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The role of variability in

perceived rhythmic complexity

Leigh VanHandel

University of British Columbia

leigh.vanhandel@ubc.ca

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2

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Rhythmic complexity

Syncopation

Gomez, Melvin, Rappaport & Toussaint 2005
Longuet-Higgins and Lee 1984

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

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

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.
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, indicating the musical 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

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

400 ms

(150 bpm)

800 ms

(75 bpm)

density

syncopation

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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.

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 at the 533 ms tempo. However, in the no-context condition, the effect of density remains stronger than syncopation until the slower 800 ms tempo, indicating that the relative strength of these parameters in relation to one another is affected by both tempo and the presence or absence of a metrical context.

We wanted to explore that tempo dependency more, 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

We wanted to systematically explore the relationship between density, syncopation, and variability. To narrow it down, we focused on the densities of 6, 7, 8, and 9 events in a 4/4 measure. [click]

We generated every possible rhythmic pattern in a 4/4 measure with a density of 6, 7, 8, and 9, and analyzed each rhythmic pattern to determine its variability and syncopation. (For information about how you can do that, go see my student Anika’s poster on Saturday’s poster session!) [click] We used that information to select a set of stimuli that systematically represented syncopation and variability levels within each density, resulting in 216 total rhythmic patterns ... [click] which, because we still had the two groups of four tempos, meant that each group consisted of 864 stimuli.

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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

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Experiment 3

533

666

*

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Experiment 3

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Experiment 3

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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

Audience participation time: what’s going on here?

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Experiment 3

Audience participation time: what’s going on here?

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Conclusions

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Perceived rhythmic complexity is strongly tempo dependent.

Relative tempos affects perceived complexity ratings.

Syncopation and density were both important factors in perceived complexity ratings.

Density is a stronger factor at faster tempi and decreases for slower tempi, whereas syncopation is a stronger factor at the slower tempi.

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

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Acknowledgements

31

Research assistants:

Zachary Lookenbill

Gerardo Lopez

Ryan Jones�Nathalie Nordan

UBC VanLab research assistants:

Claire Brillon

Tania Cheng

Yewon Hong

Rae Jourard

CC Liang and CC Liang’s dad

Leah McNeil

Risa Murakami

Lola Quinn

Jay Villanueva

Claudia Wu

Lily Xie

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The role of variability �in perceived rhythmic complexity

Leigh VanHandel

University of British Columbia

leigh.vanhandel@ubc.ca

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Tempo determination

The real-time act or process of determining what an

appropriate tempo is for an unfamiliar piece of music,

based on musical characteristics that act as cues.

34

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Tempo determination

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Preferred tempo

Tempo memory

Tempo discrimination

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Rhythmic complexity

Complexity metrics

Povel & Essens 1985

Essens 1990

Shmulevich & Povel 2000

Thul & Toussaint 2008

Syncopation

Gomez, Melvin, Rappaport

& Toussaint 2005

Longuet-Higgins and Lee 1984

Variability

Grabe & Low 2002

Patel & Daniele 2003

VanHandel 2005, 2006, 2021

Density

Eerola, Himberg, Toivianinen & Louhivuori 2006

Temperley 2019

Research on 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, other characteristics may contribute to perceived complexity, including density (the number of attacks [Eerola et al., 2006]) and rhythmic variability, measured using the nPVI (Patel & Daniele 2005).

Today I’ll report on the results of two experiments on rhythmic complexity that focus on the interaction of syncopation, density, and variability on perceived complexity ratings for rhythmic patterns. This work grew out of the work I’ve been doing on tempo determination, or the real-time act or process of determining what an appropriate tempo is for an unfamiliar piece of music, based on musical characteristics that act as cues; because of the pandemic we weren’t able to run in-person experiments so like many others we pivoted to questions that were easier to explore in an online environment.

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Musical cues

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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

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Complexity

Povel & Essens 1985

Essens 1990

Shmulevich & Povel 2000

Thul & Toussaint 2008

Syncopation

Gomez, Melvin, Rappaport & Toussaint 2005

Longuet-Higgins and Lee 1984

Variability

Grabe & Low 2002

Patel & Daniele 2003

VanHandel 2005, 2006, 2021

Density

Eerola, Himberg, Toivianinen & Louhivuori 2006

Temperley 2019

Research on 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, other characteristics may contribute to perceived complexity, including density (the number of attacks [Eerola et al., 2006]) and rhythmic variability, measured using the nPVI (Patel & Daniele 2005).

Today I’ll report on the results of two experiments on rhythmic complexity that focus on the interaction of syncopation, density, and variability on perceived complexity ratings for rhythmic patterns. This work grew out of the work I’ve been doing on tempo determination, or the real-time act or process of determining what an appropriate tempo is for an unfamiliar piece of music, based on musical characteristics that act as cues; because of the pandemic we weren’t able to run in-person experiments so like many others we pivoted to questions that were easier to explore in an online environment.

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Context vs. no-context

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No context

Context

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Experiment 1: Methods

Subjects used an Arduino spin wheel controller to manipulate the tempo of the looping rhythmic patterns in real time. They were asked to find the “right” tempo for that rhythmic pattern, thus determining the tempo from a continuous tempo spectrum.

Instructions:

“You will hear a short rhythm that will continuously repeat with no break. You will use this spin wheel to control the speed of the rhythm until you think you have found the ‘right’ speed for the rhythm. You should listen to the first few seconds of the rhythm before starting to manipulate the tempo.”

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Experiment 1: Methods

The real time tempo manipulation is controlled by a phase vocoder patch in the Pure Data environment, interfaced with a custom experimental program.

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Tempo determination �rhythm experiment

400 ms

(150 bpm)

800 ms (75 bpm)

Context

No context

Context

No context

Starting Tempos

Beat unit at 400 ms IOI (150 bpm) or 800 ms IOI (75 bpm)

Context

With metrical context, and without

Conditions:

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Rhythm Experiment 1

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35 participants drawn from university research pool

Non–musicians:

GMSI General Sophistication score: 44-68%, average 58%

Tasks:

Tempo determination tasks
Tapping task
Perceived complexity rating task

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Rhythm Experiment 1: Stimuli

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30 rhythmic stimuli:

6 rhythms drawn from Povel & Essens 1985
24 original rhythms, composed to represent a variety of variability, density, and syncopation

Recorded using a percussive sample from Ableton Live that allowed for real-time tempo changes.

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Average determined tempo

45

Slower

(75 bpm)

Faster

(150 bpm)

*

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Other rhythmic characteristics

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		density	variability	syncopation
overall		.650**	-.286**
400 ms	together	.710**	-.311*
	no context	.907**	-.408*
	context	.664**
800 ms	together	.446**
	no context	.786**
	context	.457*	-.434*

** Correlation is significant at the 0.01 level

* Correlation is significant at the 0.05 level

Because starting tempo and context were such strong predictors of determined tempo, we ran a partial correlation controlling for those factors to see what the effects of density, variability, and syncopation were on determined tempo. Results indicated that overall, across both starting tempo conditions, density had the highest correlation with determined tempo, followed by variability. Recall that “density” was a simple measurement of how many notes were present in a rhythmic stimuli. Density and variability are significantly negatively correlated with one another, of course – as the number of notes increases in a rhythmic pattern constrained by meter, variability will decrease. Syncopation did not have any significant effect overall on tempo determination.

[CLICK] We then ran correlations for each starting tempo and condition. Density remained significant in each combination, with variability occasionally reaching statistical significance. [CLICK] In each tempo condition, the effect of density was stronger in the no context condition.

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Effect of density

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Complexity ratings: Experiment 1

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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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Rhythm Experiment 1: Stimuli

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Complexity ratings

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Complexity and rhythmic characteristics

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		density	variability	syncopation
overall		.604**		.500**
400 ms	no context	.809**		.411*
800 ms	no context	.403*		.638**

** Correlation is significant at the 0.01 level

* Correlation is significant at the 0.05 level

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Complexity ratings: Experiment 2

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481 participants (330 F, 118 M, 33 Other) drawn from university research pool

Mean age = 22.6 years, self-reported with normal hearing

Generally non–musicians (GMSI range 33.3–80.9, average 58.9)

Task:

Perceived complexity rating task

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Complexity ratings: Experiment 2

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12 stimuli drawn from Experiment 1:

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Context vs. no-context

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Context

No context

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Complexity ratings: Experiment 2

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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

density

syncopation

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Complexity ratings: Experiment 2

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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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Complexity ratings: Experiment 2

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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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Complexity ratings: Experiment 2

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*

**

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Complexity ratings: Experiment 2

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*

533

666

Comparing the perceived complexity of stimuli in the 533 and 666 ms tempos across the fast and slow groups revealed a trend for stimuli in these “middle” tempos to be perceived differently based on whether they were part of the group of faster tempos or slower tempos. When these tempos were the two slower tempos as part of the ‘fast” group, the stimuli were rated as less complex than when they were the fastest two tempos in the “slow” group. This difference was significant for the 533 ms tempo; participants rated stimuli at the 533 ms tempo as significantly more complex when it was the fastest tempo in a group of slower tempi as opposed to being a slower tempo in a group of faster tempi [F=9.95, MSe=18.204, p<.002]. A similar trend is present for the 666 ms stimuli but the difference was not significant.

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Complexity ratings: Experiment 2

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Tempo		Variability	Density	Syncopation
333		-.022	.475**	.399**
400		-.034	.470**	.369**
533		.002	.368**	.364**
	Fast	-.019	.401**	.367**
	Slow	.022	.338**	.363**
666		.019	.328**	.343**
	Fast	.010	.341**	.350**
	Slow	.027	.315**	.335**
800		.051**	.242**	.334**
933		.047*	.230**	.316**

Correlations with perceived complexity and rhythmic characteristics

* Correlation is significant at the 0.05 level. ** Correlation is significant at the 0.01 level.

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Complexity ratings: Experiment 2

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As a reminder, in the first experiment, density and syncopation were significant factors in perceived complexity, with the factors’ relative strength reversed between the 400 ms and 800 ms tempo conditions; density had a stronger effect at the faster 400 ms tempo, and syncopation had a stronger effect at the slower 800 ms tempo.

The multiple tempos in this study provided some more information about this reversal, and we can determine the relative effect strength of the two factors at each tempo by comparing the standardized correlation coefficient.

We found that the effect of syncopation is actually relatively consistent across all six tempos; the effect does decline slightly but overall it was density whose role changed across tempos. For the context and no-context conditions combined, density has a stronger effect on perceived complexity at faster tempi and decreases in effect for slower tempi, falling below syncopation around the 533 ms tempo.

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Complexity ratings: Experiment 2

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Complexity ratings: Experiment 2

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Conclusions – rhythm

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	Tempo determination	Perceived complexity
Density	X	X
Syncopation		X
Variability	X

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Distribution of �determined tempo

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Rhythm Experiment 1

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Two starting tempo conditions:

800 ms (75 bpm) or 400 ms (150 bpm)

Task 3 – rating

Perceptual complexity rating at original starting tempo on a scale of 1 (not complex) to 6 (complex).

03

Task 2 (context)

Rhythmic stimuli alternates with four isochronous pulses; otherwise same procedure as Task 1.

02

Task 1 (no context)

Rhythmic stimuli plays on loop, participant manipulates tempo until it feels “correct.”

Rhythm plays on loop at determined tempo; participant taps what they felt was the beat for 20 taps.

01

There were three tasks in the study. For the first task, participants heard the rhythmic excerpts on a continuous loop and used the spin wheel controller to manipulate the tempo and confirm their determined tempo. When they confirmed their determined tempo, the stimulus continued looping at that determined tempo, and the subject was asked to use the space bar to tap what they felt was the “beat,” and their tapping rate was recorded for 20 taps.

In the second task, the stimuli appeared in the context condition, but the participant’s task was the same: manipulate the tempo of the stimuli until it sounded “correct,” and then tap along with the beat at the determined tempo.

Participants were also asked to rate the complexity of the stimuli (presented in the no-context condition) at their assigned starting tempo of 800 ms or 400 ms. They used a scale of 1 to 6, with 1 being not complex and 6 being highly complex.

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Tapping to the beat

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Tempo determination task

Tapping task

*								*								*								*
*				*				*				*				*				*				*				*
*		*		*		*		*		*		*		*		*		*		*		*		*		*		*		*
*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*

Pulse level

Subdivision

Hypermeter

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Tapping to the beat

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Tempo determination task

Tapping task

*								*								*								*
*				*				*				*				*				*				*				*
*		*		*		*		*		*		*		*		*		*		*		*		*		*		*		*
*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*

Pulse level

Subdivision

Hypermeter

*				*				*				*				*				*				*				*
*		*		*		*		*		*		*		*		*		*		*		*		*		*		*		*
*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*

75 bpm

37.5 bpm

150 bpm

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Tapping to the beat

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Tempo determination task

Tapping task

*								*								*								*
*				*				*				*				*				*				*				*
*		*		*		*		*		*		*		*		*		*		*		*		*		*		*		*
*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*

Pulse level

Subdivision

Hypermeter

*				*				*				*				*				*				*				*
*		*		*		*		*		*		*		*		*		*		*		*		*		*		*		*
*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*

160 bpm

80 bpm

320 bpm

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Tapping to the … beat?

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Tempo determination task

Tapping task

*								*								*								*
*				*				*				*				*				*				*				*
*		*		*		*		*		*		*		*		*		*		*		*		*		*		*		*
*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*

Pulse level

Subdivision

Hypermeter

*				*				*				*				*				*				*				*
*		*		*		*		*		*		*		*		*		*		*		*		*		*		*		*
*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*

160 bpm

80 bpm

320 bpm

The participant may have shifted their attention to a different metrical level, now interpreting the 80 bpm hypermetrical pulse as the pulse unit, and 160 bpm as a subdivision of that pulse unit. So to them, they may be interpreting that rhythm as only being a little bit faster than the starting tempo of 75 bpm – just with an additional layer of subdivision added.

In order to learn more about where participants were perceiving the beat unit after they had determined the tempo, we had them do the tapping task, on the theory that if they tapped to the beat, we could determine where they thought the pulse unit was relative to the original starting tempo.

Anyone who has had participants do a tapping task, or anyone who has tried to interpret the results of a tapping task, knows exactly where this is going.

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Tapping ratios

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*		*		*		*		*		*		*		*		*		*		*		*		*		*		*		*

*		*		*		*		*		*		*		*		*		*		*		*		*		*		*		*

*				*				*				*				*				*				*				*

*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*

*	*		*	*		*	*	*		*	*	*	*	*	*				*			*		*					*

beat unit

1:1 ratio (1)

1:2 ratio (.5)

2:1 ratio (2)

…?

It’s notoriously difficult to describe to a non-musician what it means to “tap to the beat”, especially when they’re listening to relatively complex rhythmic patterns. [CLICK] Sometimes they will tap to the beat… [CLICK] Sometimes they’ll tap every two beats … [CLICK] Sometimes they’ll tap the subdivision … [CLICK] … and sometimes they tap the rhythm itself, or tap inconsistently or randomly.

We calculated a ratio that consisted of the determined tempo in milliseconds divided by the average inter-tap interval. The closer a participant tapped to what the experimental program thought their determined tempo was, the closer their ratio would be to 1. If they tapped [CLICK] hypermetrically, the inter-tap interval would be longer than the determined tempo, so it would be a ratio of less than 1. A .5 ratio, for example, would indicate a hypermetrical level where two of the original pulse units are being considered as one beat hypermetrically. If they tapped a subdivision [CLICK], the ratio would be larger than 1; and if they tapped the rhythm, [CLICK] or tapped randomly, it would be whatever the ratio was between the determined tempo to the average inter-tap interval.

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Tapping ratio by starting tempo and context

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**

** Correlation is significant at the 0.01 level

* Correlation is significant at the 0.05 level

If we look at the interaction between tapping ratio, starting tempo, and context, we can see that at the 800 ms tempo, there’s no difference in tapping ratios between the no-context (tan) and context (brown) conditions; the average tapping ratio for both context conditions is close to 1, indicating that in the 800 ms condition, participants tended to tap along with the beat unit they had determined. This makes sense – the 800 ms condition is the slower starting tempo, and while participants sped up the stimuli in the slow condition, the average was still only a 689 ms IOI, or roughly 87 bpm. At that average tempo, in order to tap hypermetrically they would be tapping somewhere around 43 beats per minute, which is on the very low end of Justin London’s acceptable tempo spectrum.

Looking at the 400 ms starting condition, there’s a significant difference between the no-context (light blue) and context (teal?) conditions; the tapping ratio is significantly lower in the no-context condition, indicating that participants were more likely to be tapping slower than their determined tempo – that is, they were more likely to be tapping hypermetrically. Again, this makes sense – the 400 ms, or 150 bpm, starting tempo is relatively fast, and even though participants tended to slow the faster stimuli down slightly to 421 ms, or 143 bpm, a hypermetrical tapping strategy would have them tapping at about 71 bpm. It’s likely that the participants didn’t tap hypermetrically in the context condition because the pulse unit was being provided to them by the four isochronous pulses, so it was likely easier for them to identify the pulse unit to tap along to.

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Tapping ratio by starting tempo and context

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**

** Correlation is significant at the 0.01 level

* Correlation is significant at the 0.05 level

The remainder of this tapping data is being presented narrowing the participants down slightly. We looked at participants whose tapping was consistent, and whose tapping ratio was close to 1 – that is, between .75 and 1.25 – because we wanted to see what the “good” tappers would do. In terms of overall tapping ratio, there really isn’t much difference between this group and everyone else, but I just want to be clear that these are our good, consistent tappers we’re reporting results for.

If we look at the interaction between tapping ratio, starting tempo, and context, we can see that at the 800 ms tempo, there’s no difference in tapping ratios between the no-context (tan) and context (brown) conditions; the average tapping ratio for both context conditions is 1, indicating that in the 800 ms condition, participants tended to tap along with the beat unit they had determined. This makes sense – the 800 ms condition is the slower starting tempo, and while participants sped up the stimuli in the slow condition, the average was still only a 689 ms IOI, or roughly 87 bpm. At that average tempo, in order to tap hypermetrically they would be tapping somewhere around 43 beats per minute, which is on the very low end of Justin London’s acceptable tempo spectrum.

Looking at the 400 ms starting condition, there’s a significant difference between the no-context (light blue) and context (teal?) conditions; the tapping ratio is significantly lower in the no-context condition, indicating that participants were more likely to be tapping slower than their determined tempo – that is, they were more likely to be tapping hypermetrically. Again, this makes sense – the 400 ms, or 150 bpm, starting tempo is relatively fast, and even though participants tended to slow the faster stimuli down slightly to 421 ms, or 143 bpm, a hypermetrical tapping strategy would have them tapping at about 71 bpm. It’s likely that the participants didn’t tap hypermetrically in the context condition because the pulse unit was being provided to them by the four isochronous pulses, so it was likely easier for them to identify the pulse unit to tap along to.

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Tapping ratio and rhythmic characteristics

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		density	variability	syncopation
400 ms	together
	no context		-.387*
	context
800 ms	together		-.271*
	no context		-.419*
	context

** Correlation is significant at the 0.01 level

* Correlation is significant at the 0.05 level

We can also compare the tapping ratio to the characteristics of density, variability, and syncopation to see whether these characteristics influenced participants to tap at different metrical levels.

In the 400 ms no context condition and the 800 ms overall and no context condition, the tapping ratio negatively correlated with variability; this indicates that as variability increases, the tapping ratio decreases – and a decreasing tapping ratio means that participants tapped more slowly – which may mean that participants were more likely to tap a hypermetrical level when variability was high. The fact that this is significant in the no context condition, but not in the context condition, aligns with the observation that participants may be less likely to tap hypermetrically in the context condition because of the presence of the pulse unit.

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Tempo project – �Overall conclusions

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Melodic and rhythmic characteristics do play a role in tempo determination.

Starting tempo is important. plays an enormous role in tempo perception, and needs to be accounted for in future studies.

So much for free will …

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Future directions

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Future directions for the tempo project include following up on questions emerging from rhythm study, studying harmonic and timbral characteristics, and creating a model for how they interact.

Online version of experiment forthcoming, with a primary goal to extend to non-WEIRD populations.*

* Henrich 2020

We’ve been unable to run subjects since the world ended on March 11, 2020, because the current experimental design requires the participant to be in front of the computer with a spin wheel. We’re working on moving the experiment online so that we are not constrained by having to run in person. With this, we hope to be able to reach a broader population and see how the influence of these characteristics may change; we’re also hoping to learn about how non-Western, Educated, Industrialized, Rich, and Democratic populations approach the process of tempo determination.

For more on how complexity is more, well, complex than just syncopation, be sure to check out Zach Lookenbill’s upcoming ICMPC presentation! And for information about how GMSI scores and hypermetrical tapping interact, see Nathalie Nordan’s poster as part of this very conference,