According to the acoustic theory of speech production (Fant, 1960) it is theoretically possible to remove vocal tract length differences from our descriptions of speech acoustics (Nordström & Lindblom, 1975). Once we have measurements of the vocal tract resonant frequencies (F1-F4) we can use the talker-specific vowel formant distributions to estimate normalization factors. Several normalization methods have been proposed, with more or less reference to acoustic theory (Gerstman, 1968; Lobanov, 1971; Nordström & Lindblom, 1975; Nearey, 1978; Watt & Fabricius, 2002). The aim of acoustic vowel normalization is to make it possible to express the formant frequencies using a measurement scale that is independent of talker differences, for use in comparing dialects and languages with each other. Vowel normalization may also be a part of the cognitive process of speech perception.

Whether listeners make use of vocal tract length in speech perception is an open question (Johnson, 1997). It has been argued that perceptual compensation for talker differences must be more than simply a vocal tract length normalization, because acoustic differences between men and women (who tend to differ in vocal tract length) is language-specific; male/female differences depend in part on the language or dialect that they speak (Johnson, 2005). This language-specificity of gender differences suggests that there is more involved than just vocal tract length difference. Instead, there appears to be a performative aspect of gender that is overlaid on physical sex differences in vocal tract length. Secondly, it has been observed that perceptual talker normalization effects are conditioned to some degree on higher-level factors that are given by prior phonetic context (Ladefoged & Broadbent, 1957; Johnson, 1990), visual context (Strand & Johnson, 1996), or even experimenter suggestion (Johnson, Strand, & D’Imperio, 1999).

Despite these observations about speech perception, there are a couple of reasons to suppose that vocal tract length normalization might be a component of the speech perception process. First, the perception of a conspecific individual’s body size is important for social organization in many species (e.g., Harrington & Mech, 1979), and there is evidence that vocalizations are used to convey individual characteristics such as size (Reby & McComb, 2003). This suggests that the perception of vocal tract length may be evolutionarily prior to linguistic communication, so language processing may be overlaid on a cognitive structure that already included vocal tract length perception. Second, there are regions of the brain involved in talker perception that do not overlap with regions involved in speech perception (e.g., Van Lanker, Kreiman, & Cummings, 1989; Johnson & Sjerps, to appear), suggesting that the perceptual system may include processes that compute talker information that can be mixed with phonetic information in a stream that produces ‘talker neutral’ phonetic information.

One reason to believe that vocal tract length normalization is not a viable mechanism for speech perception is that, as usually formulated, it relies on information that is extrinsic to the vowel. That is, it is usually assumed that the estimation of vocal tract length is computed over a collection of vowel tokens—information beyond what is available intrinsically in the vowel to be classified. Common experience, and controlled experimentation confirm that isolated vowels are accurately recognized (Nearey, 1989; Strange, Jenkins, & Johnson, 1983). This suggests that each vowel contains the information that is needed for its own recognition, and this intrinsic information is usually sufficient for vowel recognition.

Lammert and Narayanan’s (2015) finding is relevant for this discussion. Building on earlier work in speech engineering (Paige & Zue, 1970; Kirlin, 1978; Wakita, 1977), they devised a regression method using the frequencies of formants F1-F4 to estimate vocal tract length over short stretches of speech. Vocal tract length normalization using four formants F1-F4 should be less reliant on the specific vowel tokens that are used to calculate normalization parameters than methods like z-score normalization (Lobanov, 1971) that normalize F1 (for example) on the basis of information about the distribution of F1 in a set of vowel measurements. Obviously, you cannot accurately estimate the distribution of a formant’s frequencies from a single token, so the Lobanov normalization method cannot be a model of how listeners identify (and normalize) isolated vowels. On the other hand, in vocal tract length normalization, information from F1-F4 determines a normalization scale factor, so each vowel contains a formant pattern within which to evaluate (and normalize) the formants.

Regardless of whether vocal tract length normalization is done in speech perception, descriptive studies of language phonetic systems rely on vowel normalization algorithms to compare speech produced by different talkers or groups of talkers (Disner, 1980; Adank, Smits, & van Hout, 2004). Methods used in these studies are sometimes based on general-purpose statistical normalization techniques such as range normalization (Gerstman, 1968), or z-score normalization (Lobanov, 1971), but more specialized methods using what could be called ‘mean normalization’ (the ratio of x to the mean of x) are also widely used (Nearey, 1978; Watt & Fabricius, 2002).

In vocal tract length normalization (Nordström & Lindblom, 1975), a single normalization scale factor is derived from an estimate of the length of the speaker’s vocal tract. An acoustic factor related to vocal tract length is ϕ, the fundamental frequency of vocal tract resonances in an unconstricted vocal tract (Paige & Zue, 1970; Kirlin, 1978; Lammert & Narayanan, 2015). The factor ϕ is equal to F1 and the other formants are on odd harmonics of this value: F2 = 3ϕ, F3 = 5ϕ, F4 = 7ϕ. ΔF is simply 2ϕ, and is an estimate of formant spacing in an unconstricted vocal tract (Reby & McComb, 2003), where F1 = ½*ΔF; F2 = 1½*ΔF; F3 = 2½*ΔF, etc. Because a single normalization scale factor is used to scale the formants (rather than a separate factor for each formant), this method is called a ‘uniform’ normalization method.

Despite its basis in the acoustic theory of speech production and the interpretation of the normalization scale factor in terms of a physical characteristic of the talker, vocal tract length normalization was rejected almost as soon as Nordström and Lindblom (1975) proposed it, because it did not seem to work very well. However, there has been significant progress in deriving more accurate vocal tract length estimates from the acoustic vowel spectrum, so a reconsideration of vocal tract length normalization is due.

This paper will introduce the ΔF method of vocal tract length normalization in Section 2. Section 3 will describe the methods used in two studies of vowel normalization techniques. The results of the studies are presented in Section 4, and the paper concludes with recommendations in Section 5. The aims of the paper are to evaluate ΔF vowel normalization as a practical technique for acoustic analysis of language data, and also to gauge the feasibility of vocal tract length normalization as a perceptual mechanism.

The data analyzed in this paper are the published American English vowel production data from Peterson and Barney (1952) and Hillenbrand et al. (1995), as distributed by Santiago Barreda in his ‘phonTools’ package for the R statistical programming language.

The data were analyzed in the R statistical programming language, and the normalization algorithms were implemented as illustrated in Table 1. As the variable names in the last four rows of the table make clear, the normalizing factor ΔF can be calculated by any method that provides an estimated vocal tract length from the acoustic vowel formants. The Watt & Fabricius method prescribes taking the mean of formants from particular judiciously selected vowel qualities to ensure that the center of the talker’s acoustic vowel space is adequately captured, to provide a cross-linguistically consistent scale. The implementation here used the mean of the entire sample as the estimate of the center of the acoustic vowel space. This works just as well for vowel classification within language, and avoids subjectivity or other mistakes in the selection of the exemplary vowel tokens.

The regression coefficients used in this paper for the Lammert & Narayanan (2015) method are different from the ones they published because the datasets used here only include the first three formants, while L&N used F1–4 to estimate vocal tract lengths. Dr. Lammert was kind enough to provide two sets of coefficients for regressions fitted from F1–3 for simulated data. Without an intercept (β₁ = 0.28, β₂ = 0.31, β₃ = 0.47) the RMS error in estimated vocal tract length is 1.91 cm. A regression formula that includes an intercept term (β₀ = 262, β₁ = 0.14, β₂ = 0.16, β₃ = 0.25) leads to a smaller error of estimated vocal tract length (1.22 cm).1 The classification studies in the next section used the second formula, the one with an intercept term. The ΔF estimates of the intercept formula tend to be regulated, drawn away from extremely short or long estimates, by the β₀ constant, as further discussed below.2

The comparison of the Lammert and Narayanan (2015) coefficients to the average spacing coefficients, formula (1), is complicated by a slight difference in how they are presented and calculated. Formula (1) is reproduced here as (4) and can be expanded as (5) in the case where we have three formants per vowel. Simplifying (5) into an expression similar to the form used by L&N, we get (7).

Lammert and Narayanan’s no-intercept expression for ΔF with three vowel formant measurements is in (8). Simplifying, we get equation (10).

Now we can compare the coefficients used in the Lammert and Narayanan (2015) calculation for three formant vowels (10) with the average spacing formula (7). The L&N formula has a larger coefficient for F3 and smaller coefficients for F1 and F2, than are found in the average spacing expression. This is likely to lead to better vocal tract length estimates when only one or two vowel tokens are available because the frequency of F3 is less variable across vowels than are F1 and F2. However, as seen in the left panel of Figure 2, with a full set of data, this method (formula 8) produces vocal tract length estimates that differ from the estimates given by the average-spacing formula (4) by only a constant (probably due to the difference between the acoustic effective length of the vocal tract versus the actual length of the vocal tract; Johnson, 2011, p. 42).

Lammert and Narayanan (2015) also fit a formula for estimating vocal tract length that includes an intercept term. This provides a slightly more accurate measure of vocal tract length with a small set of vowel tokens, by regularizing the estimated vocal tract length. This is illustrated in the right panel of Figure 2, which compares the Lammert and Narayanan VTL estimates (formula 2) using a regression formula with an intercept term (11), to the VTL estimates produced by average spacing (4) for the Hillenbrand et al. data. The main thing to notice is that the range of VTL estimates for formula (11) is smaller (min=13.6 cm, max=16.75cm) than the range given by the non-regularized formula (8).

The regression formula is simplified as (12). Notice that the coefficients for F1-3 are about half as large as in the no-intercept formula (10). If we enter F1=500, F2=1500, and F3=2500 into formula (12) we get 1074 Hz. So, about one half of the value of ΔF is determined by the formant frequencies and the rest is determined by the intercept (β₀ = 524). This will tend to shrink the range of ΔF, and with it the range of estimated vocal tract lengths. The formula in (11) results in better vowel classification when vocal tract length is estimated from only a few vowel tokens and was used in the analyses reported here. Unsurprisingly, using Lammert and Narayanan’s no-intercept estimate of ΔF (formula 8) resulted in classification accuracy that was almost identical to that reported for the average-spacing ΔF method.

For each normalization method, normalization factors (ΔF, log(mean), SD, etc.) were calculated over all vowel tokens for a talker, and then the talker’s vowel formant frequencies were normalized using the code in Table 1. After all of the vowel formant frequencies in the dataset were normalized, support vector machines (SVM, Cortes & Vapnik, 1995) were used to classify the vowels. SVM is a supervised machine learning technique that, in this case, can be used to find optimal classification boundaries between the vowel categories given the data. The results of SVM classification can be used to infer an objective estimate of the level of vowel separation in the data. The classifiers built in this study used a radial basis function with gamma = 0.5. These are the default parameters for the R function svm(). Classification performance was evaluated by using the trained model to predict the category membership of each item in the training set and the percent correct vowel classification was computed from these predictions.

The normalized vowels were also used to build SVM classifiers to identify the talker group (male, female, child) of each token. Assuming that the goal in normalization is to remove talker differences, then a lower percent correct talker classification is an indication of better normalization performance.

The studies in this paper neglect two sources of vowel intrinsic information that have been shown to be useful both in automatic speech recognition and human speech perception—fundamental frequency of voicing (Fujisaki & Kawashima, 1968), and vowel inherent spectral change (Nearey & Assman, 1986). Thus, the classification results here are a minimal baseline.

As in study 1, the vowel formant data to be classified in the second study were the full Hillenbrand et al. (1995) or the Peterson and Barney (1952) datasets. However, where in study 1 the normalization factors were calculated over all of the tokens for a talker, in this study the vowels were normalized with scale factors that were computed from randomly selected subsets of the available vowel data.

Different tests were conducted with normalization based on different numbers of vowel tokens per talker. Subsets of size 1, 2, 4, 6, or 9 tokens per talker were tested in the Hillenbrand data set, and subsets of size 1, 2, 5, 10, 15, and 20 tokens per talker were tested in the Peterson and Barney data set. In each test, a random subset of vowel tokens was selected and then normalization scale factors for each of the several normalization methods were computed from the randomly selected tokens. These normalization scale factors were then used to normalize the full set of vowel tokens (all of the tokens in the corpus), and as in study 1, SVM vowel classifiers were then built and the classification accuracy over the test data set was noted. This process was repeated 50 times in each test to give an estimate of the variability of the accuracy scores for a given normalization subset size.

When the subset was a single token, the formant ‘mean’ that was used in the non-uniform normalization methods was set equal to the value of that formant in the selected token. The standard deviation in the Lobanov method was also set equal to the formant value when only a single token was the basis for normalization parameters.

Two additional tests were conducted. In one, a single token of the vowel [ə] was used to calculate normalization factors, and in the second, formants of the corner vowels [i, u, ɑ] were used to define the normalization statistics.

As shown in Table 2, vowel classification with ΔF (average-spacing or Kirlin’s VTL method) is close to state of the art, with vowel classification accuracy approaching 90% correct for the Peterson and Barney (1952) data set, and about 80% correct for the Hillenbrand et al. (1995) data set. Also, reduction of talker information in vowels was comparable to the state of the art achieved by the non-uniform normalization methods proposed by Nearey (1978), Lobanov (1971), and Watt & Fabricius (2002). The Lammert & Narayanan (2015) method of VTL estimation (with intercept) is less accurate. The Lammert & Narayanan no-intercept method (not shown in the table) is virtually the same as the average spacing method.

Vowel classification with average-spacing ΔF normalization is much better than with Nordström & Lindblom’s (1975) method of VTL estimation, but not quite as good as the best non-uniform normalization methods. The ΔF method is also quite comparable with the other models in removing much of the talker group information (e.g., man, woman, child). It is worth noting, however, that chance ‘talker group’ identification (calculated with 1000 random permutations of the datasets) is 40% correct in the Peterson and Barney (1952) corpus, and 31% correct in the Hillenbrand et al. (1995) corpus, so none of the methods fully ‘normalize out’ this information. This is a neglected point in vowel normalization studies (but see the recent insightful discussion in Barreda & Nearey, 2018, who reference a discussion in Hindle, 1978). We ‘normalize’ vowel formant measurements, but gender and age differences are not fully removed. Knowing when a method ‘over-normalizes’ and removes talker variability that actually ‘should’ remain because it is sociolinguistically significant, is an important problem. This highlights the need for a principled approach, grounded in acoustic theory, and also means that regardless of the normalization method used, the residue of talker variation left behind by normalization should be statistically modeled, because we can be sure that some talker information remains.

Figure 3 shows plots of normalized and unnormalized vowel formants for the Hillenbrand et al. (1995) data set. These plots show that vocal tract length normalization using the ΔF method results in a normalized vowel space that is remarkably similar to spaces obtained with non-uniform methods that require more, and less interpretable, parameters. Note that the center of the vowel space is marked by the horizontal and vertical lines. For the Nearey, Lobanov, and Watt & Fabricius methods these lines mark the mean F1 and F2; for the ΔF methods they mark the resonances of the uniform tube.

This analysis indicates that vocal tract length normalization using a single interpretable normalization parameter (ΔF—an estimate of formant spacing in a vocal tract with no constrictions) is comparable to other vowel normalization methods. The talker-independent dimension that is used in this normalized representation is derived from both the formants being normalized (F1 and F2), as well as higher formants (F3 in this case, and also F4; Lammert & Narayanan, 2015), which are less likely to vary as a function of the particular inventory of vowels found in a language. Higher formants, F3 and F4, are sometimes not measured. This study suggests that they should be, and that this additional information about the talker’s vocal tract could be extremely valuable in interpreting the F1/F2 vowel space.

Because the normalization factor ΔF is directly interpretable in terms of a physical property of the talker, vocal tract length normalization is valid for cross-linguistic comparison of vowel spaces. In addition, it is remarkable that the state of the art in vowel formant normalization is almost entirely reducible to normalization in terms of the talker’s vocal tract length. This has not been observed before because phoneticians have not used a measure calibrated to vocal tract length in our vowel normalization schemes. Arguably, Nearey’s (1978) uniform normalization technique, with mean logF*, uses a measure that reflects vocal tract length, although because it is not calibrated to vocal tract length it is a problematic measure, giving incomparable measurement scales for values normalized over mean logF* (F1..F4) versus values normalized over mean logF* (F1..F3). The ΔF measurement scale is the same whether ΔF is estimated from three formants or four.

Beyond the practicalities of having a workable vowel normalization scheme for comparing dialects and languages, the discovery that vocal tract length normalization is a powerful method for reducing some of the talker variability found in speech leads one to wonder whether vocal tract length normalization might play a role in speech perception. Study 2 tested this by limiting the amount of information available to the normalization algorithms in tests of vowel classification using the Hillenbrand et al. (1995) and Peterson and Barney (1952) datasets. Limiting the information available for normalization creates a situation that Barreda and Nearey (2018) call ‘type B’ over-normalization which is “due to noise in the estimated speaker parameters used for normalization.”

Figure 4 shows the results of study 2 on the Hillenbrand et al. (1995) data set, with normalization based on different numbers of randomly selected vowels (1, 2, 4, 6, or 9 tokens), or basing vowel normalization on the corner vowels [i], [u], and [ɑ], on schwa [ə], or on the entire set of observations from each talker (12 tokens). For these later models (12 tokens, schwa, or corner vowels) there was no repeated random selection of a basis for extrinsic normalization, so only one SVM was fit for each normalization method.

Figure 5 shows analogous results for the Peterson and Barney (1952) dataset. With both the HIllenbrand et al. (1995) data and the Peterson and Barney (1952) data, the results show that uniform scaling techniques (Nearey’s uniform scaling method, the ΔF methods—Kirlin, Ave-spacing, and Lammert & Narayanan) improve vowel classification accuracy over unnormalized classification even with a very small random sample of speech. While non-uniform methods, i.e., those that scale each formant using information in the corpus about that formant (z-score normalization [Lobanov], mean ratio [Watt & Fabricius], or log mean difference [Nearey], non-uniform), are more dependent upon the particular vowel tokens that are used to calculate the vowel normalization factors, and need either a carefully chosen sample, or a large sample. Random selection of vowel tokens, as done here, has a catastrophic effect on the non-uniform methods if only a few vowel tokens are taken to represent the talker. In practice, where a large corpus of vowel measurements is available, this does not matter, and in fact the non-uniform methods may reduce talker differences better than the uniform methods in corpus analysis. But these methods are brittle and are generally inappropriate as models of the perceptual process.

Testing the Watt & Fabricius method with randomly selected vowel tokens, as done here, especially goes against the spirit of that method because its main feature is a judicious selection of vowel tokens to represent the full possible range of F1 and F2 for a talker. However, random selection of tokens is justified in this study because it is designed to evaluate the plausibility of extrinsic normalization as a component of speech perception. Listeners are not presented with a judicious selection of vowel tokens, but have to deal with whatever the talker says. It is worth noting that even without judicious selection of vowel tokens, Watt & Fabricius’ mean ratio representation has very good vowel classification performance when a large sample of tokens is taken; however the normalization scale may depend on the specific vowel inventory. Regardless, the Watt & Fabricius method was not designed as a model of perceptual vowel normalization and is clearly not a feasible one.