Beating is the sensation that typically occurs when two sounds with similar frequencies mutually interfere, giving rise to a waveform with a rhythmic oscillation in amplitude. Following the fundamental contribution of Helmholtz’s treatise, On the Sensation of Tone (1954), first published in 1863, contemporary explanations of consonance are grounded in the notions of beating and complex tones—i.e., sounds displaying a broad array of sinusoidal components (harmonics).

Roederer (2008, p. 35) provides an illuminating classification of the effects of superposing two pure tones depending on where in the listener’s auditory system the sounds become entangled. The above mentioned beating is labeled by Roederer as “first-order beating,” because it is processed mechanically in the cochlear fluid and along the basilar membrane. Evidence of the physiological basis of first-order beating stems from the fact that its effect disappears when sounds are played separately in different ears—i.e., dichotically. Another kind of first-order beating effect is known as combination tones, which are produced by the non-linear interaction of waves in narrow spaces, such as the body of musical instruments or the inner ear. Combination tones can be considered as the product of two sine waves. A common example is the terzo suono theorized by Giuseppe Tartini (see Lohri, 2016). If a and b are two frequencies with a > b, then the terzo suono is a tone at frequency a–b that is discernible only by the listener, because it is produced inside the inner ear rather than being caused by external air vibrations. Combination tones can be heard across the octave at sound pressure levels (SPLs) of 80 dB or higher, and across part of the octave at 50 dB SPL and above.

At 80 dB (or higher) while maintaining the interval around the octave, a distinct beating can be perceived. This disappears when f₂ = 2f₁ (where f₁ and f₂ represent the two frequencies) and reappears as long as the octave becomes mistuned by a factor 𝜀 (i.e., f₂ = 2f₁ + 𝜀). The beating frequency turns out to be 𝜀 (Plomp and Levelt, 1965). Beating “is created by the relatively quick changes produced by modulation frequencies in the region between about 15 to 300 Hz” (Fastl and Zwicker, 2006, p. 257). Unlike first-order beats, the beating persists when tones are fed dichotically, implying that, in this case, beat perception is the result of neural processing. Hence, they are defined as “second-order beats” (Roederer, 2008). Second-order beating shows a modulation in the vibration pattern, i.e., a periodic change in phase difference between the two sounds that form the interval (Roederer, 2008, p. 49), although no amplitude modulation is present. Second-order beats are also called “beats of mistuned consonances” because they are audible when pure tones are superposed to form a fifth (Plomp, 1976). In fact, whereas the vibration pattern of a correctly tuned fifth (f₂ = 3/2 f₁) or fourth (f₂ = 4/3 f₁) is static, the mistuned cases f₂ = 3/2 f₁ + 𝜀 and f₂ = 4/3 f₁ + 𝜀 cause the vibration pattern to change periodically in form, but not in amplitude. From the octave to the fifth and to the fourth, the second-order beats become faster (beating frequency being 𝜀 for the octave, 2𝜀 for the fifth, and 3𝜀 for the fourth) as the vibration pattern grows in complexity (see Figure 1).

Their neural origin makes second-order beats an excellent phenomenon for investigating the link between the mathematical description of the signals and their neural processing, and consequently allows us to shed light on their perceived “pleasantness.” To achieve a consistent picture of second-order beats, it is fundamental to overcome the frequency–time space representation trade-off and the related problem of non-stationary signal characteristics.

Graphic representations of sound typically plot the course of amplitude over time or report the relative amplitudes of the different frequencies computed by the Fourier Transform. Thus, there is no mention of time in the latter, and no mention of frequency in the former. However, in the actual hearing process, time and frequency are strictly intermingled, because specific frequencies are processed at specific moments. This fact suggests that we should focus on the simultaneous analysis of time/frequency dimensions (Roads, 2001).

To determine the frequency of an oscillatory phenomenon, we must count the number n of vibrations that occur within a set time interval Λt. As n is an integer, the minimum error in measuring the frequency is one, thus generating a kind of uncertainty principle in the form Λf ≥ 1/Λt. Increasing the precision of the frequency reclaims a wider window in which to count the time, thus increasing the indetermination of the instant in which the specific frequency occurs.

It is possible to neglect the explicit consideration of time and visualize tone relationships within the octave by computing the ratio of two simultaneous frequencies and then plotting the interval ratio against the amplitude. This is achieved by forming a linear combination of two pure tone waves, a glissando from the unison (f₁) to the octave (2f₁) and a firm wave at frequency f₁. Similar stimuli were previously adopted by Helmholtz (1954) and Kameoka and Kuriyagawa (1969a,b). More recently, Piana (2007) provided a phenomenological explanation of consonance and dissonance when moving from the glissando and ruling out intervals and harmonics. In the following, we propose a numerical approach to Helmholtz’s glissando. Note that this approach maintains the time dimension in terms of the determined sequence of interactions between glissando and the fixed frequency. Focusing on these interactions allows us to overcome the trade-off in the frequency–time representation. For this purpose, we base our approach on the concept of recurrence, a simpler and more fundamental property of the signals with respect to the oscillation frequency (Eckmann et al., 1987; Marwan et al., 2007). The degree of recurrence of a series is estimated by the number of times a signal comes back to an already visited state (see section “Materials and Methods”), and can be computed by the application of recurrence quantification analysis (RQA) (Marwan et al., 2007). Estimating the recurrence rate avoids any stationarity assumption, as the estimate is obtained by a “computation window” sliding along the signal; the result is a profile of recurrence values relative to subsequent epochs along the original signal. This provides a model-free, discrete, and local estimation of the recurrent properties of the series, enabling a quantitative description of second-order beats. The recurrence peaks exactly match the values of the interval ratios corresponding to just intonation and are proportional to the order of consonance of the intervals, thus providing a link between consonance and the dynamical features of the signal.

A non-stationary signal exploring all interval combinations within the octave can be generated by merging the course of two sounds into a single waveform. The first sound is set at constant frequency f₁ for the full duration of the course, while the second follows an ascending glissando from f₁ to f₂ = 2f₁. Figure 3 shows an instance of the above procedure.

The most conspicuous singularity (recurrence peaks, see below) in the graph occurs when lines cross themselves, i.e., when f₂ = f₁ (unison, interval ratio of 1:1). A second relevant case occurs at the interval ratio of 2:1, which corresponds to the octave. Less evident events occur at 3:2 (fifth) and 4:3 (fourth), as can be seen in the zoomed inset in Figure 3. Singularities in the waveform are thus localized where the frequency ratios are expressed by lower integers and with an apparent amplitude (or degree of singularity) matching the accepted ranking of consonance. In our representation, second-order beats appear as a zone of relative calm centered in rational numbers, surrounded by the tempestuous region of irrationals that Roederer (2008) called “beat holes.”

Following the numerical solution of Helmholtz’s glissando, we explore the glissando/constant frequency signal through an RQA windowing procedure called Recurrence Quantification of Epochs (RQE). RQE performs a scansion of the whole signal by sequentially selecting small windows—specifically episodes of 480 points—in which the RQA algorithm (with the consequent computation of recurrence rate for each episode) is applied. The subsequent windows are shifted by 48 points and the process is repeated throughout the entire file. For each iteration, we retain both the recurrence value and the interval ratio in which this value occurs, calculated as the mean of the interval ratios in the window. Figure 4 represents the degree of recurrence along the continuum of interval ratios within the octave.

Emergent features of the glissando are evident in Figure 4. Firstly, the higher peaks exactly correspond to the places of just intonation (see Trulla et al., 2005), thus establishing a link between pleasantness and the dynamical features (i.e., recurrence) of the signal. As expected from the numerical model, all peaks correspond to rational numbers. Secondly, it is worth noting the symmetry of the peaks around the perfect fifth. Moreover, the correlation between the extent of recurrence and the rank order of consonance derived from the literature evidences the link between the present model based on signal analysis and results from psychological approaches (see Schwartz et al., 2003, i.e., U > P8 > P5 > P4 > M6 > M3 > m3 > m6 > m7 > M7, in decreasing order of consonance; see Table 1).

In summary, RQA allows us to establish a natural link between the signal properties and the consonance judgment of the listeners without any a priori hypothesis or frequency estimation. The reasons why integer numbers play such an important role in harmony has recently been addressed in the literature, with many different recipes presented for calculating the simplicity of the intervals. We use the consonance index provided by Frova (1999) to demonstrate the close relationship between the proposed recurrence index and the bare numerical characteristics of the intervals. If m/n is the rational number in its lowest terms, Frova’s index is (m+n)/(m × n) (Frova, 1999, p. 178). Figure 5 illustrates the correlation of this index with the notion of simplicity (i.e., degree of recurrence).

Whereas Frova’s index is derived from the energy of the partials forming a complex sound, the percentage recurrence is a purely bottom–up phenomenological descriptor of a pure tone signal, relating recurrence (and consonance) to secondary beating and thus providing a natural (albeit roughly phenomenological) link between the signal properties and neural processing.

Note that the computation of recurrences gives very similar results with respect to models based on primary beating, such as the Plomp and Levelt model reported in Figure 6.

In this paragraph, we relate the self-similar appearance of the recurrence graph in Figure 4 to the mathematical fractal structures generated by physical processes. Figure 7 shows the empirical cumulative recurrence distribution (obtained by adding consecutive points) and a formal Devil’s staircase in the [1, 2] interval: the similarities between the two graphs are remarkable.

The Devil’s staircase pattern is a fingerprint of dynamical systems characterized by the mode-locking phenomenon (Schroeder, 1990, p. 171), which is crucially important in both music generation and perception. In the 17^th century, Christian Huygens studied mode-locking and discovered the phenomenon of resonance. He noticed that, after a time, the pendulums of two clocks fixed on the same mounting swung synchronously. The synchronization of two coupled oscillators starting from (slightly) different frequencies is called resonance. A more general case of resonant behavior appears when a specific constant frequency is periodically driven by an external power to oscillate at a different frequency; the so-called Devil’s staircase pattern refers to the behavior of forced quasilinear oscillators. In the glissando, the constant frequency is the intrinsic frequency and the glissando the external periodic force. Every plateau in the Devil’s staircase relates to a particular phase-locked solution (stable state), and its relative width forms a hierarchy that follows the explained propriety of rational numbers. The mathematical model for this case is the circle sine map (McCauley, 1994).

where k is a coupling strength parameter that controls the degree of non-linearity. Without coupling (k = 0), the behavior of the system is expressed by the ratio p/q (often called Ω, the bare winding number). When k > 0, the system locks into rational frequency ratios, preferably with small denominators. In this case, the long-term description of the system corresponds to w, the dressed winding number. For the critical value k = 1, the infinite number of locked frequency intervals corresponding to all the rational numbers between 0 and 1 cover the entire Ω range.

In our terms, Ω is the cumulative recurrence and w is the interval ratio. In other words, the system is locked at any rational number—indicated as the interval ratio—but the width or extent of the lock comes from the cumulative recurrence. Thereby, most relevant consonances have extended areas around the lowest rationales—like the unison or octave—and a strong attraction exists toward these exact ratios. This is perfectly sound in terms of music theory.

The above considerations can be summarized in three main points:

Numerous studies have confirmed the adequacy of concepts from non-linear dynamics for music perception and construction (e.g., Cartwright et al., 2001, 2002, 2010), and for the study of synchronization among sound sources (Abel et al., 2009). Additionally, neuroscientific research has adopted non-linear dynamical models to describe phase-locked neural populations (Bidelman and Krishnan, 2009; Large and Almonte, 2012) and build in silico neuronal models (Lots and Stone, 2008).

Taken together, our work and previous results support the idea that the production and perception of sound are intimately linked, the perceived pleasantness of intervals being an intrinsic property of the signal (in terms of the degree of recurrence), and not only a secondary effect of the signal on the listener. In turn, this allows us to speculate on the auditory system. Second-order beats have been attributed to the central auditory nervous system, and neuronal webs are known to support phase-locking, as in the mammalian auditory system, in which neural activity in areas including the cochlear nucleus, inferior colliculus, and primary auditory cortex is phase-locked to the stimulus waveform (Large and Tretakis, 2005). The mode-locking model was proposed by Lots and Stone (2008) as the basis for musical consonance, leading to the development of a dynamical system model based on stylized neural oscillators producing both synchronization and mode-locking. These results support the idea that both parts of the communication system (the sender and the receiver of sounds) are similarly “wired.” Bidelman and Heinz (2011) applied a waveform to a computational model of the acoustic nerve and, after deriving the autocorrelation function for the nerve fibers, generated the pitch salience profile for the different intervals, giving rise to a distribution that could be superimposed onto the recurrence rate (Figure 4). Using an artificial neural network model, Pankovski and Pankovska (2017) recently demonstrated that a specific auditory spectral distribution caused by non-linearities and Hebbian neuroplasticity are sufficient phenomena for a system to generate the consonance pattern.

In line with the literature on music perception (Benade, 1973; Roederer, 2008), we believe that the link between music generation and perception could rely on the fact that the vibrating elements of musical instruments undergo mode-locking into stationary complex vibration patterns. In turn, these can be recognized as the “best fit” to a harmonic template (resident in a properly wired neural circuit). Though this explanation stems from empirical correlations, we are convinced that the simplicity and versatility of the RQA approach could pave the way for neuro-psychological studies with the great advantage of considering the acoustic signal and the perceiver from the same mathematical perspective.

The origins of the distinction between consonance and dissonance have been hotly debated in recent years. As the phenomenon of consonance represents a key element of Western music theory, this has mainly been investigated in terms of Western science (i.e., mathematics, physics, psychoacoustics, and neuroscience). For this reason, Parncutt and Hair (2011) called for studies on the use of consonance and dissonance in non-Western cultures to be conducted in terms of local indigenous musicians, rather than in terms of Western science. In this direction, a relevant study published in Nature by McDermott et al. (2016) compares the harmonic preferences of people who have different degrees of exposure to Western music. An indigenous population from Bolivia (the Tsimané) was assumed to have no exposure to Western music, and their preferences were compared with groups of city residents in Bolivia and the United States with different degrees of exposure to Western music. The results show that the subjective preferences of Tsimané participants differ from those of the comparison groups; in particular, they failed to rate consonance as being more pleasant than dissonance. The authors state that, as the Tsimané are able to hear the acoustic distinctions associated with consonance and dissonance, the lack of a measurable preference for consonance appears to reflect difference in their aesthetic response to this contrast (McDermott et al., 2016, p. 549). Correctly, they state that the observed cross-cultural variation suggests that consonance preferences are unlikely to be innate, and so preference is probably acquired. However, the fact that the preference for consonance co-varies with presumptive exposure to Western culture is not sufficient to conclude that consonance perception is not biologically determined. Though preferences vary with cultures, the discrimination of consonance is a prerequisite for preference and has a biological basis, as supported by a large number of neurobiological studies (Koelsch and Mulder, 2002; Koelsch et al., 2005; Minati et al., 2008; Perani et al., 2010; Park et al., 2011; Wang, 2013). Investigating whether consonance perception is biologically determined or shaped by culture is likely to be misleading, as it conceives enculturation as a non-biologically constrained process. Harmonic intervals are a consequence of the entrainment of the nervous system with the sound excitation. This forms a universal biological foundation under any musical culture, determining the distinction between acoustic consonance and dissonance and leaving it to each culture to determine exactly how to employ these acoustic distinctions. However, the existence of different musical cultures and systems does not imply the lack of a shared natural/biological basis for music production. The interaction between nature and culture is much more complex, and cross-cultural variations in musical systems only show that biology does not rigidly determine music aesthetics. Similar considerations have led to a more adequate definition of music as a “biocultural phenomenon” (Cross, 2003).

The main contribution of this paper stems from the numerical solution of Helmholtz’s glissando. Though the standard modern theory of consonance is based on first-order beating, we have shown that similar results can be obtained starting from second-order beats. The recent interest in second-order beating has been fruitful for models of pitch recognition or neural circuitry (see Roederer, 2008), but not for theories on consonance.

Scholars have started to consider music from the perspective of dynamical systems, both in neurobiological and physical terms, showing that mode-locking models can explain how the nervous system manages sound and is engaged in the ranking of consonances. The resemblance between the formal Devil’s staircase model and the cumulative recurrence distribution strengthens this idea.

From a methodological perspective, the main contribution of this work is to provide neuroscience scholars with an extremely simple and model-free tool (RQA) that approaches the acoustic signal and the listener’s perception system with the same mathematical method. Different RQA applications have been reported in research on otoacoustic emission (see, for example, Zimatore et al., 2002, 2003). We are therefore confident that the use of a simple statistical approach will foster interactions between music theory and neuro-psychological approaches.

Finally, our results support the idea of natural roots of consonance perception, and are thus in line with several studies published in recent years (see, for example, Wang, 2013; Bowling and Purves, 2015; Nikolsky, 2015; Foo et al., 2016; González-García et al., 2016; Di Stefano et al., 2017). However, as proved by McDermott et al. (2016), the role of perception in the formulation of aesthetic judgment remains unclear. Therefore, musical consonance and dissonance remains a hotly debated topic (see Bowling et al., 2017), in need of further research to merge different approaches into a consistent theory.

LT originally conceived the idea of the paper, elaborated the stimuli, provided all the figures, and significantly contributed to the results and discussion. NDS prepared the manuscript, co-authored the introduction and the results with LT, contributed to the discussion, wrote the conclusion, and finally revised the entire draft. AG wrote the section “Recurrence Quantification Analysis,” reviewed the entire manuscript, and suggested useful ideas for the discussion. All authors equally contributed to the revision of the manuscript before agreeing on the final version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.