Rhythmic Analysis in Animal Communication, Speech, and Music: The Normalized Pairwise Variability Index Is a Summary Statistic of Rhythm Ratios

Abstract

Rhythm is fundamental in many physical and biological systems. Rhythm is relevant to a broad range of phenomena across different fields, including animal bioacoustics, speech sciences, and music cognition. As a result, the interest in developing consistent quantitative measures for cross-disciplinary rhythmic analysis is growing. Two quantitative measures that can be directly applied to any temporal structure are the normalized pairwise variability index (nPVI) and rhythm ratios (r_k). The nPVI summarizes the overall isochrony of a sequence, i.e., how regularly spaced a sequence’s events are, as a single value. Meanwhile, r_k quantifies ratios between a sequence’s adjacent intervals and is often used for identifying rhythmic categories. Here, we show that these two rhythmic measures are fundamentally connected: the nPVI is a summary static of the r_k values of a temporal sequence. This result offers a deeper understanding of how these measures are applied. It also opens the door for creating novel, custom measures to quantify rhythmic patterns based on a sequence’s r_k distribution and compare rhythmic patterns across different domains. The explicit connection between nPVI and r_k is one further step towards a common quantitative toolkit for rhythm research across disciplines.

1. Introduction

1.1. Rhythm in Speech, Music, and Bioacoustics

Most physical and biological systems exhibit transient behavior that changes over time. From simple oscillators to complex emergent behaviors, the temporal evolution of these systems and their components may carry information on their underlying mechanisms. Some complex systems produce discrete events in time: a clock ticking, a dog barking, and a bassist plucking individual strings all produce distinct events with non-random temporal structure. These temporal sequences and their resulting patterns may be described as rhythms. Living organisms are particularly apt at producing and perceiving rhythms because their nervous systems seem to rely on them for various processes, including physiology, locomotion, fine movement, and communication [1,2].

Rhythmic behaviors span wildly different spatial and temporal scales and are of interest to a broad range of scientific disciplines [3,4,5,6,7,8,9,10,11,12]. Three of these disciplines may seem strange bedfellows at first yet feature deep theoretical and methodological commonalities: animal bioacoustics, speech science, and music cognition are all interested in the structural properties of sound as one possible window to study the underlying cognitive mechanisms. Animal bioacoustics focuses on biotic sounds, while the other two disciplines typically focus on human-generated sounds [13]. In all three disciplines, rhythmic structure is most commonly sought in a range spanning millisecond to seconds when zooming in on temporal properties. Because these areas share the common goal of finding rhythm in sound-based systems, their quantitative tools and frameworks are often applicable domain-generally and sometimes exchanged between fields [14].

Theoretical definitions of rhythm abound because of its multidimensionality [15,16,17,18]. One pragmatic approach to rhythm is to use measures that capture the rhythmic properties of a signal; a behavior is then quantified as more or less rhythmic depending on the outcome of the used measure(s). To define and characterize rhythmic behavior, contributions from multiple fields are needed. It is important, among other things, to establish mathematical and statistical characterizations of rhythm. After identifying those measures applicable in animal bioacoustics, speech sciences, and music cognition, a good question is whether some of these measures are orthogonal or unexpectedly related.

1.2. Describing Rhythm Across Disciplines

Speech science, music cognition, and animal bioacoustics share a common interest in the structural properties of sound [19]. A common approach across these domains can support our understanding of quintessentially human behaviors like music, dance, and speech [19,20]. Such an approach can also shed light on the evolutionary origins of music and speech by examining their fundamental building blocks, with the rhythmic structure being pivotal [21,22,23].

Measurement tools applicable to all these fields are crucial to facilitate a cross-disciplinary approach. As such, recent years have seen growing attention towards the development of rhythmic analyses that can be applied to music cognition and speech sciences as well as to animal bioacoustics [14,23,24,25,26,27,28,29,30]. From this perspective, a quantitative measure should be directly applicable to any rhythmic structure, regardless of the context in which rhythm is expressed.

A good starting point for cross-disciplinary rhythmic measures is the basic definition of rhythm as “patterns of events in time” [16]. Adopting such a basic definition is important to avoid assumptions about the structure of language, music, and animal vocalizations, which for the latter often has yet to be established [31,32]. By stripping down a sequence—whether it originates from human speech, music, or animal vocalizations—to its basic temporal structure, one can quantify its rhythmic properties, and draw meaningful parallels across fields [23,24]. One outstanding challenge lies in measuring a multidimensional temporal sequence in a way that captures the rhythmic properties of interest.

Different studies and fields collect drastically different data types, from time series of continuous sensor measurements to sequences of observed behaviors [33,34,35,36]. To apply the type of rhythmic analyses and measures presented here, the first step is identifying the events and intervals of interest to the research question under consideration [14]. This could consist, for example, of finding peaks or abrupt changes in a sensor time series or annotating the onset of different vocalizations in an audio recording. The resulting sequences of events should capture the temporal properties of interest in the original dataset and be stripped of other aspects deemed irrelevant for the current analysis. Existing general rhythmic measures and analyses can then be applied to the resulting temporal sequences. While this approach should by no means exclude other, more specialized analyses of the original data, this transformation into simple temporal sequences allows for the same rhythmic analyses to be applied across a broad range of scientific fields and enables cross-disciplinary comparisons of the results [14].

Below, we provide a short overview of the typical approach to measure the rhythmic properties of such sequences. Note that this is not meant as a complete, in-depth methodological framework for rhythmic analysis; rather, we introduce two commonly used rhythmic measures, and we focus on the relationship between them.

1.3. Measuring Rhythm

Starting from a sequence of discrete events in time, most approaches for rhythmic analysis rely on the intervals between subsequent events (see Figure 1A). If the events have a duration, typically inter-onset intervals (IOIs; i.e., the interval between their onsets) are considered. Working with the duration of inter-event or inter-onset intervals is intuitive, as it highlights temporal patterns rather than the absolute time at which they occur. Additionally, a typical assumption borrowed from music perception studies is that the events’ duration is less relevant for temporal structure than the intervals between their onsets [37,38,39]. Based on the interval durations, other rhythmic measures (such as rhythm ratios, r_k; see Figure 1B and below) can be calculated to quantify the intervals’ relationship.

Investigating the distribution of interval durations is often the first step of any rhythm analysis. As illustrated in Figure 1A,C, such a visualization can already reveal initial differences between different types of rhythm and shows whether interval durations are clustered around certain values or continuously distributed. The interval distribution can reveal how different durations are used throughout a sequence, yet it cannot capture any sequential patterns as it neglects the intervals’ order. For example, summary measures of intervals—e.g., their mean, representing the average tempo—reveal little about the detailed rhythmic structure of the sequence. Random drumming or a clock ticking can share the exact same mean interval duration. Consequently, we need measures that capture the relationship between different interval durations and quantify the rhythmic structure in more detail.

Among the various existing rhythmic measures, we focus here on two in particular: the normalized pairwise variability index (nPVI; [40]) and rhythm ratios (r_k; [41]; see Figure 1B). As these two measures are generically applicable to any temporal sequence or dataset with multiple sequences, they are both relevant in a cross-disciplinary context. Additionally, as we show below, these two measurements are fundamentally connected, providing deeper insight into the rhythmic properties they measure.

1.4. The Normalized Pairwise Variability Index

As a rhythmic measure, the normalized pairwise variability index (nPVI) focuses on the relationship between pairs of adjacent intervals. More specifically, given the $n-1$ pairs of adjacent interval durations ($i_k$ and $i_{k+1}$) in a sequence, the nPVI is defined as

$$nPVI={\displaystyle \frac{200}{n-1}}\sum _{k=1}^{n-1}\left|{\displaystyle \frac{i_k-i_{k-1}}{i_k+i_{k+1}}}\right|$$

(1)

Broadly speaking, the nPVI value of a sequence measures the level of isochrony, irrespective of tempo, i.e., how close the sequence is to having perfectly equally spaced—or isochronous—events (see Section 2.1 for a detailed explanation of the formula). The smaller the nPVI, the more isochronous a sequence is (e.g., [25]).

The nPVI was initially introduced in linguistics to obtain a more reliable comparison of rhythmic patterns across languages. Grabe and Low [40] proposed the index after finding that, when used to measure variability in durations of adjacent vowels, it strongly correlates to the classic classification of spoken languages into stress-, syllable-, and mora-timed [42,43]. The nPVI is however not specific to linguistic analysis. In fact, by replacing the vowel durations with the duration of intervals between general events, nPVI becomes a rhythmic measure for a broad range of acoustic signals.

In the field of music cognition, this approach has uncovered significant parallels between linguistic and musical rhythm [44]. An analysis of nPVI values in composers’ native languages and their musical works showed that rhythmic variability in music closely mirrors that in language [40], establishing a link between linguistic and musical rhythm [45,46]. Later, this measure was fully adopted to quantify rhythmic changes in musical data in relation to historical trends [47,48], national influences [45,49,50], and compositional styles [51,52].

Similarly, the nPVI has also been applied to analyze complex acoustic signals in animals. It has proven a particularly useful measure for distinguishing more isochronous patterns—marked by low variability between IOIs—from more complex or random ones [23,25]. For example, studies using the nPVI revealed different levels of isochrony in the sounds produced by different fish species [26]. Similarly, the measure was used to quantify isochronous flipper slapping patterns in harbor seals [53].

Despite its utility as a simple, cross-disciplinary, rhythmic measure, the nPVI also has limitations. The resulting value is an average over one or multiple temporal sequences and can oversimplify the complexities of rhythmic patterns and lose potentially crucial within-sequence variation. This is especially important when interval durations follow several distinct categories, e.g., in the context of music. In such cases, the information measured by nPVI can often be summarized using simpler methods, such as counting the proportion of identical successive intervals [54]. However, such methods make more assumptions and are thus less likely to be applicable across contexts and disciplines. All in all, these caveats are part of a necessary tradeoff when capturing the temporal regularity of a sequence in a single and generic metric [23].

1.5. Rhythm Ratios

A second, more recently proposed measure of rhythmic structure is the rhythm ratio (r_k; [41]). Similarly to the nPVI, it is calculated based on two adjacent intervals $i_k$ and $i_{k+1}$:

$$r_k={\displaystyle \frac{i_k}{i_k+i_{k+1}}}$$

(2)

Contrary to the nPVI, each $r_k$ represents the relationship between two intervals (see Section 2.1) and does not summarize a temporal sequence as a single value. Instead, a temporal sequence will result in a sequence of r_k values, and subsequent analyses typically focus on the observed distribution of these values. Figure 1B,C presents an example of the r_k values and their associated distributions resulting from four different types of temporal sequences. An r_k value of ½ (i.e., 0.5) indicates two successive intervals of equal length (i.e., isochrony); smaller values result from a longer interval following a shorter one, and vice versa. The r_k rhythm ratio is particularly useful for certain questions in bioacoustics and music cognition research, where the precise durations of intervals are often less important than the rhythmic patterns that emerge from them. It has also proven useful for characterizing rhythmic behaviors in related disciplines (e.g., language pathology; [55]).

The r_k ratio was proposed by Roeske et al. [41] to explore the presence of rhythmic categories in human music and birdsong. They tested the resulting r_k distributions for peaks around small-integer ratios; e.g., an r_k value of 0.33 corresponds to a pair of intervals where the second one has double the duration of the first (i.e., with intervals following a 1:2 ratio). A significant peak in the distribution around this r_k value of 0.33 suggests the presence of two rhythmic categories according to this 1:2 ratio.

The normalization in the r_k formula ensures that the measure is tempo-independent, allowing researchers to identify rhythmic categories within temporal sequences and acoustic signals. These categories may correspond to fundamental units that structure speech and music, such as syllables or hierarchical rhythmic units. Specifically, most music genres show interval relationships coinciding with small-integer ratios [56,57,58,59]. Because of this, after its introduction by Roeske et al. [41], the r_k measure has been adopted in a broadly cross-disciplinary range of studies. It was applied to test for the presence of small-integer ratios in singing primates [60,61,62,63], non-human apes [64,65,66], and other taxa [53,67,68]. Overall, this method continues to be applied to a large range of animal behaviors, from songs and calls to percussive signals [69] and locomotion patterns [70,71].

2. Results and Discussion

2.1. The Relationship Between the nPVI and Rhythm Ratios

The nPVI and the distribution of r_k values originate from different research contexts and are typically applied to complementary research questions. As we outlined above, the nPVI (Equation (1)) typically serves as a simple measure of isochrony, whereas r_k values (Equation (2)) are most often applied in search of rhythmic categories. However, looking at the two formulas, we can actually notice some similarities. This is not an accident: here, we mathematically show that the nPVI is a summary statistic of the r_k rhythm ratio distribution.

To better understand this, let us revisit the two formulas. Formally, consider a temporal sequence of $n+1$ events (or event onsets), at times $t_1,t_2,\dots ,t_{n+1}$. Logically, there are $n$ intervals between these onsets, $i_1,i_2,\dots ,i_n$ with $i_k=t_{k+1}-t_k$.

The nPVI [40] considers the relative difference between two adjacent intervals $i_k$ and $i_{k+1}$ in a temporal sequence: the absolute difference between durations $\left|i_k-i_{k+1}\right|$ is divided by their average duration $(i_k+i_{k+1})/2$. The nPVI then equals the average over all $n-1$ adjacent intervals pairs, multiplied by 100:

$$nPVI={\displaystyle \frac{100}{n-1}}\sum _{k=1}^{n-1}{\displaystyle \frac{\left|i_k-i_{k+1}\right|}{\left(i_k+i_{k+1}\right)/2}}={\displaystyle \frac{200}{n-1}}\sum _{k=1}^{n-1}\left|{\displaystyle \frac{i_k-i_{k+1}}{i_k+i_{k+1}}}\right|$$

(3)

As described earlier, the lower an nPVI measure, the higher the degree of isochrony: in the case of perfect isochrony, all intervals $i_k$ are equal, rendering the whole equation equal to zero. For any other sequence, nPVI can never equal 200 or higher. Values close to 200 are reached by alternating intervals that differ immensely in duration (e.g., if $i_k\gg i_{k+1}$, then ${\displaystyle \frac{i_k-i_{k+1}}{i_k+i_{k+1}}}\approx {\displaystyle \frac{i_k}{i_k}}=1$); when this holds for all adjacent interval pairs in a sequence, nPVI will be almost 200.

The same sequence of $n$ intervals results in $n-1$ r_k values, as given by Equation (2) [41]:

$$r_k={\displaystyle \frac{i_k}{i_k+i_{k+1}}}$$

Any r_k value lies between 0 and 1 [41,72]. Two intervals with the same duration $i_k=i_{k+1}$ will result in $r_k={\displaystyle \frac{1}{2}}$, and so, a perfectly isochronous sequence will have a distribution of r_k values concentrated at 0.5. Moreover, just like the nPVI, r_k values are normalized and independent of the overall tempo and focus on the relative relationship between adjacent intervals.

It is no coincidence that the central part of the nPVI formula, ${\displaystyle \frac{i_k-i_{k+1}}{i_k+i_{k+1}}}$, looks similar to the rhythm ratio formula ${\displaystyle \frac{i_k}{i_k+i_{k+1}}}$. With a few algebraic manipulations, we can rewrite ${\displaystyle \frac{i_k-i_{k+1}}{i_k+i_{k+1}}}$ as $2\left({\displaystyle \frac{i_k}{i_k+i_{k+1}}}-{\displaystyle \frac{1}{2}}\right)$ and find a straightforward mathematical connection between these two measures:

$$nPVI={\displaystyle \frac{400}{n-1}}\sum _k\left|r_k-{\displaystyle \frac{1}{2}}\right|$$

(4)

In other words, we find that the nPVI is in fact a summary statistic of a sequence’s r_k values. More precisely, the nPVI quantifies in a single value how much, on average, the measured r_k values in a sequence diverge from 0.5 (i.e., isochrony).

Consequently, any distribution of empirical or theoretical r_k values has exactly one corresponding nPVI value. On the contrary, based on a specific nPVI value, it is not possible to determine the exact r_k distribution of the original sequence. In other words, the distribution of rhythm ratios cannot be directly reconstructed from the nPVI value. Figure 2 illustrates how different r_k distributions can result in the same nPVI value, as well as how the nPVI captures the r_k distribution’s spread around 0.5. Also note how both the nPVI and as well as the distribution of r_k values disregard the internal order of the ratios within a sequence.

2.2. Interpretation of the nPVI and Rhythm Ratios

The relationship between the nPVI and r_k values provides insights into both rhythmic measures. Most importantly, our results show that the distribution of r_k values is a more comprehensive measure of rhythmicity than the single nPVI summarizing that distribution. While this might not come as a surprise—a distribution of values potentially contains much more information than a single value—our results highlight that the two measures represent and process rhythmic sequences the same way, since they normalize the data in the same way (i.e., dividing by total duration of two adjacent intervals).

Recent work has investigated the mathematical assumptions underlying the r_k formula, showing that r_k values are uniformly distributed between 0 and 1 for random sequences generated by the highly random Poisson point process [72]. Many analyses of r_k values have implicitly assumed this as the baseline against which to compare rhythmic patterns, even though this process might be considered too random in some cases. This finding can now be translated into new insights about the nPVI. A uniform r_k distribution between 0 and 1 can be shown to have an expected nPVI of 100. Consequently, the expected nPVI value of a sequence generated by a Poisson process is 100. This means that any nPVI value lower than 100 is closer to isochrony than one generated by a Poisson process. Similarly, higher nPVI values (i.e., between 100 and 200) correspond to sequences that are less isochronous than random, i.e., sequences that alternate more than random between short and long intervals. However, note that many different r_k distributions can correspond to the same nPVI. Consequently, an nPVI close to 100 does not necessarily indicate a random temporal sequence. It is equally possible that it results from a highly rhythmic sequence with an intermediate to high amount of alternation (e.g., consisting only of 1:3 and 3:1 integer ratios).

Overall, the r_k distribution has the potential to provide a more complete view of the rhythmic structure of a sequence. Conversely, the found mathematical link between the two measures shows that the r_k values, while being the more novel approach of the two, do not capture a wholly different rhythmic property. The more recent r_k results can be interpreted in a larger context of studies stretching back to almost 20 years before the r_k approach by Roeske et al. [41]. This unification of the two measures’ interpretation has the potential of making them even more useful for cross-disciplinary research.

2.3. Alternative and Complementary Summary Statistics of r_k

Interpreting the nPVI as a summary statistic also opens the door for alternative measures based on a sequence’s r_k distribution. The nPVI is suited for capturing certain rhythmic properties but not others. For example, Figure 2 shows that there is not a perfect correlation between randomness and the nPVI. Complementing the nPVI through the use of different summary statistics of the r_k distribution can be more useful for describing rhythmic properties in a more complete manner.

For r_k values, we can use existing summary measures for characterizing distributions. For example, the standard deviation of r_k values can be calculated to capture the spread of the r_k values similarly to the nPVI (i.e., $nPVI={\displaystyle \frac{400}{n-1}}\sum |r_k-{\displaystyle \frac{1}{2}}|$ vs. ${\sigma }_r=\sqrt{{\displaystyle \frac{1}{n-1}}\sum {\left(r_k-{\mu }_r\right)}^2}$). Note that ${\sigma }_r$ measures a subtly different property compared to the nPVI: it calculates the spread around the mean r_k value, rather than around an r_k of 0.5 (i.e., isochrony), as is the case for the nPVI. Analogously, the mean or median r_k can give an indication of the “average acceleration or deceleration” within a sequence. These common summary statistics have the advantage that their properties and interpretation are thoroughly understood, enhancing their potential for interdisciplinary use.

An altogether different summary of a probability distribution is the entropy. Entropy measures how concentrated or how spread out the r_k values are. The main difference to the nPVI or standard deviation is that it does not measure the spread in comparison to a single point (i.e., 0.5 or the mean r_k); instead, it characterizes the inherent concentration of values within a distribution. Such a measure can, for example, be used to quantify how strongly the r_k distribution displays distinct categories. In practice, the entropy can be calculated either by calculating the Shannon entropy of the discretized distribution or by estimating a continuous version of entropy (such as differential entropy [73]).

There are plenty of other ways in which the r_k distribution could be summarized into one or a couple of values. The insight gained in this study allows for finetuning future analyses to more precisely measure the relevant rhythmic properties in temporal data. However, it is essential to remember that alternative rhythmic measures are not meant to replace but to complement the nPVI. Reporting the nPVI—and where possible, the full r_k distribution—is still an essential practice to support cross-disciplinary comparisons between studies.

2.4. nPVI and r_k Are Inherently Local Measures

Importantly, the nPVI and any summary measure based on r_k values are local quantifications of rhythmic structure. These measures only compare adjacent intervals and therefore do not capture any higher-order patterns that cannot be deduced from successive intervals. This is often a desirable tradeoff: r_k values and the nPVI preserve some information about the order in which the intervals appear, which is not the case for direct summary measures of interval durations. Additionally, the denominator in both formulas (i.e., $i_k+i_{k+1}$) normalizes for tempo and removes tempo drift (Figure 1). Independence of tempo is often a desirable property of this approach.

Nevertheless, the approach is a tradeoff, and r_k and nPVI are not the only rhythm analyses available. For example, the coefficient of variation (CV) is a measure of dispersion calculated by normalizing the standard deviation by the mean. When applied to the distribution of interval durations (i.e., i_k), the CV captures how close to isochrony a sequence is, independent of overall tempo. It thus allows for a direct comparison of interval duration variability across different datasets. The main difference between the nPVI and CV is that the latter is globally normalized by the average tempo of a sequence or dataset rather than locally, per pair of adjacent intervals. In contrast to the nPVI, tempo drift across a sequence will affect the CV of interval durations. Figure 3 demonstrates the differences between the nPVI and CV by calculating them for the four different example sequences of Figure 1.

Another related rhythmic measure is beat precision (previously known as universal goodness of fit, or ugof; [24]). Beat precision measures how close each event is to a hypothetical underlying isochronous grid and normalizes by the constant interval duration of the hypothesized grid. Only the distance between events and the grid are measured: a non-isochronous sequence consisting of multiples of the grid’s interval duration is considered beat-precise. While this is a potentially important advantage over the nPVI and CV, beat precision comes with its own tradeoffs and is, for example, impacted more by tempo drift or minor rhythmic irregularities. Similarly, the Fourier transform is a well-known mathematical technique that, in the context of rhythmic analysis, can serve to quantify the global presence of different frequencies in a continuous signal [24,25,26,74].

Different rhythmic measures quantify subtly different rhythmic properties. A thorough understanding of these subtleties is required when choosing which measures to use in which contexts. When in doubt, multiple measures should be used such that they complement each other. As an example, if the nPVI, CV, and beat precision all result in low values, one can be confident about the amount of isochrony in a sequence. If one of the measures indicates differently, this creates a potential for understanding which types of sequence would generate such results. In general, using different methods can provide a more comprehensive understanding of the data, ultimately leading to more robust and reliable conclusions.

3. Conclusions

Many approaches exist to quantify the rhythmic structure in a temporal sequence. Here, we have described a fundamental relationship between two commonly used measures to quantify rhythmic structure, concluding that the normalized pairwise variability index (nPVI) is a summary statistic of rhythm ratios (r_k).

This methodological result provides a more nuanced understanding of the two measures and unifies them. For example, insights into rhythm ratios (e.g., [72]) directly translate into intuition about the nPVI: random sequences generated by a Poisson point process result in a uniform distribution of r_k ratios, which has an nPVI value of 100. Additionally, regarding this methodological perspective, past studies using the nPVI can now be interpreted in the context of rhythm ratios, and vice versa (e.g., [61,75]).

Our goal in this study was not to infer general mechanisms governing rhythmic structures at the physical, biological, or cognitive levels. Rather, we established an explicit mathematical link between two commonly used quantitative tools. This, in turn, should be used to build a shared quantitative toolkit for rhythm research.

There exists no perfect universal rhythmic measure. Every measure is a tradeoff between succinctly summarizing a sequence to one or a few values on the one hand, and retaining as much temporal information as needed on the other. In general, the full distribution of r_k values will result in a more comprehensive understanding of a sequence’s rhythmic structure [14]. However, the insight that nPVI and r_k are conceptually related allows researchers to fine-tune future analyses to their precise research question. Summary statistics (e.g., the standard deviation) of r_k values complement the nPVI when describing rhythms and quantify subtly different rhythmic properties. The nPVI remains an appropriate choice for quantifying the amount of isochrony in a sequence and enables a cross-disciplinary comparison with previous and future results in the literature.