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Article

Study of Non-Linearities in Humpback Whale Song Units

by
Yann Doh
1,2,
Dorian Cazau
3,
Giulia Lamaj
3,
Eduardo Mercado
4,
Joy S. Reidenberg
5,
Jeff K. Jacobsen
6,
Christina E. Perazio
4,
Beverley Ecalle
2 and
Olivier Adam
7,8,*
1
Terre-Mer-Veille, 97125 Bouillante, Guadeloupe, France
2
ABYSS NGO, 1 Rue du Quai Berthier, 97420 La Réunion, France
3
ENSTA, Lab-STICC, UMR 6285, Institut Polytechnique de Paris, 29200 Brest, France
4
Department of Psychology, University at Buffalo, The State University of New York, Buffalo, NY 14260, USA
5
Center of Anatomy and Functional Morphology, Mount Sinai School of Medicine, New York, NY 10029, USA
6
VE Enterprises, McKinleyville, CA 95719, USA
7
Sorbonne Université, CNRS, Institut Jean Le Rond d’Alembert, 5005 Paris, France
8
Université Paris-Saclay, CNRS, UMR 9197, Institut des Neurosciences Paris-Saclay, 91400 Saclay, France
*
Author to whom correspondence should be addressed.
J. Mar. Sci. Eng. 2025, 13(2), 215; https://doi.org/10.3390/jmse13020215
Submission received: 25 November 2024 / Revised: 31 December 2024 / Accepted: 10 January 2025 / Published: 23 January 2025
(This article belongs to the Special Issue Recent Advances in Marine Bioacoustics)
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Abstract

:
Unique in mammals, the vocal generator of mysticete species comprises membranes covering the two arytenoid cartilages that vibrate as the airflow passes through the trachea from the lungs to the laryngeal sac. By adjusting the airflow as well as the spacing and orientation of the two cartilages, mysticetes control the vibrations and vary acoustic qualities of the produced sounds, including the duration, amplitude, and frequency modulation of vocalizations. Humpback whales control sound production in this way to construct a complex vocal repertoire, including vocalizations with or without harmonics as well as pulsed sounds. Some vocalizations within humpback whale songs, called units, exhibit non-linearities such as frequency jumps and chaos. Here, we further describe non-linear features of units, including two additional non-linearities: subharmonics and biphonation. Subharmonics within units are probably due to higher air flow rates and to the acoustic modes of internal resonators. Biphonic vocalizations are likely generated either by an asymmetric opening of the arytenoid cartilages or by the passage of the air flow at two separate positions along the membranes. Our analyses revealed acoustic non-linearities in vocalizations emitted by six different singers during multiple breeding seasons and from populations in different oceans, suggesting that singing humpback whales often produce units with non-linear features.

1. Introduction

Adult male humpback whales (Megaptera novaeangliae) sing mainly during breeding seasons. The objective is possibly to attract females and/or for male–male interactions (identification and localization of the individuals) during sexual competition. Vocalizations, called a song unit (SU), are separated by silences and are emitted in a temporal order [1]. These SUs are organized in time into subphrases, and successive subphrases become phrases. A sequence of similar phrases becomes a theme, and a sequence of themes becomes a song that can last about 10 to 20 min (see review in [2]). These vocalizations were highly studied, and recently, it has been noticed that some of their emitted vocalizations present acoustic non-linearities (the features of the generated acoustic wave do not vary linearly with the air pressure variations applied to the vocal vibrator). Are they rare? What are they? Can it be explained anatomically? Answering these questions would allow a better understanding of the purpose of these specific vocalizations and whether they have a function in communication. For this study, we extracted humpback whale vocalizations exhibiting acoustic non-linearities and related these vocal characteristics to the anatomy of their vocal generator (Figure 1). As in other mammals, the respiratory system of humpback whales consists of paired nasal tracts, a common nasopharynx, larynx, trachea, and paired lungs. Mysticetes (baleen whales, including humpback whales) do not possess phonic lips (also called monkey lips or dorsal bursae) in the nasal region, as odontocetes do (toothed whales, including dolphins and porpoises) [3,4]. Instead, their acoustic vibrator has been identified in the larynx as the homolog of mammalian vocal folds (also called vocal “cords”), consisting of the membranes covering the arytenoid cartilages (also called the U-fold or thyroarytenoid fold) [5,6,7].
At rest, the voice generator (larynx) is totally sealed with the closure (apposition) of the two symmetrical arytenoid cartilages that press hermetically against each other. The arytenoid cartilages can be raised dorsally, where they contact a thick fat tissue pad called a cushion (or cricoid cushion), situated in the midline on the ventral surface of the cricoid cartilage [7,8]. Pressing the arytenoids against this cushion reinforces the seal (Figure 1). Then, when the whale wants to emit a vocalization, it will break the seal by opening the two cartilages and either lowering the arytenoid [6] or tensing the cushion with a muscle (musculus pulvini) that pulls it cranially while flattening it [7]. The pair of arytenoids regulate the gap that connects the trachea to the laryngeal sac (located on the ventral aspect of the larynx). The parted arytenoids allow airflow to circulate between the lungs and the laryngeal sac and, passing over the arytenoid cartilages, will cause the membranes that cover them to vibrate [9,10]. Positioned in parallel and connected to each other by a thick interarytenoid ligament, the paired arytenoids (and their membrane coverings) form a U-shape and thus are also referred to as the U-fold (or U-shaped fold) [5]. They have a symmetrical shape but they have non-uniform characteristics along their length, with an L shape at the midsection (where each arytenoid is fused with a corniculate cartilage) and a more rounded shape at the distal section (solely arytenoid cartilage) [6] (Figure 2). The membranes also have variable volume and stiffness (density): on the midsection, the membrane is smooth and thin on the dorsal and medial surfaces of the arytenoids, but on the distal section, it is thick and loose [6].
Theoretically, the production of harmonic vocalizations could result from the vibrations of the membranes covering the arytenoid cartilages due to the pressure of the airflow [5,9] and may also generate vibrations as air flows across the narrow gap between each arytenoid and the overlying cricoid cushion as the arytenoids are lowered or the cushion is tensed [7]. In the case of pulsed sounds (a single vocalization is made up of several successive pulsations), the arytenoid membranes are initially pressed against each other and the cushion, and when the air pressure is high enough, the membranes open to let air through. The air pressure then drops instantly and the membranes return to their initial closed position. The pulses are created from this rapid succession of openings and closings of the membranes.
It is possible to classify vocalizations by analyzing acoustic features extracted from narrow-band spectrograms, like the time duration, fundamental frequency, harmonics, or time-frequency shape (for ex., [11]). Since the songs are made with a large variety of vocalizations, acoustic analyses of the main differences between similar SUs emitted by different singers were performed. Some SUs exhibit two non-linearities, such as frequency jumps and chaos [12]. These features were common in songs, occurring in 35–40% of the analyzed vocalizations. It was also shown that chaos and frequency jumps usually occurred at different times within SU.
In this current article, we show that humpback whales emit two other types of vocal non-linearities: biphonation and subharmonic regime.
Note the following:
-
Frequency jumps: the fundamental frequency may change suddenly in the vocalization. This may happen more than once in the same vocalization.
-
Chaos: the chaotic mode is like an acoustic noise inside the vocalization.
-
Biphonation: vocalizations that simultaneously exhibit two fundamental frequencies.
-
Subharmonics: they occur at fractional intervals of the fundamental frequency.
Non-linearities were also found for animal vocalizations [13], including for other mammals [14], including North American wapiti [15], Asian elephants [16,17,18,19], chacma baboon [20], squirrel monkey [21], dog-wolf [22], dholes [23,24], and manatees [25]. For cetaceans, only eight species out of ninety were identified to emit sounds with non-linearities: short-finned pilot whales (biphonation described in [26]), bottlenose dolphins (biphonation described in [27]), killer whales (biphonation described in [28,29]), short-beaked common dolphin (subharmonics, frequency jumps, biphonation, and deterministic chaos in [30]), Bryde’s whales (biphonation in [31]), North Atlantic right whales (subharmonics, deterministic chaos, biphonation, and frequency jumps described in [28]), bowhead whales (biphonation described in [32]), and humpback whales (frequency jumps described in [33]); frequency jumps and chaos described in [12].
Even if vocal non-linearities can be the result of dysfunction (paralysis, injury), abnormality (atrophy), disease, fatigue, or a lack of control, several studies suggest that they could also be emitted intentionally to convey information. They can characterize an individual physical state, like, for example, in common chimpanzees [19]. They can provide information about the size or even the identification of individuals [34]. They can be indicators to show urgency, singularity, and virtuosity. They could show that the individual has great power, better health, or a specific skill. For humpback whales, three hypotheses have been put forward to explain these vocal non-linearities: an anatomical problem at the level of their vibrator, poor control of the vocal generator due to the whale’s age (insufficient learning in young individual or loss of sensitivity for old singers), and/or a communication strategy allowing them to transmit individual information and attract the attention of conspecifics [12].
In this paper, we suggest links between these acoustic characteristics and the functional anatomy of the humpback whale vocal generator to investigate the possible acoustic sources and mechanisms involved in the production of these specific sounds.

2. Materials and Methods

2.1. Collection of Underwater Acoustic Recordings

In order to collect high-quality humpback songs, with no or superimposed vocalizations from other whales in the background soundscapes, isolated humpback singers were targeted and recorded at around a 100 m distance to ensure a higher acoustic level of vocalizations emitted from this singer compared to the potential presence of other singers in a nearby geographic area estimated at less than 1 km.
The typical singer’s behavior is to surface quickly for breathing and a resumption of singing as soon as the diving began. The motorboat was placed at around a 100 m distance from where the animal was seen to breathe at the sea surface for the last time before diving. The engine was turned off during the acoustic recordings to avoid disturbing the cetaceans and adding noise to the acoustic recordings. The hydrophone was deployed at about a 10–15 m depth. Passive acoustics is a non-invasive method that does not influence the behavior and the vocalization patterns of whales. Recordings continued at least until the next surfacing in order to capture a complete theme of the song. To avoid truncating a song, the protocol we defined was to continue recording the targeted whale for at least 30 min more after re-surfacing from the initial dive. The proximity with the singer ensures a high signal-to-noise ratio (SNR), but sometimes, during the acoustic recordings, the boat slowly drifted due to a local water current and/or wind, inevitably moving away from the singer and slightly decreasing the SNR.
The hydrophones and the digital recorders were chosen because their technological characteristics (sensitivity, bandwidth) were suitable for recording the acoustic characteristics of humpback whale songs. Therefore, as an example, to record the singer off Samaná Bay (Dominican Republic), we chose the hydrophone Aquarian h2d with a sensitivity of 172 dB re: 1 V/µPa (±4 dB for 20 Hz–4 kHz), corresponding to the acoustic features of humpback whale songs [35]. The hydrophone was connected to an F3 Zoom recorder. The samples were digitalized at a 96 kHz sample rate and were coded in a 32-bit float format. The complete sequence lasts more than 48 min to be sure that the recordings include at least two cycles of the song.

2.2. Acoustic Dataset

For this study, acoustic recordings of isolated humpback singers with a high signal-to-noise ratio were selected from datasets collected during the breeding seasons in different oceans:
-
In the Caribbean Sea, Dominican Republic (DR), off Samaná Bay, 2024. The recording time was 48 min 15 s.
-
In the North Pacific. Mexican Pacific, off Isla de Socorro (IS), 2001. The recording lasted 40 min 34 s.
-
In the Indian Ocean, Madagascar, off the Sainte Marie Channel, 2022. The 4 recordings M1, M2, M3, and M4, respectively, lasted 30 min 11 s, 28 min 28 s, 28 min 50 s, and 35 min 2 s.
The material used and other characteristics of the dataset are summarized in Table 1.

2.3. Data Analysis

The same method was applied to all the selected acoustic recordings in order to detect the SUs, to annotate them, and to extract the acoustic features. The long-term spectrogram (LTS) was used to highlight the complete structure of the recorded sound production.
An example of this time-frequency representation for the recording off the Dominican Republic is given in Figure 3. The LTS with a sliding time window of 42 ms was computed, with a 75% overlap and Hamming window. Unlike what is classically performed for LTSs, representing the averaged spectra for each frame, here, only the most energetic frequencies in five frequency bands are displayed: low frequencies (LF): 190–500 Hz, medium frequencies 1 (MF1): 500–800 Hz, medium frequencies 2 (MF2): 800–1050 Hz, high frequencies 1 (HF1): 1050–2500 Hz, and high frequencies 2 (HF2): 2500–20,000 Hz. In Figure 3a, the repetitive pattern indicates two song cycles of the respective durations of 852 s and 890 s.
Because of the proximity with the targeted singer, the signal-to-noise ratio was positive: higher than 20 dB. The targeted singer’s sound production was therefore clearly distinguishable from the background underwater noises and the vocalizations emitted by far away singers. This vocal production was investigated, labeled, and classified aurally using headphones and visually by inspecting the spectrograms. Sequences of repetitive SUs were observed, determined, and categorized by comparing them to the vocalizations already identified and classified into catalogs, as available in the scientific literature [11,36,37,38,39,40,41,42,43]. When the SUs were not previously cataloged, new names were given based on onomatopoeia or on frequency variations to provide an easier mental representation of the acoustic characteristics of these sounds. The sequences can be made up of the same SU or of a combination of different SUs. The LTS of the complete annotated first cycle is represented in Figure 3b. The detection and identification of vocal non-linearities were performed manually by scanning the spectrograms of the vocalizations (Figure 4).

3. Results

For each analyzed song, we extracted the SUs, and then we focused on them with the following non-linear phenomena: frequency jumps, subharmonics, biphonation, and deterministic chaos (Table 1). All these non-linearities were found in the analyzed dataset. The proportion of observed non-linearities among the annotated SUs is represented in the bar graph in Figure 5. Chaos is found in more than 60% of the SUs. This feature is the most frequent for all the singers and all the sites. Biphonation appears to be the least representative, with less than 20% for all cases. The proportion of the subharmonic regime and frequency jump seems to vary more from one individual to another.

Examples of Non-Linear Acoustic Features in SU

Some SUs can simultaneously exhibit one or more non-linear acoustic features. Biphonation was detected in SUs categorized as “upsweep creaking” (Figure 6a) and “downsweep creaking” (Figure 6a). The SU in Figure 6 has a fundamental frequency (F0) around 950 Hz and seven harmonics. In the first 0.4 s, the SU has a modulated frequency (3500–3000 Hz) component G0 harmonically independent of F0. Vibratory coupling is observed with harmonics of F0 centered around G0. Later in the SU, a constant component in H0 is produced at 1200 Hz independently of F0 and without this coupling. Biphonation was regularly found in phrases containing these SUs.
Phrase repetitions provide an opportunity to analyze the variability (intra-individual) of non-linear features. In Figure 6b, three successive cycles of “downsweep creaking” production have been concatenated. The red dot line marks the fundamental frequency when starting the SU. The value of F0 is the most constant characteristic and present in all the SU productions. On nine SUs, the mean F0 is 964 Hz (std = 36 Hz). On the contrary, G0 max peaks represented by white dot circles can be modulated in Song 1, from 2980 to 3859 Hz. G0 can also disappear completely as in Song 2’s last vocalization. H0 max peaks, marked as the gray dot line, are only seen in Song 1. For the three occurrences, values of the H0 max peak are, respectively, 1172, 1129, and 1113 Hz.
Upsweep creaking (Figure 7a) is a 1 s duration SU presenting a modulated (125–5000 Hz) fundamental frequency F0 and its harmonics. This is the first SU of the song cycle. The SU shows a biphonation G0 (0.4 s duration) with a constant component around 1640 Hz and with no coupling with F0. The fundamental frequency is abruptly modulated by frequency jumps in a “stairway” structure (giving the creaking door sound). There are successively three frequency jumps with respective frequency variations of 2420, 600, and 1220 Hz. There is a harmonic relationship between frequency jumps (4ΔF, ΔF, 2ΔF). This SU shows a third non-linear feature: 0.4 s of chaos on 0–3000 Hz bandwidth.
Long-short upsweep creaking is the sixth SU of the song cycle. The long upsweep creaking exhibited a combination of three non-linear acoustic features: subharmonics, deterministic chaos in the LF bandwidth, and multiple frequency jumps (Figure 7b). As in the previous one, this SU also presents a discontinuing upsweep modulation of the fundamental frequency, but for a shorter frequency range between 500 and 675 Hz. The SU shows three similar frequency jumps of 50 Hz. This SU has first a 1.5 s subharmonics part (F0/4) starting from the fifth harmonic until almost 7 kHz, then followed by a second 0.6 s subharmonics part (F1/2) between 2.9 kHz and 5.1 kHz. This SU has multiple frequency jumps, all in the order of 50 Hz. At the middle of the SU, chaos is observed during 2 s between 0 and 2000 Hz.
Within a song, the evolution of F0, G0, frequency variation for frequency jumps, and chaos bandwidth for the first cycle of upsweep creaking are given in Figure 8. Like previously observed, over 13 consecutive SUs, the fundamental frequency is a stable characteristic with a minimum, on average, at 98 Hz (std = 11 Hz) and a maximum, on average, at 349 Hz (std = 27 Hz). G0 has more variations over time with a mean frequency at 1579 Hz (std = 93 Hz). While ΔF2 and ΔF3 seem to evolve jointly, the first frequency jump ΔF1 seems more correlated to the maximum frequency band for chaos. Regarding the harmonic relationship between frequency jumps, for this sequence, the average ratio and std for ΔF2 and ΔF3 as a function of ΔF1 are, respectively, (4.1 ± 1.8) ΔF1 and (1.67/−0.70) ΔF1. The little number of SUs does not allow more complex analyses, such as ANOVA or other statistical tests.
The HF upsweep creaking has a high fundamental frequency around 1.3 kHz and also combines two non-linearities: over this 1.6 s SU, there is a 160 Hz frequency jump followed by a subharmonics part for 0.3 s in F1/8 between 1.3 and 5.5 kHz (Figure 9a). The downsweep creaking II (Figure 9b) associates a 160 Hz jump in frequency in the first part with a biphonation G0 (at 3200 Hz), coupled with F0 at 600 Hz in the last part of the SU.

4. Discussion

Mysticete whales have a unique respiratory system that they use to breathe, vocalize, and air recycle (enabling continued sound production during apnea) [5] and to adjust buoyancy [44], similarly to Odobenus [45]. The humpback whale vocal source (U-fold) is made up of two cartilages covered with membranes that are homologous to the vocal folds of other mammals. The airflow between the lungs and the laryngeal sac can cause the membranes covering the arytenoid cartilages to vibrate and produce sounds [7,9,46].
Unlike the vocal folds of terrestrial mammals, the mysticete U-fold’s membrane is attached to the arytenoid cartilages, limiting them to a fixed length. As the ligament is restricted to the bottom of the U-fold only, it cannot lengthen or contract along the length of the U-fold in a manner similar to the ligament in true vocal folds of other mammals to modulate frequency. Tension or relaxation of the membrane may be controlled by small muscles that attach it to the arytenoids, thus adjusting the stiffness of the vibrating free edge.
Moreover, as these cartilages can be moved in different positions, the emitted sounds could have features with more or less acoustic complexity, constituting a large repertoire of song units, which should not be limited to six or nine classification categories [11,35]. In fact, the types of emitted sounds depend both on the distance the arytenoid cartilages are spaced from each other and on the rate and the pressure of the airflow on the covering membranes. In addition, because of the cartilage anchor points, the whale has at least two degrees of freedom to open them in three different orientations: longitudinally (apposed in the midline), in a U-shape, and in a vertical V-shape. The orientations may allow the whale to choose which sections of the membranes (midsection or distal section—Figure 2) and their volume to vibrate. The section parts potentially can be controlled with fine accuracy. Thus, the whale will produce pulsed sounds and low frequency vocalizations by passing the airflow through the total or near-total length of these membranes with the longitudinal or V-shape, or by channeling air to the thicker part of the membranes located in the distal section (Figure 2). For medium- and high-frequency vocalizations, the whale could reduce the airflow, bring the cartilages closer together, and choose to vibrate a smaller volume of the membranes, probably with the thin part located in the midsection. This mechanism may explain the singer’s capacity to produce a wide repertoire of vocalizations [6,9,10]. Note that the role of the cushion is to seal the laryngeal sac, but it could also be involved in the production of sounds [7,47].
Whales should precisely adjust the volume and speed of the airflow and regulate the size of the opening between the two parallel arytenoid cartilages, and they can also manage the different positions of these cartilages. Some vocalizations exhibit non-linear acoustic features. Two out of four were already reported: frequency jumps and chaos [12,33]. In this paper, we confirmed the presence of these two acoustic non-linearities and found two additional ones: subharmonics and biphonation. These acoustic non-linearities were detected in recorded song units, confirming the great diversity of SUs that singers can emit. Since non-linearities only appear in certain vocalizations and are repeated in the same way several times in the songs, they are probably intentional and are not due to an anatomical defect of the membranes (tumor, atrophy, scar) or a disease (partial paralysis). Moreover, these non-linearities were detected in the song units emitted by different singers and were also found in populations residing in different oceans. In previous studies and in this current study, it was shown that the features are neither rare nor isolated. These acoustic non-linearities are likely due to the non-coupling between the vibrator and the propagation conduits, as well as to the asymmetries of the anatomical characteristics, including the non-perfectly identical morphological characteristics of the membranes along the entire length and the non-uniformity of the texture in the trachea and in the laryngeal sac. Acoustic characteristics such as high wave amplitude or non-uniform pipes can contribute to or increase turbulence.
Nevertheless, whales may retain the capacity to choose when to emit these sounds. Additionally, they may add non-linearities by actively controlling their vocal generator, which is consistent with the Mercado and Perazio’s study of vocal production in humpback whales [48] and also in other mammalian species [14,15,34]. It means that this control should be based on the precise adjustment of the air circulation between the lungs and the laryngeal sac and also on the diversion of this airflow to specific regions of the arytenoids’ vibrating membranes (e.g., different sound qualities may correlate with different membrane segments). The fact that non-linearities were systematically found in some units, being repeated in a certain order to form phrases, provides support for the active control of these features by singers. If the production of non-linear elements requires special management of the vocal generator, they may require precision to manage their vibrator in coordination with the pressure and direction of the airflow. In the songs we analyzed, we did not note any differences in the time distribution of non-linearities between the beginning and the end of the acoustic recording. Furthermore, it is not possible to know for how long the singer had been singing and also whether the singer continued to sing for long after we stopped the acoustic recording. Furthermore, it is not possible to hypothesize about the link between the distribution of non-linearities and the potential fatigue of the vocal generator of humpback whales, as has been demonstrated in the primate indri (Indri indri) [49]. This may be due to the nature of the vibrator, since the ligaments of the indris are more susceptible to fatigue than the membranes of the arytenoid cartilages of humpback whales. In addition, the way in which their vocal generator is used is completely different: the indri suddenly starts, for a short moment, to scream as loudly as possible at the extreme maximum limit of their vocal vibrator. Conversely, humpback whales produce their song over time with regular amplitude modulations that continually use less force on their vibrator and therefore create less fatigue of their vocal generator.
Also, the number of non-linearities is not higher just before the humpback whale comes to the surface to breathe, which suggests that the appearance of non-linearities does not seem to be related to the duration of their apnea. However, evolution in the number of these non-linearities in the song units might change from one year to the next for the same individual. The fact that (1) all singers share a common structure of NL proportion distribution and (2) there is no evidence that the four Madagascar singers share a common structure differentiable from the Rep Dom or Socorro singers (see Figure 8) suggests that the non-linearities are determined by physiological aspects of the humpback whales and are then tuned by the individual (morphological and/or intentionality) rather than by regional or seasonal cultural innovation in the ‘song composition’.
Our results provide new insights into the four types of non-linearities:
The ascending or descending frequency jumps within harmonic SUs appear with discontinuity or rapid changes in pitch. The jump is observed for the fundamental frequency and also for the harmonics. It can also be provided by a sudden modification of the position of the arytenoid cartilages, affecting which membrane section vibrates. More than one frequency jump can be detected in the same vocalization, showing that the whale can move the position of these cartilages very quickly.
For the chaos and the subharmonic modes, the whale could perhaps emit sounds at the acoustic limits (e.g., excessive resonance and saturation) of its vocal generator. Thus, deterministic chaos could be provided by the subglottal air pressure exceeding a certain threshold relative to the spatial spacing of the membranes, causing a temporary short loss of control of the original harmonic vibration, as sometimes happens in speech production [50]. Generally, the whale promptly adjusts this pressure before the end of the SU, causing the chaotic vibrations to dissipate and the original harmonic vibration to reappear.
Regarding the subharmonic mode, the presence could also be explained by irregularities in the oscillations of the arytenoid membranes. Subharmonics appear when the response of the vibrator is not directly proportional to the pressure of the air flow, especially at the lower limit of the possible pressure for the vibratory system [47]. Subharmonics arise through bifurcation, where the membranes suddenly shift from one type of oscillations to another. This can happen due to changes in the air pressure or from various resonances of the vibratory system above the larynx related to the position of the paired nasal plugs within the paired bony nasal cavities and/or of the tension of the laryngeal sac filled with air [12], as suggested in pharyngeal pouches for Odobenus [45].
A second vibrating element is necessary to explain biphonation. For example, elephants can biphonate using both their larynx and the tip of their trunk [17]. There are several possibilities to explain biphonation in humpback whales. More than one of the two sections of the arytenoid membranes may simultaneously vibrate (Figure 2). The whale may open its arytenoid cartilages in a position that allows airflow passage across two separate regions along the entire cartilages, separated by an immobile zone (for example, one volume of the membranes in the midsection and one in the distal section). A second mechanism involves independent vibrations on both sides of the larynx. Airflow may pass simultaneously between the left arytenoid and the fat cushion, as well as between the right arytenoid and the fat cushion [47,51]. Another explanation could be that the membranes that cover the cartilages are not used in a completely symmetrical way. The membranes may vibrate asynchronously if the cartilages are not oriented on the same exact axis, if the whale has the muscular control capacity to put more tension on one side than the other, and/or if the air flow is intentionally guided more towards one membrane than the other (the volumes of the vibrating parts of the membranes are different on each arytenoid cartilage). In this case, the air pressure will not be equivalent on the two arytenoid membranes, causing them to vibrate at different frequencies. This technique is known to bassoonists who use different forces on the double reed with their lips to make the two reeds vibrate asymmetricall [52,53]. A third possibility is that there is yet another sound-producing region of the larynx. The large flaps of tissue attached to the paired corniculate cartilages (fused cranially to the paired arytenoid cartilages) may vibrate as air passes between them while simultaneously vibrating the membranes attached to the arytenoids [5]. The functional role(s) of these non-linearities could be to enable emitter identification if consistent across the lifespan and different contexts [12,34,54,55] and/or to improve the efficiency of mating calls, navigation, social interactions, or indications of physical conditions [34]. It is possible that some specific sounds with non-linearities are specifically emitted by a single individual, while others are shared by several whales. Further work will be needed to clarify this point. Analyses of non-linearities in song units could perhaps be useful for identifying specific singers and, when they are not too distorted by losses due to acoustic propagation, could be added to other acoustic features to extract their individual acoustic signature. If these non-linearities could also be detected from social sounds during social interactions between adults and even between mothers and calves, future work could be devoted to the ontogeny of non-linearities and perhaps to the cultural transmission of vocal repertoires.

5. Conclusions

Singing humpback whales emit vocalizations by precisely controlling their vocal generator, suggesting that these sounds are intentionally emitted, obtained via different volumes and speeds of the airflow circulating in their respiratory system. Some of the vocalizations exhibit acoustic non-linearities. Their production could correspond to a specific configuration of the vibrator, taking into account the management of the volume of air already contained in the lungs and the laryngeal sac, the guidance and the intensity of the airflow, its pressure on the chosen part of the arytenoid membranes, and the control of the time-varying position of the arytenoid cartilages.
In many species, acoustic non-linearities play crucial roles in communication between individuals, including partner attractiveness, individual identification, and alert transmission. For humpback whales, it is important to study, from an ethological point of view, why they use them in their songs with such diversity and quantity. To do this, it is crucial to systematically consider these non-linearities in future analyses on songs, in particular to specify their prevalence and consistency. Such analyses will facilitate cross-singer comparisons and, in particular, analyses of individual differences in song production, with the final objective being to extract an acoustic individual signature. It would also be important to investigate whether the emission of these non-linear features is related to the behavior of the singer or to the context in which songs are used (singer depth, presence of other singers, or presence of calf). A future analysis could also be extended to the songs of different humpback whale populations from other oceans and also of other mysticete species.

Author Contributions

Conceptualization, O.A., D.C., Y.D. and J.S.R.; methodology, O.A., D.C., Y.D. and E.M.; software, Y.D. and G.L.; validation, O.A., D.C., B.E., J.K.J., G.L., E.M., C.E.P. and J.S.R.; formal analysis, O.A., Y.D. and G.L.; investigation, O.A., Y.D., D.C., E.M. and J.S.R.; resources, O.A., Y.D., J.K.J. and J.S.R.; data curation, Y.D. and J.K.J.; writing—original draft preparation, O.A. and Y.D.; writing—review and editing, O.A., D.C., Y.D., B.E., J.K.J., G.L., E.M., C.E.P. and J.S.R.; visualization, O.A., Y.D. and J.S.R.; supervision, O.A.; project administration, O.A.; funding acquisition, O.A. and Y.D. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data can be accessible on request to the contact author.

Acknowledgments

The authors thank, for the song recorded off Samaná bay (Dominican Republic), Aurélie Twarog and Florent Defay from the Murmure des Oceans Team, and for the song recorded off Ste Marie channel (Madagascar), Anjara Saloma, François-Xavier Meyer, Henry Bellon, Schédir Marchesseau, Ifaliana Andriamananjara, Misaina Ramanantsoa, Ny Ando Rahagalala, Cédric Faucher, and Kim Poupon from the Cétamada Team. Songs recorded in 2001 off Isla Socorro, one of the most remote islands in the Mexican Pacific, were made possible by the generous support and friendship of the Mexican Navy, NAVSOC, with principal field recording assistance by Erin Andrea Falcone. We also want to thank Diane Crespi (Instagram: animals.interest) for Figure 1.

Conflicts of Interest

Author Jeff K. Jacobsen was employed by the company VE Enterprises. The remaining 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.

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Figure 1. The different elements of the respiratory system of humpback whales (dimensions are not respected).
Figure 1. The different elements of the respiratory system of humpback whales (dimensions are not respected).
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Figure 2. Description of the arytenoid cartilages and the membranes covering them in 2 sections from ventral and transverse views.
Figure 2. Description of the arytenoid cartilages and the membranes covering them in 2 sections from ventral and transverse views.
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Figure 3. Long-term spectrogram of (a) the complete acoustic recording and (b) zoom in the first cycle of the song with the type of extracted SUs.
Figure 3. Long-term spectrogram of (a) the complete acoustic recording and (b) zoom in the first cycle of the song with the type of extracted SUs.
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Figure 4. Spectrograms of the 13 SUs sequences, extracted and named during a song cycle recorded off Samaná Bay, Dominican Republic. Spectrogram parameters: Hamming time window of 186 ms, windows overlap of 70%.
Figure 4. Spectrograms of the 13 SUs sequences, extracted and named during a song cycle recorded off Samaná Bay, Dominican Republic. Spectrogram parameters: Hamming time window of 186 ms, windows overlap of 70%.
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Figure 5. Bar graph of the proportion of non-linearities encountered in the whole annotated dataset.
Figure 5. Bar graph of the proportion of non-linearities encountered in the whole annotated dataset.
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Figure 6. Downsweep creaking with (a) biphonation and (b) concatenation of 3 cycles of downsweep creaking and stability of the frequency components: F0 in red, H0 in gray, and G0 in white with the corresponding value of the starting frequency.
Figure 6. Downsweep creaking with (a) biphonation and (b) concatenation of 3 cycles of downsweep creaking and stability of the frequency components: F0 in red, H0 in gray, and G0 in white with the corresponding value of the starting frequency.
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Figure 7. SU with 3 non-linear acoustic features: (a) upsweep creaking and (b) long-short upsweep creaking.
Figure 7. SU with 3 non-linear acoustic features: (a) upsweep creaking and (b) long-short upsweep creaking.
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Figure 8. Evolution in time of non-linear features of upsweep creaking (# = 13). In black, the frequency band of chaos; in the red line, F0; in the red square G0; in green, the frequency variations involved in successive frequency jumps. Mean frequency and std are given for each descriptor.
Figure 8. Evolution in time of non-linear features of upsweep creaking (# = 13). In black, the frequency band of chaos; in the red line, F0; in the red square G0; in green, the frequency variations involved in successive frequency jumps. Mean frequency and std are given for each descriptor.
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Figure 9. Acoustic non-linearities in (a) HF upsweep creaking and (b) downsweep creaking II.
Figure 9. Acoustic non-linearities in (a) HF upsweep creaking and (b) downsweep creaking II.
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Table 1. Overview of the proportion of non-linearities encountered among the annotated SUs and data collection characteristics including 3 different locations: Dominican Republic, Isla Socorro, and Madagascar, and 4 different singers at the same location: Madagascar during the same season (2022).
Table 1. Overview of the proportion of non-linearities encountered among the annotated SUs and data collection characteristics including 3 different locations: Dominican Republic, Isla Socorro, and Madagascar, and 4 different singers at the same location: Madagascar during the same season (2022).
SiteRecorderHydrophoneFs (kHz)Code (bits)DeploymentRecording Time (min s)Mean Cycle Duration (min)Number of Annotated SUs% with Subharmonic Regime% with Frequency Jump% with Chaos% with Biphonation
DRZoom F3Aquarian H2d9632Boat, 10 m cable48 1114.528425.744.483.819.3
ISSony TC5-DM cassette recorderCRT H11616Boat, 15 m cable40 349.219716.38.763.53.5
M1Zoom H5Aquarian H2d9624Boat, 10 m cable30 1113.37762223.766.710.9
M2Zoom H5Aquarian H2d9624Boat, 10 m cable28 281064531.827.373.64.3
M3Zoom H5Aquarian H2d9624Boat, 10 m cable28 5011.657858.438.979.910.5
M4Zoom H5Aquarian H2d9624Boat, 10 m cable35 0216.285843.628.972.58.6
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MDPI and ACS Style

Doh, Y.; Cazau, D.; Lamaj, G.; Mercado, E.; Reidenberg, J.S.; Jacobsen, J.K.; Perazio, C.E.; Ecalle, B.; Adam, O. Study of Non-Linearities in Humpback Whale Song Units. J. Mar. Sci. Eng. 2025, 13, 215. https://doi.org/10.3390/jmse13020215

AMA Style

Doh Y, Cazau D, Lamaj G, Mercado E, Reidenberg JS, Jacobsen JK, Perazio CE, Ecalle B, Adam O. Study of Non-Linearities in Humpback Whale Song Units. Journal of Marine Science and Engineering. 2025; 13(2):215. https://doi.org/10.3390/jmse13020215

Chicago/Turabian Style

Doh, Yann, Dorian Cazau, Giulia Lamaj, Eduardo Mercado, Joy S. Reidenberg, Jeff K. Jacobsen, Christina E. Perazio, Beverley Ecalle, and Olivier Adam. 2025. "Study of Non-Linearities in Humpback Whale Song Units" Journal of Marine Science and Engineering 13, no. 2: 215. https://doi.org/10.3390/jmse13020215

APA Style

Doh, Y., Cazau, D., Lamaj, G., Mercado, E., Reidenberg, J. S., Jacobsen, J. K., Perazio, C. E., Ecalle, B., & Adam, O. (2025). Study of Non-Linearities in Humpback Whale Song Units. Journal of Marine Science and Engineering, 13(2), 215. https://doi.org/10.3390/jmse13020215

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