l ' s t h o u g h t s - - a n e v e n t u a l i t y l ik e l y t o d e s t r o y t h e
s p o n t a n e i t y a n d s i g n i f ic a n c e o f l ife . B u t t h e r e a r e n o
a p r io r i
r e a s o n s f o r
s u p p o s i n g t h a t i t is i m p o s s i b l e t o d e v e lo p s c i e n t i f ic t h e o r ie s o f g e n e r a l
p s y c h o l o g i c a l a b i l i t i e s .
[ Joh nso n-L a i rd , 1983 ; p . 12 ]
h e c o m p u t a t io n a l t h e o r y o f a n e x p r e s s io n s y s t e m
T h e s e t w o is s u e s d i s c u s s e d a b o v e , o f r e p r e s e n t a t i o n a n d p r o c e s s , a r e
c e n tr a l to a n y i n f o r m a t i o n - p r o c e s s i n g t y p e a p p r o a c h t o c o g n i t i o n a n d
c o g n i t iv e m o d e l l i n g . O u r m a i n t a sk , t h e r e f o re , i n t h e c o n s t r u c t i o n o f s u c h
a m o d e l is t o m a k e e x p li c it , in t h e f o r m o f a n a l g o r i t h m , t h e p r o c e s s o f
p e r f o r m a n c e a n d i ts i n p u t . H o w e v e r , a s D a v i d M a r r (1 98 2) h a s s a i d s u c h
a s y s t e m c a n b e v i e w e d f r o m t h r e e l e v e l s o f e x p l a n a ti o n :
Downl
oad
ed
By:
[Ingenta
Content
Di
strib
uti
on
Psy
Press
Ti
tl
es]
At:08
:065
Decemb
er2009
A
computational model
of rubato 71
words it does not describe the process of performance. Whilst it is
reasonable to
suppose that
the performer can hold the whole structure in
long-term memory, indeed a musicians's ability to memorise is quite
remarkable, it seems implausible
that
the
performer could access
the
whole structure at
anyone
time. In the early model the computations
were
done
for each
component and then
added together. In
an
actual
performance the computat ions are
done as each phrase is accessed
in turn
and
the
components superposed
note by note.
The obvious answer, and this is the second premise of the new model,
is that in order to describe the process of performance the model
needs
to
be formulated in computational terms
and implemented
in a suitable
high-level language
such
as Lisp. In particular, what is
important
here is
the idea that a process should be cast in terms of an effective procedure
(Longuet-Higgins, 1978, 1981; Johnson-Laird, 1983),
thus
enabling the
theory to be precise and testable.
he indeterminism
o
individual performances
Whilst
such
a theory does make predictions, given a certain input, the
goal of the theory is not the prediction of individual performances as
such, but the principled explanation of performance data. This is so in
psychology in general, and music psychology in particular, because
i
the
theory
were
completely deterministic it would negate the creative aspect
of performance. Johnson-Laird
1983)
has expressed this indeterminism
of individual performances in the language of
computer
science:
If human beings
are
at least
as complicated
as Turing machines and their
individual processes of thought
differ
as a result of
their
genes
and experience
then their behaviour
is
most unlikely
to become
wholly predictable because
there
is no effective procedure that
can
predict the
behaviour
of an arbitrary
Turing machine. There is thus little danger ofcreating a psychology
capable
of
modelling an
individual's thoughts - an eventuality
likely to destroy
the
spontaneity
and significance
of
life.
But
there
are no
a priori reasons for
supposing that
it
is impossible to develop scientific theories of general
psychological abilities.
Uohnson-Laird, 1983; p. 12]
he computational theory
o
an expression system
These two issues discussed above, of representation and process, are
central to
any
information-processing type
approach
to cognition
and
cognitive modelling. Our main task, therefore, in the construction of such
a model is to make explicit, in the form of
an
algorithm, the process of
performance and its input. However, as David Marr 1982) has said such
a system can be viewed from three levels of explanation:
A
computational model
of rubato 71
words it does not describe the process of performance. Whilst it is
reasonable to
suppose that
the performer can hold the whole structure in
long-term memory, indeed a musicians's ability to memorise is quite
remarkable, it seems implausible
that
the
performer could access
the
whole structure at
anyone
time. In the early model the computations
were
done
for each
component and then
added together. In
an
actual
performance the computat ions are
done as each phrase is accessed
in turn
and
the
components superposed
note by note.
The obvious answer, and this is the second premise of the new model,
is that in order to describe the process of performance the model
needs
to
be formulated in computational terms
and implemented
in a suitable
high-level language
such
as Lisp. In particular, what is
important
here is
the idea that a process should be cast in terms of an effective procedure
(Longuet-Higgins, 1978, 1981; Johnson-Laird, 1983),
thus
enabling the
theory to be precise and testable.
he indeterminism
o
individual performances
Whilst
such
a theory does make predictions, given a certain input, the
goal of the theory is not the prediction of individual performances as
such, but the principled explanation of performance data. This is so in
psychology in general, and music psychology in particular, because
i
the
theory
were
completely deterministic it would negate the creative aspect
of performance. Johnson-Laird
1983)
has expressed this indeterminism
of individual performances in the language of
computer
science:
If human beings
are
at least
as complicated
as Turing machines and their
individual processes of thought
differ
as a result of
their
genes
and experience
then their behaviour
is
most unlikely
to become
wholly predictable because
there
is no effective procedure that
can
predict the
behaviour
of an arbitrary
Turing machine. There is thus little danger ofcreating a psychology
capable
of
modelling an
individual's thoughts - an eventuality
likely to destroy
the
spontaneity
and significance
of
life.
But
there
are no
a priori reasons for
supposing that
it
is impossible to develop scientific theories of general
psychological abilities.
Uohnson-Laird, 1983; p. 12]
he computational theory
o
an expression system
These two issues discussed above, of representation and process, are
central to
any
information-processing type
approach
to cognition
and
cognitive modelling. Our main task, therefore, in the construction of such
a model is to make explicit, in the form of
an
algorithm, the process of
performance and its input. However, as David Marr 1982) has said such
a system can be viewed from three levels of explanation:
-
8/11/2019 A computational model of rubato (Todd).pdf
5/21
72 Neil Todd
A t o ne e x t r e m e t he top le v e l i s t he abs t r ac t c om pu t a t i on a l t he or y o f t he de v i c e
i n w h i c h t he pe r f o r m anc e o f t he de v i c e is c har ac t e ri z ed a s a m a pp i ng f r om one
k i nd o f i n f o r m a t i on t o ano t he r . . . . I n t he c e n t r e i s t he c hoi ce o f r e pr e s e n t a t i on
f o r t h e i n p u t a n d o u t p u t a n d t h e a l g o r it h m t o be u s e d t o tr a n s f o r m o n e i n to t h e
o t h er . A n d a t t h e o t h e r e x tr e m e a r e th e d e ta i ls o f h o w t h e a l g o r i th m a n d t h e
representa t ion are rea l i zed phys ica l ly .
[Marr, 1982; p. 24]
At the level of computational theory then, is useful to express the
various processes of music performance in symbolic terms. Let N stand
for the music notation or score, P for performance, and 9 for the internal
representation. Thus we can think of the process of performance as a
mapping:
9 :
v ~ ~ p 1 . a )
where the map ping is carried out by a pe r f o r m anc e p r oc e dur e or f u n c t i o n ~ .
In the same way the process of sight-reading can be thought of as a double
mapping:
N---~ 9 ~ P (1.b)
Conversely, we can think of the process of perception as the mapping:
A :P--~ ~ (2.a)
where the mapping is carried out by a
l i s t e n i ng p r oc e dur e
or
f u n c t i o n A .
Again in the same manner the process of notation can be thought of as:
P---~ 9 ~ N (2.b)
So, at the algor ithmic level then , our task is twofold: a) to find a suitable
representa tion for ~; and b) to make explicit an algorithm for performing
the mapping 9 --~ P.
ethodology
The methodology adopted in order to implement the twofold task
outlined above is threefold:
(a) a n a l y si s - - which involves finding a value for ~, either from the score
or the data;
(b) s y n t h e s i s - - which involves taking the value for 9 and using it as an
input to a performance algorithm which generates an output in the
form of a graph or list of numbers; a nd
(c) e v a l u a t i o n - - which involves the comparison of data with algorithm
output.
This metho d is, of course, similar to the analysis-by-synthesis me tho d
of Sundberg and his co-workers (Fryden & Sundberg, 1984) but pe rhaps
more closely related to the method of Risset and Wessel in their work on
timbre (Risset & Wessel, 1982). The differences with the Sundberg
Downl
oad
ed
By:
[Ingenta
Content
Di
strib
uti
on
Psy
Press
Ti
tl
es]
At:08
:065
Decemb
er2009
72 Neil
Todd
t one extreme the top
level is
the
abstract
computational theory of
the
device
in which
the
performance of
the
device is characterized
as
a mapping
from
one
kind of information
to another
In the centre is
the
choice
of
representation
for the
input
and
output
and
the
algorithm
to
be
used
to
transform
one
into
the
other.
nd
at
the
other extreme are
the
details
of
how
the
algorithm and
the
representation are realized physically.
[Marr,
1982; p.
24]
At the level of computational theory then, is useful to express
the
various processes of music performance
in
symbolic terms. Let N
stand
for
the
music notation
or
score, P for performance,
and
I for the internal
representation. Thus
we
can think of the process of performance as a
mapping:
l.a)
where
the mapping
is carried
out
by a performance procedure or function
1T.
In
the same way the process of sight-reading can be thought of as a double
mapping:
l.b)
Conversely,
we
can think of the process of perception as the mapping:
A ~
I 2.a)
where the
mapping
is carried
out
by
a
listening
procedure
or
function
A.
Again in the same manner the process of notation can be thought of as:
P ~ I ~ N 2.b)
So,
at
the algorithmic level
then our
task is twofold: a) to find a suitable
representation for
1 ; and b)
to make explicit
an
algorithm for performing
the
mapping
I P.
ethodology
The methodology
adopted
in order to implement the twofold task
outlined above is threefold:
(a)
analysis - which involves finding a value for 1 , either from the score
or the data;
(b) synthesis - which involves taking the value for I
and
using it as
an
input to a performance algorithm which generates
an output in
the
form of a
graph or
list of numbers;
and
(c)
evaluation
- which involves the comparison of ~ t with algorithm
output.
This method is, of course, similar to the analysis-by-synthesis
method
of Sundberg
and
his co-workers (Fryden Sundberg, 1984) but
perhaps
more closely related to the
method
of Risset
and
Wessel
in
their work on
timbre (Risset Wessel, 1982). The differences with the
Sundberg
72 Neil
Todd
t one extreme the top
level is
the
abstract
computational theory of
the
device
in which
the
performance of
the
device is characterized
as
a mapping
from
one
kind of information
to another
In the centre is
the
choice
of
representation
for the
input
and
output
and
the
algorithm
to
be
used
to
transform
one
into
the
other.
nd
at
the
other extreme are
the
details
of
how
the
algorithm and
the
representation are realized physically.
[Marr,
1982; p.
24]
At the level of computational theory then, is useful to express
the
various processes of music performance
in
symbolic terms. Let N
stand
for
the
music notation
or
score, P for performance,
and
I for the internal
representation. Thus
we
can think of the process of performance as a
mapping:
l.a)
where
the mapping
is carried
out
by a performance procedure or function
1T.
In
the same way the process of sight-reading can be thought of as a double
mapping:
l.b)
Conversely,
we
can think of the process of perception as the mapping:
A ~
I 2.a)
where the
mapping
is carried
out
by
a
listening
procedure
or
function
A.
Again in the same manner the process of notation can be thought of as:
P ~ I ~ N 2.b)
So,
at
the algorithmic level
then our
task is twofold: a) to find a suitable
representation for
1 ; and b)
to make explicit
an
algorithm for performing
the
mapping
I P.
ethodology
The methodology
adopted
in order to implement the twofold task
outlined above is threefold:
(a)
analysis - which involves finding a value for 1 , either from the score
or the data;
(b) synthesis - which involves taking the value for I
and
using it as
an
input to a performance algorithm which generates
an output in
the
form of a
graph or
list of numbers;
and
(c)
evaluation
- which involves the comparison of ~ t with algorithm
output.
This method is, of course, similar to the analysis-by-synthesis
method
of Sundberg
and
his co-workers (Fryden Sundberg, 1984) but
perhaps
more closely related to the
method
of Risset
and
Wessel
in
their work on
timbre (Risset Wessel, 1982). The differences with the
Sundberg
-
8/11/2019 A computational model of rubato (Todd).pdf
6/21
A comp utational model ofrubato 73
method are that the starting point here is actual performances, rather
than performer intuitions, a nd the evaluation process involves the direct
comparison of data and model, rather than the subjective rating of
generated output.
A n a l y s i s : s c o r e > r e p r e s e n t a t i o n v s . data ~ r e p r e s e n t a t i o n
We ne ed to find a value for the internal representa tion ~. A distinction is
made here between three possible representations. First, the ana l y s t s
r e pr e se n t a ti on ~ A , which is determined directly from the score; second, the
per former s r epresenta t ion ~p, which is also determined from the score but
which is unobservable; and third, the da ta de t e r m i ne d r e pr e se n t a ti on ~ m . So,
we can represent the computational theory at this stage thus:
~trp --~ Pp ~ at D
N ~ (3)
~A
To find a value for ~A involves taking the score of the piece of music und er
investigation and production an anlysis of the grouping or phrase
structure. At the mome nt t he most useful analytic meth od is that
developed by Lerdahl and Jackendoff (1983) despite its deficiencies
(Slawson Peel, 1985; Clarke, 1986; Baker, in press). After the analysis is
complete the grouping is converted to a Lisp representation which
becomes the input to an algorithm for generating a duration structure.
(see figure 1).
( s e t q t s r ( ( A) ( B)
( s e t q A ( (a ) ( a ) ) )
( s e t q B ( (b ) ( b ) ) )
( s e t q a ( 3 1 2 i ) )
( s e t q b ( 3 1 2 I ) )
A ) ) )
Figure 1 A Lisp representationof Lerdahland Jackendoff'sbracket notationfor grouping.
At the top level there are two groups A and B arranged symmetrically n the order ABA.
Group A contains he sub-group a repeated and group Bcontains he sub-group b repeated.
The integers represent the metricalstrength of a beat.
A value for ~D determined by analysing the data from actual
performances. The basic idea is that a slowing indicates a group
bounda ry. This can be done systematically using an algorithm l i s t e n
which takes that data as input and returns a Lisp representation of the
grouping t s r (Todd, in press).
Downl
oad
ed
By:
[Ingenta
Content
Di
strib
uti
on
Psy
Press
Ti
tl
es]
At:08
:065
Decemb
er2009
A computational model o
rubato 73
method
are
that the
starting point here is actual performances, rather
than
performer intuitions,
and
the evaluation process involves the direct
comparison of data
and
model, rather
than the
subjective rating of
generated
output.
Analysis: score _ representation vs. data _ representation
We
need
to find a value for
the
internal representation
It.
A distinction is
made
here between three possible representations. First, the analyst s
representation qt
A
which is determined directly from the score; second, the
performer s representation It
p,
which is also determined from the score but
which is unobservable;
and
third,
the
data determined representation ltD So,
we can represent the computational theory at this stage thus:
3)
To find a value for ItA involves taking the score of the piece of music
under
investigation
and
production
an
anlysis of the
grouping or
phrase
structure. At the
moment
the most useful analytic
method
is
that
developed
by
Lerdahl
and
Jackendoff
1983)
despite its deficiencies
(Slawson Peel, 1985; Clarke, 1986; Baker, n press). After the analysis is
complete the
grouping
is converted to a Lisp representation which
becomes the input to
an
algorithm for generating a duration structure.
(see figure 1).
(setq t s r A)
(B) (A)
(setq A a )
( a)
(setq B
b )
(b)
(setq
a
(3
1 2
1
(setq b (3 1 2 1
Figure 1 A Lisp representation of Lerdahl and Jackendoff s bracket notation for grouping.
At the top level there are two groups A and B arranged symmetrically
in
the order ABA.
Group
A contains the sub-group
a
repeated
and
group B contains the sub-group
b
repeated.
The integers represent the metrical strength of a beat.
A value for qtD determined by analysing the data from actual
performances. The basic idea is
that a slowing indicates a group
boundary. This can be done systematically using
an
algorithm
listen
which takes that data as input and returns a Lisp representation of the
grouping
tsr
(Todd, in press).
A computational model o
rubato 73
method
are
that the
starting point here is actual performances, rather
than
performer intuitions,
and
the evaluation process involves the direct
comparison of data
and
model, rather
than the
subjective rating of
generated
output.
Analysis: score _ representation vs. data _ representation
We
need
to find a value for
the
internal representation
It.
A distinction is
made
here between three possible representations. First, the analyst s
representation qt
A
which is determined directly from the score; second, the
performer s representation It
p,
which is also determined from the score but
which is unobservable;
and
third,
the
data determined representation ltD So,
we can represent the computational theory at this stage thus:
3)
To find a value for ItA involves taking the score of the piece of music
under
investigation
and
production
an
anlysis of the
grouping or
phrase
structure. At the
moment
the most useful analytic
method
is
that
developed
by
Lerdahl
and
Jackendoff
1983)
despite its deficiencies
(Slawson Peel, 1985; Clarke, 1986; Baker, n press). After the analysis is
complete the
grouping
is converted to a Lisp representation which
becomes the input to
an
algorithm for generating a duration structure.
(see figure 1).
(setq t s r A)
(B) (A)
(setq A a )
( a)
(setq B
b )
(b)
(setq
a
(3
1 2
1
(setq b (3 1 2 1
Figure 1 A Lisp representation of Lerdahl and Jackendoff s bracket notation for grouping.
At the top level there are two groups A and B arranged symmetrically
in
the order ABA.
Group
A contains the sub-group
a
repeated
and
group B contains the sub-group
b
repeated.
The integers represent the metrical strength of a beat.
A value for qtD determined by analysing the data from actual
performances. The basic idea is
that a slowing indicates a group
boundary. This can be done systematically using
an
algorithm
listen
which takes that data as input and returns a Lisp representation of the
grouping
tsr
(Todd, in press).
-
8/11/2019 A computational model of rubato (Todd).pdf
7/21
74 Neil Todd
S y n t h e s i s : r e p r e s e n t a t i o n ---> p e r f o r m a n c e
Having obtained
t s r
we need to make explicit the procedure ,rr for
mapping representation into the performance. We can represent the
computational theory at this stage thus:
~P ---> PP ~ ~ tY D - ' - > PD
~ffA ~ > PA
4)
such that each representation ~i has its corresponding performance Pi.
The performance is modelled using an algorithm p l a y See Append ix 1).
The basic heuristic of the algorithm is to look-ahead and plan the phrasing
of a group at a given level then move d own to the next sub-group, look-
ahead and plan, and so on recursively. The planned phrasings are
superposed onto an outp ut plan see
o u t p u t ,
Appendix 1) which
continuous ly evolves as the performance unfolds. Whe n a surface-group
is reached the first element of the ou tpu t plan is printed and discarded,
and so on. Whe n the surface-group is completed the program backtracks
to the next level and so on until all the surface groups are played. The
output from the program is a list of durations, which could easily be
adapted to be sent to a synthesiser given a suitable sys tem see figure 2).
The precise durations within a phrase are det ermined by a parabolic
function PB embedded within the performance procedure. This function
has the following form:
a2 { t a 4 - 1 ) } 2
= a6 , t = 1 , 2 ..... T 5)
PB t,
a i
a l J r ( 1 - - - - a 6 )
a 3
as
where t is metrical time and a i is a vector of parameters such that:
a l = t e m po ,
a2 = am p l i t ude ,
a3 = l e ng t h o f phr as e ,
a 4 ~ b o u n d a r y s t r e n g t h ,
a s = u p p e r l i m i t o f b o u n d a r y s t r e n g t h ,
a6 = o f f se t o f parabo la m i n i m um .
1 -
a6) -2 is a normali sation factor such that if the b oundary st rength a4 =
1 and t = a3 i.e. at the e nd of the phrase) the n a 2 represents the true
ampl itude Todd, 1985). As for the values of the parameters, al and
a2 are input at the start of the algorithm p l a y see functions s t a r t and
s e t _ u p _ v a t s , Appendix 1); a3 and a 4 are computed for each group as the
program runs see functions
p l a n
and
r u b a t o ,
Appendix 1); and as and a6
are set with in the program with a6 = 0.52. In Todd 1985) as = 11 but in the
new model as = 3 because the number of possible bound ary strengths is
reduced see function s e t _ u p _ v a t s , Appendix 1).
Downl
oad
ed
By:
[Ingenta
Content
Di
strib
uti
on
Psy
Press
Ti
tl
es]
At:08
:065
Decemb
er2009
74 Neil Todd
Synthesis: representation performance
Having obtained
tsr
we
need
to make explicit the procedure
1T
for
mapping
representation into
the
performance. We can
represent
the
computational theory at this stage thus:
4)
such
that
each representation
l i has
its corresponding performance Pi
The performance is modelled using an algorithm
play
See ppendix 1).
The basic heuristic of the algorithm is to look-ahead and plan the
phrasing
of a
group at
a given level
then
move
down
to
the
next sub-group, look
ahead and plan, and so
on
recursively. The planned phrasings are
superposed
onto
an output
plan
see output,
ppendix
1) which
continuously evolves as
the
performance unfolds. When a surface-group
is reached the first
element
of the output plan is printed and discarded,
and so on. When
the
surface-group is completed
the program
backtracks
to
the
next level and so on until all
the
surface
groups
are played. The
output from the program is a list of durations, which could easily be
adapted to be
sent
to a synthesiser given a suitable system see figure 2).
The precise
durations
within a
phrase
are
determined
by
a parabolic
function
P
embedded within
the
performance procedure. This function
has
the
following form:
_ a2
{t a4
- 1 } 2 _ )
PB t,ai)-a1+ 2 -a6 t-1,2, ... T 5
1
-
a6) a3
as
where t is metrical time and ai is a vector of parameters such that:
a1 tempo,
a2
amplitude,
a3
length of
phrase,
a4
boundary strength,
as upper limit ofboundary strength,
a6 offset ofparabola minimum.
1 - a6)-z is a normalisation factor
such
that i the boundary strength a4
1 and t
=
a3 i.e. at the
end
of the phrase) then a2 represents the true
amplitude Todd, 1985). As for
the
values of
the
parameters, al and
az are input at the start of the algorithm
play
see functions
start and
set_up_vars,
ppendix
1); a3
and
a4
are
computed
for each
group
as
the
program
runs
see functions
plan
and
rubato,
ppendix
1);
and
as
and
a6
are set within
the
program with
a6 =
0.52. In
Todd
1985) as
= 11 butin the
new
model
as = 3 because the
number
of possible
boundary strengths
is
reduced see function
set_up_vars,
ppendix 1).
74 Neil Todd
Synthesis: representation performance
Having obtained
tsr
we
need
to make explicit the procedure
1T
for
mapping
representation into
the
performance. We can
represent
the
computational theory at this stage thus:
4)
such
that
each representation
l i has
its corresponding performance Pi
The performance is modelled using an algorithm
play
See ppendix 1).
The basic heuristic of the algorithm is to look-ahead and plan the
phrasing
of a
group at
a given level
then
move
down
to
the
next sub-group, look
ahead and plan, and so
on
recursively. The planned phrasings are
superposed
onto
an output
plan
see output,
ppendix
1) which
continuously evolves as
the
performance unfolds. When a surface-group
is reached the first
element
of the output plan is printed and discarded,
and so on. When
the
surface-group is completed
the program
backtracks
to
the
next level and so on until all
the
surface
groups
are played. The
output from the program is a list of durations, which could easily be
adapted to be
sent
to a synthesiser given a suitable system see figure 2).
The precise
durations
within a
phrase
are
determined
by
a parabolic
function
P
embedded within
the
performance procedure. This function
has
the
following form:
_ a2
{t a4
- 1 } 2 _ )
PB t,ai)-a1+ 2 -a6 t-1,2, ... T 5
1
-
a6) a3
as
where t is metrical time and ai is a vector of parameters such that:
a1 tempo,
a2
amplitude,
a3
length of
phrase,
a4
boundary strength,
as upper limit ofboundary strength,
a6 offset ofparabola minimum.
1 - a6)-z is a normalisation factor
such
that i the boundary strength a4
1 and t
=
a3 i.e. at the
end
of the phrase) then a2 represents the true
amplitude Todd, 1985). As for
the
values of
the
parameters, al and
az are input at the start of the algorithm
play
see functions
start and
set_up_vars,
ppendix
1); a3
and
a4
are
computed
for each
group
as
the
program
runs
see functions
plan
and
rubato,
ppendix
1);
and
as
and
a6
are set within
the
program with
a6 =
0.52. In
Todd
1985) as
= 11 butin the
new
model
as = 3 because the
number
of possible
boundary strengths
is
reduced see function
set_up_vars,
ppendix 1).
-
8/11/2019 A computational model of rubato (Todd).pdf
8/21
00
W
X
I
J
Q
00o
0
~
0
0
i
I
|
|
|
0
0
0
0
0
C
,
0
0
0
W
I
0W
O
[
N
(
s
N
O
I
V
~
3
7
V
3
N
Downloaded By: [Ingenta Content Distribution Psy Press Titles] At: 08:06 5
VI
E
z
0
{
a
>
0
U J
: ;
--
0
U J
: ;
--
-
8/11/2019 A computational model of rubato (Todd).pdf
9/21
76 NeilTodd
Ev alu atio n: PA ~ PP ~ PD?
H a v i n g g e n e r a t e d PA o r PD w e n e e d t o c o m p a r e t h e m w i t h a n a c tu a l
p e r f o r m a n c e
Pp.
I n T o d d (1 98 5) t h e d a t a a n d m o d e l ( ie P p a n d
PA
w e r e
c o m p a r e d v i s u a l ly u s i n g t h e c r it er ia ; a ) t h e p o s i t io n o f p e a k s o r p o i n t s o f
s l o w i n g ; b ) t h e r e l a t iv e h e i g h t s o f th e p e a k s . W h i l st t h is m e t h o d i s u s e f u l
i t i s u n s a t i s f a c t o r y fo r a n u m b e r o f r e a s o n s . F i rs t, t h e c o m p a r i s o n o f
r e l a ti v e h e i g h t s i s o n l y q u a l i t a ti v e . S o , o b v i o u s l y a m o r e s y s t e m a t i c a n d
q u a n t i t a t iv e t e s t i s r e q u i r e d . H i e r a r c h i c a l c l u s t e r i n g ( J o h n s o n , 1 9 6 7 ) i s
s u c h a m e t h o d w h i c h h a s b e e n s u c c e s s f u ll y a p p l i e d i n a n a l y s i n g s p e e c h
( G r o s je a n a n d G e e , 19 83) a n d I a m c u r r e n t l y w o r k i n g o n w a y s o f a p p l y i n g
t h is t o m u s i c p e r f o r m a n c e .
A s e c o n d p r o b l e m l ie s i n t h e i n d e t e r m i n i s m o f i n d i v i d u a l p e r f o r m a n c e s
a s d i s c u s s e d a b o v e . W h i l s t i t i s o f t e n p o s s i b l e t o o b s e r v e c o n s i d e r a b l e
a c r o s s - p e r f o r m e r a g r e e m e n t ( S h a f fe r & T o d d , 1 98 7) t h e r e a r e a ls o m a n y
d i f fe r e n c e s . A l so , t h e r e i s n o r e a s o n w h y t h e a n a l y s t 's i n t e r p r e t a t io n
~ t r
s h o u l d b e t h e s a m e a s t h e p e r f o r m e r ' s ~ v s in c e th e r e is n o s u c h t h i n g a s a
s i n g le c o r r e c t g r o u p i n g . I t is f o r t h e s e r e a s o n s t h a t t h e i n p u t u s e d is t h e
r e p r e s e n t a t i o n d e r i v e d f r o m t h e d a t a ~ D w h i c h i s o b t a i n e d v i a t h e
a l g o r i t h m listen. T h is p r o c e d u r e i s c e r ta i n ly n o t i n t e n d e d t o g i v e li c en c e
t o a d j u s t t h e t h e o r y po st hoc t o fit e a c h s e t o f d a t a - - o n t h e c o n t r a r y t h e
s a m e p e r f o rm a n c e m a p p i n g p l a y is u s e d i n e a c h c a se . R e m e m b e r t h e g o a l
o f a t h e o r y o f p e r f o r m a n c e i s t h e p r i n c ip l e d e x p l a n a t io n o f p e r f o r m a n c e
d a t a. S o , w h i ls t w e c a n n o t p r e d i c t a p e r f o r m a n c e w i t h a n y c e r t a in t y w h a t
w e c a n s a y f o r e a c h p e r f o r m a n c e i s t h a t i f t h e f o ll o w i n g t h r e e a s s u m p t i o n s
p r o d u c e a g o o d m a t c h b e t w e e n P p a n d PD t h e n t h e a s s u m p t i o n s
c o n s t i t u t e a re a s o n a b l e e x p l a n a t i o n :
(a) t h e p e r f o r m e r h a s u s e d s l o w i n g t o i n d ic a t e a g r o u p i n g b o u n d a r y ;
(b ) t h e p e r f o r m e r ' s g r o u p i n g a n a l y s is c o r r e s p o n d s t o tsr;
(c) t h e p e r fo r m e r ' s m a p p i n g p r o c e d u r e c o r r e s p o n d s to play .
T h e m o d e l t h e n i s re a l ly a n a n a l y ti c al t h e o r y o f p e r f o r m a n c e r a t h e r t h a n a
p r e s c ri p ti v e t h e o r y o f p e r f o r m a n c e . H o w e v e r , t h e r e a r e n o r e a s o n s , if
e n o u g h t d a t a i s a m a s s e d , w h y p r o b a b i l i t y w e i g h t i n g s c o u l d n o t b e
a s s i g n e d t o v a r io u s p e r f o r m a n c e s a s a f u n c t i o n o f s ty l e a n d i n s t r u m e n t .
E x a m p l e s
P r e s e n t e d i n F i g u r e s 3 a n d 4 a r e t w o e x a m p l e s o f d a t a f r o m a c t u a l
p e r f o r m a n c e s c o m p a r e d a g a i n s t t h e m o d e l . T h e f i r s t e x a m p l e i s t a k e n
f r o m a p e r f o r m a n c e o f t h e A d a g i o f r o m t h e H a y d n S o n a t a in B -F lat w h i c h
w a s a ls o u s e d i n T o d d (1 98 5) s o th a t c o m p a r i s o n w i t h t h e o l d m o d e l is
p o s si b le . T h e s e c o n d e x a m p l e is t a k e n f r o m t w o p e r f o r m a n c e s o f t h e
C h o p i n p r e l u d e i n F # M i n o r (S h a f fe r a n d T o d d , 1 9 87 ). T h e d a t a w e r e
o b t a i n e d u s i n g t h e m e t h o d o f S h a f f e r (1 98 1).
Downl
oad
ed
By:
[Ingenta
Content
Di
strib
uti
on
Psy
Press
Ti
tl
es]
At:08
:065
Decemb
er2009
76 Neil Todd
Evaluation PA
e p e
P
D
?
Having generated P
A
or P
D
we need to compare them with
an
actual
performance
P
p
In
Todd 1985)
the
data
and
model (ie
p
and
P
A
were
compared visually using the criteria; a)
the
position of peaks or points of
slowing; b) the relative heights of the peaks. Whilst this
method
is useful
it is unsatisfactory for a
number
of reasons. First,
the
comparison of
relative heights is only qualitative. So, obviously a more systematic
and
quantitative test is required. Hierarchical clustering Gohnson, 1967) is
such
a
method
which
has been
successfully applied in analysing speech
(Grosjean
and
Gee, 1983)
and
I am currently working
on
ways of applying
this to music performance.
A second problem lies in
the indeterminism of individual performances
as discussed above. Whilst it is often possible to observe considerable
across-performer agreement (Shaffer Todd, 1987) there are also
many
differences. Also, there is
no
reason
why
the analyst's interpretation
qr
should be the same as
the
performer's
qrp
since there is
no
such thing as a
single correct grouping.
t
is for these reasons
that the
input
used
is the
representation derived from the data
qrD
which is obtained via
the
algorithm
listen
This procedure is certainly
not
intended to give licence
to adjust the theory
post hoc
to fit each set of data - on
the
contrary
the
same performance
mapping pl y
is
used
in each case. Remember the goal
of a theory of performance is the principled explanation of performance
data. So, whilst
we
cannot predict a performance
with any
certainty
what
we
can say for each performance is
that
if the following three assumptions
produce a good match between
p and
P
D
then
the
assumptions
constitute a reasonable explanation:
a) the
performer
has used
slowing to indicate a grouping boundary;
(b)
the
performer's grouping analysis corresponds to
tsr;
c)
the performer's
mapping
procedure corresponds to
play
The model
then
is really
an
analytical theory of performance rather than a
prescriptive theory of performance. However, there are
no
reasons,
if
enought
data is amassed, why probability weightings could not be
assigned to various performances as a function of style
and
instrument.
Examples
Presented
in
Figures 3
and
4 are two examples of data from actual
performances compared against the model. The first example is taken
from a performance of the Adagio from
the Haydn
Sonata
in
B-Flat which
was
also
used in
Todd 1985) so
that
comparison
with
the
old model is
possible. The second example is taken from two performances of the
Chopin prelude in
F
Minor (Shaffer
and
Todd, 1987). The data were
obtained using the
method
of Shaffer (1981).
76 Neil Todd
Evaluation PA
e p e
P
D
?
Having generated P
A
or P
D
we need to compare them with
an
actual
performance
P
p
In
Todd 1985)
the
data
and
model (ie
p
and
P
A
were
compared visually using the criteria; a)
the
position of peaks or points of
slowing; b) the relative heights of the peaks. Whilst this
method
is useful
it is unsatisfactory for a
number
of reasons. First,
the
comparison of
relative heights is only qualitative. So, obviously a more systematic
and
quantitative test is required. Hierarchical clustering Gohnson, 1967) is
such
a
method
which
has been
successfully applied in analysing speech
(Grosjean
and
Gee, 1983)
and
I am currently working
on
ways of applying
this to music performance.
A second problem lies in
the indeterminism of individual performances
as discussed above. Whilst it is often possible to observe considerable
across-performer agreement (Shaffer Todd, 1987) there are also
many
differences. Also, there is
no
reason
why
the analyst's interpretation
qr
should be the same as
the
performer's
qrp
since there is
no
such thing as a
single correct grouping.
t
is for these reasons
that the
input
used
is the
representation derived from the data
qrD
which is obtained via
the
algorithm
listen
This procedure is certainly
not
intended to give licence
to adjust the theory
post hoc
to fit each set of data - on
the
contrary
the
same performance
mapping pl y
is
used
in each case. Remember the goal
of a theory of performance is the principled explanation of performance
data. So, whilst
we
cannot predict a performance
with any
certainty
what
we
can say for each performance is
that
if the following three assumptions
produce a good match between
p and
P
D
then
the
assumptions
constitute a reasonable explanation:
a) the
performer
has used
slowing to indicate a grouping boundary;
(b)
the
performer's grouping analysis corresponds to
tsr;
c)
the performer's
mapping
procedure corresponds to
play
The model
then
is really
an
analytical theory of performance rather than a
prescriptive theory of performance. However, there are
no
reasons,
if
enought
data is amassed, why probability weightings could not be
assigned to various performances as a function of style
and
instrument.
Examples
Presented
in
Figures 3
and
4 are two examples of data from actual
performances compared against the model. The first example is taken
from a performance of the Adagio from
the Haydn
Sonata
in
B-Flat which
was
also
used in
Todd 1985) so
that
comparison
with
the
old model is
possible. The second example is taken from two performances of the
Chopin prelude in
F
Minor (Shaffer
and
Todd, 1987). The data were
obtained using the
method
of Shaffer (1981).
-
8/11/2019 A computational model of rubato (Todd).pdf
10/21
I
I
l
l
I
I
I
I
I
I
O
0
0
0
0
0
I
0
0
0
c
0
I
I
I
(
o
0
)
c00
o
d
c
O
0
0
oo
o(
1
i
~
I
i
0
~
r
O
~
~
.
~
(
~
)
N
O
I
L
V
H
A
O
H
V
H
Downloaded By: [Ingenta Content Distribution Psy Press Titles] At: 08:06 5
-
'
-
z
5000
4000
a
1
a2
3200ms
300ms
HAYDN
SONATA
B-FLAT
MAJOR
V\ \ :
:VJ
\
A \NVi l
a
. .
:; .
.,
; . j /
::
0
cc 3000