synchronization in duet performance: testing the two-person...

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Dirk Vorberg

Institut für PsychologieTechnische Universität Braunschweig

Braunschweig, Germany

RPPW2005, Alden Biesen

Synchronization in duet performance:Testing the two-person

phase error correction model

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Overview

1. How do ensemble players manage to remainsynchronized?

2. Sensorimotor synchronization, tapping alongperfect metronome.

Synchronization is achieved by linear phaseerror correction.

3. Extend model to duet performance.Major advantage: Use computer to simulateone of the duet partners.

4. Experimental study.Preliminary data.

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task: tap in close synchrony with the metronome

metronome

overtresponses

In In+1

An An+1

Definition of interresponse intervals and synchronizationerrors

interresponse intervals

synchronization errors („asynchronies“)

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In In+1

An An+1

overtresponses

metronomeCn Cn+1

The phase-correction model

(Vorberg & Wing, 1994, 1996; Vorberg & Schulze, 2002; Schulze & Vorberg, 2003)

Mn Mn+1 Mn+2

Tn Tn+1timercommands

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Mn Mn+1 Mn+2

An An+1

In In+1

Tn Tn+1timercommands

overt responses

metronomeCn Cn+1

The two-level timing model augmented by phase errorcorrection

1. basic assumption: Tn* = Tn + (1– α)An

1. testable consequence:An+1 = (1– α)An + (Tn+Mn+1-Mn) - Cn

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Model predictions I:response to experimental perturbations

-30

-20

-10

0

10

20 IRI synch err

Sync

h Er

r; IR

I - T

empo

(ms)

10 20

-30

-20

-10

0

10

20

Sync

h Er

r; IR

I - T

empo

(ms)

Response and interval no.10 20

Response and interval no.

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Results (Antje Fuchs, 2003)

-2 -1 p1 p2 p3 p4 p5 p6 1 2 3 4

-40

-30

-20

-10

0

10

20

30

TimekeeperTriple

session 1-3 session 4-6

IRI a

nd S

ynch

Err

[ms]

-2 -1 p1 p2 p3 p4 p5 p6 1 2 3 4

-40

-30

-20

-10

0

10

20

30

TimekeeperDuple


-2 -1 p1 p2 p3 p4 p5 p6 1 2 3 4

-40

-30

-20

-10

0

10

20

30 Motor DelayTriple


IRI a

nd S

ynch

Err

[ms]

Position within Period-2 -1 p1 p2 p3 p4 p5 p6 1 2 3 4

-40

-30

-20

-10

0

10

20

30

Motor DelayDuple


Position within Period

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Results (Antje Fuchs, 2003)

6 12 18 24

-30

-20

-10

0

10

Triple


IRI &

Syn

c Er

r (m

s)

6 12 18 24

-30

-20

-10

0

10Duple

6 12 18 24

-30

-20

-10

0

10

Triple

IRI &

Syn

ch E

rr (m

s)

Position within Period6 12 18 24

-30

-20

-10

0

10

Duple

Position within Period

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Model predictions II:serial or auto-covariance function (acvf)

AnAn-1...…Ai+1AiAi-1…A3A2A1

serial variance = acvf at lag 0 = acvf(0)

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Auto-covariance function (acvf)

AnAn-1....Ai+1AiAi-1…A3A2A1


lag 1 auto-covariance = acvf(1)

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Auto-covariance function (acvf)

An-1An-2..Ai+1AiAi-1…A3A2A1


lag 2 auto-covariance = acvf(2)

auto-correlation function acf(lag) = acvf(lag) / acvf(0)

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Predicted asynchrony acf (as a function of lag)

0 < α < 1 1 < α < 2

Note:

Synchronization performance is unstable if α outsidethis range.

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Basic assumption: Each player serves as metronome for theother one.

Parameters:Player A (subject)

timekeeper variance σT²motor variance σM²error correction α

Player B (metronome)timekeeper variance σU²motor variance σN²error correction β

Extension of the model to duet performance

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Predicted 2-person asynchrony acvf

var(A) = [(σT²+σU²)+2(α+β)(σM²+σN²)]

/ [1-(1-(α+β))²]cov(An,An+k) =

[1-(α+β)]k-1[var(A)(1-(α+β)) – (σM²+σN²)]

Predicted 1-person asynchrony acvf

var(A) = [( σT² ) + 2( α )( σM² )]

/ [1-(1-( α ))²]cov(An,An+k) =

[1-( α )]k-1[var(A)(1-( α )) – ( σM² )]

Two-person phase synchronization model: Main result

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Predicted asynchrony acf for two-person model:

1. Synchronization performance is unstable if α+β outside thisrange.

2. Predictions:Stable but oscillatory acf for β positive .Unstable synchronization for β negative.

0 < α+β < 1 1 < α+β < 2

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Experiment: Conditions

1. tempoIOI=450 ms / 300 ms

2. meterduple / triple / quadruple

3. metronome gain factorβ=0β=.4 / .8β=-.25 / -.50

4. seven subjects6 one hour sessions18 sequences/condition

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Results

1. Exemplary time series after six hours of practiceasynchroniesinterresponse intervals, IRI (subject)interonset intervals, IOI (metronome)

2. Auto-correlation functions, acf

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subject an: asynchronies(x-axis: tap no. 1 – 48; y-axis: tap-metronome asynchrony in ms)

β=0 β=.4 β=.8 β=-.25 β=-.50

slow

fast

50 ms

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subject an: IRIs (top) and IOIs (bottom)(x-axis: tap no. 1 – 48; y-axis: deviation from nominal IOI, in ms)

β=0 β=.4 β=.8 β=-.25 β=-.50

100 ms

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subject an: acf.s for slow (top) and fast tempi (bottom) (x-axis: lag 0 to 6; y-axis: correlation size)

β=0 β=.4 β=.8 β=-.25 β=-.50

duple triple quadruple

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subject bv: asynchronies slow (top) and fast (bottom)

β=0 β=.4 β=.8 β=-.25 β=-.50

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subject bv: IRIs (top) and IOIs (bottom)

β=0 β=.4 β=.8 β=-.25 β=-.50

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subject bv: acf.s for slow (top) and fast (bottom) tempi

β=0 β=.4 β=.8 β=-.25 β=-.50


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subject eh: asynchronies, slow (top) and fast (bottom)

β=0 β=.4 β=.8 β=-.25 β=-.50

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subject eh: IRIs (top) and IOIs (bottom)

β=0 β=.4 β=.8 β=-.25 β=-.50

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subject eh: acf.s for slow (top) and fast (bottom) tempi

β=0 β=.4 β=.8 β=-.25 β=-.50


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Empirical asynchrony acf.s (all subjects)

β=0 β=.4 β=.8 β=-.25 β=-.50

dupletriplequadruple

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Empirical asynchrony acvf.s (average across subjects)(x-axis: lag 0 to 6; y-axis: autocovariance at lag k)

β=0 β=.4 β=.8 β=-.25 β=-.50

dupletriplequadruple

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Summary and conclusions

1. Two-person model is in qualitative agreement withobservations.

As predicted, acf becomes oscillatory as metronomegain β increases.For negative gain β, performance is unstable for mostsubjects.

2. Subjects can to adapt their phase-correction strategy to that of the duet partner.

3. Next step: Quantitative model fit.4. Model-based experimental paradigm is a promising tool for

studying duet synchronization. The model is easilyextended to musically more challenging conditions.

synchronization in duet performance: testing the two-person...

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