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Not just syncopation: rhythmic complexity is … complex.

Leigh VanHandel

University of British Columbia

leigh.vanhandel@ubc.ca

Society for Music Theory, Jacksonville FL, November 9, 2024

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WEIRD

estern

ducated

ndustrialized

ich

emocratic

-Henrich et al, 2010

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Rhythmic complexity

Lempel and Ziv 1976
Povel and Essens 1985
Shmulevich and Povel 2000
Fitch and Rosenfeld 2007
Gomez, Thul, and Touissant 2007
Thul 2008
Thul and Touissant 2008
Fleurian et al 2015
Gündüz 2023
Senn et al 2023

Cooper and Meyer 1963
Yeston 1974
Forte 1983
Berry 1985
Marvin 1991
Hayes 1995
Hasty 1997
Temperley 2004
London 2012
Cox 2016

Discussions of rhythmic complexity have spanned the music theoretical and music cognition and perception literature, with many attempts to describe or quantify what affects the perception of rhythmic complexity. These attempts have taken many different forms – quantitative, qualitative, algorithmic, experimental, or even combinations of those.

Thul 2008 – measuring the complexity of musical rhythm

Fleurian et al 2015: measures based on compression algorithms (entropy rate and Kolmogorov complexity)

Gunduz: Shannon entropy theory

Attempts to define rhythmic complexity has often used syncopation as a primary factor in, or as a proxy for, perceived complexity (Longuet-Higgens & Lee 1984, Povel & Essens 1985, Gomez et al. 2005, Thul & Toussaint 2008). However, [click] we’ve found that other characteristics contribute to perceived complexity, including density and rhythmic variability.

Today I’ll report on the results of two experiments that focus on the interaction of syncopation, density, and variability on perceived complexity ratings for rhythmic patterns. I’m giving a bit of short shrift to the background lit section, and a whirlwind summary of the first experiment, so that I can spend more time on the most recent one.

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Rhythmic complexity

“We encountered significant heterogeneity in the measurement and conceptualization of rhythmic complexity. … In order to move forward, more agreement is needed regarding measures and notions of complexity.”

-Cossavella et al, 2024

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Musical cues for tempo determination

Melodic

Rhythmic

Harmonic

Sonic

Ornamentation
Contour change
Interval size

Attack rate
Complexity
Syncopation

Change rate
Progressions
Repetition

Register
Timbre
Loudness

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Rhythmic complexity

Syncopation

Longuet-Higgins and Lee 1984
Povel and Essens 1985
Gomez et al. 2005
Smith and Honing 2006

Density

Eerola, Himberg, Toivianinen & Louhivuori 2006
Temperley 2019

Variability

Grabe & Low 2002
Patel & Daniele 2003
VanHandel 2005, 2006, 2021, 2023

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Rhythmic complexity

Syncopation

Longuet-Higgins and Lee 1984
Povel and Essens 1985
Gomez et al. 2005
Smith and Honing 2006

Density

Eerola, Himberg, Toivianinen & Louhivuori 2006
Temperley 2019

Variability

Grabe & Low 2002
Patel & Daniele 2003
VanHandel 2005, 2006, 2021, 2023

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Rhythmic complexity

Syncopation – events that occur on relatively weak beats in an established metrical structure that obscure or displace an expected strong beat

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Rhythmic complexity

Variability: nPVI – normalized Pairwise Variability Index

Density – number of events per measure

density = 4

density = 10

variability = 0

variability = 55.6

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The story so far …. Experiment 1

Context

400 ms

(150 bpm)

No context

Context

333 ms

(180 bpm)

No context

Context

533 ms

(113 bpm)

No context

Context

666 ms

(90 bpm)

No context

Context

800 ms

(75 bpm)

No context

Context

933 ms

(64 bpm)

No context

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The story so far …. Experiment 1

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The story so far …. Experiment 1

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The story so far …. Experiment 1

400 ms

(150 bpm)

800 ms

(75 bpm)

density

syncopation

When we controlled for the effect of tempo, the perceptual complexity ratings for the stimuli were strongly positively correlated with both the density measurement and the syncopation measurement, indicating that as density and syncopation increase, the complexity rating also increased. We found mixed results for variability and couldn’t make any conclusions about that parameter in this experiment.

We found that the effect strength of density and syncopation switched places between the 400 ms and 800 ms stimuli; in the faster 400 ms stimuli, density influenced the complexity ratings more, and in the slower 800 ms stimuli, syncopation was more highly correlated with higher complexity ratings, confirming that individual factors that lead to perceptual complexity may be also tempo dependent.

We wanted to explore that tempo dependency more, so we ran our second experiment.

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The story so far …. Experiment 2

Context

400 ms

(150 bpm)

No context

Context

333 ms

(180 bpm)

No context

Context

533 ms

(113 bpm)

No context

Context

666 ms

(90 bpm)

No context

Context

800 ms

(75 bpm)

No context

Context

933 ms

(64 bpm)

No context

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The story so far …. Experiment 2

Context

No context

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The story so far …. Experiment 2

Context

400 ms

(150 bpm)

No context

Context

333 ms

(180 bpm)

No context

Context

533 ms

(113 bpm)

No context

Context

666 ms

(90 bpm)

No context

Context

800 ms

(75 bpm)

No context

Context

933 ms

(64 bpm)

No context

Fast Context (FC) group

Slow Context (SC) group

Fast No Context (FN) group

Slow No Context (SN) group

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The story so far …. Experiment 2

*

**

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The story so far …. Experiment 2

*

533

666

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The story so far …. Experiment 2

When we controlled for the effect of tempo, the perceptual complexity ratings were strongly positively correlated with both the density and syncopation metric, indicating that as density and syncopation increase, the complexity rating also increases. We found mixed results for variability and couldn’t make any conclusions about that parameter.

The standardized coefficient allows us to compare the relative effect strength of density and syncopation; these charts show the standardized coefficient for density and syncopation at each of the six tempos in both the metrical context and no-context condition.

In both conditions, the effect of syncopation appears to be relatively consistent across all six tempos. However, density’s role changes dramatically across tempos; it is more strongly correlated with perceived complexity ratings at the faster tempos, and less strongly correlated at slower tempos.

What was interesting was how the context/no-context condition affected the relative tempo where that crossing happened. In the context condition, shown on the upper left, the effect of density is strongest for the fastest two tempos and then drops below syncopation for the four slower tempos. However, in the no-context condition, the effect is reversed; density remains stronger than syncopation for the four faster tempos, and is only weaker than syncopation for the two slowest tempos. This indicates that both tempo and the presence or absence of a metrical context affects the relative strength of these parameters.

We wanted to explore that tempo dependency more, and also to get more information about the role of variability, so we ran our third experiment.

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Experiment 3

Context

400 ms

(150 bpm)

No context

Context

333 ms

(180 bpm)

No context

Context

533 ms

(113 bpm)

No context

Context

666 ms

(90 bpm)

No context

Context

800 ms

(75 bpm)

No context

Context

933 ms

(64 bpm)

No context

Fast group

Slow group

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Experiment 3

Context

No context

Context

No context

Context

No context

Context

No context

Context

No context

Context

No context

Fast group = 864 stimuli

Slow group = 864 stimuli

6

7

8

9

8008

11,440

12,870

11,440

50

56

54

Possible rhythmic patterns

Selected rhythmic patterns

Total # of rhythmic patterns

216

x 4 tempos per group

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Experiment 3

Context

No context

Context

No context

Context

No context

Context

No context

Context

No context

Context

No context

Fast group = 864 stimuli

Slow group = 864 stimuli

Fast group – 273 participants

Slow group – 235 participants

508 usable responses

150

956 participants total

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Experiment 3

Context

No context

Context

No context

Context

No context

Context

No context

Context

No context

Context

No context

Fast group = 864 stimuli

Slow group = 864 stimuli

Fast group – 273 participants

Slow group – 235 participants

508 usable responses

150

956 participants total

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Experiment 3

Context

No context

Context

No context

Context

No context

Context

No context

Context

No context

Context

No context

Fast group = 864 stimuli

Slow group = 864 stimuli

Fast group – 273 participants

Slow group – 235 participants

508 usable responses

150

956 participants total

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Experiment 3

Context

No context

Context

No context

Context

No context

Context

No context

Context

No context

Context

No context

Fast group = 864 stimuli

Slow group = 864 stimuli

Fast group – 273 participants

Slow group – 235 participants

508 usable responses

150

956 participants total

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Experiment 3

90

bpm

64

bpm

113

bpm

150

bpm

180

bpm

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Experiment 3

533

666

*

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Experiment 3

90

bpm

113

bpm

150

bpm

180

bpm

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Experiment 3

90

bpm

64

bpm

113

bpm

75

bpm

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Experiment 3

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Experiment 3

F

S

F

S

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Experiment 2 -- reminder

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Experiment 3

In Experiment 3, we once again found the significant positive correlations between perceived complexity and density and syncopation, including the tempo-based shift in relative importance between the two, with density still having a stronger effect at faster tempos and syncopation having a stronger effect at slower tempos. Whereas the previous experiment showed syncopation having a relatively consistent effect across tempos, in this study we do see more of an actual switch in relative importance with density, with the two crossing right at the 533 ms tempo.

We were also able to identify a weaker but mostly significant contribution of variability for all tempos except the slowest 933 ms (64 bpm) tempo. It appears that variability contributes more to perceived complexity at the faster tempos, and that effect disappears as tempos slow down.

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Experiment 3

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Experiment 3

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Post-hoc investigations involving syncopation categorization

1 e 2 e 3 e 4 e

Stronger effect at slower tempos than faster tempos:

Strongest effect overall:

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Post-hoc investigations involving syncopation categorization

1 & 2 & 3 & 4 &

1 e 2 e 3 e 4 e

Stronger effect at slower tempos than faster tempos:

Strongest effect overall:

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Post-hoc investigations involving syncopation categorization

1 & 2 & 3 & 4 &

Stronger effect at slower tempos than faster tempos:

Strongest effect overall:

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Post-hoc investigations involving syncopation categorization

1 & 2 & 3 & 4 &

1 & 2 &

1 &

We suspect the rating difference between the faster and slower tempos may have to do with hypermetrical interpretation; if a rhythmic pattern with a syncopation on beat 2 is heard in a fast tempo that allows for hypermetrical hearing, that syncopation would occur in a weaker metrical position in that hearing and would thus perhaps not even be perceived as a syncopation.

The two fastest tempos, 180 bpm and 150 bpm, are fast enough to allow for hypermetrical interpretations, possibly even at different hypermetrical levels. The slower tempos would be more likely to preserve the metrical structure at the notated tempo.

This interpretation is supported by the fact that if we recode the syncopations to reflect a hypermetrical hearing in the two fastest tempos, the effect of the syncopations on 2 and 4 more closely resemble the results from the slower tempos. Unfortunately, this is just a conjecture as we can’t determine for certain whether our participants were hearing hypermetrically; that is something we’ll need to follow up on.

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Conclusions

Perceived rhythmic complexity is strongly tempo dependent, including relative tempos.

Syncopation and density are both important factors in perceived complexity ratings; density has a stronger effect at faster tempos and syncopation has a stronger effect at slower tempos.

Variability plays a small but significant role in perceived complexity, but is still presenting complications in interpretation.

Current/future research: why variability’s role is different in density 7, and how the location of syncopations and hypermetrical interpretation affect perceived complexity.

A summary:

The strongest factor in perceived complexity is tempo – rhythms at faster tempos are rated as more complex. This means that any metric for perceived complexity needs to take tempo into account.

The strength of the effect of tempo means that even relative tempos affected perceived complexity ratings – the same stimuli were rated as more complex when they were the fastest tempos in the slow set as opposed to when they were the slower tempos in the fast set.

Density and syncopation are significant factors in perceived complexity, though their effect is also highly tempo dependent.

Rhythmic variability plays a small but significant role in perceived complexity, at least at the faster tempos, but is still presenting complications in interpretation that we are in the middle of following up on, so stay tuned for those results.

We are also planning to further explore the effect of the location and type of syncopation, as well as of hypermetrical interpretation.

So, in conclusion, rhythmic complexity is not just about syncopation… it’s, well, complex.

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Acknowledgements

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MSU Research assistants:

Zachary Lookenbill

Gerardo Lopez

Ryan Jones�Nathalie Nordan

UBC VanLab research assistants:

Claire Brillon

Tania Cheng

Yewon Hong

CC Liang and CC Liang’s dad

Risa Murakami

Konrad van Heukelom

Jay Villanueva

Claudia Wu

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