A proposal for epithelial dominance in extremely high fundamental frequency vocalizations

Abstract

In this hypothesis article, we explore the upper limit of the fundamental frequency in vocalization. Most mammalian vocalizations are produced by airflow-induced, self-sustained vibration of vocal folds, with fundamental frequency being determined by multiple tissue layers in the folds, including muscle, ligament, and epithelial tissues. These layers contribute to vocal fold length, depth of vibration, and viscoelasticity needed for oscillation. While current vocal fold models explain a large range of frequencies, some extremely high-frequency vocalizations (e.g., whistle voice in humans) remain unexplained based on known tissue properties. We hypothesize that the thin layers near the epithelial surface become primary contributors to elasticity at high frequencies. Anatomical studies indicate weak allometric scaling in the epithelium, i.e., number of epithelial cell layers and thickness of the epithelium scale weakly with body size. This could allow species to produce frequencies outside the typical size-dependent spectral range if this layer dominates. Computational simulations using tissue property data support this hypothesis. We propose a model in which epithelial cells combined with collagen fibers in the lamina densa form structures capable of generating fundamental frequencies in the kilohertz range with minimal depths of vibration.

I. INTRODUCTION

Two soft tissue masses, known as vocal folds, located inside the mammalian larynx or the avian syrinx (Longtine et al., 2024), can be adducted and abducted to act as a versatile airflow valve. For vocalization, the vocal folds are positioned to leave an opening called the glottis. When the appropriate airflow passes through the glottis, it can cause the vocal folds to self-sustain vibration, generating sound by rapid changes in airflow. Our understanding of how the structural design of the vocal folds relates to the acoustic features of the produced voice is still evolving. One of the most puzzling questions in vocal communication is the generation of very high fundamental frequency voices. The question is perplexing because current concepts of airflow-induced self-sustained vibration rely on the ability of muscle and vocal ligament tissues to sustain high mechanical stresses, yet experimental data on the elastic properties of these tissues have not predicted extremely high frequencies (Titze et al., 2016). Here, we present data supporting a new hypothesis called the epithelial dominance hypothesis.

All mammalian vocal folds investigated to date follow an allometric scaling of relaxed vocal fold length. Using a fiber-gel model, Titze et al. (2016) outlined the relationship between vocal fold length, vocal fold tensile stress, and fundamental frequency $(f_{\mathrm{o}})$ range. By actively varying the length of the vocal folds through the action of intrinsic laryngeal muscles, which lengthen or shorten the folds and adjust the tension in muscle and ligament tissue, a range of vocal frequencies spanning several octaves was predicted. Vocal fold mass and shape of cellular and extracellular matrix composition exhibit diverse structural adaptations that do not adhere to the same allometric scaling as vocal fold length (Julias et al., 2013; Riede and Goller, 2014). Changes in those features represent structural adaptations that result in shifts to the predicted $f_o$ range, which is based on vocal fold length only.

What is an “extremely high $f_o$”? At this stage, it refers to frequencies that cannot be generated by longitudinal stress in muscle and ligament tissues. Fibers in these tissue layers can explain the human $f_o$ range of 50–1600 Hz. Video stroboscopic imaging and magnetic resonance imaging (MRI) of vocal fold vibrations in female singers has confirmed this upper limit with self-sustained vocal fold oscillations (Švec et al., 2008; Kato et al., 2023; Echternach et al., 2024). Frequencies beyond the approximate 1600 Hz threshold in humans are currently unexplained mechanistically. Figure 1 shows spectrograms of three examples of human high-frequency vocalization. We see a straight tone and vibrato sequence around 2.4 kHz in sequence 1, a downward gliding $f_o$ from 5.5 kHz in sequence 2, and rising $f_o$ to 3.8 kHz in sequence 3.

FIG. 1.

Three singing sequences by the famous “whistle” voice performer Georgia Brown are analyzed. In sequence 1, the fundamental frequency is around 2400 Hz in the non-vibrato segment, followed by a lower frequency with vibrato. A strong second harmonic is present, suggesting some vocal fold contact may occur. In sequence 2, the fundamental frequency starts at around 5500 Hz and decreases in steps. In sequence 3, the fundamental frequency increases smoothly to about 3.8 kHz.

Similarly, non-human mammals that produce sounds via self-sustained vocal fold oscillation have $f_o$ ranges that exceed what is predicted by the known nonlinearity of vocal fold ligament stiffness. For example, in North American elk, tissue properties explain an $f_o$ range of up to 1.5 kHz, but $f_o$ in advertisement calls during mating season can reach up to 2.4 kHz (Riede and Titze, 2008; Titze and Riede, 2010). In other species, while vocal fold tissue properties have not been directly measured, available data suggest that they also produce extremely high $f_o$. Dolphins can emit signature calls with $f_o$ up to 20 kHz (Madsen et al., 2012), bats generate songs and echolocation calls with $f_o$ above 20 kHz (Hartley and Suthers, 1988; Håkansson et al., 2022), and some rodent species produce long-distance calls with $f_o$ up to 24 kHz (Riede et al., 2024).

In singing pedagogy, vocal register terminology refers to chest register, falsetto register, and whistle register. However, those terms are not uniformly agreed upon and should be linked to underlying mechanisms (Echternach et al., 2024). Three distinct mechanisms of vocal fold vibration have been labeled $M1$, $M2$, and $M3$ (Roubeau et al., 2009) but the anatomical correlates of those labels remain unclear. To address this discrepancy, we will show here that these mechanisms may be related to dominance in muscle, ligament, and epithelium control of fundamental frequency.

Extremely high $f_o$ values have not been successfully modeled because the required fiber stresses in the vocal ligament and vocal muscle tissues are an order of magnitude higher than direct measurements have recorded (Perlman et al., 1984; Perlman and Titze, 1988). We propose that a third tissue layer, the epithelium with a thin collagen fiber substrate in the basement membrane known as the lamina densa, may play a crucial role in this process. No direct measurements of vocal fold epithelial stiffness are available, and obtaining such measurements continues to be challenging (Gómez-González et al., 2020). We argue that existing data on (a) the mechanical properties of other types of epithelial tissues, (2) phenotypic diversity in vocal fold epithelium morphology between species, and finally (3) a modeling approach, suggest that this thin outer layer could support extremely high $f_o$ if it becomes dominant in self-sustained oscillation.

We begin by discussing how the epithelial dominance hypothesis can be integrated with the body-cover theory of self-sustained vocal fold vibration, and by reviewing available data on the mechanical properties of other epithelial tissue. We will address differences in mechanical properties between epithelium, vocal ligament, and the vocal fold (TA) muscle. Then, we present two sets of data. First, we conducted a morphometric analysis of the epithelium in vocal folds from 21 mammalian species, ranging from 30 g mice to 5000 kg giraffes, illustrating a considerable diversity of vocal fold epithelia. Second, we performed a computational analysis of a vibrating epithelium coupled to a lamina densa and a portion of a superficial lamina propria layer.

II. BACKGROUND

A. Body-cover theory

Hirano’s body-cover theory proposes that the vibrating vocal fold tissue can be divided into a stiff (minimally vibrating) body layer and a pliable (maximally vibrating) cover layer (Hirano, 1974; Kurita et al., 1983) [Fig. 2(A)]. These layers can vary in thickness, meaning that the anatomically defined layers (muscle, ligament, mucosal layer, and epithelium) can be variably recruited into a body or cover function [Fig. 2(A)]. An anatomical variation of the body-cover is the addition of a vocal membrane [Fig. 2(B)], an epithelial double layer most often positioned on the free edge of the vocal fold, and present in some species of bats (Hartley and Suthers, 1988), nonhuman primates (Harrison, 1995), and rodents (Pasch et al., 2017; Riede et al., 2022). A vocal membrane alters the pressure-flow distribution in the glottis (Mergell et al., 1999) and may be able to vibrate somewhat independently of the main vocal fold.

FIG. 2.

Vocal fold layer structure. (A) and (B) Coronal section through vocal folds showing multiple tissue layers. (A) Typical human morphology, (B) vocal membrane extension as in some bats, primates, and rodents. (C)–(E) Gradual shift from muscle dominance (C) to ligament dominance (D) to epithelial dominance (E) as a target fundamental frequency range is increased. Hypothesized tissue layer dominance (shown in dark) for low $f_{\mathrm{o}}$ (C), mid-to-high $f_{\mathrm{o}}$ (D), and extremely high $f_{\mathrm{o}}$ (E).

Here, we investigate how self-sustained oscillation changes when vocal fold vibration is confined to the epithelium cells, a lamina densa of the basement membrane, and a portion of the superficial layer of the lamina propria. In this scenario, the ligament and the TA muscle lie dormant, being involved only in the length and shape control of the vocal folds. While an ultimate goal is to incorporate an epithelial layer into a finite-element model that includes all layers of tissue, as recently demonstrated by Deng and Peterson (2023), it is prudent to first explore the possibility of epithelial tension dominance with a simple ribbon model. This approach avoids the complexity and computational instability associated with sub-millimeter finite elements. In the Deng and Peterson study, creating a finite-element mesh structure for the epithelium was a significant task for relatively low fundamental frequencies (less than 500 Hz). The computational challenge increases when targeting frequencies an order of magnitude higher.

The proposed epithelial dominance hypothesis is illustrated in Figs. 2(C)-2(E). Each anatomical layer can assume dominance, with a gradual shift from TA muscle dominance at low $f_o$ (human: 60–400 Hz) to ligament dominance at mid-to-high $f_o$ (human: 400–1500 Hz), to epithelial dominance at extremely high $f_o$ (human: 1500–5000 Hz). As the $f_o$ increases, the depth of vibration into the tissue decreases, meaning that higher frequencies involve more cover and less body tissue in vibration.

Current computational and physical models (Alipour et al., 2000; Zhang, 2016; Thomson et al., 2005; Palaparthi et al., 2019; Titze and Hunter, 2004; Titze et al., 2016) have incorporated most of these tissue properties, including variable depths of the layers, fiber nonlinearity, gel elasticity and viscosity, and some degree of nonuniformity along the length and thickness. Boundary conditions have also been included. However, the specific properties of the epithelium have only begun to be investigated (Deng and Peterson, 2023). This very thin outer layer (less than 100 μm) has generally been combined with the superficial gel layer and treated as one mucosal layer (Alipour et al., 2000; Zhang, 2016). Here, we divide this mucosal layer into two components, a tension-bearing epithelium and a portion of the gel-like superficial lamina propria.

B. Elasticity of epithelial tissue

The integrity of the vocal fold epithelium is of great importance for its function as a barrier between the outside and inside environment (Gray, 2000; Levendoski et al., 2014; Lungova et al., 2022), but also for its function as vibrating tissue mass during voice production. Both functions are intricately linked, especially at high sound intensities and high $f_o$ (Rousseau et al., 2011; Kojima et al., 2014). The measured viscoelastic properties of the vocal fold lamina propria (Perlman et al., 1984; Riede et al., 2011) cannot be directly applied to the epithelium due to anatomical complexities within the basement membrane attachment of epithelial cells to the superficial layer (Gray et al., 1994). The lamina propria is made up of both cellular and non-cellular components, with the organization of extracellular proteins—such as elastin, collagen, and hyaluronic acid—determining its viscoelastic properties (Gray et al., 1999; Chan et al., 2007). In contrast, the vocal fold epithelium consists of multiple layers of individual epithelial cells connected by junctional complexes. Epithelium by itself does not have an extracellular matrix because epithelial cells are tightly packed together with very little space between them. An extracellular matrix component associated with epithelium is the specialized basement membrane, which lies beneath the epithelial cell layer, attaching it to the underlying connective tissue. It is hypothesized that a thin basement membrane layer (a lamina densa) provides the structural integrity of the epithelium under high tension [Fig. 3(A)].

FIG. 3.

(A) Basement membrane structure located beneath the vocal fold epithelium, (B) stress-strain curves of three dominant tissues, redrawn after Titze (2000). The epithelium curve is an extrapolation from a mucosal layer.

The epithelium cells are classified as stratified and squamous, meaning that there are several flattened layers (Gray, 2000; Fisher et al., 2001; Gill et al., 2005); however, differences between layers have been proposed (Dowdall et al., 2015). The human vocal fold epithelium contains 5–10 layers of epithelial cells (Arens et al., 2007). Non-epithelial cells have been found within the epithelium (Hubbard et al., 2024). Cell junctions play a crucial role in providing structural support by linking adjacent epithelial cells and sealing the paracellular space. These junctions consist of protein complexes that enable adhesion and communication between cells or between a cell and the basement membrane (Knight and Holgate, 2003; Gill et al., 2005). The specific organization and distribution of junctions determines stress anisotropy of the epithelial layer (Marín-Llauradó et al., 2023). However, there is considerable fluid transport between the cells (Tanner et al., 2010), suggesting that the cell matrix is porous and that the cells alone cannot support extremely high directional stresses.

There is also much stress relaxation in layers of epithelial cells. Rheological studies suggest that cells and cell-association behave as poro-elastic material, i.e., the cytoplasm is treated as a biphasic material consisting of a porous elastic solid meshwork (cytoskeleton, organelles, macromolecules) bathed in an interstitial fluid (cytosol) (Moeendarbary et al., 2013). According to their study, the deformation of the solid is affected by the fluid flow, i.e., the rate of cellular deformation is limited by the rate at which intracellular water can redistribute within the cytoplasm. Since the cytoplasm is the largest part of the cell by volume, its rheology sets the rate at which cellular shape changes can occur. A recent study suggests that epithelial cells increase their tension based on this strain rate (Safa et al., 2024). For example, cadherin bonds exhibit rate-dependent strengthening, i.e., increased strain rates result in elevated stress levels at which cadherin bonds fail (Esfahani et al., 2021).

The mechanical properties of isolated epithelial cells are primarily governed by the intracellular cytoskeleton at low strain and by the inextensibility of the plasma membrane at high strain (Pegoraro et al., 2017). During large deformations, such as vocal fold elongation, the mechanical stress is mainly influenced by the plasma membrane and inter-cellular protein connections, which produce stresses only in the kilopascal (kPa) range (Khalilgharibi et al., 2019). We speculate that the collagen fibers in the basement membrane zone are the important load carriers if larger stresses in the megapascal (MPa) range are required for high-frequency oscillation. The basement membrane is formed by a mixture of molecules including type IV collagen and laminin. Stress in isolated collagen type IV fibrils can reach up to 100 MPa at less than 30% strain (Shen et al., 2008). For a group of fibrils forming a matrix structure in the basement membrane, the stress depends on the orientation and density of fibrils. With intra-fibril proteoglycans and fibril cross-bridging, the overall lamina densa stress is likely to be somewhere in the 1–100 MPa range (Sun, 2021; Andriotis et al., 2023), but no measurements have been conducted.

The exact anterior and posterior boundary conditions of the vocal fold epithelium are also not entirely clear. The vocal ligament terminates by fusing into the macula flavae. These, in turn, attach to the vocal process of the arytenoid cartilage posteriorly, and to the thyroid cartilage anteriorly. In contrast, the epithelium covers the macula flavae and is continuous inside the laryngeal lumen. If the boundaries were only partially firm, the effective length of vibration would vary. If completely free boundary conditions were to exist, which is unlikely, the epithelium could develop a narrow belly region due to cell contraction when linearly stretched. Assuming the cell tissue to retain its volume with internal cell contraction, free or weak anterior or posterior boundary attachment would offer little resistance to effective shortening of the epithelium with only a dominant lateral boundary constraint. A protruding vocal membrane could develop. However, the vocal membrane in non-human species is not yet modeled with variable boundary constraints (Mergell et al., 1999; Kanaya et al., 2022).

For the current computational study, the epithelial layer will be treated as a vibrating ribbon with unidirectional stress in the anterior–posterior (AP) direction. Fixed AP boundaries will be imposed. Measured stresses in ligament and muscle layers of human and canine vocal folds have been in the kPa range, not in the MPa range (summarized in Titze et al., 2016). Figure 3(B) shows a summary of typical stress measurements obtained on human vocal fold tissues for positive strains. The epithelium curve was an extrapolation based on mucosae measurements of various gel thicknesses. It remains speculative whether the epithelium curve extended into the 20%–30% strain region could reflect stresses in the MPa range.

A simple vibrating string or ribbon formula (discussed mathematically in Methods) predicts that a stress of 0.4 MPa is required with a vocal fold length of 1.0 cm to reach 1000 Hz. To double that frequency to 2000 Hz, either the stress would need to quadruple to 1.6 MPa (keeping vocal fold length constant) or the length would need to be cut in half (Titze et al., 2016). A frequency of up to 1600 kHz for soprano singing was confirmed with the full length of the vocal folds in vibration (Echternach et al., 2024). Computational modeling showed that the vocal ligament could sustain a high enough stress. Svec et al. (2008) also showed that full-length vibration is observed at frequencies up to 1600 Hz.

Vocal ligament morphology involves a trade-off between collagen density and fiber thickness to support high stresses (Klemuk et al., 2011), while also maintaining the pliability needed for high-frequency tissue vibrations. However, there seems to be a limit to how much collagen density can increase in the vocal ligament, given that it cannot sustain stresses much beyond 1.0 MPa within the 0%–30% strain range. Therefore, we hypothesize that both the complex interaction between (a) the lamina densa and (b) the specific response of the epithelial layer to high strain contribute to supporting the necessary tissue stresses.

III. METHODS

A. Morphological data

The goal of this morphological analysis of vocal folds from 21 species was to explore the phenotypic variability of the vocal fold epithelium. Hirano (1974) previously noted similarity between the morphology of skin and vocal fold layers. The epidermis layer of skin exhibits important protective adaptations to sustain different mechanical stresses and other functions depending on its location on the outer body (Yousef et al., 2020). Whether the vocal fold epithelium exhibits similar species-specific adaptations is unknown. We examined this variability across diverse species to determine whether these differences suggest adaptations of the epithelium to various laryngeal functions, including vocalization. Species differences could provide insights into how specific phenotypes support particular vocal behaviors. Currently, available data on vocal fold epithelium are limited to humans, house mice, and rabbits (Gray, 2000; Lungova et al., 2015; Mizuta et al., 2017). By broadening our analysis to include additional species, we hope to gain a deeper understanding of the evolutionary biology of this often-overlooked layer of the vocal fold.

Vocal fold tissue was sourced from zoological gardens, commercial vendors, or wild-caught specimens (details in Table I). Tissue was collected as soon as possible after death and fixed in 10% neutral buffered formalin. Mid-membranous coronal sections (5 μm thick) were stained with haematoxylin–eosin for a general overview, Masson’s Trichrome for collagen fiber stain, and Elastica-van-Gieson for elastic fiber stain. Histological sections were digitized with an Aperio CS 2 slide scanner and processed with IMAGESCOPE software (v. 8.2.5.1263; Aperio Tech.).

TABLE I.

Raw data of vocal fold measurements from 21 species. Vocal fold measurements were taken from tissue samples collected from three sources (BTNR, MZB, RL) or from published references. Larynges were harvested from animals euthanized for clinical reasons related to old age by zoo veterinarians at Omaha’s Henry Doorly Zoo. Hirano (1981) and Gray and Titze (1988). BTNR, Boys Town National Research Hospital Institutional Animal Care and Use Committee (Protocol Numbers 04-04, 07-04 and 10-04); MZB, Maryland Zoo in Baltimore; RL, collected as part of research efforts in the Riede lab; N, sample size; BM, body mass in kg; VFL, vocal fold length in mm; VF, vocal fold; LP, lamina propria; E, epithelium.

Common species name	Scientific name	Source	$N$	BM (kg)	Log 10 BM	VFL (mm)	VF thickness (mm)	TA thickness (mm)	LP thickness (mm)	$E$ thickness	$\mathit{E}$ layers
Human	Homo sapiens	Hirano (1981)	—	75	1.88	15	6	3	1	25	7
Chimpanzee	Pan troglodytes	MZB	1	100	2.00	25	6.3	4.5	1.8	35	5
Lemur	Lemur spec.	MZB	1	5	0.70	5	3.7	3	0.5	19	5
Rhesus monkey	Macaca mulatta	WPC	1	6	0.78	4	3.9	1.9	0.4	15	4
Bengal tiger	Panthera tigris (ID: 5380)	BTNZ	1	120	2.08	32	14.4	11.7	4	41	5
Amur (Siberian) tiger	Panthera tigris altaica (ID: 6220)	BTNZ	1	120	2.08	32	12.5	9.4	3.8	31	6
Lion	Panthera leo (ID: 5333)	BTNZ	1	120	2.08	28	15.8	6.9	8.8	64	7
Red fox	Vulpes vulpes	MZB	2	6	0.78	5	5.6	3.4	1.9	34	7
Leopard	Panthera pardus pardus	MZB	1	60	1.78	11	8.8	5.8	3	30	6
Domestic dog	Canis domesticus	Gray and Titze (1988)	—	10	1.00	6	4	2.5	0.5	20	4
Giraffe	Giraffa camelopardalis reticulata	MZB	1	5000	3.70	40	10.1	4.2	4.9	18	5
Sitatunga	Tragelaphus spekii	MZB	3	80	1.90	30	5.4	3.9	1.4	20	4
Sheep	Ovis aries	MZB	1	42	1.62	15	3.1	1.6	1	24	5
Reindeer	Rangifer tarandus	RL	1	155	2.19	18	5.6	2.9	0.9	15	6
Pronghorn	Antilocapra americana	RL	2	90	1.95	18	3.4	1.6	1.3	33	6
North American elk	Cervus canadensis	RL	5	250	2.40	25	9.6	5	3	29	6
Mule Deer	Odocoileus hemionus	RL	5	85	1.93	19	3.7	2.8	0.6	31	10
House mouse (CD 1)	Mus musculus	RL	8	0.03	−1.52	1	0.5	0.3	0.08	15	3
Laboratory rat (Sprague Dawley)	Rattus norvegicus	RL	8	0.2	−0.70	1.5	0.7	0.3	0.07	20	4
Grasshopper mouse	Onychomys leucogaster	RL	8	0.06	−1.22	0.6	0.6	0.4	0.05	15	3
California mouse	Peromyscus californicus	RL	8	0.06	−1.22	0.8	0.6	0.5	0.06	10	3

Five variables were measured. Cadaveric vocal fold length was measured before tissue fixation between the vocal process and the thyroid cartilage. All other measurements were taken at a point close to the mid-membraneous position. Vocal fold thickness was measured as the shortest distance between vocal fold epithelium and inner surface of the thyroid cartilage. Thyroid muscle thickness was measured as the shortest latero-medial distance. Lamina propria thickness was measured as the shortest distance between medial TA muscle surface and basal cell layer of the epithelium. Epithelium thickness was measured as the distance between basal cell layer and outermost cell layer. The number of epithelial cell layers was counted with each layer representing a distinct arrangement of cells in the epithelium.

B. The basics of a computational ribbon model

In computational vocal fold modeling, the typical approach has been to model the mucosa as a homogeneous layer, combining the dense cellular epithelium tissue with the gel-like superficial tissue. Given that the elastic properties of the vocal fold epithelium have not been measured in isolation and given that the epithelium as a separate layer requires extremely small elements in finite-element modeling, the approach here is to first make an analytical $f_o$ estimation with a simple vibrating ribbon equation. This ribbon model will then be discretized and embedded into a digital airway construct for self-sustained oscillation.

Consider the epithelium to be a flexible ribbon of length, $L$, vocal fold thickness, $T$, and a very small medial–lateral depth (a “skin” depth in vibration science). For a longitudinal (AP) applied stress, the strained length follows that of an elongated string,

$$\mathrm{L}={\mathrm{L}}_{\mathrm{o}}(1+\epsilon ),$$ (1)

where ${\mathrm{L}}_{\mathrm{o}}$ is the unstained vocal fold length and $\epsilon$ is any given strain. The stress $\sigma$ becomes exponential as cells and connective tissue are stretched,

$$\sigma ={\sigma }_{\mathrm{o}}{\mathrm{e}}^{\mathrm{B}(\epsilon -\epsilon \mathrm{o})},$$ (2)

where ${\epsilon }_0$ is the strain at which the stress–strain curve becomes exponential, usually from a somewhat linear region at lower strains. Further, ${\sigma }_{\mathrm{o}}$ is the stress magnitude when the curve becomes exponential, and B is an exponent that defines the curvature (steepness) of the exponential curve. A simple prediction of the fundamental frequency, based on the vibrating string formula but equally applicable to a stretched ribbon, is

$${\mathrm{f}}_{\mathrm{o}}=(1∕2\mathrm{L}){(\sigma ∕\varrho )}^{1∕2},$$ (3)

where $f_o$ is fundamental frequency, $\mathrm{L}$ is vocal fold length, $\sigma$ is the stress, and $\varrho$ is tissue density.

Figure 4(A) is a semi-logarithmic plot of three stress–strain curves calculated with Eq. (2) using a matlab script. Exponential curves become straight lines on this semi-logarithmic scale. The predicted fundamental frequencies from Eq. (3) are shown in Fig. 4(B), also on a semi-logarithmic scale. Depending on the steepness of the exponential curves [factor $B$ in Eq. (2)], the magnitude of the term ${\sigma }_{\mathrm{o}}$, and the exponential strain onset ${\epsilon }_0$, the curves can overlap to exchange tissue layer dominance. For low strains up to about 0.1, the TA muscle can dominate with active contraction, for mid-to-high strains (0.1–0.25) the ligament dominates, and for very high strains (0.25–0.4), the epithelium dominates. The same three regions are identifiable in the fundamental frequency curves. For the curves shown, the parameters are given in Table II. The TA muscle was given an active stress of 50 kPa in addition to the passive fiber stress, which gave the TA muscle its low-frequency dominance. These parameters are adopted from Titze et al. (2016), who summarized existing empirical data on stress–strain relationships. The ultimate stress at which vocal ligaments begin to rupture was, across different mammalian species, around 1 MPa.

FIG. 4.

(A) Mathematically developed semi-logarithmic stress–strain curves for three vocal fold tissue layers, contracted muscle, ligament, and epithelium. Note the overlap for stress dominance in the low-strain, mid-strain, and high-strain regions. (B) Fundamental frequency predictions for the regions shown in (A), also plotted semi-logarithmically.

TABLE II.

Nonlinear stress–strain parameters for vocal fold tissue layers.

Parameter	Muscle	Ligament	Epithelium
B	6.5	9.0	18
${\epsilon }_0$	−0.5	−0.5	0.0
${\sigma }_{\mathrm{o}}$	5000	5000	50 000

C. Computational simulation with a ribbon model embedded in an airway

For a computer simulation of self-sustained oscillation, the epithelium plus a thin superficial gel layer were modeled as a combined-layer ribbon under tension. It became the airway wall in the glottis, one element deep. Vibratory amplitudes were expected to be very small, less than a millimeter. Therefore, the ligament and the TA muscle were assumed not to be in vibration. With an approximate 1 cm vocal fold length, a tissue density of 1.0 g/cm³, and an epithelial stress in the range of 1.0–10 MPa over a strain range of 0–0.3, the self-oscillation frequencies were expected to reach the range of 1500–5000 Hz, following the natural frequencies calculations in Fig. 4.

Assume $D_{\mathrm{e}}$ and $D_{\mathrm{s}}$ are the depth of the epithelial and superficial layers into the wall, ${\sigma }_{\mathrm{e}}$ and ${\sigma }_{\mathrm{s}}$ are the respective longitudinal stresses, and $\mathrm{T}$ is the ribbon thickness glottal entry to exit. Then, summing the tensions in the two layers allows a net ribbon stress to be determined,

$${\sigma }_{\mathrm{e}}{\mathrm{D}}_{\mathrm{e}}\mathrm{T}+{\sigma }_{\mathrm{s}}{\mathrm{D}}_{\mathrm{s}}\mathrm{T}=\sigma ({\mathrm{D}}_{\mathrm{e}}+{\mathrm{D}}_{\mathrm{s}})\mathrm{T}=\sigma \text{DT},$$ (4)

where $\sigma$ is now the ribbon stress and $D$ is the ribbon depth, namely,

$$\mathrm{D}={\mathrm{D}}_{\mathrm{e}}+{\mathrm{D}}_{\mathrm{s}}.$$ (5)

Solving for the ribbon stress

$$\sigma =({\sigma }_{\mathrm{e}}-{\sigma }_{\mathrm{s}}){\mathrm{D}}_{\mathrm{e}}∕\mathrm{D}+{\sigma }_{\mathrm{s}}.$$ (6)

Now consider the ribbon to have an effective mass and stiffness when driven medio-laterally by glottal airflow. Let $M=\varrho D$ be the wall mass per unit area LT (ribbon length times thickness), let $E$ be the Young’s modulus of the wall, and let $K=\mathrm{E}∕\mathrm{D}$ be the wall stiffness per unit area. The natural fundamental frequency of a mass-spring wall element can then be equated to the ribbon natural frequency,

$${(\mathrm{K}∕\mathrm{M})}^{1∕2}∕(2\pi )={(\sigma ∕\varrho )}^{1∕2}∕(2\mathrm{L}).$$ (7)

It is now possible to use the mass-spring wall parameters to probe self-sustained oscillation with aerodynamic excitation. In terms of the longitudinal stress described earlier, the stiffness per unit wall area becomes

$$\mathrm{K}=\sigma \mathrm{D}{(\pi ∕\mathrm{L})}^2.$$ (8)

A four-section lumped-element model of the glottal wall was embedded into a digital model of the airway for self-sustained oscillation. Thus, computation of airflow and acoustic wave propagation were the same in the glottis as anywhere else in the airway The airway model has been developed and tested over several decades. Fluid transport and wave propagation in the airways (trachea to lips) were produced by a computationally efficient Navier–Stokes solution (Titze et al., 2014). In this solution, the pressure variations transverse to the axis of flow and wave propagation were computed analytically, leaving the solution along the axis one-dimensional for digital implementation. For all sections along the axis (trachea to lips), viscous resistances, acoustic inertances, and air compliances were computed. At the section interfaces, kinetic pressure losses were computed from empirical equations (Scherer and Guo, 1990; Pelorson et al., 1994; Scherer et al., 2001; Scherer et al., 2010). The computations were embedded in a fourth-order Runge–Kutta loop such that pressure and airflow computations were simultaneous across all sections. Unlike the often-used wave reflection algorithm, this one-dimensional Navier–Stokes solution can easily accommodate variable section lengths for small geometric airway detail. It also includes the transport of steady airflow through the vocal tract, which was needed for fluid–structure interaction and self-sustained oscillation of the vocal fold walls.

IV. RESULTS AND DISCUSSION

A. Morphological data

We examined how vocal fold morphology scales with body size. Univariate linear regressions revealed that several vocal fold traits scale significantly with log₁₀-transformed body mass across species (Table III). Vocal fold length and vocal fold thickness showed the strongest positive association with body size, indicating robust allometric scaling. TA muscle thickness and lamina propria thickness also displayed significant positive relationships, though with lower explained variance. Epithelium thickness and the number of epithelial cell layers were also positively associated with body mass, suggesting that even epithelial features scale modestly with size. Collectively, these findings support that structural components of the vocal fold exhibit scaling relationships with body mass, though the strength of these associations varies across tissue layers.

TABLE III.

Univariate regression results for six vocal fold traits used as dependent variables predicted by log₁₀ body mass. Bolded p-values indicate statistical significance at p < 0.05, p < 0.01, or p < 0.001. VFL, vocal fold length; VF-T, vocal fold thickness; TA-T, thyroarytenoid muscle thickness; LP-T, lamina propria thickness; E-T, epithelium thickness; ECL, number of epithelial cell layers).

Trait	Adjusted R2	Slope $\beta$	Intercept	p
VFL	0.71	7.67	6.23	<0.001
VF-T	0.47	2.30	3.05	<0.001
TA-T	0.32	1.29	1.99	<0.01
LP-T	0.31	0.91	0.72	<0.01
E-T	0.16	3.97	20.94	<0.05
ECL	0.31	0.72	4.38	<0.01

Figure 5(A) shows a schematic of the epithelial layer. Figures 5(B)-5(E) show histological sections of vocal fold epithelia from four different species. The large cat exhibited a wave-like interface between the epithelium and superficial lamina propria, resembling dermal papillae in skin, which provide structural support [Fig. 5(B)], possibly an adaptation to the large amplitude vibration during very loud low-frequency vocalizations (Klemuk et al., 2011). The ruminant artiodactyls had some of the thickest epithelial layers, possibly an adaptation to protect against the frequent passage of regurgitated digesta during rumination [Fig. 5(C)]. While some ruminants possess 3–5 layers of keratinized, flattened epithelium on top of cuboid-shaped basal cells, this feature was absent in the North American elk, a species known for its high-pitched bugle calls. The red fox exhibits four to five layers of uniform epithelial cells [Fig. 5(D)]. Rodents have only two or sometimes three layers of relatively flat epithelial cells. Cricetid rodents are known for having vocal membranes [Fig. 5(E)]. These vocal membranes consist of a thin epithelial double layer resting on a robust, thick TA muscle, with a loosely organized lamina propria in between.

FIG. 5.

(A)–(E) Morphology of vocal fold epithelium. (A) Schematic of a mid-membraneous vocal fold section illustrating muscle layer, three lamina propria layers, and epithelium. Between epithelium and lamina propria exists a basal membrane. (B)–(E) Histological mid-membranous sections of vocal fold epithelium from four nonhuman mammals. For each species, a schematic illustration was generated to capture species–specific features. Note the wave-like interface between the epithelium and the superficial layer underneath in the African lion (B), or the extremely thin epithelial layer consisting only of two layers in the California mouse (E). Stains: H&E, hematoxylin and eosin; TRI, Masson’s trichrome; EVG, Elastica-van-Giesson.

The anatomical results suggest that epithelial cell design varies moderately between species. While other vocal fold measurements scale with body size, the thickness of the entire epithelium is weakly associated with body size. All species exhibited between 2 and 10 epithelial layers. It is tempting to speculate that the weak allometric scaling makes the epithelium a strong candidate to help escape the mostly size-dependent fundamental frequency determination.

B. Computational model results

With the steep stress–strain curves and high expected fundamental frequencies, computational stability was an issue. With a sampling frequency of 8 × 44 00 Hz, a fundamental frequency as high as 3.3 kHz was reached. Figure 6(A) shows a time snapshot of the diameters of the airway from the entry of the trachea (left) to the lips (right). Color in the airway indicates pressure at one moment in time (red being positive, green being negative, relative to atmospheric pressure). These pressures change dramatically with every time step as a result of source excitation and wave propagation in both directions. The supraglottal vocal tract was chosen to be a uniform tube with a narrow larynx canal. As mentioned before, there is no vowel identity in a human whistle voice; hence, a uniform (neutral) vocal tract was a reasonable choice. The total length of the supraglottal airway was 14.28 cm, typical for a human female. The larynx canal portion was 1.59 cm long, with a cross sectoral area of 0.4 cm². The remainder of the supraglottal tube had a 3.0 cm² cross sectional area. With this configuration, the supraglottal airway resonances were 612 Hz and odd multiples thereof (1838, 3063, 4288, 5513, …), modified slightly by the narrow larynx canal. In isolation, the larynx canal would have its own resonance frequency at 5470 Hz as a short closed–open tube, but coupled to the vocal tract, it modifies the third, fourth, and fifth supraglottal resonance frequencies slightly.

FIG. 6.

(A)–(C) Modeling production of extremely high $f_{\mathrm{o}}$. (A) Discretized airway for computer simulation. Colors show pressures in the airways at one moment in time (red is positive pressure and green is negative pressure). These pressures vary with every time step. (B) Fundamental frequency glide with a 0.1 mm epithelial depth, a 0.25 mm superficial (gel) depth, a 0.1 damping ratio, and a lung pressure of 2.5 kPa. The epithelial stress was increased from 8 to 12 MPa and then decreased to 6 MPa. (C) Fundamental frequency glide with a reduced 0.2 mm superficial (gel) depth, reduced damping ratio (0.05), and slightly higher lung pressure (3.0 kPa). The epithelial stress was increased from 3 to 15 MPa and then decreased back to 3 MPa.

Two vocal ribbons (left and right) formed elliptical cross sections in the glottis. With four sections (each 0.1 cm–wide) from glottal entry to exit, the ribbons could flex in the flow direction, producing both convergent and divergent glottal shapes. The medial surfaces of the vocal folds are not parallel but have a convex shape when the vocal folds are brought together for vibration. The glottis is wider at entry and exit, and narrower in the center. We gave the ribbon this convex shape with four sections from bottom to top.

The major diameters of the four elliptical glottal cross-sections were kept at 1.0 cm, the vocal fold ribbon length. The minor diameter of each section was adjusted for self-sustained oscillation. Throughout all ranges of frequencies chosen, oscillation occurred when the four minor diameters were 0.10, 0.04, 0.03, and 0.06 cm, giving the ribbon the desired curvature. The damping ratios for the four vertical sections were 0.1, 0.1, 0.1, and 0.6, glottal entry to exit, following Ishizaka and Flanagan (1972).

As a simulation example, the lung pressure was chosen to be 2.5 kPa, the epithelium depth was 0.1 mm, and the superficial gel depth was 0.25 mm. Figure 6(A) shows an up–down frequency glide for a relatively small pitch range around 2500 Hz. The $f_o$ glide was achieved by gradually increasing the epithelium stress from 8 to 12 MPa and then decreasing it downward to 6 MPa. There were strong harmonics in the spectrum because collisions occurred between the left and right ribbons in section 3 (bottom to top). This third glottal section had the greatest amplitude due to the tightest adduction and highly fluctuating driving pressures (positive and negative due to vocal tract interaction).

For a wider frequency range, 1500–3300 Hz as shown in Fig. 6(C), the $f_o$ glide was achieved by gradually increasing the epithelium stress from 3 to 15 MPa and then decreasing it back to 3 MPa. The lung pressure was raised from 2.5 to 3.0 kPa. The epithelium depth was kept at 0.1 mm, but the superficial gel depth $D_{\mathrm{s}}$ was reduced from 0.25 to 0.20 mm to increase the average ribbon stress [recall Eqs. (5) and (6)]. The damping ratios were cut in half to 0.05, 0.05, 0.05, and 3.0 for the four ribbon sections. The glottal half-widths were kept at 0.05, 0.02, 0.015, and 0.03 cm, and the larynx canal areas were kept at 0.4 cm² each. Note that there are bifurcations in the oscillation regime, owing to greater lung pressure and less tissue damping.

With increased source energy, there was more interaction with the vocal tract. In the first 100 ms of the signal, the harmonic frequencies are altered by the resonances of the vocal tract. The first interaction (a frequency uncertainty) occurs when the fundamental frequency passes through the second resonance frequency of the vocal tract (about 1800 Hz on the upglide). A smaller interaction occurs when the fundamental passes through the third resonance frequency at about 3000 Hz. Thereafter, as the natural frequency of the ribbons is raised, the oscillation frequencies rose according to the programmed natural frequencies. However, another bifurcation occurred when the second harmonic crossed the resonance of the larynx canal (5470 Hz).

This entrainment pulled all harmonic frequencies upward for a few milliseconds, after which the harmonics resumed their original trajectory. On the downward glide, the smaller bifurcation was not evident, suggesting some kind of hysteresis.

C. Discussion

While a large inventory of simulations could be presented here, more detail of the biomechanical and anatomical parameters of the airway would be needed. Nevertheless, the computations have shown that high fundamental frequencies are attainable with surface vibrations in an epithelium-like tissue layer less than 1 mm in depth. The hypothesis that neither the ligament nor the TA muscle need to be engaged in vibration has received some validity. For research on human singing, a biomechanical explanation of whistle voice (Mechanism 3, or M3) is hereby proposed. For general mammalian vocalization, while size is the main determinant of ${\mathrm{f}}_o$ range, epithelial dominance may assist in escaping from the upper limit. The vocal range can be expanded.

Morphological features assist in elevating fundamental frequency beyond values predicted by allometric scaling of larynx size. For example, length reduction has been implemented as a surgical approach for trans-female alteration (Wendler, 1990; Mastronikolis et al., 2013). The procedure has been modeled with computation (Titze et al., 2021). By shortening the vibrational length by 50%, a fundamental frequency increase of 30%–40% was reported. An exact inverse relation between $f_o$ and vibrating length was not achieved. The most plausible explanation is that the combined stiffness of multiple tissue layers does not scale in a simple proportion to vocal fold length. Thus, vibrating only a portion of the vocal fold (i.e., like damping a portion of a violin string with a finger) is not likely to produce multiple octaves in the kilohertz range. Hence, dominance of a yet poorly described tissue layer, the lamina densa of the epithelium, was proposed here as a more likely mechanism for extremely high-pitched vibration.

Humans appear to be able to produce extremely high $f_o$ easier at a younger age (Herzel and Reuter, 1997) and women are the demographic for the production of “whistle voice” (Walker, 1988; Švec et al., 2008; Garnier et al., 2012). Assuming that ligament and muscle properties are not yet fully developed prior to adulthood, the system may favor the transfer of tension control to the epithelium. The shorter vocal fold length in females and children is also an obvious advantage for high $f_o$ if a vibrating ribbon or string model is assumed.

A comparison can be made between high-pitched sounds produced using vocal membranes (seen in animals like bats, mice, and nonhuman primates) versus the epithelial dominance hypothesis. Vocal membranes are small structures found along the edge of the vocal folds. They are made of two layers of epithelium with some connective tissue in between and remain visible even after the tissue is preserved [see Fig. 5(E)]. These membranes help produce louder and higher-pitched sounds by lowering the pressure needed to start vocal fold vibration (Mergell et al., 1999; Kanaya et al., 2022).

On the other hand, the epithelial dominance hypothesis suggests that during very high-pitched sounds in humans, only the outermost layers of the vocal fold—specifically the epithelium—are gradually recruited to vibrate, with the epithelium and a thin superficial gel layer vibrating on their own at the highest pitches. Since very high frequencies are rare in human voices, not everyone may be able to achieve this. Building on Hirano’s body-cover theory, we suggest that strong activation of the cricothyroid (CT) muscle, with low or moderate activation of the thyroarytenoid (TA) muscle, could cause the epithelium to bulge outward and take over the vibration (Hirano, 1974; Vahabzadeh-Hagh et al., 2018). Unlike in vocal membranes, this bulging could be temporary and reversible.

There are also biomechanical differences that require further exploration. Vocal membranes are likely to have a greater rotational component in vibration, being bounded laterally to a smaller surface of inner vocal fold tissue layers. The anterior and posterior boundary conditions are not well known for either the membrane or the epithelium. Given that the epithelium lines the entire surface of the vocal folds, its boundary conditions for the vibrating portion cannot be determined a priori.

A permanent or temporal vocal membrane-like extension may represent phenotypic variants of the epithelial layer, which support or hinder different types of vocal behavior. A vocal membrane-like extension sometimes co-occurs with the sulcus vocalis, a fine longitudinal furrow along the medial edge of the vocal fold (Nakayama et al., 1994; Hsiung et al., 2000). In one study of human vocal folds, sulcus vocalis was found in 50% of patients with laryngeal cancer and in 20% of a control group without known laryngeal pathology, suggesting vocal membranes and sulcus vocalis may also occur as physiological variants (Nakayama et al., 1994).

D. Limitations of this study

The study revealed only weak correlations between epithelial thickness and body size. However, it is important to note that species used in microscopic morphometry were represented by only one or a few individuals, which raises concerns about the small sample size. Epithelial morphology can be influenced by factors such as age (Filho et al., 2003), meaning our findings may not fully capture the range of vocal fold epithelial phenotypes within species. Future studies should also incorporate molecular components, characterize proteins in the subepithelial extracellular matrix, and conduct rheological investigations to better understand the role of intercellular junctions, the actomyosin cytoskeleton, and intermediate filaments, in determining how stress is distributed across the epithelial layer. To better understand the stress–strain relationships of the basement membrane, parameters similar to ${\sigma }_{\mathrm{o}}$, $B$, and ${\epsilon }_0$ (as listed in Table II for muscle, ligament, and mucosa) should be expanded to include the epithelial layer across multiple species for further empirical validation.

Finally, computational models should consider varying boundary conditions at the end points of the vocal folds, as not all fibers terminate similarly at the arytenoid and thyroid cartilages. While finite-element models with sub-millimeter element sizes will require high-fidelity computations, such efforts are most productive once viscoelastic and geometric parameters are known with greater accuracy.

V. CONCLUSIONS

While mean fundamental frequency in vocalizing species is highly correlated with body size, and more specifically with vocal fold length, the range of $f_o$ varies greatly with the layered structure of vocal fold tissues. Periodic vocal fold vibration is possible if one layer dominates with its natural frequency. Here, we have shown that the thyroarytenoid muscle, the vocal ligament, and the epithelium can all produce the tissue stresses for dominance in three separate frequency regions.

This study was a first attempt to explain a possible mechanism by which humans (and perhaps other species) can produce fundamental frequencies well into the kilohertz range with vocal fold lengths on the order of 1.0 cm. It follows a logical connection between depth of vibration and frequency in surface waves on solids and liquids. As depth of vibration shrinks, the tissue closest to the surface becomes dominant in stiffness. Roubeau et al. (2009) suggested the labels M1, M2, and M3 for distinct mechanisms of vocal fold oscillation in humans. The M3 mechanism was not described in detail mechanically, but perceptually, it is known as the “whistle” register. With further investigations, it may become clear that M1 is characterized by TA muscle dominance, M2 by ligament dominance, and M3 by epithelium dominance.

DATA AVAILABILITY

The data that support the findings of this study are available from the authors. Voice samples are publicly available.