Musical tension is affected by metrical structure dynamically and hierarchically

Abstract

As the basis of musical emotions, dynamic tension experience is felt by listeners as music unfolds over time. The effects of musical harmonic and melodic structures on tension have been widely investigated, however, the potential roles of metrical structures in tension perception remain largely unexplored. This experiment examined how different metrical structures affect tension experience and explored the underlying neural activities. The electroencephalogram (EEG) was recorded and subjective tension was rated simultaneously while participants listened to music meter sequences. On large time scale of whole meter sequences, it was found that different overall tension and low-frequency (1 ~ 4 Hz) steady-state evoked potentials were elicited by metrical structures with different periods of strong beats, and the higher overall tension was associated with metrical structure with the shorter intervals between strong beats. On small time scale of measures, dynamic tension fluctuations within measures was found to be associated with the periodic modulations of high-frequency (10 ~ 25 Hz) neural activities. The comparisons between the same beats within measures and across different meters both on small and large time scales verified the contextual effects of meter on tension induced by beats. Our findings suggest that the overall tension is determined by temporal intervals between strong beats, and the dynamic tension experience may arise from cognitive processing of hierarchical temporal expectation and attention, which are discussed under the theoretical frameworks of metrical hierarchy, musical expectation and dynamic attention.

Introduction

Human beings are deeply obsessed with composing and appreciating music, whenever and wherever they are. A widely accepted reason for this is the great emotional value of music (Huston et al. 2015). Tension-resolution patterns are the key elements of emotional experience induced by music (Huston et al. 2015). As an emotional state being opposite to release or relaxation, tension is built on a very basic cognitive mechanism, which is associated with the processes of generating, fulfilling or violating musical expectations (Lehne et al. 2013; Lehne and Koelsch 2015).

Given that music is highly-structured according to a set of rules (Lerdahl and Jackendoff 1996), it has been suggested that tension is determined by the structures of music, through the expectations generated based on these structures when listening to music (Meyer 1956; Margulis 2005; Huron 2006; Lerdahl and Krumhansl 2007). Abundant studies investigated tension generation by manipulating the violation of musical structures, in which the musical sequences were designed intentionally to break the preceding contexts and violate the original expectations for upcoming events (Bigand et al. 1996; Krumhansl 1996; Bigand and Parncutt 1999; Steinbeis et al. 2006; Koelsch et al. 2008a, b; Lehne et al. 2013; Sun et al. 2020a, b). The findings from these studies have reached a consensus that the effects of musical structures on tension are achieved substantially via musical expectations, the future-directed expectations about what or when the next event would happen as music unfolds (Meyer 1956; Margulis 2005; Huron 2006; Juslin and Vastfjall 2008; Rohrmeier and Koelsch 2012; Juslin 2013; Lehne and Koelsch 2015).

Compared to the concerns about the effects of harmonic and melodic structures, the important roles of temporal structures in tension generation seem to be overlooked. Music is an art of time, the temporal dimension is indispensable for structuring music and inducing emotion (Large and Kolen 1994; Fitch 2013; Vuust and Witek 2014; Sun et al. 2020c). With its unique temporal characteristics on various time scales (Lerdahl and Jackendoff 1996), metrical structure (meter) is an ideal tool to investigate the effects of musical temporal structure on tension. Metrical structure is a regular pattern of alternating strong and weak beats, and conceptualized as metrical grid of beats at various hierarchical levels (Lerdahl and Jackendoff 1996; Large and Jones 1999; London 2002; Fitch 2013; Levitin et al. 2018). In metrical structures, beats are organized into measures (also called as bars), with one strong beat and one or several subsequent weak beats within a temporal interval between two adjacent strong beats (as exemplified in Fig. 8A). The metrical structure is a cognitive representation derived from a regular sequence of measures.

Fig. 8.

A Frequency spectrum of sound envelope of each meter. B EEG spectrum of each meter, averaged across FCz and Cz electrodes. The thick vertical arrows indicate the expected beat frequency, the thin vertical arrows indicate the fundamental meter frequencies and harmonic frequencies. C Group mean amplitudes of SS-EPs elicited by each meter, SS-EP amplitudes different significantly from zero are marked by *p < 0.05, **p < 0.01, or ***p < 0.001, vertical lines indicate ± SE

This experiment aimed to investigate how musical tension is affected by metrical structures on multi-time scales, and what cognitive mechanisms are responsible for these effects. According to dynamic attending theory (Jones and Boltz 1989), perception of metrical structure is essentially the processes of generating and maintaining expectations in temporal dimension, sharing a similar cognitive mechanism with tension experience. On a large time scale, when presented with long meter sequences, listeners could extract periodicities from the initial events and form expectations for the temporal properties of subsequent events. Therefore, we assumed that different meter sequences may elicit different overall degrees of expectations, which might be indicated by neural activities obtained on large time scale. On a small time scale, a measure could be defined as a perceptual unit built on time-varying temporal expectations (London 2002). More specifically, beats at different levels in metrical hierarchy could generate different degrees of temporal expectations, that is, strong beats located at higher level of hierarchy might generate stronger expectations towards subsequent beats than weak beats located at lower level (London 2002; Fujioka et al. 2015). Based on these, we further assumed that within measures different types of beats may induce expectation in a hierarchical way, which might be indicated by neural signal fluctuations on small time scale (Fujioka et al. 2015).

Critically, Margulis (2005) proposed a melodic expectation model to quantify tension and encouraged this model to be integrated with temporal expectancies. Based on dynamic attending theory she stated that “metric-level” temporal expectations which “predict the pitch to occur at a certain level within the metric hierarchy” also needed to be addressed. Therefore, we assumed that temporal expectations in different time scales might be the underlying cognitive mechanisms linking the processing of metrical structure to the corresponding tension experience.

Theories have been proposed to define and model harmonic tension (Lerdahl and Jackendoff 1996; Lerdahl 2001; Lerdahl and Krumhansl 2007), however it seems that metric tension has not be well modeled or empirically studied, which may because that the temporal structures of music are complex and are not as easy to quantify as the tonal structures (Farbood 2012). Therefore three simple meters are adopted in our experiment: 2/4 meter, 3/4 meter, and 4/4 meter, which reflect the basic temporal features of metrical structures. These meters have different temporal organizational patterns of sequential strong and weak beats, the latter were generated by manipulating the loudness of beats (see Fig. 1 and description about stimuli in Materials and Methods). Therefore from the perspective of temporal structure, these three types of meters are mostly distinguished by the temporal interval between adjacent strong beats (Palmer and Krumhansl 1990). We hypothesized that such difference would be the underlying structural factors which influence the meter perception and the corresponding tension via temporal expectations.

Fig. 1.

Samples of the original meter sequences in A Major. For each type of original sequence, three types of meters are only different in their organizational patterns of beats. In 2/4 meter, there are two beats in each measure and each beat is quarter note, with the pattern as “strong–weak”. In 3/4 meter, there are three beats in each measure and each beat is quarter note, with the pattern as “strong–weak- weak”. In 4/4 meter, there are four beats in each measure and each beat is quarter note, with the pattern as “strong–weak-middle-weak”. Note that the notes did not exactly stand for the real durations of each beats (600 ms)

In the processing of these meters, temporal expectations are directed to known and specific events (i.e., beats) without any uncertainty or violation, since the organizational patterns of strong and weak beats in a measure are regular and fixed. Therefore in the present experiment, the tension induced by regular meters may be classified as expectancy-tension according to Margulis’s tension model (Margulis 2005), which arises during the occurrence of current events that trigger expectations towards future subsequent events and is positively related to the strength of such unambiguous expectations. As introduced above, most of the previous studies have focused on the tension that arises when unexpected events occur and violate the original expectations, that is, denial-tension (Margulis 2005), however seem to ignore expectancy-tension, a common tension experience when we are “holding our breath” and expecting some known subsequent notes to appear while listening to a familiar music. This experiment therefore may provide some insights into expectancy-tension and further a more comprehensive understanding of tension.

In this experiment, the subjective tension was rated continuously and EEG responses were recorded during music listening. Growth curves of tension ratings were fitted to reveal the dynamics of tension. Both the low-frequency (δ band: 2 ~ 4 Hz) steady-state evoked potentials (SS-EPs in short) and high-frequency induced periodic modulations (β-band event-related desynchronization, β-ERD in short) were analyzed to elucidate the relevant neural activities. These neural activities have been admittedly associated with metrical temporal expectations (Jones and Boltz 1989; Fujioka et al. 2009, 2012, 2015; Henry and Herrmann 2014; Nozaradan 2014; Merchant et al. 2015; Chang et al. 2016; Fujioka and Ross 2017). Furthermore, the low-frequency entrainment extracted from long time-span meter sequence are expected to be associated with overall tension experience on large time scale, while the high-frequency periodic modulations are assumed to be related to dynamic tension fluctuations within a measure on small time scale.

Materials and methods

Participants

Thirty-five right-handed participants (twenty-one females; mean age: 20 years, age range: 18–23 years) were recruited. Given that the processing of long sequence temporal structure, especially the meter structuring (Celma-Miralles and Toro 2019), may be difficult for non-musicians (Sun et al. 2018), the participants were all professional performing musicians. They all had received formal western music instrumental training (mean years: 12 years, range: 8–17 years; average daily practice time: 3 h, range: 1–6 h), playing piano, violin, viola, cello, electronic organ, saxophone, French horn or trumpet. None of them reported any history of hearing, neurological, or psychological disorders. The study was approved by the Ethics Committee of Institute of Psychology, Chinese Academy of Sciences, and the written informed consent was provided to participants before experiment running.

Stimuli

First of all, three original sequences were composed by a professional musician in a fixed pitch and in A major, which differed in the occurring orders of several simple notes, including quarter notes, eighth notes, and dotted quarter notes. Then, the notes in sequences were assigned with different loudness in order to generate different types of beats. Three types of metrical sequences were organized based on their orders, as shown in Fig. 1. The loudness of the strong beat, the middle beat and the weak beat were set as “120”, “100” and “60” on the volume scale respectively in Cubase 5 (100%, 83%, 50% if normalized to strong beat, see Fig. 8A). The duration of each beat was set as 600 ms. In order to encourage subjects to concentrate on music stimuli, reducing the monotony or boredom they might feel during the experiment, we created seven modified versions for each original sequences, by transposing it from A Major into six other pitch classes (Ab/G/Gb/F/E/Eb). Thus, sixty-three sequences were generated totally (3 meter types × 3 rhythmic patterns × 7 pitch classes) with twenty-one sequences for each meter. Sound files of the sequences were created using Sibelius 7.5 and Cubase 5, and played in Taiko drum timbre at a tempo of 100 beats per minute, the spectrum characteristics of Taiko drum was shown in Fig. 2.

Fig. 2.

The spectral characteristics of Taiko drum timbre, extracted from the audio waveform

For 2/4 meter, each sequence lasted for 19.2 s, with 31 beats organized into 16 measures, and the every second beat was acoustically accented as a strong beat (the last measure consisted of a half note as cadence). For 3/4 meter, each sequence lasted for 19.8 s, with 31 beats organized into 11 measures, and the every third beat was the strong beat (the last measure consisted of a dotted half note as cadence). For 4/4 meter, each sequence lasted for 19.2 s, with 29 beats organized into 8 measures, the every fourth beat was a strong beat (the last measure consisted of a whole note as cadence). Stimuli were presented in a pseudo-random manner.

It should be noted that the rhythmic patterns of the sequences formed by different notes were quite simple to avoid any additional cognitive consumption, and repeated from measure to measure randomly. There were no strong contrasts in the complexity or note density between these simple rhythmic patterns within or between metrical sequences. In other words, these rhythmic patterns were controlled to be consistent across three types of metrical structures. Therefore according to our experimental design, the possible effects of rhythmic patterns on tension could be balanced out across three meter conditions.

Procedures

In a soundproof room, participants were comfortably seated in a chair to keep their head and body still, while keeping their eyes fixated on a computer screen in front. The instructions were as follows: during the experiment, you will hear a number of musical meter sequences via a pair of high-quality stereo headphones, with one piece of musical meter sequence in a trial. At the beginning of a trial, you need to click the mouse to start the music. Please listen to the music carefully, and simultaneously to rate the subjective musical tension you feel while listening by moving the mouse to monitor the visual slider shown on the computer screen. The slider should be moved within a scale ranging from 0 to 100. You should sit still and not move the rest of your body during the experiment, in order to avoid any unnecessary motor activities.

Behavioral ratings were recorded using the E-prime 2.0 interface at the sampling rate of 500 Hz, and EEG data were recorded simultaneously with the rating. Each trial lasted for around 24 s. Three blocks were contained in the experiment, each one consisted of 21 trials. Whole experiment lasted around 30 min for each subject.

EEG recording

The EEG was recorded by Neuroscan SynAmps Amplifier with 64 Ag/AgCl electrodes in International 10–20 system scalp locations at the sampling rate of 500 Hz, with 0.05 Hz low cut off filtering and 100 Hz high cut-off filtering. The left mastoid electrode served as online reference, and the arithmetic mean value of the bilateral mastoid electrodes as offline reference. The ground electrode was located between Fz and FPz. The vertical electro-oculogram (EOG) was recorded by electrodes placed superior and inferior to the left orbit, and the horizontal EOG was recorded by the electrodes placed 1 cm to the left and right external canthi. The impedance was kept below 5 KΩ throughout the experiment.

Behavioral data analysis

Large time scale analysis

On large time scale of meter sequences, the difference in overall tension ratings during whole sequences across three types of meters were estimated. The tension ratings obtained in our experiment are time series data. As shown in Fig. 3, tension ratings for all three types of meters fluctuate over time, but notably their dynamic features are complex and different. These dynamics should be taken into consideration when comparing the tension ratings across three meters. Therefore we adopted growth curve analysis (GCA for short) (Mirman 2014) to fit the continuous tension ratings over time into smooth curves using multi-order orthogonal polynomials. GCA is frequently used to estimate time series dynamic data, and could capture and compare the dynamic features of fitted curves across three meters. We set meter and time as fixed factors, subject and subject-meter interaction as random factors, and the tension ratings at successive timepoints as the meter sequences unfold were taken as dependent variables. Fourth-order orthogonal polynomial was used to characterize the overall dynamics of fitted tension curves. For each meter, two time windows were used to analyze the overall tension. Firstly, the entire stage of the sequence (from the onset of the sequence to the onset of the last cadence beat in 2/4 and 3/4 meter; from the onset of the sequence to the half of the last cadence beat in 4/4 meter, 0 ~ 18 s); secondly, stable stage of the sequence (from the onset of 6th beat to the end of entire stage, 3.6 ~ 18 s), when the tension fluctuations tended to be stable and regular based on visual inspection of tension curves in Fig. 3.

Fig. 3.

Group mean of subjective tension ratings for three meters (top). Group mean of subjective tension ratings for 2/4 meter (left, bottom), 3/4 meter (middle, bottom) and 4/4 meter (right, bottom) respectively. Symbols on the curves mark the time onsets of each beat. The shaded areas indicate the 95% confidence interval of the group mean

Further, we compared the overall tension ratings for the same types of beats across meters. Over the time window of stable stage of the sequence, we firstly fitted the tension ratings for successive strong beats of three meters respectively into fourth-order polynomial curves, and then compared the fitted curves across meters using GCA (the parameter settings were the same to the above GCA on overall tension during whole sequences). The same analysis was conducted on the last beats of 3 meter (which are weak beats in 2/4 meter, second weak beats in 3/4 meter and second weak beats in 4/4 meter). The above GCA analyses were carried out by R 4.0.0 using the lmerTest package.

Small time scale analysis

On small time scale of measures, to compare the tension ratings for different beats within measures for each meter, we firstly segmented the data during stable stage (3.6 ~ 18 s) in terms of the 600 ms beat interval. For the 2/4 meter two-beat interval as a segment (segments were time-locked to the onsets of the 4th to the 15th strong beats, yielding 12 measures). For the 3/4 meter three-beat interval as a segment (segments were time-locked to the onsets of the 3rd to the 10th strong beats, yielding 8 measures). For the 4/4 meter four-beat interval as a segment (segments were time-locked to the onsets of the 3rd to the 7th strong beats, yielding 5 measures). Then, for each type of meter, the data were averaged across segments to derive the tension curves within measures, as shown in Fig. 8B. Within measures, the tension rating for each beat was calculated as follows: mean ratings over 600-ms beat duration were regarded as the tension induced by the corresponding beats, except for the last weak beats in 3/4 and 4/4 meter. Given that in real-life musical performance, when existing successive weak beats (or middle beats) within metrical units (like 3/4 and 4/4 meter), the approximate second halves of the last weak beats are normally the period for breath switching and preparation for the next strong beats. Therefore, averaging the whole 600-ms beat duration might cause confusion and instead we took mean ratings over 300-ms after the onsets of these beats as their corresponding tension. Further, we compared the tension ratings for the same types of beats within measures across three meters, namely the strong beats and the last beats within measures.

Following statistical tests were performed to compare the tension induced by beats within measures for each meter: a paired-samples t test for 2/4 meter, one-way repeated-measures ANOVAs for 3/4 meter (three-level independent variable) and 4/4 meter (four-level independent variable). Besides, a one-way repeated-measures ANOVAs (three-level independent variable) were performed to compare the tension induced by the strong beats and the last beats within measures across three meters. All the statistical test were performed by SPSS 25.

Here we found it critical to clarify that the loudness we used as an objective parameter, in order to directly manipulate the type of beats and indirectly distinguish different meter, is distinct from the subjective tension rated by listeners, given that the latter is driven by expectation and could be influenced by more high-level structural processing of meter. However it is also well-known that loudness of the beats as an low level acoustic factor can potentially influence tension (Granot and Eitan 2011; Farbood 2012). Thus, in order to separate the contributions of loudness and meter type on tension difference, a linear mixed-effects model (LMM) was performed. We set meter and loudness as fixed factors, subject and item as random factors (the item here refers to each meter sequence). While in growth curve analysis continuous time series data of tension was dependent factor, in LMM tension rating for each beat was set as dependent factor, and were calculated in the same way as the above analysis for each subject and each trial. The LMM analysis was carried out by R 4.0.0 using the lmerTest package.

EEG data analysis

Preprocessing

The preprocessing of EEG data were performed using EEGLAB (EEGLAB 14.1.1b, https://sccn.ucsd.edu/eeglab), an open-source toolbox running in the MATLAB environment. Continuous EEG data were re-referenced offline to the bilateral mastoids and filtered with a band-pass filter of 0.1 ~ 30 Hz. Epochs lasting 21 s were obtained by segmenting the data from − 1 to + 20 s relative to the onsets of the auditory sequence, thus yielding 63 epochs for each subject. The above processing steps were performed by using the tools in EEGLAB.

Artifacts produced by eye blinks or eye movements were removed using a validated method based on the Independent Component Analysis (ICA) (Jung et al. 2000), using the runica algorithm (Bell and Sejnowski 1995; Makeig et al. 1995) incorporated in EEGLAB. Epochs in any electrode exceeding ± 150 μV were rejected as artifacts and excluded from data analysis. The data of two participants were excluded due to their excessive rejected epochs (more than 30% of the total). After preprocessing, the data of 33 subjects were retained, which in total amounts to 659 epochs for 2/4 meter, 661 epochs for 3/4 meter and 663 epochs for 4/4 meter respectively.

The following EEG processing steps were performed using MATLAB (The MathWorks), EEGLAB (EEGLAB 14.1.1b, https://sccn.ucsd.edu/eeglab) and FieldTrip (http://fieldtrip.fcdonders.nl) (Oostenveld et al. 2011). The statistical tests were performed with SPSS 25, and were all carried out at subject level.

Frequency-domain analysis

Frequency-tagging method was widely adopted to measure the so-called selective beat- and meter-related neural entrainment, by estimating steady-state evoked potentials (SS-EPs) in EEG frequency spectrum (Nozaradan et al. 2012a; Nozaradan 2014), and to detect peaks at specific frequencies in spectrum as signs of SS-EPs (Nozaradan et al. 2011, 2012b, 2015, 2016; Li et al. 2019). To investigate SS-EPs at the frequencies of beat and meter, we adopted this method for further processing.

Each epoch (− 1 to + 20 s relative to the onset of the auditory sequence) was firstly re-segmented from + 2 to + 20 s relative to the onset of the stimulus, removing the data during the first 2 s of each epoch, in order to remove the transient auditory evoked potentials. For each subject and condition (three types of meter, as shown in Fig. 1), EEG epochs were averaged across trials in the time domain in order to reduce the contribution of non-phase locked activities and to enhance the signal-to-noise ratio. The obtained average waveforms were then transformed into the frequency domain using Fast Fourier Transform (FFT) algorithm, yielding a frequency spectrum of signal with amplitude (μV) ranging from 0 to 250 Hz with a frequency resolution of 0.056 Hz.

To obtain valid estimates of SS-EPs, the contribution of noise (i.e., due to spontaneous EEG activities, muscle activities, or eye movements), at each frequency bin of frequency spectra, was minimized by subtracting the averaged amplitude across neighboring bins (second to fourth on both sides, namely 3 bins ranging from − 0.22 to − 0.11 Hz and 3 bins from + 0.11 to + 0.22 Hz relative to each bin). For each subject and condition, noise-subtracted spectra of FCz and Cz electrodes were then averaged, because SS-EPs were predominant in frontocentral area as shown in previous studies (Nozaradan et al. 2011).

In addition, the frequency spectra of sound envelopes of auditory stimuli were extracted using a Hilbert function incorporated in the MIRToolbox running in the MATLAB environment (see Fig. 7A). Based on the envelope spectra and stimulus properties, following frequencies were regarded as target frequencies: beat frequency (1.67 Hz), the fundamental meter frequencies and their harmonic frequencies for three meters. Specifically, for 2/4 meter, 0.83 Hz is its fundamental meter frequency and 0.42 Hz is harmonic frequency; for 3/4 meter, 0.56 Hz is its fundamental meter frequency and 1.11 Hz is harmonic frequency; and for 4/4 meter, 0.42 Hz is its fundamental meter frequency and 0.83 Hz is harmonic frequency. The magnitudes of SS-EPs for each condition and subject were estimated by taking the noise-subtracted amplitude at the exact (or the approximate for 0.42 Hz only) target frequencies (Lenc et al. 2018) (the frequent resolution is 0.056 Hz, thus for 0.42 Hz SS-EP: amplitude at 0.44 Hz among 0.39, 0.44 and 0.50 Hz; for 0.56 Hz SS-EP: amplitude at 0.56 Hz among 0.50, 0.56 and 0.61 Hz; for 0.83 Hz SS-EP: amplitude at 0.83 Hz among 0.78, 0.83 and 0.89 Hz; for 1.11 Hz SS-EP: amplitude at 1.11 Hz among 1.06, 1.11 and 1.17 Hz; for 1.67 Hz SS-EP: amplitude at 1.67 Hz among 1.61, 1.67 and 1.72 Hz). The spectra were shown in Figs. 6 and 7.

Fig. 7.

Group mean of beat- and meter-related SS-EPs. The EEG spectrum from 0 to 8 Hz, averaged across FCz and Cz electrodes (bottom). The EEG spectrum within a frequency range containing the beat frequency (1.67 Hz) and meter frequencies for each meter (middle).The topographical maps of EEG signals at 0.42 Hz, 0.56 Hz, 0.83 Hz, 1.11 Hz, 1.67 Hz obtained in each meter (top)

Fig. 6.

Group mean tension ratings for strong beats and the last weak beats within measures and the comparison results across three meters. *p < 0.05, **p < 0.01, or ***p < 0.001, vertical lines indicate ± SE

One-sample t tests were performed for each meter to determine whether the amplitude measured at the target frequencies was significantly different from zero (μV). Accordingly the signal amplitudes at the non-target frequencies were considered close to zero. One-way repeated-measures ANOVAs were performed to compare the magnitudes of SS-EPs at each target frequency across three meter conditions. Critically, one-way repeated-measures ANOVA was performed to compare the magnitudes of SS-EPs at the fundamental meter frequency of three meters respectively.

Auditory ERPs analysis

The preprocessing procedures for auditory ERPs analysis remained the same as for the frequency-domain analysis, except that the baseline correction was performed according to − 200 to 0 ms of each epoch. For each subject, all epochs in the same condition were averaged in the time domain (namely the length of each epoch: −1 to + 20 s relative to the onset of the auditory stimulus), and then averaged across all subjects to obtain grand averaged waveforms. Furthermore, the waveforms were then segmented and averaged in the same way as the behavioral analysis to obtain ERP waveforms within the timescale of a measure in three meters respectively, as shown in Fig. 8C.

Time–frequency analysis

The ERPs were computed by averaging signals in time-domain, preserving the information both time-locked and power-locked to the event. In contrast, the endogenous event-related desynchronization (ERD) preserves information time-locked but not phase-locked, namely induced neural responses (Pfurtscheller and Lopes da Silva 1999). The time course of β-ERD is more tightly coupled with changes in processing of temporal expectations than the time course of ERPs (Tzagarakis et al. 2010). To obtain such induced oscillatory activities, we conducted time–frequency analysis as follows.

The preprocessing procedures were the same as for the frequency-domain analysis. In order to reduce the evoked exogenous neural activities and retain solely the induced endogenous event-related desynchronization (Pfurtscheller and Lopes da Silva 1999), for each subject and meter condition, the time-domain averaged waveform across all epochs obtained from the above ERPs analysis were subtracted from each single-trial waveform (Fujioka et al. 2015; Fujioka and Ross 2017). The time–frequency representation (TFRs) of each epoch was computed using Morlet Wavelet decomposition with 58 evenly-spaced bins from 2 to 30 Hz. The half-maximum width of the wavelet was adjusted across the frequency range to contain three cycles at 2 Hz and five cycles at 30 Hz with 4 ms as time resolution. Then, the signal power of each epoch was normalized to the mean power across 18 s succeeding the onset of auditory stimulus (approximately the entire meter sequence), and expressed as the signal power change (“relchange” as the calculation method of baseline correction).

For each condition, the baseline-corrected TFRs were firstly averaged across trials and subjects, and then averaged across all electrodes over scalp to exclude any electrode selection bias. Then the TFRs of stable stage (3.6 ~ 18 s) was segmented in the same way as small time scale behavioral analysis. The time–frequency representations were averaged across segments for each meter, TFRs are shown in Fig. 8D. It is noteworthy that the power suppression (shown as the color changing from green to blue or red to green in the TFRs) was not only found in β-band (around 13 ~ 25 Hz) as expected, but also in high α-band (> 10 Hz). It has been suggested that top-down attention, mainly encoded in α-band power, could be guided by temporal expectation to sharpen the dynamic attention to enhance expectation in turn for future events (Rohenkohl and Nobre 2011), in line with the classical dynamic attending theory (Jones and Boltz 1989).

For the β-band activities, based on the previous findings of the strongest β-ERD at around 20 Hz (Fujioka et al. 2015; Fujioka and Ross 2017) and the visual inspection of TFRs in our study, we considered 13 ~ 25 Hz as the frequency range of β-band for analysis (exact range is 13.3 ~ 24.6 Hz given that the frequency resolution of TFR is 0.49 Hz); and 10 ~ 13 Hz for α-band activities (exact range is 10.4 ~ 12.8 Hz). The grand averaged time courses of β-band and α-band activities for each meter were shown in Fig. 8E. The magnitudes of β-ERD and α-ERD at around 200 ms following beat onsets were computed as the averaged power over the time window of ± 20 ms around the grand-averaged peak latency for each beat (Graber and Fujioka 2020).

Previous studies estimated induced β-ERDs in metrical hierarchy and reported that the magnitudes of β-ERDs differentiated significantly across beats of different loudness during perception intervals and physically identical beats during imagery intervals (Fujioka et al. 2015). To examine whether the strong beats and weak beats induced different levels of temporal expectation and attention, as indicated by different magnitudes of β-ERD and α-ERD, statistical tests were performed to compare the β-ERD and α-ERD of beats respectively within measures: a paired-samples t test for 2/4 meter, one-way repeated-measures ANOVAs for 3/4 meter (three-level independent variable) and 4/4 meter (four-level independent variable).

For all the behavioral and EEG statistical tests, degrees of freedom were corrected with the Greenhouse–Geisser correction for violations of sphericity. Size effects were expressed by the partial η². The p-values of further paired comparisons were corrected using a FDR procedure and the significance level was set at p < 0.05.

Results

Tension ratings on large and small time scales

Large time scale results

On large time scale, the original tension curves formed by continuous tension ratings for three meter sequences were shown in Fig. 3, and the GCA results on overall tension during the whole sequences were shown in Fig. 4. The curves fitted from tension ratings for successive different beats of three meters were shown in Fig. 5.

Fig. 4.

Observed data (symbols, vertical lines indicate ± SE) and growth curve model fits (lines) of subjective tension ratings for each meter

Fig. 5.

The fourth-order polynomial curves fitted from group mean tension ratings for different beats of three meters

From the GCA results for overall tension during the whole sequences, we found that the main effect of meter on the intercept term was significant over the entire stage (F_(2,70) = 5.13, p < 0.01) as well as over the stable stage (F_(2,70) = 7.88, p < 0.001). The intercept term of growth curve indexes the average height of the curve over the analysis window (Mirman 2014), which corresponds to the overall average of tension ratings. Paired comparisons revealed that overall-averaged tension of 2/4 meter sequence was higher than that of 4/4 meter sequence significantly over the entire stage (t ₍₇₀₎ = 3.14, p < 0.01), as well as over the stable stage (t ₍₇₀₎ = 3.97, p < 0.001). The overall-averaged tension of 2/4 meter sequence was also higher than that of 3/4 meter sequence over two stages, and both contrasts were marginally significant (entire: t ₍₇₀₎ = 2.12, p = 0.057; stable: t ₍₇₀₎ = 1.92, p = 0.059). The overall-averaged tension of 3/4 meter sequence was higher than that of 4/4 meter sequence during the entire stage but did not reach significant level (p = 0.308), however during the stable stage the contrast was marginally significant (t ₍₇₀₎ = 2.05, p = 0.059). The main effect of meter on the linear term was also significant over the entire stage (F_(2,70) = 10.88, p < 0.001). The linear term of growth curve indexes the mean linear slope over the analysis window (Mirman 2014), which corresponds to the overall increase rate of the tension ratings over the entire stage. Paired comparisons revealed that the overall increase rate of tension of 2/4 meter sequence was faster than that of 4/4 meter sequence (t ₍₇₀₎ = 4.36, p < 0.001), and also faster than that of 3/4 meter sequence but did not reach significant level (p = 0.46) The overall increase rate of tension of 3/4 meter sequence was faster than that of 4/4 meter sequence (t ₍₇₀₎ = 3.62, p < 0.001). The results of paired comparisons were shown in Table 1.

Table 1.

The results of growth curve analyses

	Time window (s)	2/4 meter vs. 3/4 meter				2/4 meter vs. 4/4 meter				3/4 meter vs. 4/4 meter
		Est	SE	t	p	Est	SE	t	p	Est	SE	t	p
A
Intercept	0 ~ 18	0.85	0.40	2.12	0.057	1.26	0.40	3.14	0.002	0.41	0.40	1.03	0.31
	3.6 ~ 18	0.87	0.45	1.92	0.059	1.80	0.45	3.97	< 0.001	0.93	0.45	2.05	0.059
Linear	0 ~ 18	0.96	1.29	0.74	0.46	5.63	1.29	4.36	< 0.001	4.67	1.29	3.62	< 0.001
B
Intercept	3.6 ~ 18	1.86	0.07	26.01	< 0.001	2.93	0.07	40.86	< 0.001	1.07	0.07	14.85	< 0.001
Linear	3.6 ~ 18	0.55	0.50	1.12	0.26	3.56	0.50	7.10	< 0.001	3.00	0.5	5.98	< 0.001

(A) The differences in the intercept and linear terms of fitted curves from tension ratings during whole sequences across three types of meter sequences. (B) The differences in the intercept and linear terms of fitted curves from tension ratings of successive last beats across three types of meter sequences. The time window of 0 ~ 18 s represents the entire stage of meter sequence, and the time window of 3.6 ~ 18 s represents the stable stage of tension perception. Bold p-values represent significance or marginal significance level

From the GCA results on overall tension for successive strong beats, no significant main effect of meter were found (p = 0.11).

From the GCA results on overall tension for successive last weak beats, we found that the main effect of meter on the intercept term was significant (F_(2,4970) = 855.53, p < 0.001) over the stable stage. Paired comparisons revealed that overall-averaged tension for the last weak beats of 2/4 meter sequence was higher significantly than both of 3/4 meter sequence and 4/4 meter sequence (2/4–3/4, t ₍₄₉₇₀₎ = 26.01, p < 0.001; 2/4–4/4, t ₍₄₉₇₀₎ = 40.86, p < 0.001). The overall-averaged tension for last weak beats of 3/4 meter sequence was higher than that of 4/4 meter sequence (t ₍₄₉₇₀₎ = 14.85, p < 0.001). The main effect of meter on the linear term was also significant (F_(2,4970) = 29.11, p < 0.001). Paired comparisons revealed that the overall increase rate of tension for last weak beats of 2/4 meter sequence was faster than that of 4/4 meter sequence (t ₍₇₀₎ = 7.10, p < 0.001), and also faster than that of 3/4 meter sequence but did not reach significant level (p = 0.26). The overall increase rate of tension for last weak beats of 3/4 meter sequence was faster than that of 4/4 meter sequence (t ₍₇₀₎ = 5.98, p < 0.001). The results of paired comparisons were shown in Table 1B.

Small time scale results

On small time scale, the tension ratings of beats within measures were compared for each meter. In the 2/4 meter, strong beat induced higher tension than weak beat (t ₍₃₄₎ = 5.75, p < 0.001); in the 3/4 meter and 4/4 meter, the main effects of both beat types on tension were significant (3/4 meter: F_(2,68) = 42.25, p < 0.001, η² = 0.554; 4/4 meter: F_(3,102) = 37.85, p < 0.001, η² = 0.527). In the 3/4 meter, follow-up paired comparisons revealed that tension induced by strong beat (S for short) was higher than both two weak beats (W1 and W2 for short; S-W1: t ₍₃₄₎ = 7.00, p < 0.001; S-W2: t ₍₃₄₎ = 6.62, p < 0.001), and tension induced by W1 was higher than W2 (W1-W2: t ₍₃₄₎ = 5.09, p < 0.001). In the 4/4 meter, follow-up paired comparisons revealed that tension induced by strong beat was higher than three weaker beats (W1, M and W2 for short; S-W1: t ₍₃₄₎ = 6.81, p < 0.001; S-M: t ₍₃₄₎ = 7.16, p < 0.001; S-W2: t ₍₃₄₎ = 6.49, p < 0.001), and tension induced by W1 was higher than both M and W2(W1-M: t ₍₃₄₎ = 4.63, p < 0.001; W1-W2: t ₍₃₄₎ = 4.67, p < 0.001), and tension induced by M is also higher than W2 (t ₍₃₄₎ = 3.04, p < 0.01). The results of paired comparisons were shown in Fig. 9B.

Fig. 9.

A Relative loudness distributions of beats within measures of three meters, normalized to strong beat. SOA represents the interval of isochronous beats. B Group mean of subjective tension ratings within measures of each meter. The shaded areas indicate the 95% confidence interval of the group mean. C Group-level transient auditory ERPs elicited within measures of each meter, averaged across frontocentral electrodes (F1/Fz/F2/FC1/FCz/FC2/C1/Cz/C2). D Group-level TFRs within measures of each meter, averaged across all electrodes. E Time courses of β- (13 ~ 25 Hz) and α- (10 ~ 13 Hz) band activities within measures. The blue bars mark the time windows (± 20 ms around the grand-averaged trough latency) of β-ERDs and the red bars mark the time windows of α-ERDs. F The topographical maps of β-ERDs and α-ERDs (in the corresponding time windows) obtained within measures. The dashed vertical lines mark the time onset of each beat within measures. S represents the strong beat, W represents the weak beat, W1 and W2 represent the first and the second weak beat of 3/4 and 4/4 meter, M represents the middle beat of 4/4 meter

The tension ratings for strong beats and the last beats within measures were also compared across three meters. The results for strong beats did not found any significant main effects of meter (p = 0.87). The results for last beats revealed significant main effect of meter (F_(2,68) = 15.23, p < 0.001, η² = 0.309). Follow-up paired comparisons revealed that tension induced by the last weak beats within measures of 2/4 meter were significantly higher than those of both 3/4 meter and 4/4 meter (2/4–3/4, t ₍₃₄₎ = 5.33, p < 0.001; 2/4–4/4, t ₍₃₄₎ = 4.61, p < 0.001), and tension induced by the last weak beats within measures of 3/4 meter was higher than that of 4/4 meter but did not reach significant level (p = 0.27). The results were shown in Fig. 6. The results from LMM analysis revealed significant main effects of both the meter (F_(2,49921) = 3.53, p = 0.029) and loudness (F_(2,49921) = 1062.07, p < 0.001) on tension ratings for beats. Follow-up paired comparisons revealed that the tension ratings for all beats (without distinguishing the types of beats) of 2/4 meter sequences were higher than those of 4/4 meter (t_(2,49921) = 2.66, p = 0.024), and also higher than those of 3/4 meter but did not reach significant level (t_(2,49921) = 1.29, p = 0.20). The tension ratings for all beats of 3/4 meter sequences were higher than those of 4/4 meter but did not reach significant level (t_(2,49921) = 1.53, p = 0.19). It is noteworthy that such pattern of difference is similar to that of the results of GCA analysis on large time scale. Additionally, regardless of which type of meter sequence the beats belong to and in which sequential order they appear, the paired comparisons revealed that the tension ratings of all strong beats with the highest loudness were larger than that of all weak beats (t_(2,49921) = 44.29, p < 0.001), and also larger than that of middle beats (t_(2,49921) = 26.22, p < 0.001). The tension ratings of all weak beats were larger than that of middle beats (t_(2,49921) = 5.2, p < 0.001). Such pattern of difference is similar to the results of the analysis on small time scale.

Taking together the results on large time scale across three meters and the results on small time scale both within and across meters would lead us to following summaries: (1) tension induced by strong beats of three different meters had similar level and dynamics on both time scales. (2) Tension decreased continuously and significantly while beats appearing sequentially within measures for all three meters. (3) Tension induced by the last weak beats of three different meters tended to show such difference pattern on both time scales: 2/4 > 3/4 > 4/4 meter. Given that the overall tension level and increase rate also showed a difference pattern of 2/4 > 3/4 > 4/4 meter, we could infer that strong beats anchored the maximum level of tension; the longer intervals between successive strong beats left more time for tension decrease, leading to lower tension of last weak beats within measures and eventually lower overall tension over whole sequence. Critically, the beats with same loudness in different meters (the last weak beats across three meters) and within meters (two successive weak beats in 3/4 meter) induced different levels of tension. The LMM analysis also suggested that with the effect of beat loudness on tension being regressed out, there still existed significant main effect of meter types on beat tension These indicate that the effects found here may not be simply explained by low-level acoustic processing of beats, instead the metric contexts to which the beats belong showed significant influence on tension perception.

Steady-state evoked potentials

As shown in Figs. 6 and 7, all three meter conditions elicited clear EEG signal peaks at 1.67 Hz as beat-related SS-EPs. As for the meter-related SS-EPs, the 2/4 meter elicited peaks at 0.83 Hz and 0.42 Hz; the 3/4 meter elicited peaks at 0.56 Hz and 1.11 Hz; the 4/4 meter elicited peak at 0.83 Hz but not at 0.42 Hz. The scalp topographies of beat- and meter-related SS-EPs reached maximal over frontocentral regions (see Fig. 6).

Beat-related SS-EPs

The magnitudes of the beat-related SS-EPs were higher significantly than zero in all three conditions (2/4 meter: t ₍₃₂₎ = 7.87, p < 0.001; 3/4 meter: t ₍₃₂₎ = 8.30, p < 0.001; 4/4 meter: t ₍₃₂₎ = 7.56, p < 0.001), and not different significantly across three conditions (F_(2,64) = 2.99, p = 0.057, η² = 0.086).

Meter-related SS-EPs

In the 2/4 meter, the magnitudes of the meter-related SS-EPs at 0.83 Hz and 0.42 Hz were both higher significantly than zero (0.83 Hz: t ₍₃₂₎ = 6.26, p < 0.001; 0.42 Hz: t ₍₃₂₎ = 2.50, p = 0.018). In the 3/4 meter, the magnitudes of the meter-related SS-EPs at 0.56 Hz and 1.11 Hz were higher significantly than zero (0.56 Hz: t ₍₃₂₎ = 3.86, p < 0.001; 1.11 Hz: t ₍₃₂₎ = 3.36, p < 0.01). In the 4/4 meter, the magnitude of the meter-related SS-EP at 0.42 Hz was not higher significantly than zero (0.42 Hz: t ₍₃₂₎ = 0.04, p = 0.971), while that at 0.83 Hz was higher than zero and the difference was slightly significant (0.83 Hz: t ₍₃₂₎ = 2.09, p = 0.045). The results of t tests were shown in Fig. 7.

Besides, the magnitudes of the meter-related SS-EPs at 0.56 Hz, 0.83 Hz, 1.11 Hz differed significantly across three meter conditions (0.56 Hz: F_(2,64) = 19.34, p < 0.001, η² = 0.377; 0.83 Hz: F_(2,64) = 9.93, p < 0.001, η² = 0.237; 1.11 Hz: F_(2,64) = 7.09, p < 0.01, η² = 0.181). Paired comparisons revealed that at 0.56 and 1.11 Hz the magnitudes of the SS-EPs were larger significantly in 3/4 meter than both in 2/4 meter (0.56 Hz: t ₍₃₂₎ = 4.43, p < 0.001; 1.11 Hz: t ₍₃₂₎ = 2.83, p < 0.01) and 4/4 meter (0.56 Hz: t ₍₃₂₎ = 4.83, p < 0.001; 1.11 Hz: t ₍₃₂₎ = 2.9, p < 0.01), and the comparisons between 2/4 and 4/4 meter did not reach significance (0.56 Hz: t ₍₃₂₎ = 1.13, p = 0.26; 1.11 Hz: t ₍₃₂₎ = 0.81, p = 0.42). While at 0.83 Hz the magnitudes of the SS-EPs were larger significantly in 2/4 meter than both in 3/4 meter (t ₍₃₂₎ = 4.35, p < 0.001) and 4/4 meter (t ₍₃₂₎ = 3.02, p < 0.01), the comparisons between 3/4 and 4/4 meter did not reach significance (t ₍₃₂₎ = 1.43, p = 0.16). The magnitudes of the meter-related SS-EPs at 0.42 Hz were not different significantly across three conditions (F_(2,64) = 2.46, p = 0.094, η² = 0.071).

Critically, the comparison of respective fundamental meter-related SS-EPs across meters showed a significant main effect of meter (F_(2,64) = 9.25, p < 0.001, η² = 0.224). Paired comparisons revealed that the magnitudes of the SS-EP of 2/4 meter at 0.83 Hz was larger significantly than 4/4 meter at 0.42 Hz (t ₍₃₂₎ = 3.91, p < 0.01), and the magnitudes of the SS-EP of 3/4 meter at 0.56 Hz was larger significantly than 4/4 meter at 0.42 Hz (t ₍₃₂₎ = 3.65, p < 0.01). There was no significant difference between 2/4 meter at 0.83 Hz and 3/4 meter at 0.56 Hz (p = 0.959).

In sum, the results of steady-state evoked potentials revealed that all three meter conditions elicited significant beat-related SS-EPs. However, there existed difference in meter-related SS-EPs. Both the 2/4 meter and 3/4 meter elicited evident peaks of spectrum at their fundamental meter frequencies and harmonic frequencies, and larger than other meters did at those frequencies, while the 4/4 meter did not elicit evident fundamental meter-related SS-EPs. Importantly, at their respective fundamental meter-related frequency, both the 2/4 meter and 3/4 meter elicited larger SS-EPs than 4/4 meter. As introduced before, low frequency SS-EPs obtained on large time scale may be the neural index of expectation generating during whole sequences. The difference in the meter-related SS-EPs across three meters suggest different levels of overall expectations generated from metrical sequences processing and may be related to overall tension difference.

Auditory event-related potentials

As shown in Fig. 8C, within measures of three meters, the onset of the strong beats elicited clear ERPs (N1 and P2 peaks), which were maximal at frontocentral electrodes, but not for the weaker beats. These ERPs are typical primary auditory related components and were also found by previous researches (Nozaradan et al. 2011; Fujioka et al. 2015). The auditory N1 and P2 were associated with early perceptual processing, as they were elicited in response to stimulus onsets and were considered as exogenous, stimulus-driven evoked responses (Naatanen and Picton 1987; Rosburg et al. 2008).

Induced power modulations

The Fig. 8D and E demonstrated how each type of metrical structure led to dynamic changes in β- and α-band activities within measures. Based on the TFRs shown in Fig. 8D, the time courses of β-band and α-band signal changes (expressed as the percentage change) were obtained by averaging the power in 13 ~ 25 Hz and 10 ~ 13 Hz bands respectively, as shown in Fig. 8E. The β-band and α-band activities decreased (ERD) sharply after the onset of each beat, then reached the minimum around 200 ms latency (sometimes later), and rebounded with a shallow slope (ERS) before the next beat. However, these periodic modulations of β-ERD and α- ERD seemed to be much more prominent after strong beat than after other subsequent beats, as the troughs of β-ERD and α- ERD were noticeably deeper after strong beat.

In the 2/4 meter, there were larger β-ERD and α-ERD after strong beat than weak beat (β-ERD: t ₍₃₂₎ = 2.25, p = 0.031; α-ERD: t ₍₃₂₎ = 2.05, p = 0.048).

In the 3/4 meter, the troughs of β-ERD and α-ERD after strong beat were deeper than two weak beats within measures. The main effects of beat type on the magnitudes of both β-ERD and α-ERD were significant (β-ERD: F_(2,64) = 3.65, p = 0.032, η² = 0.102; α-ERD: F_(2,64) = 10.17, p < 0.001, η² = 0.241). For β-ERD, paired comparisons revealed that the main effect was mostly contributed by the comparison between strong and two weak beats, and both contrasts were marginally significant(S vs. W1: t ₍₃₂₎ = 2.11, p = 0.064; S vs. W2: t ₍₃₂₎ = 2.28, p = 0.064). There was no significant difference between W1 and W2 (p = 0.923). For α-ERD, the main effect was also mostly contributed by the comparison between strong and two weak beats (S vs. W1: t ₍₃₂₎ = 4.19, p < 0.001; S vs. W2: t ₍₃₂₎ = 3.47, p < 0.01). There was no significant difference between W1 and W2 (p = 0.593).

In the 4/4 meter, like 3/4 meter, the troughs of β-ERD and α-ERD after strong beat were also deeper than three weaker beats within measures. The main effects of beat type on the magnitudes of both β-ERD and α-ERD were significant (β-ERD: F_(3,96) = 2.95, p = 0.037, η² = 0.084; α-ERD: F_(3,96) = 6.52, p < 0.001, η² = 0.169). For β-ERD, the main effect was mainly contributed by the comparison between S and W1, and the contrast was marginally significant(t ₍₃₂₎ = 2.64, p = 0.076). No other significant difference was found (ps > 0.1). For α-ERD, the main effect was also mainly contributed by the comparison between strong and three weaker beats (S vs. W1: t ₍₃₂₎ = 3.21, p < 0.01; S vs. M: t ₍₃₂₎ = 3.52, p < 0.01; S vs. W2: t ₍₃₂₎ = 3.18, p < 0.01). No other significant difference was found (ps > 0.2). The results of all the above paired comparisons were shown in Fig. 9C and D.

Fig. 10.

A Representative measure of each meter. B Group mean of subjective tension ratings of each beat within measures of each meter. Group mean amplitudes of C β-ERDs and D α-ERDs induced by each beat within measures of each meter. p < 0.08, *p < 0.05, **p < 0.01, or ***p < 0.001, vertical lines indicate ± SE

In sum, The β-ERD and α-ERD were always larger after strong beats than the weaker beats, and the ERDs after the weaker beats were not significantly different from one another, and such difference patterns were consistent across three meters.

Further, in order to explore the potential relationship between the tension ratings and the magnitudes of induced β-ERD and α-ERD, correlation coefficient analyses were conducted for each subject. The behavioral and EEG data were taken from the small time scale behavioral analysis and time–frequency analysis described above respectively. We put 231 pairs (35 subjects × 7 types of beats: S and W of 2/4 meter, S and W of 3/4 meter, S, M and W of 4/4 meter) of behavioral and EEG data into each correlation analysis. As shown in Table 2, significant negative correlations were found between tension ratings and the magnitudes of both β- and α-ERDs. These results indicate that the deeper the power troughs of β- and α-band neural activities induced by a particular beat, the higher the tension ratings for that beat. The magnitudes of β-ERDs may be neural indexes of temporal expectation induced by specific beats, and α-ERDs may indicate the degrees of attention deployment during beat processing. These high frequency neural activities suggest temporal expectation and attention may underlie the dynamic tension perception on small time scale.

Table 2.

The results of correlation analysis

	Tension (N = 231)
Beta-ERD	− 0.142*
Alpha-ERD	− 0.148*

Correlation coefficients between subjective tension ratings of beats and magnitudes of induced β-ERDs and α-ERDs

*p < 0.05, **p < 0.01

Discussion

In the present study, we investigated the tension induced by three types of metrical structures and explored the relevant neural activities. To this end, the behavioral ratings and EEG responses were collected and analyzed in two temporal scales: large time scale (entire sequences) and small time scale (within measures). Our behavioral and EEG results provided consistent evidence for the dynamic effects of metrical structures on tension. The results were discussed in more detail below.

The overall tension is determined by temporal intervals between strong beats

Our results of GCA across three meters on large time scale suggest that tension induced by strong beats of different meters are the same. Critically, the last weak beats of 2/4 meter induced higher level and faster increase rate of tension than those of 3/4 and 4/4 meter did, and such difference pattern across three meters was also found in overall tension height and increase rate. The meter sequences in our study differ mainly in the temporal intervals between adjacent strong beats. Strong beats are the initial and the loudest beats in meter sequences and function as temporal anchors in metrical structures, while the other weaker beats merely fill in the time periods between strong beats (Lerdahl and Jackendoff 1996; Large and Jones 1999; Large 2002; Fitch 2013). Listeners can spontaneously track the temporal structures when listening or dancing to the music by detecting the periodicity of strong beats, which is the core process of metrical perception (Iversen et al. 2009; Geiser et al. 2014; Leow and Grahn 2014). Given that our findings indicate that on large time scale different meter sequences generate the same maximum level of tension (see strong beats), and on small time scale within measures tension decrease continuously between strong beats, therefore the meter sequence inducing lower overall-averaged tension may due to its longer intervals between strong beats which give rise to lower minimum level of tension (see last weak beats).

In the classical quantitative model to predict real-time tension proposed by Farbood (2012), various musical parameters were taken into account, including temporal features (i.e., meter, rhythmic regularity). Although in her study meter and rhythmic regularity did not influence tension significantly, the results indicated that the metrical structures with strong beats occurring more frequently would induce higher tension. In our study, strong beats occurred the most frequently in 2/4 meter , followed by 3/4 meter and then 4/4 meter. Thus, the different overall tension experience may mainly result from temporal intervals of strong beats, given that other musical parameters of the metrical structures are well controlled. Moreover, it is known from our daily experiences that musical meters could be used to express various emotional tones and to forge different musical styles. Duple meter such as 2/4 meter always appears in march to boost intense emotions due to more frequent occurrence of strong beats; while triple meter (3/4 meter) and quadruple meter (4/4 meter) could often be found in dance music (i.e., waltz) or cantilena (i.e., lullaby), and convey more gentleness and lyricism for their longer period of strong beats (Cooper and Meyer 1960). The higher overall tension induced by 2/4 meter may also result from its greater ability to elicit high-arousal emotions, compared with 3/4 and 4/4 meter.

The low-frequency steady-state evoked potentials (SS-EPs) on large time scale may further explain the behavioral findings above. Our data replicated previous findings as to δ- (2 ~ 4 Hz) band neural activities (Nozaradan et al. 2011, 2012b, 2015, 2017; Stupacher et al. 2016) but with further valuable observations. In line with existing studies, the 2/4 meter and 3/4 meter both elicited SS-EPs at their fundamental meter frequency and additional harmonic frequencies, however 4/4 meter only elicited slightly significant SS-EPs at harmonic frequency but not at its fundamental meter frequency (Fig. 7). These results reflect that beat extraction is effortless for all three types of meters. But metrical construction, during which the beats are organized into hierarchical structure, might require more cognitive efforts (Grahn 2012; Levitin et al. 2018; Celma-Miralles and Toro 2019; Li et al. 2019), especially for 4/4 meter due to its relatively lower meter frequency for entrainment. As previous studies have proposed, subjective tension ratings could be positively predicted by visceral entrainment, the physical internal sense of bodily felt entrainment (Trost et al. 2011, 2015; Labbé and Grandjean 2014). Therefore, we further assume that in our study the degrees of entrainment to meter, indexed by meter-related SS-EPs, may be a predictor of overall tension ratings.

It should be noted that the SS-EPs not only reflects the entrainment to metrical periodicity, but also serve as a neural basis for temporal expectations (Jones and Boltz 1989; Merchant et al. 2015; Obleser and Kayser 2019). More critically, musical expectations are thought to be positively correlated with expectancy-tension, specifically, musical events that trigger more stable and stronger expectations generate higher expectancy-tension (Margulis 2005; You et al. 2021). Thus, in our study, the consistent patterns of difference in meter-related SS-EPs (elicited at fundamental meter frequencies) and in overall tension across three meter conditions may imply a potential relationship between these two factors. Taken together, it would be reasonable to speculate that the metrical sequence with shorter interval of strong beats (higher meter frequency) could generate stronger temporal expectation indicated by more significant meter-related SS-EPs, further result in higher overall height and faster increase rate of expectancy-tension during whole sequence (Margulis 2005; You et al. 2021).

Importantly, the complementary evidence from LMM analysis suggested that the effects of different metrical sequences on dynamic tension experience may not simply attributed to low-level acoustic perception, i.e., the sense of loudness changing, but substantially come from high-level cognitive processes, including forming cognitive organizational patterns of strong and weak beats, and generating different degrees of overall temporal expectations.

Based on the above results from large time scale analyses, we intend to attribute the difference in overall tension to the difference in the interval between strong beats across three metrical structures, in other words, the shorter the temporal intervals between adjacent strong beats, the higher the overall tension. This interpretation verifies the important roles of strong beats in metrical structures (Large and Jones 1999; Large 2002), and conforms to the real-life experience when listening to music of different styles.

Dynamic tension experience is induced by hierarchical temporal expectation and attention

On small time scale, the tension curves within measures of three types of metrical structures show similarities in their dynamic features. The tension curve rises up mildly soon after the onset of the strong beat, reaches to the maximum approximately at the midpoint between strong beat and (the first) weak beat, followed by a decline continuing toward the midpoint of (the last) weak beat, and then ends with an evident rise before the onset of the next strong beat (Fig. 8B). These periodic fluctuations of tension curves, together with the comparison of tension induced by beats within measures, suggest that tension would rise up before the onsets of strong beats, and release continuously as the weaker beats appear subsequently between two adjacent strong beats.

These findings on small time scale indicate that within a measure metrical structure could influence tension in a hierarchical way: strong beat anchors the maximum values of tension, while the sequential weaker beats just play subsidiary roles to fill in the “release space” between two adjacent strong beats. Comparisons across meters on small time scale found that strong beats of three meters could induce the same level of tension, while the last weak beats farther away from the strong beats in time induce lower tension. Taking these together with the results on large time scale discussed above, it can be concluded that shorter intervals between strong beats may imply shorter duration for tension decrease within measures, hence result in higher overall tension height across whole sequence of measures. Our findings verify that temporal interval between strong beats play significant roles in the overall height and the dynamics of tension curves.

The dynamic modulations of induced β-ERDs were found to be well periodic within measures, and the significant correlation between tension ratings and magnitudes of β-ERDs suggested that larger β-ERD induced by a beat may predict higher tension rating for this beat. It is widely accepted that periodic β-band power fluctuations could reflect active generation of endogenous temporal expectation (Large and Snyder 2009; Fujioka et al. 2012; Arnal et al. 2015; Merchant et al. 2015; Herrmann et al. 2016). Significantly, (Kilavik et al. 2013) reviewed a large body of studies investigating the roles of β-band power fluctuations, and concluded that β-band activities may reflect multifaceted cognitive processes, not only the generation of temporal expectations. Additionally, uncertainty about the features of forthcoming events was also found to influence β-ERDs substantially: the less the uncertainty of the task, the more the β-band power decreased during the motor preparation (Tzagarakis et al. 2010, 2015). Therefore, our findings might indicate that strong beats could generate stronger temporal expectation towards subsequent (weaker) beats with less uncertainty, and thus induce higher expectancy-tension during the occurrence of strong beats than weaker beats do. This finding fit with the view that high expectancy-tension arises at the moment when an event occurs if that very event triggers strong expectation for subsequent event without uncertainty (Margulis 2005; You et al. 2021).

Furthermore, induced α-band activities are also found to fluctuate in a pattern similar to β-band, and the depths of α-ERDs are more prominent on strong beats than weaker beats. Following the classical dynamic attending theory (Jones and Boltz 1989), during meter processing, dynamic modulation of attention generates temporal expectation, and the latter in turn gathers more attention resources to enhance temporal processing of forthcoming beats. In fact, it is suggested that expectation and attention could jointly modulate temporal processing (Todorovic et al. 2015; Graber and Fujioka 2020). The fluctuations of both β- and α-band activities were found to be induced by regular stimuli sequences (Arnal et al. 2015; Breska and Deouell 2016), and α-ERD is attributed to the increased attention (Praamstra et al. 2006), and interacts with temporal expectation (Rohenkohl and Nobre 2011; Breska and Deouell 2014; Herrmann et al. 2016). A recent study also found that lower α-band power, reflecting deep engagement of attention resources, is directly associated with higher tension induced by tonal structural violation (Sun et al. 2020b).

In our study, the sequential decrease of α-ERDs within a measure may suggest gradual attention diffusion, a process of tension resolution (Fig. 9B). However, this inference should be drawn with caution considering that predictability of metrical timing was found to directly influence N1 component (Foldal et al. 2020), an evoked response in low theta-alpha band. Foldal et al. (2020) manipulated rhythmic predictability by varying the probability of rhythm violation and found that N1 was attenuated when rhythm predictability was high, and this effect was found to be independent of attention. Therefore the decrease of α-ERDs might not only be considered as the marker of attention diffusion but also be influenced by the increased predictability of stimulus timing, which are correlated with growth in prior experience and knowledge of metrical timing.

Therefore, within measures of three types of metrical structures, the potential roles of temporal expectation, dynamic attention and possibly timing experience in music tension generation are indicated by the periodic modulations of β- and α-ERDs. More directly, the correlation between the tension ratings and magnitudes of β- and α-ERDs suggested that tension might be predicted by the β- and α-band neural activities, the larger β-ERD and α-ERD induced by a specific beat may lead to higher tension rating for this beat. Specifically, strong beats could increase attention and generate higher temporal expectation, leading to higher tension, which in turn interactively facilitate the attention towards the future events (Jones and Boltz 1989; Margulis 2005). Taken as a whole, expectation and attention reflected by the dynamic modulations of high-frequency activities may function collaboratively in tension induced by metrical structures.

Conclusions and future directions

In conclusion, the current study showed the dynamic tension perception induced by three different metrical structures. From the perspective of structural features, strong beats at higher level of metrical hierarchy anchored the maximum level of tension, and the temporal intervals between them determined the overall height and slope of tension curves, with the sequential weaker beats filled in such intervals forming the space for continuous tension reduction. The relationship between tension and neural activities, both the low-frequency entrainment and high-frequency periodic modulations, suggested that temporal expectation and attention may be important cognitive bases which trigger and influence tension.

By exploiting the characteristics of internal meter perception, that is, generating hierarchical expectations in time dimension, we induced external metric tension in a hierarchical and dynamic manner. The current study suggests that without deliberate violation in expectations, the generation of dynamic expectation itself could elicit tension (i.e., expectancy-tension). The findings extend our understanding of tension by investigating expectancy-tension, which is more common exhibited in real-life music listening experience, thereby providing more ecological validity.

There still exist a few limitations of this study, which should be addressed in future studies. Firstly, our experimental design lacked control conditions. In fact musical sequences without regular metrical structure (that is, sequences in which all the beats are strong or weak beats) may be considered as the baseline for the comparisons between different types of meters. However, our concern was that even when listening to a sequence of identical beats, listeners would still tend to perceive it into a binary (more often) or ternary meter spontaneously, because participants in our study were all experienced musicians. Therefore we assumed that setting such a control condition may interfere with performance under other conditions and should be done with great cautions, hopefully our follow-up work could find an optimal way to settle this issue. A maximum entropy paradigm could be an option, in which we remove the musical characteristics and the predictability of the stimuli, but remain the physical dimension of the stimuli. Secondly, the musical materials used in our study were relatively simple compared to real music. These may impair the practical implications of the present study. Therefore in the future real music with more complex and various features can be adopted. Thirdly, the tension ratings method adopted in the current experiment measured tension in a subjective way. Additionally, some objective physiological indexes were also found to be effective measurements of tension. For instance, skin conductance response and heart rate were measured in response to unexpected chords to study tension experience physiologically (Koelsch et al. 2008b). Future work could consider combining both the subjective and objective measurements to cross validate the rating of tension, in order to assess tension in a more comprehensive way.

Furthermore, from a computational viewpoint, several studies had proposed quantitative models of musical tension which took into account the dynamic, temporal features of music (Farbood 2012; Barchet et al. 2022) and various specific tonal musical parameters (Navarro-Caceres et al. 2020). Those models were proved to effectively predict and describe mostly the tonal tension. It would be interesting to also adapt such model to predict metrical tension, considering the analytical descriptions of metrical parameters of meter sequences, and test the model by assessing the correlation with the empirical data from current study. Further work can be done to fulfil this goal. Last but not the least, the neural substrates that we can detected by EEG in current study are limited, in the future studies MEG, fMRI or other physiological indicators should be used to investigate the brain regions and connectivities on more microscopic scales, for instance the neural activities in the lateral orbitofrontal cortex and superficial amygdala which were found to be associated with tension (Lehne et al. 2014; Lehne and Koelsch 2015) or the mCBGT loop (motor cortico-basal ganglia-thalamo-cortical) involved in motor and predictive processing in meter perception (Merchant and Honing 2013; Proksch et al. 2020), in order to reveal a profound neurocognitive mechanisms underlying musical tension induced by metrical structures.

Data availability

The datasets generated during and analyzed during the current study are available from the corresponding author on reasonable request.

Declarations

Conflict of interest

The authors declare no conflict of interest.