Transcript A Model of Expressive Timing in Tonal Music

A Model of Expressive Timing
in Tonal Music
Neil Todd
Presented by Xiaodan Wu
Goal
 Combine the generative musical theory
with the principle of phrase-final
lengthening to generate a duration
structure corresponding to the rubato
in a performance.
Introduction
 Duration and Intensity
 Music is organized hierarchically.
Mozart - A Major Sonata K331
Phrase-Final Lengthening
 Phenomenon
 Boundary Marker
Generative Music Theory
 Grouping structure
 Metrical structure
 Time-span reduction
 Prolongation reduction
Unit Time Span
 It was introduced when we need to
compare the real-time duration with
metrical duration.
Structural Endings and Embedding Depth
C  (c1 ,, cn )
E  (e1 ,, en )
e j  Nb j  Nc j
• C is an ordered set of time
spans.
• cj is the time span containing
the jth structural ending.
• n is the total number of
structural endings.
• E is an ordered set of numbers
with one-one correspondence
with the elements of C.
• ej is the embedding depth of
the jth structure ending.
Tree diagram of a time-span reduction
Red for Cadence
[b1]
[b1] [b1]
4
[b1] [b1]
8
[b1] [b1]
12
[b1]
16
C = (4, 8, 12, 16)
E = (1, 3, 1, 5)
That is, E = [(0+1), (1+2), (0+1), (3+2)]
Duration
 Timing is organized on about 3 levels:
 Global component
 Intermediate components
 Local components
A model for
Intermediate Components
 Formulate it as a parabola
t ij
(e j 1 )
1 2
D(t ij )  4m( 
 ) A
ej
12
2
m is rubato amplitude constant
A is tempo constant
A hypothetical performance duration structure generated by
the model with the given TSR
[b1]
[b1] [b1]
4
[b1] [b1]
8
[b1] [b1]
12
[b1]
16
D
m
A
4
8
12
16 t
The Application
 Mozart A Major Sonata K.331
 Haydn Sonata 59 Adagio
 Chopin Trois Nouvelles Etudes No.3
Conclusion
 Is a approximation only.
 Should consider harmonic structure or
prolongation reduction in the future model.
 Should consider intensity and other
secondary expressive variables in the
future model.
Question?