Brownian dynamics for the vowel sounds of human language

We present a model for the evolution of vowel sounds in human languages, in which words behave as Brownian particles diffusing in acoustic space, interacting via the vowel sounds they contain. Interaction forces, derived from a simple model of the language-learning process, are attractive at short range and repulsive at long range. This generates sets of acoustic clusters, each representing a distinct sound, which form patterns with similar statistical properties to real vowel systems. Our formulation may be generalized to account for spontaneous self-actuating shifts in system structure which are observed in real languages, and to combine in one model two previously distinct theories of vowel system structure: dispersion theory, which assumes that vowel systems maximize contrasts between sounds, and quantal theory, according to which nonlinear relationships between articulatory and acoustic parameters are the source of patterns in sound inventories. By formulating the dynamics of vowel sounds using interparticle forces, we also provide a simple unified description of the linguistic notion of push and pull dynamics in vowel systems.

Figure 1

Formant data for the vowel sounds of 76 speakers of American English from the original study by Peterson and Barney [6], where speakers were recorded reading the words heed, hid, head, had, hod, hawed, hood, who'd, hud, and heard. Axes have been reversed so that the positions of vowels correspond to those in the IPA vowel chart [5]. The approximate shape of this chart has been superimposed on the formant data, to illustrate the correspondence between mean formant positions (large points), and the arrangement of the traditional chart. Front vowels: blue $\mathtt{\text{i}}$, orange $\text{ɪ}$, green $\text{ɛ}$, red $\text{æ}$. Back vowels: light green $\mathtt{\text{u}}$, gray $\text{ʊ}$, pink $\text{ɔ}$, purple $\text{ʌ}$, brown $\text{ɑ}$. Central vowel: turquoise $\text{ɜ}$.

Figure 2

Early stage configuration of 100 particles (words) in a chamber representing acoustic ($F_1,F_2$) vowel space. Range of attractive force $\alpha =1$, cloud radius $\sigma =1$, diffusion coefficient $D=0.1$. Blue arrows show attractive forces, red show repulsive.

Figure 3

Later stage configuration ($t=0.5$) of 100 particles (words) in a chamber representing acoustic ($F_1,F_2$) vowel space. Range of attractive force $\alpha =1$, cloud radius $\sigma =1$, diffusion coefficient $D=0.1$.

Figure 4

The articulatory to acoustic map $\mathit{g}$. In order to utter a target sound in acoustic space, a speaker must arrange her articulatory apparatus (tongue, lips, laryngeal structures, etc.) into the correct position $\mathit{y}$. The uttered sound will be $\mathit{x}=\mathit{g}\left(\mathit{y}\right)$.

Figure 5

The set of utterances for two different frames whose vowels are part of the same phoneme. The template mean is the average location of the frames, and since frames are attracted toward their template mean, they are attracted to other frames in the same phoneme.

Figure 6

Approximate distribution of $F_1$ values for female front vowels, derived from Peterson and Barney vowel data [6]. Curves are normal densities having means and standard deviations equal to those of the first formants of female front vowels. Color coding matches that in Fig. 1: blue $\mathtt{\text{i}}$, orange $\text{ɪ}$, green $\text{ɛ}$, red $\text{æ}$.

Figure 7

Approximate distribution of $F_1$ values for female back vowels, derived from Peterson and Barney vowel data [6]. Curves are normal densities having means and standard deviations equal to those of the first formants of female back vowels. Color coding matches Fig. 1: light green $\text{u}$, gray $\text{ʊ}$, pink $\text{ɔ}$, purple $\text{ʌ}$, brown $\text{ɑ}$.

Figure 8

The set of utterances for two different phonemes, each consisting of at least one frame. When two phonemes are near to each other in acoustic space, utterances of one may sound like the other increasing the likelihood that these utterances will be not be accepted as valid templates by language learners. Such utterances are represented as open circles in the above diagram.

Figure 9

Shift in the exemplar mean away from the frame position ${\overline{\mathit{x}}}_i-{\mathit{x}}_i$ for one phoneme (modeled by unit variance Gaussian) induced by another identically sized phoneme. Red: exact calculation using acceptance probability (23) when $\sigma =1,\gamma =0.5,R=3$. Black: approximation with the same parameter values (30). Blue: approximation with larger clouds $\sigma =1.5$ and lower functional load $\gamma =0.3$.

Figure 10

Average formant values of the peripheral vowels $\mathtt{\text{i,e,ɛ,a,ɑ,o,ɔ,u}}$ for female speakers of Northern Standard Dutch [51]. Marked formant frequencies give the positions of the extreme vowels $\mathtt{\text{/i,a,u/}}$. Orange region shows the typical frequency range $F_2\in [1400,1700]$ for the second subglottal resonance in female speakers [4] (see Sec. 6).

Figure 11

Symmetric trapezoidal vowel space used in our model, together with the IPA symbols and approximate positions of the nine most common short vowels (see Table 1). Orange region is the periphery of vowel space, purple is the interior. Speakers of English may be surprised by the absence of low back vowels $/\text{ɑ}/$ (7%) or $/\text{ɒ}/$ (4%) which appear in words like $\langle \mathrm{pot}\rangle$. These are rather rare worldwide.

Figure 12

The 15 most common vowel system types following categorization of PHOIBLE repository vowel data. Percentages give frequency amongst 3020 surveys, and integers give system size.

Figure 13

Red dots show positions of standard vowels used for classification. Each colored dot shows the position of a single phoneme obtained by mean shift cluster analysis of quasistable state of $n=150$ frame system. Parameter values: $D=0.025,\sigma =1.5,\gamma =0.5,\alpha =1$. Plot superimposes 20 different simulated vowel systems.

Figure 14

Average number of spontaneously formed phonemes vs template range, for a system of $n=150$ frames. Parameter values: $D=0.025,\sigma =1.5,\gamma =0.5$. Dots show underlying data with darker shading indicating repeated data points.

Figure 15

Phoneme locations for 10 simulations of $n=150$ using low functional load ($\gamma =0.1$). Other parameter values: $\sigma =1.5,\alpha =1.25,D=0.05$. Each set of symbols represent the phonemes from a single simulation.

Figure 16

Average number of spontaneously formed phonemes vs functional load, for a system of $n=150$ frames. Parameter values: $\sigma =1.5,\alpha =1.25$. Blue curve $D=0.025$. Orange curve $D=0.05$.

Figure 17

Nonlinear relationship between articulatory and acoustic parameters. The nonlinear region II divides acoustic/articulatory space into two quantal regions I and III, where speech output is less sensitive to changes in articulation.

Figure 18

Distortion of the acoustic frame/phoneme cloud as a subglottal resonance at the origin is approached (red and green distributions). Each cloud is normal in the articulatory parameter (with $\sigma =1$), and away from the nonlinear region it remains normal (purple cloud). The parameters of $g_1$, given by Eq. (65), are $j=1,w=0.5$.

Figure 19

Solid blue curve shows shift in exemplar mean with threshold $R=2\sigma$, generated by jump in $g_1$ of magnitude $j=1$ and width $w=0.5$, defined in Eq. (65). Solid red curve shows exemplar shift for a sharper discontinuity ($w=0.2$). Dashed curves show least-squares fits of approximate quantal force function (68). Fitting parameters: blue, $Q=0.34,{\omega }_x=1.53$; red, $Q=0.60,{\omega }_x=1.45$. In both cases we have assumed we are at the top of vowel space of $x_2=x_2^{*}$.

Figure 20

A phonemic configuration matching the Yoruba and Italian languages [47], which is highly stable in the presence of a quantal force, but only marginally stable without. This is the second most common vowel system cross linguistically, accounting for 13.4% of language surveys in the PHOIBLE database [53]. Parameter values $\alpha =1,\sigma =1.5,D=0.025,\gamma =0.5,Q=2,{\omega }_x=2,{\omega }_y=2$.

Figure 21

Simulation of a vertical vowel system where vowel length is the allophonic parameter and $\alpha =1,\beta =0.5,{\sigma }_x=1,{\sigma }_z=0.5,D=0.02,\gamma =1$.

Figure 22

The mechanics of push and pull chains, adapted from Labov [17]. Pink and blue dots are utterances of two different phonemes, $A$ and $B$. Dots circled in red have a high probability of rejection as tokens used for learning. Orange circling indicates a lower probability of rejection. Green circle dots are unlikely to be rejected. Mystery walls are required to hold the phonemes together before the push chain begins.

Figure 23

An example of what is commonly called a pull or “drag” chain. Starting from randomized frame locations, four phonemes spontaneously formed. The highest is removed from the system at $t=25$. Parameter values $\alpha =1,\sigma =1.5,\gamma =1,D=0.025$. One time unit = a single speaker's expected lifetime. No momentum.

Figure 24

Black line shows mean location of 100 frames, starting from position $x_i=15$, as diffusion coefficient is increased from zero over $T=5000$ lifetimes. System size $L=30$. Red and blue-green lines show 70th (30th) and 90th (10th) percentile frame locations, which indicate distribution of frames within the system. Vertical olive band shows theoretical location of critical diffusion coefficient $D_c={\alpha }^2/3$ where phoneme disintegrates. Other parameter values $\alpha =2,\sigma =1,\gamma =0$.

Figure 25

Darker green and red lines show locations of two phonemes starting from positions $x=1,5$ in a system of size $L=10$. Lighter lines give 90th (10th) percentile frame locations for these two phonemes, as an indication of size. Inset plot shows age vector sensitivity as a function of time. Horizontal red line shows estimated critical sensitivity $b\approx 1.1$. Model parameter values $\tau =1,\alpha =1,\sigma =1,\gamma =1,D=0.025,\theta =0.5$. Parameters for mean reversion (Ornstein-Uhlenbeck) process followed by momentum sensitivity are $b^{*}=0.9,a=0.05,{\sigma }_b=0.1$.

Figure 26

The interaction of two phonemes (black and red), each with two frames. Solid lines: high-frequency frames ($f=0.45$); dashed lines: low frequency ($f=0.05$). Parameter values $\sigma =1,\omega =5,\gamma =1,\tau =1$. Initial age vector of black frame $\mathrm{\Delta }=2$. Momentum-response parameters $b=0.7,\theta \to \infty$.

Figure 27

Dots: estimated $\lambda$ values (obtained by automated peak identification) when $\alpha =0.5,\gamma =0$. Circles: $\gamma =1$. Initial fluctuations given by high-frequency sine wave. Dashed lines: analytical predictions. Inset: solution to (105) for $\gamma =0.5,D=0.02$, giving $\lambda =1.39$ (cf. prediction ${\lambda }^{*}=1.33$).

Figure 28

Contributions to the net mode growth rate from diffusion (red), attraction (green), and repulsion (blue), when $D=0.02,\alpha =0.5,\gamma =1$. Black dotted curve shows overall mode growth rate $\omega f\left(\omega \right)$ with maximum circled.