A Model of Expressive Timing A Model of Expressive Timing in Tonal Musicin Tonal Music

Neil ToddNeil Todd

Presented by Xiaodan Wu

GoalGoal

Combine the generative musical theory Combine the generative musical theory with the principle of phrase-final length with the principle of phrase-final lengthening to generate a duration structure ening to generate a duration structure corresponding to the rubato in a perforcorresponding to the rubato in a performance.mance.

Introduction Introduction

Duration and Intensity Duration and Intensity Music is organized Music is organized hierarchicallyhierarchically..

Mozart - A Major Sonata K331

Phrase-Final LengtheningPhrase-Final Lengthening

PhenomenonPhenomenon Boundary Marker Boundary Marker

Generative Music TheoryGenerative Music Theory

Grouping structureGrouping structure Metrical structureMetrical structure Time-span reductionTime-span reduction Prolongation reduction Prolongation reduction

Unit Time SpanUnit Time Span

It was introduced when we need to It was introduced when we need to compare the real-time duration with compare the real-time duration with metrical duration. metrical duration.

Structural Endings and Embedding DepthStructural Endings and Embedding Depth

),,( 1 nccC

),,( 1 neeE jjj NcNbe

• 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 reductionTree diagram of a time-span reduction

[b1] [b1] [b1] [b1] [b1] [b1] [b1] [b1]4 8 12 16

C = (4, 8, 12, 16)

E = (1, 3, 1, 5)

That is, E = [(0+1), (1+2), (0+1), (3+2)]

Red for Cadence

DurationDuration

Timing is organized on about 3 levels:Timing is organized on about 3 levels: Global componentGlobal component Intermediate componentsIntermediate components Local componentsLocal components

A model for A model for Intermediate ComponentsIntermediate Components

Formulate it as a parabolaFormulate it as a parabola

Ae

e

tmtD j

j

ijij

2

1

)2

1

12

)((4)(

m is rubato amplitude constant

A is tempo constant

A hypothetical performance duration structure generated by A hypothetical performance duration structure generated by the model with the given TSRthe model with the given TSR

[b1] [b1] [b1] [b1] [b1] [b1] [b1] [b1]4 8 12 16

A

t

D

m

4 8 12 16

The ApplicationThe Application

Mozart A Major Sonata K.331Mozart A Major Sonata K.331 Haydn Sonata 59 AdagioHaydn Sonata 59 Adagio Chopin Trois Nouvelles Etudes No.3Chopin Trois Nouvelles Etudes No.3

ConclusionConclusion

Is a approximation only.Is a approximation only. Should consider harmonic structure or Should consider harmonic structure or

prolongation reduction in the future model.prolongation reduction in the future model. Should consider intensity and other Should consider intensity and other

secondary expressive variables in the secondary expressive variables in the future model.future model.

Question?Question?