*2.5. Size-Rank Law*

The size-rank law mathematically connects the Brevity and Zipf's laws, indicating that the words of larger rank tend to have larger size [4]. Results for the case of words are depicted in Figure 7. Interestingly, despite the precision problems of Glissando for short durations already described previously, the size-rank law holds more robustly than the brevity law for both Catalan and Spanish. The slight variations in the exponent *θ* of Catalan (0.06) and Spanish (0.058), with respect to English (0.07), are here a consequence of the variations in the *λ* exponents of brevity law (Table 3).

**Figure 7.** Size-rank law for words. Linear-log representation of word size - versus rank of all words (blue dots denote binned data) in Catalan (**left**) and Spanish (**right**). The black dashed line is a fit of raw data (light grey dots) to the size-rank law (see Table 1), i.e., the fit of this law is not done to the binned data, however its agreemen<sup>t</sup> is excellent.

#### *2.6. Menzerath–Altmann's Law (MAL)*

The results of fittings of the Catalan and English corpus to MAL for different scales are depicted in Figures 8 and 9. For the scale of BGs vs words (Figure 8), MAL holds well when the size of the constituent is measured in physical units of time duration (outer panels) and it is either poorly or not fulfilled when the size is measured in symbolic units such as number of letters or number of phonemes per word (inset panels). Coefficients of determination *R*<sup>2</sup> = 0.47 for Catalan and *R*<sup>2</sup> = 0.84 for Spanish when size is measured in time duration, to bee compared with Catalan *R*<sup>2</sup> = 0.23 (Catalan, characters), *R*<sup>2</sup> = 0.11 (Catalan, phonemes), *R*<sup>2</sup> = 0.04 (Spanish, characters) and *R*<sup>2</sup> = 0.08 (Spanish, phonemes). These results are in agreemen<sup>t</sup> with the case of English [4]. Overall, better agreemen<sup>t</sup> to MAL is found for Spanish than for Catalan in time duration. Results and agreemen<sup>t</sup> to MAL also hold at the word vs phoneme scale. In fact, these results are new clear evidence in favor of the acoustical origin of the law [21] and the physical model explained in [4]. Note that while the size of the BGs are not large enough to reach to observe the range where MAL is inverted (at *b*/*c* ≈ 34 words [4]), the value of the exponents (see Tables 3 and 4) certifies that such regime inversion indeed exists.

**Figure 8.** Menzerath–Altmann law: BG vs words Representation of BG size measured in number of words versus the mean size of those words for Catalan (**left**) and Spanish (**right**), where the size of the words can be measured in physical magnitudes (**main panel**) or symbolic units (phonemes or number of characters, inset panels). Each grey point represents one BG, whereas blue circles are the mean duration of BGs. MAL holds in physical magnitudes (with coefficient of determination *R*<sup>2</sup> = 0.47 for Catalan and *R*<sup>2</sup> = 0.84 for Spanish), while it is poorly fulfilled when the size is measured symbolically (Catalan: *R*<sup>2</sup> = 0.23 for character units and *R*<sup>2</sup> = 0.11 for phoneme units; Spanish: *R*<sup>2</sup> = 0.04 for character units and *R*<sup>2</sup> = 0.08 for phoneme units). Fitted parameters *a*, *b*, *c* are reported in Table 3.

**Figure 9.** Menzerath–Altmann law: words–phonemes relation between the word size measured in number of phonemes versus the size of those phonemes in physical magnitudes. Orange squares represent the mean size of each word. Fitted parameters are shown in Table 4 coefficient of determination for these are *R*<sup>2</sup> = 0.75 for Catalan and *R*<sup>2</sup> = 0.9 for Spanish.
