Predicting Achievable Fundamental Frequency Ranges in Vocalization Across Species

Abstract

Vocal folds are used as sound sources in various species, but it is unknown how vocal fold morphologies are optimized for different acoustic objectives. Here we identify two main variables affecting range of vocal fold vibration frequency, namely vocal fold elongation and tissue fiber stress. A simple vibrating string model is used to predict fundamental frequency ranges across species of different vocal fold sizes. While average fundamental frequency is predominantly determined by vocal fold length (larynx size), range of fundamental frequency is facilitated by (1) laryngeal muscles that control elongation and by (2) nonlinearity in tissue fiber tension. One adaptation that would increase fundamental frequency range is greater freedom in joint rotation or gliding of two cartilages (thyroid and cricoid), so that vocal fold length change is maximized. Alternatively, tissue layers can develop to bear a disproportionate fiber tension (i.e., a ligament with high density collagen fibers), increasing the fundamental frequency range and thereby vocal versatility. The range of fundamental frequency across species is thus not simply one-dimensional, but can be conceptualized as the dependent variable in a multi-dimensional morphospace. In humans, this could allow for variations that could be clinically important for voice therapy and vocal fold repair. Alternative solutions could also have importance in vocal training for singing and other highly-skilled vocalizations.

Fig 1. Schematic representation of vocal fold tissues, indicating three main layers: epithelium, lamina propria, and muscle.

The lamina propria is further differentiated into a superficial, an intermediate and a deep layer. The intermediate and deep layers constitute the vocal ligament.

Fig 2. Two contrasting cases for obtaining a similar f_o range.

(a) A steep stress-strain curve with small elongation, and (b) a shallow stress-strain curve with large elongation.

Fig 3. Relations between logarithmic fundamental frequency ratio (high/low) and the respective vocal fold length ratio (long/short) for vocal fold tissue characterized by different B-values (from Eq 3).

A value of L₁ = 0.7L₀ was assumed.

Fig 4

Empirical data from Table 1, (a) cadaveric vocal fold length L₀ versus body mass, (b) minimum and maximum fundamental frequency versus body mass, (c) derived minimum and maximum fundamental frequency versus cadaveric vocal fold length L₀, (d) B-value versus cadaveric vocal fold length L₀; the trend line was calculated without one outlier (rhesus monkey).

Fig 5. Contour plot of predicted fundamental frequency range (high/low, f_o2/f_o1 ratio) for morphological variables B and L₂/L₁.

The range depends on two important factors: the rotational flexibility of the laryngeal framework, which facilitates L₂/L₁; and the B value that quantifies the tissue stress response to elongation. For a given B value, a larger fundamental frequency range can be achieved with greater rotational flexibility. For a given L₂/L₁ ratio, a larger frequency range can be achieved with a greater B value. Note that the changes in the B value are not large to achieve a larger frequency range for a given a given L₂/L₁ ratio.