Introduction

Languages evolve in a way similar to species (Mufwene, 2001). This idea has been applied in modelling language evolution, using languages as species, words as genes, and translation equivalents as alleles of a common genetic loci. This approach, known as phylogenetic linguistics, has provided useful insights into the history of language evolution (Bowern, 2018; Zhang et al., 2019; Sagart et al., 2019).

Nevertheless, it relies on two assumptions: (1) one human individual holds one lexical form (allele) for one concept (loci), and (2) interbreeding is necessary for cross-linguistic lexical transmission. These assumptions may be valid when studying core-words (Swadesh, 1955) that are etymological cognates (Zhang et al., 2019; Sagart et al., 2019). However, to fully understand language evolution, we must consider co-evolution.

Derived from ecology, the term "co-evolution" describes the phenomenon of closely associated species influencing each other and resulting in reciprocal changes (Thompson and Rafferty, 2020). In the context of language evolution, here we adopt this term to denote the interactions between different linguistic varieties that mutually influence and bring about reciprocal changes.

As human languages evolve, they inherently influence one another due to how humans cognitively process language. This is evident in people’s ability to possess multiple lexical forms from different languages for the same meaning (Kroll and Sholl, 1992; Dijkstra and Van Heuven, 1998), as well as their capacity to learn and borrow words and sounds, which allows for instantaneous transmission across related linguistic varieties (Wu et al., 2021). Consequently, words and pronunciations are not only passed down through generations, but also regularly pass over between different languages, like the spread of parasitic features (Mufwene, 2001). Thus, the change in lexical alignment between co-evolving languages likely involves mechanisms distinct from those involved in intergenerational transmission of human genomes.

In terms of cross-linguistic lexical alignment, in addition to the widely acknowledged similarity-based mechanism of language contact (Thomason, 2011), such as English "computer" being borrowed into German as "computer", a well-known evolutionary linguistic phenomenon is that across many languages there is a systematic correspondence between semantically related words and their phonemes (Dyen, 1963, p. 634; Meillet and Ford, 1967; Schleicher, 1967). For instance, English “d” typically corresponds with German “t”, as demonstrated by word pairs such as “deed–Tat”, “deep–tief”, etc. (Grimm, 1967; Verner, 1967) The systematic correspondences between two languages can be inherited from a common ancestral source, but can also be constructed through various processes of language contact, such as lexical borrowing (Jacobson, 1971; Poplack et al., 1988; Thomason, 2011) and analogical spread of sounds (e.g., uvular rhotics across Europe, Trudgill, 1974; and superimposed sound changes in Chinese dialects, Wang, 2010). Systematic correspondence has been studied extensively and has been used to uncover the past connections between languages (Beekes and Vaan, 2011). Despite its vital importance, it remains unclear which general mechanisms are regulating the shifting relationship of systematic correspondence between co-evolving languages, especially when lexical borrowing and sound spreading are also taken into consideration. To our knowledge, few studies have made explicit predictions regarding this topic, except for Dixon’s Punctuated Equilibrium Model (Dixon, 1997), which posits that co-evolving languages tend to converge on a prototype, until the split of peoples interrupts this process. However, it remains unclear how this mechanism applies to systematic correspondence and whether the changes in systematic correspondence are regulated by other linguistic ecological subtleties known to influence the orderly heterogeneous evolution of various linguistic varieties (Labov, 1963) and creole evolution (Mufwene, 2001).

Given the existing understanding of language evolution, three potential theories may be proposed to explain the change of systematic correspondence that occur across co-evolving languages. (1) The theory of attrition predicts that due to the overlaying of phonotactic constraints in neogrammarian sound changes (e.g., Grimm, 1967; Verner, 1967), the residues of lexical diffusion (Wang, 1969), and the accumulation of other exceptions (Mazaudon and Lowe, 1993), the vocabularies of co-evolving languages will drift apart and become less systematically aligned. (2) Instead, a generalisation of Martinet’s (1952) integration theory may suggest that, due to similar external pressure, continuous mutual influences, analogical sound changes (Anttila, 1977), as well as the need to reduce the cognitive cost for maintaining two vocabularies in one mind (Bialystok, 2009), systematicity between corresponding vocabularies would increase over time. (3) Alternatively, the theory of self-regulated adaptation may propose that, the relationship between co-evolving languages are restructured (Mufwene, 2001) to adapt to the changing linguistic ecology. In some cases, a loss of systematicity in one aspect would be compensated in another aspect (Labov, 1994).

Moreover, since similarity-based phonological influences have been widely accepted as a key factor in language contact, particularly in the process of lexical borrowing (Weinreich, 1953; Poplack et al., 1988), it is necessary to ensure that mechanisms based on systematic correspondences are not simply a result of similarity-based influences in language contact.

Please note that systematic correspondence and cross-linguistic similarity in the pronunciation of related words are related concepts but have distinct meanings. To illustrate this, let’s suppose we have language A and language B. In language A, words A1, A2, A3, A4… belong to the same lexical tone class and all have rising tones. In language B, the translation equivalents of these words, B1, B2, B3, B4… also belong to the same tone class, but with falling tones. In this case, we can identify a tonal systematic correspondence "rule" between language A and language B, where A1… and B1… are considered tonally corresponding words. However, it’s important to note that despite their correspondence, A1… and B1… have different tonal contours—one is rising while the other is falling, indicating that they are not similar.

In the current body of international literature on language studies, there is still a lack of clarity regarding the general mechanisms that govern the fluctuating dynamics of systematic correspondence between co-evolving languages, particularly when the influence of lexical borrowing and sound spreading is taken into consideration. As a novel contribution, this paper aims to elucidate and compare these mechanisms to provide a deeper understanding of the intricate relationship between co-evolving languages. This study suggests a vector-based approach to measure systematic correspondence and evaluates the related theories using two co-evolutionary lexical datasets.

The two datasets were chosen to encompass distinct scenarios of language co-evolution. One dataset focuses on the interplay between two related national languages of equal social status, while the other dataset examines the co-evolution of non-literal local sub-dialects alongside a regional high variety and a national high variety. As mentioned earlier, the theory of attrition suggests that in both datasets, there will likely be a decrease in lexical systematic correspondence over time. Conversely, the theory of integration implies that systematic correspondence may increase in both cases. However, with the incorporation of the theory of self-regulated adaptation and Dixon’s Punctuated Equilibrium Model, contrasting patterns may emerge. Specifically, the former dataset, characterised by a split population, is expected to display an absence of consistent directions in the evolution of sound systems during changes in systematic correspondence. On the other hand, in the latter dataset, where two distinct prototypes are identifiable within the intermixed population, lexical pronunciations of the sub-dialects are expected to converge towards these prototypes. Nevertheless, existing research does not offer explicit predictions regarding the varying influences between a more distant national standard and a more similar regional high variety.

Considering the availability of data, we have selected a range of English-German related words spanning from Old English and Old High German to recent English and German pronunciations, to represent the former scenario. Furthermore, to depict the latter scenario, we have selected thirty-year sliced morphemic transcriptions documenting the Chinese dialects spoken in Shanghai. Here we offer comprehensive backgrounds for both lines of study. For more pragmatic data processing details, please consult the “methods” section.

Old to recent English-German related words

English and German, originally closely related West Germanic languages, have a shared history that is characterised by geographic and political separation, as well as complex social interruptions (Hickey, 2012).

Their common roots can be traced back to Proto-Germanic, which in turn can be linked to Proto-Indo-European. Evidence supporting this connection can be found in the systematic correspondence of cognate pronunciations, as observed by the 19th-century historical linguists (e.g., Grimm, 1967; Schleicher, 1967). This relation yield related word pairs such as “deaf-toub” in Old English and Old High German, which became “deaf-taub” in Modern English and German.

Additionally, both languages have been influenced by a higher Church Latin superstratum throughout the Middle Ages. This yield related word pairs such as “tiwesdæg-ziestag” in Old English and Old High German, which became “Tuesday-Dienstag” in Modern English and German.

These etymological factors contribute valuable data to our current investigation. Nevertheless, following the settlement of Germanic tribes such as the Angles, Saxons, and Jutes in the British Isles, who spoke Old English (which included a relatively standardised form in the late 9th century), there are few records indicating regular visits between them and their mainland relatives. On the mainland, German dialects, including documented Old High German, developed alongside some direct contact with Old French. In contrast, the linguistic ecology in the British Isles was shaped by the influences of Viking invasions in the 9th and 10th centuries, as well as the Norman Conquest since 1066. These influential events led to the transformation of Old English into Middle English. Throughout the Middle Ages, English and German likely coexisted and exerted some influence on each other through trade and cultural (primarily missionary) interactions (Gneuss, 1990; Hayden, 2017). However, the traces left on the vocabularies appear to be primarily linked to their shared influence from (Old) French. This can be seen in word pairs like "check-Scheck" and "accent-Akzent" in Modern English and German.

Closer contact between English and German likely resumed in the late Middle Ages, thanks to various factors such as the Age of Discovery, Mercantilism, Protestant Reformation, the introduction of Germanic clergies (although predominantly Dutch-speaking), and later the Enlightenment Movement. These developments played a significant role in the non-Latin national literacy advancements in both regions and ultimately formed the foundation of Modern English and Modern German (Machan, 2012; Hayden, 2017). This period left its mark on the vocabularies of both languages, as seen in word pairs like "coffee-Kaffee" and "smuggle-schmuggeln". However, specific statistics regarding individual proficiency in both languages during this period remain uncertain.

After World War II, there was a notable increase in direct influence between the vocabularies of English and German. However, English likely has a greater impact on the German-native speakers, who ranked among the top 10 in English proficiency (EF Education First, 2022), while the influence of German on English remains restricted.

Despite the extensive co-evolutionary history of British English and German vocabularies, it is crucial to emphasise that these two languages have always been spoken by distinct populations under separate regimes, each maintaining its linguistic standards. Their relatedness does not imply one language serving as a standard for the other. Therefore, the English-German dataset serves as a representative example illustrating the interaction between two related national languages of equal social status.

To the best of our knowledge, no direct statistical study has been conducted to analyse whether the previous systematic correspondence between the two vocabularies has diminished, strengthened, or undergone more complex fluctuations during the evolution of the two languages from their older versions to their modern forms, including their recent pronunciations. By contrasting this scenario with the upcoming dialectal case to be presented in the subsequent section, we can gain valuable insights for evaluating the three hypotheses that we reviewed earlier.

Related morphemes in the Chinese dialects spoken in Shanghai

The ecology of Chinese dialects exhibits certain similarities to that of European languages. The linguistic varieties within each language family can be traced back to a common ancestor, possess etymologically related vocabularies, and demonstrate a notable variation in mutual intelligibility (Tang and Van Heuven, 2009; Gooskens et al., 2018). However, throughout the shared history of Chinese dialects, two distinctive features have consistently endured, making them particularly intriguing for the current research.

First, Chinese regional dialects have almost always coexisted with a national standard (Confucius, 551BC–479BC), actively encouraged and supported by the ancient Chinese empire (GUO Pu, 276–324a; 276–324b). This multi-dialectal ecology has been compared to the diglossia observed in medieval Europe (Ferguson, 1959), where a privileged few used classical Latin alongside the vernacular languages, while the majority remained monolingual. However, evidence from Missionary documents reveal that by the 16th century, there was already widespread individual bi-dialectism involving the national standard, even among the least educated Chinese populace in certain regions (Ricci, 1552–1610). This extensive and endurant influence of the national standard has left a lasting impact. Modern dialectology studies often uncover historical superstratum of standard Chinese integrated in the lexical phonology of Chinese dialects (e.g., Wang, 2010).

The second notable characteristic of the Chinese dialectal ecology is the use of shared ideographic characters (known as "Zi") to represent related monosyllabic morphemes across dialects. Traditional Chinese rhyming books, such as those for the national standard (e.g., see Pulleyblank, 1998) and regional dialects (e.g., Li, 2019), organise their entries based on these related morphemes. Furthermore, throughout history, there are documentations showing both literate and non-literate Chinese speakers discussing the different pronunciations of certain characters across Chinese dialects (Wang, 2023), demonstrating their understanding of the cross-dialectal relationship between these linguistic units.

These two features make the co-evolution of Chinese dialects an ideal testing ground for investigating the scenario of linguistic varieties co-evolving with identifiable common prototypes and a sizeable and stable bi-dialectal population.

These features likely apply to the language ecology in Shanghai, a thriving migration city situated in the prosperous Lower Yangtze plains. The majority of urban Shanghainese (a Wu dialect) speakers are proficient in Standard Chinese (Putonghua, a standardised common speech with pronunciation based on the Beijing dialect, according to Standing Committee of the National People’s Congress, 2000). While there is no direct data on mutual intelligibility between the two, it has been established that a closely related dialect of Shanghainese, Suzhou Wu Chinese, is challenging for monolectal Beijing Mandarin speakers to comprehend (only 26% intelligibility in isolated words, according to Tang and Van Heuven, 2009). Shanghainese and Standard Chinese only partially overlap in terms of sound inventory and phonotactics. While most translation equivalents between them have etymological connections, some do not. Additionally, the range of similarities among their related morphemes can be extensive.

Chinese researchers have studied the sound changes in central urban Shanghainese over the past 160 years (Qian, 2003; Chen, 2019). They observed the influence of national standard on the reorganisation of phonological categories in Shanghainese. For example, characters originally pronounced with /ʑ/ initials (Edkins, 1853; Zhao, 1956) began to split into /ʑ/ and /ʥ/ categories in the 1960s (Jiangsu, 1960; Xu and Tang, 1988), depending on their pronunciations in Standard Chinese (Chen, 2019). This suggests a strengthening systematic correspondence between Shanghainese and Standard Chinese, which appears to support the integration hypothesis.

Although Standard Chinese and urban Shanghainese are the predominant dialects in Shanghai, there are also distinct sub-dialects within the city (You, 2013). These sub-dialects were inherited from historical prefectures like Suzhou-Fu, Songjiang-Fu, and Jiaxing-Fu. It remains unclear whether these sub-dialects, coexisting with the two dominant varieties, follow the same evolutionary trajectory as the central urban variety. Further investigation is necessary to clarify the extent to which the integration hypothesis applies and the influence of social factors on the collective changes of these sub-dialects.

Furthermore, previous studies have reported similarity-based influences of Standard Chinese observed in urban Shanghainese. For example, the pronunciation of 全 (whole) in Shanghainese has shifted from /ʑi/ (Edkins, 1853; Zhao, 1956) to /ʥyøn/ (Jiangsu, 1960; Xu and Tang, 1988), resembling the Standard Chinese pronunciation /ʨhyæn/. Then with more characters undergoing similar shifts (Chen, 2019), a new mapping rule formed between the two systems and hence influence the relation of correspondence.

Despite the existing research, it is necessary to go beyond specific examples and explore general mechanisms that can explain the two divergent scenarios and shed light on the relationship between correspondence-based and similarity-based mechanisms.

Quantified analyses on systematic correspondence

To investigate systematic correspondence in a quantitative manner that allows for statistical modelling and hypothesis testing, researchers need to define two types of units: the unit of data entries and the unit of systematic consideration. Previously, linguists commonly used mono-morphemic words, e.g., Latin "pater"~Gothic "faran(ire)" as used to support Grim’s Law (Grimm, 1967), or single morphemes, e.g., Chinese monosyllabic morphemes as used in the Chinese dialectological studies, as data entries, while levels of phonological units like consonants (e.g., Grimm, 1967; Chen, 1973) and vowels (Jespersen, 1909), as well as initials, rhymes, and tones (Karlgren, 1915–1916) were considered systematically. This prevailing method involved listing and enumerating word or morpheme pairs that correspond at a specified phonological level, following practices used by scholars like Grim, who was born in 1785. Grim enumerated Latin-Gothic examples to verify connections between phonological units (Grimm, 1967). Recent studies have continued this approach.

However, this approach does not account for the possibility that observed correspondence may occur by chance, especially when analysing small sound inventories like lexical tones. It may also overlook correspondences between small categories with limited data entries. To address similar limitations, Baxter (1992) applied Bayesian statistics to examine the rhyme categories of Old Chinese by studying the rhyming relationships between characters found in ancient Chinese poetry collections. This probabilistic strategy mitigates the impact of chance occurrences and addresses data scarcity in smaller categories.

By adopting a similar approach, we can use the Chi-square test to investigate systematic correspondence between sound inventories of languages A and B. The Chi-square test helps examine the association between categorical variables like consonant inventories. Additional Chi-square tests were conducted for this study, and the reports can be found in the supplementary information file.

While the Chi-square test typically supports the existence of systematic correspondence by rejecting the null hypothesis, it’s valuable to measure correspondence on a more nuanced scale. This involves analysing data at a category-by-category or word-by-word level, providing a comprehensive understanding of the strength and patterns of correspondence between phonological units that may not be captured by the Chi-square test alone.

Previous studies have made limited attempts to address the research question at hand. One such attempt utilised Linear Mixed Effect Modelling (LME) (Bates et al., 2013) to explore the relationship between distance matrices representing tonal categories in Standard Chinese and the Euclidean distances between pitch contours in a northern Mandarin Chinese dialect. This study found that later birth years are related to increasing correlation between the matrices, and further uncovered the influences from cross-dialectal similarities in pronunciation, as well as participants’ literacy education and auditory working memory (Wu et al., 2016). These results suggest alignment with the theory of self-regulated adaptation. However, the use of distance matrices as both independent and dependent variables introduced autocorrelation issues, potentially impacting the reliability of the models. Additionally, we recently became aware of an unpublished work by non-professional language enthusiasts who attempted to quantify phonological correspondence by multiplying data entries proportions for each mapping (Gu, 2023).

These endeavours reflect initial efforts to associate linguistic variation and microscopic language evolution with human lexical processing. However, all of these statistical approaches rely on the assumption that lexical retrieval events are independent, analogous to drawing coloured balls from a box (Baxter, 1992). Yet, in reality, human lexical processing involves complex simultaneous activation and competition within bilingual lexicons (Dijkstra and Van Heuven, 1998). Moreover, these studies did not account for the potential cognitive competition between mapping rules and the weight of each sound category within the overall sound inventory. Therefore, it may be more appropriate to envision the lexical processing of related linguistic varieties as drawing magnetic balls from a complex magnetic field. Hence, careful consideration of the competition between data entries and mapping rules is crucial when exploring the co-evolutionary mechanisms of related vocabularies. Furthermore, the comparability of measurements across different phonological units warrants further investigation.

Methods

The current measurement: weighted cosine systematicity

Here, we propose a vector-based measurement called Weighted Cosine Systematicity (sys_cos_w) to quantify systematic correspondence. This method represents sound inventories as multi-dimensional vectors, with each dimension corresponding to a specific sound category’s number of data entries. By employing this approach, we can effectively capture the competition among sound categories in the relationship between the two vocabularies. Moreover, the value assigned to each dimension reflects the quantity of data entries (such as morphemes) involved in this competition.

The weighted cosine systematicity accounts for three crucial factors. Firstly, a higher number of categories and/or data entries in competition with the mapping relationship indicates weaker systematic correspondence. Secondly, as the mapping under consideration involves a greater number of data entries, it implies stronger systematic correspondence. Additionally, the measurement enables the evaluation of the relative importance of the two units directly involved in the mapping by analysing their respective occurrence proportions within the total number of word pairs. This assessment provides insights into the significance of these units within their respective vocabularies.

As demonstrated in Fig. 1, for two linguistic varieties A and B, the systematic correspondence between unit a (from A) and unit b (from B), a~b, is divided into two mapping relations: a→b (indexed as “ab”) and b→a (indexed as “ba”). The mapping a→b shows the sufficiency of a for b and the necessity of b for a, which depends on the alternatives of b given a, and vice versa for the mapping b→a. Therefore, sys_cosab and sys_cosba are each mathematically defined as the cosine similarity between a target vector v = (0, …, npairthis, …, 0 m) and a reference vector vref = (napair1,…npairm). Then sys_cosab and sys_cosba are weighted with the proportions of a and b occurrences within the total number of word pairs, weighta and weightb, respectively, resulting in sys_cos_wab and sys_cos_wba. Consequently, systematic distances sys_distab and sys_distba can be calculated by subtracting sys_cos_wab and sys_cos_wba from one.

Fig. 1: Calculating weighted cosine systematicity (sys_cos_w) and systematic distance (sys_dist).
figure 1

Words clustered and labelled according to phonemic units (red for a, light-blue for b). Upper plot edges show pairings with direction of mapping. Lower plots show vectors v (red) and vref (black) with included angle and Euclidean distance. To calculate sys_cos_w, each mapping is represented as two vectors (v and vref), each of length equal to the count of this mapping’s related mappings. Vector elements are pair counts. Sys_cos_w is cosine of included angle (v-vref) multiplied by proportion of involved pairs (npair) to total number of pairs (n).

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Figure 1 is based on a dual-lexical system with 3079 Etymologically Related Translation Equivalent (ETE) pairs (n = 3079). Regarding the mapping between the units a (the rhyme /uai/ in the linguistic variety A) and b (the rhyme/uA/ in the linguistic variety B), there are eleven ETE word pairs (npairthis = 11) involved.

  1. (1)

    The unit a is involved in nineteen ETE word pairs (npaira = 19), mapped to four categories in B (m = 4, which involves one (/uai/~/A/), two (/uai/~/E/), eleven (/uai/~/uA/), and five (/uai/~/uE/) ETE word pairs, respectively. Accordingly, for the mapping a→b, the reference vector vref is (1, 2, 11, 5), the target vector v is (0, 0, 11, 0), and the cosine similarity calculated between vref and v is taken as the systematicity quantified for mapping a→b (sys_cosab = 0.895). Furthermore, by calculating the proportion of word pairs involved with unit a in the whole system, weighta = npaira/n = 0.006, the importance of mapping a→b can be weighted (sys_cos_wab = weighta × sys_cosab = 0.005). Then the systematic distance can be calculated by subtracting the weighted systematicity from one (sys_distab = 1 – sys_cos_wab = 0.995).

  2. (2)

    Similarly, the unit b is involved in twenty-two ETE word pairs (npairb = 22), mapped to two categories in A (m = 2), which each involves eleven (/uai/~/uA/ and /uA/~/uA/) ETE word pairs. Accordingly, for the mapping b→a (or mapping a←b), the reference vector vref is (11, 11), the target vector v is (11, 0), and the cosine similarity calculated between vref and v (sys_cosba = 0.707) is taken as the systematicity quantified for mapping b→a. Also, by calculating the proportion of word pairs involving unit b in the whole system, weightb = npairb/n = 0.007, the importance of mapping b→a can be further weighted (sys_cos_wba = weightb × sys_cosba = 0.005). Then the systematic distance can be calculated by subtracting the weighted systematicity from one (sys_distba = 1 – sys_cos_wba = 0.995).

Note that, different from earlier practices using crosstabs and/or Chi-square statistics (e.g., Chen, 1973), the current method provides a more fine-grained and integrated measurement: (1) an exact value is calculated for each specific ETE pair; (2) both phonemic frequency and interlingual lexical neighbourhood are incorporated, as the weight and the reference vector respectively; (3) comparisons can be further made across different linguistic levels, e.g., later we can see in the Shanghai dataset that phonemes are at higher order of magnitudes of sys_cos and sys_dist as compared to syllables.

In comparison to the Chi-square approach, our method avoids using a straw-man null hypothesis. Unlike the approach based on two distance matrices, our method directly measures systematic correspondence and does not rely on the secondary interpretation of complex statistical models. While our approach overlaps with the proportion-based approach in certain aspects, it further integrates the consideration of the number of competing mapping relations and the significance of phonemic units. In summary, the weighted cosine systematicity offers a comprehensive approach by considering the competition between sound categories, the quantity of involved data entries, and the importance of the units in the mapping relationship.

Datasets

As introduced earlier, two datasets are used in this work to represent the two scenarios of language co-evolution: (1) the interplay between two related national languages and (2) multi-dialectal interactions with two strata of high varieties. The following datasets are then chosen given the consideration of available data.

  1. (1)

    The dataset old to recent English-German related words (Flippo, 2018; Kroch, 2020; Wiktionary, 2022) includes 1913 sets of related lexical forms from Old English (<1100 AC), Old High German (<1100 AC), Modern English (<1700 AC, but reflecting earlier pronunciations < 1400), and Modern German (<1700 AC), as well as the recent pronunciations for the Modern English and Modern German lexical forms (annotated according to Cambridge Dictionary; Collins Dictionary; Conjugator; Harper, 2022; Linas, 2022; Verbix, 2022) and additionally more recently related English and German words (e.g., /kӕŋɡəˈruː/-/ˈkɛŋɡuru/ kangaroo-Känguru, 1770 by Capt. Cook), yielding 5362 sets of consonant clusters (see Table 1 for examples) and 4625 sets of vowel polyphones (see Table 2 for examples).

    Table 1 Data examples of consonant clusters (5362 sets, the column for notes not included).
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    Table 2 Data examples of vowel polyphones (4625 sets, the column for notes not included).
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    Before the modelling, word pairs/sets with missing data were excluded from consideration. In the modelling of Modern English and Modern German, words emerged after 1700 AC were excluded. In the modelling of recent English and German pronunciations, obsolete words for which no definite pronunciations can be found were excluded. When the modelling involves Old English, Old High German, Modern English or Modern German, the positions of phonemes in words were considered according to Modern English and Modern German. When the modelling only involves recent English and German pronunciations, the positions of phonemes in words were considered according to the recent pronunciations (See supplementary materials for further word-wise details).

  2. (2)

    The dataset thirty-year sliced morphemic transcriptions for Chinese dialects in Shanghai (You, 2013) includes IPA transcriptions for 3151 Chinese morphemes from twenty-two local Chinese dialectal varieties spoken in Shanghai collected in 2008, within which ten dialectal varieties (sub-dialects) were sampled twice, in the 1980s (o: old) and in 2008 (n: new) respectively, and the Shiqu (central urban) sub-dialect was additionally sampled in the 1990s (m: median). Additionally, the corresponding pronunciations in Standard Chinese (SC) were marked by the first author (PSC level 1-B). See Table 3 for examples.

    Table 3 Examples of IPA transcriptions for Chinese morphemes spoken in (3151 entries).
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Each syllabic transcription was split into its respective segmental combinations, onset consonants, rhymes, vowels, final consonants, and tone classes and then organised into tables of pairs, as exemplified in Table 4. It is important to note that the denoting numbers for tone classes hold a significant connection, as they correspond to the medieval Chinese tones. These tones consist of Yiping (1), Yangping (2), Yishang (3), Yangshang (4), Yinqu (5), Yangqu (6), Yinru (7), and Yangru (8) (Pan and Zhang, 2015). These names are widely employed to indicate related tonal categories in different modern Chinese dialects (Ho, 2015), and they also imply a similarity in tonal realisations across various sub-dialects of Shanghai.

Table 4 Examples of pair tables for syllables (Syl), segmental combinations (Seg), onset consonants (Ons), rhymes (Rhy), vowels (Vow), final consonants (Fin), and tone classes (Ton) between SC and the Shiqu_o (old central urban) Shanghainese variety.
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The treatment of the datasets involves the following noteworthy practices:

First, rather than confining the analysis to only core-words with established common origins, as previous studies in historical linguistics and modelling have done (Swadesh, 1955; Zhang et al., 2019; Sagart et al., 2019), the current approach considers all available lexical entries that are related. These entries may have derived from shared origins, historical or recent borrowing, or even borrowing from a third language, thus encompassing a broad spectrum of related vocabularies.

Second, our methodology diverges from classical historical practices that try to match language strata across related languages or dialects, as exemplified by previous studies (e.g., Wang, 2010; Chen, 2019). Specifically, we treat each pronunciation variant for a word as a distinct data entry and cross-reference it with the corresponding word in the other language under examination. For instance, if in language A the word "a" has two pronunciation variants, "a1" and "a2," and its counterpart "b" in language B also has two pronunciation variants, "b1" and "b2," we consider four data entries for modelling: "a1~b1," "a1~b2," "a2~b1," and "a2~b2." We employ this approach because we cannot verify or guarantee that public knowledge, such as "a1" mapping to "b1" but not to "b2," holds true in the speaker populations being examined. Conversely, anecdotal evidence from bilingual/bi-dialectal individuals seems to suggest that they frequently possess knowledge of divergent mappings.

Analyses

We utilised the weighted cosine systematicity (sys_cos_w) and systematic distance (sys_dist) measurements on both datasets. In relation to the English and German dataset, we examined vowel and consonant sys_cos_w and sys_dist overall, as well as at the beginning, middle, and end of words. As it comes to the Shanghai dataset, we tested the two measurements across various combinations of time-slices and locations at different levels including syllables (Syl), segmental combinations (Seg), onset consonants (Ons), rhymes (Rhy), vowels (Vow), final consonants (Fin), and tone classes (Ton). Moreover, for comparison purposes, we estimated pronunciation distances by calculating and analysing Optimal String Alignment (OSA) (van der Loo, 2014) among related lexical/morphemic forms.

Three sets of analyses were then applied on each dataset.

  1. (1)

    The weighted cosine systematicity (sys_cos_w) data from both datasets regarding all the phonemic units conforms to a long-tailed Poisson distribution or a Power Law distribution, so they were multiplied by 1000 and natural log-transformed before being fed into Mixed-Linear-Effect models (nevertheless, many subsets of the corrected data may still not conform to normal distribution). The LME models (Bates et al., 2013) were fit with time-slices and the directions of mapping as the fixed predictors, as well as mappings as the random predictors.

  2. (2)

    The average sys_dist data were calculated for each linguistic level and linguistic variety, and the average OSA distance for each linguistic variety. Subsequently, both the systematic distance and the OSA distance data were subjected to Multi-Dimensional Scaling (MDS) analysis (Venables and Ripley, 2002), followed by further analysis.

    In terms of sys_dist MDS for the dialects in Shanghai, we initially examined the relationship between the density of the old and new sub-dialects’ neighbourhoods and their distance from the statistical centroid.

    To explore the co-evolution direction within this dialectal dataset, we conducted paired t-tests on the mean systematic distances between ten local sub-dialects’ old and new variations in relation to (1) the statistical centroid, (2) the regional high variety Shiqu, and (3) the national high varieties SC. Additionally, we performed by-sub-dialect paired t-tests on the old and new mean OSA distances to the same three centres, for comparison.

    Furthermore, Pearson and Spearman correlations were employed to assess the correlation between the local sub-dialects’ distances to the central urban dialect and their geographic and commuting distances to the city centre.

  3. (3)

    To investigate the relationship between the initial sys_cos_w values of individual words and the subsequent changes they undergo, we utilised a custom-built multi-line regression function (DOI 10.17605/OSF.IO/56VT8). The function aims to capture the relationship between two variables with multiple straight lines. It takes in a set of data points with x, y coordinates. Given the largest count of the lines intended for it to fit, and the number of bins for the x coordinates, it applies the k-means partition (Hartigan and Wong, 1979; R Core Team, 2022) on the y coordinates of each bin, based on a Silhouette criterion to decide the optimal number of partitions, and fits straight lines with the centroids of aligned partitions across these bins. The coefficients and intercepts of these fitted straight lines are then repeatedly adjusted after assigning each point to its nearest line until it reaches a given number of iterations.

Results

English-German lexical co-evolution

First, we applied the weighted cosine systematicity and systematic distance measurements on the dataset of old to recent English-German related words. Figure 2 presents the obtained sys_cos_w data as network diagrams.

Fig. 2: Network diagrams of systematic correspondences.
figure 2

a1–b2 Sanky flow diagrams showing the relationship between Old English (OldE), Modern English (ModE) and Recent English (RecE), and Old (High) German (OldG), Modern German (ModG) and Recent German (RecG) for vowels (a1, b1) and consonants (a2, b2). c1–e2 The weighted cosine systematicity between English and German units for vowel (c1, d1, e1) and consonant (c2, d2, e2) correspondence networks, in their Old (Old, c1, c2), Modern (Mod, d1, d2), and Recent (Rec, e1, e2) relationships, with English units represented in red and German units in yellow. The width of flow in a1–b2 and the width of links in c1–e2 represents Weighted Cosine Systematicity (sys_cos_w), with dark blue indicating links from left to right (ab) and light-blue indicating links from right to left (ba) in a1–b2 (Allaire et al. 2017).

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Based on the butterfly-scatter-box-violin plots and MDS spaces presented in Fig. 3, no consistent increase or decrease in systematicity and systematic distance nor any clear, long-term pattern was observed in the MDS spaces over time. The trend varies with phoneme types and positions.

Fig. 3: Weighted cosine systematicity and systematic distances between English and German.
figure 3

a1–a4, c1–c4 Butterfly-scatter-violin-box plots show the weighted cosine systematicity multiplied by 1000 and natural log-transformed, across different time-slices and mapping directions. LME results are marked on conditions with positive effects, with main effects annotated in the centre of each plot. Left: sys_cos_ab_w for the mapping English→German; right: sys_cos_ba_w for the mapping German→English. From bottom to top: old, modern, recent. b1–b4, d1–d4 MDS plots based on mean sys_dist measurements illustrate the co-evolutionary trajectories of English and German varieties. Arrows are plotted from old varieties to modern varieties and from modern varieties to recent varieties. Labels: “oldE” for old English, “modE” for Modern English, “recE” for recent English, “oldG” for Old High German, “modG” for Modern German, “recE” for recent German. a1–b4 For vowels. c1–d4 For consonants. a1, b1, c1, d1 In general. a2, b2, c2, d2 At word initial positions. a3, b3, c3, d3 At inter-syllabic positions within words. a4, b4, c4, d4 At word final positions.

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Nevertheless, regarding the OSA distances across related words, LME modelling showed that pronunciations of English and German related words are significantly diverging across generations, told = −16.50, p < 0.0001, trec = 5.22, p < 0.0001 (with modern OSA as the baseline). See Fig. 4c for its OSA MDS space.

Fig. 4: Multi-dimensional scaling (MDS) visualisation of mean optimal string alignment (OSA) distances.
figure 4

a, b MDS plot for OSA distances between syllables across the dialectal varieties spoken in Shanghai, with Standard Chinese (SC) included (a) and excluded (b). An arrow is drawn from each sub-dialect’s old version to its new version, and labels and points are colour-coded according to dialectal varieties. c MDS plot for OSA distances between lexical forms across old (oldE and oldG), modern (modE and modG), and recent (recE and recG) English and German varieties. An arrow is drawn from each language’s old variety to modern vary and from modern variety to recent variety.

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Dialectal co-evolution in Shanghai

Then we applied the weighted cosine systematicity and systematic distance measurements to the dataset of thirty-year sliced morphemic transcriptions for Chinese dialects in Shanghai, which represents a set of non-literal local sub-dialects co-evolving with a regional high variety (Shiqu, central urban Shanghainese) and a national high variety (SC, Standard Chinese).

As shown in Fig. 5, in the MDS spaces, the dialectal varieties distribute in a similar way as magnets in magnet fields, with a sub-dialect’s neighbourhood density decreasing with the increase of its distance to the statistical centroid. The regional and national high varieties (Shiqu & SC) are located at or closely to the centre (except for SC regarding the whole-tonal syllables).

Fig. 5: Modelling of the systematic distances (sys_distm) in the Shanghai dataset.
figure 5

a1–a7 Multi-dimensional scaling (MDS) plots of the mean sys_dist for each dialectal variety spoken in Shanghai, with arrows connecting each sub-dialect’s old version to its corresponding new version and statistical centroids annotated at the top-left of each plot. b1–b7 Triangles representing the relationship between one variety’s mean sys_dist to the statistical centroid (the horizontal coordinate) and its mean neighbuorhood sys_dist (the vertical coordinate, n_neigh = 3, taking three closest neighbours into consideration); Points representing the other varieties’ mean sys_dist (the vertical coordinates) to this variety. c1–c7 Points representing the relationship between one variety’s mean sys_dist to the statistical centroid (the horizontal coordinate) and its mean neighbourhood density ( = 1 – mean neighbourhood sys_dist, the vertical coordinate), again with arrows connecting each sub-dialect’s old version to its corresponding new version. d1–d7 Paired box plots for the range, quartiles, and median between old (o, in the left) and new (n, on the right) varieties’ mean sys_distm to the Shiqu variety, with results of paired t-tests and Wilcox-tests annotated at the top of each plot. These plots are vertically arranged according to the phonemic units: syllables (Syl: a1, b1, c1, d1), segmental combinations (Seg: a2, b2, c2, d2), onset consonants (Ons: a3, b3, c3, d3), rhymes (Rhy: a4, b4, c4, d4), vowels (Vow: a5, b5, c5, d5), final consonants (Fin: a6, b6, c6, d6), and tone classes (Ton: a7, b7, c7, d7). The labels and points are colour-coded consistently.

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The local sub-dialects’ old versus new mean systematic distances and mean OSA distances to the (1) statistical centroid, (2) the regional high variety Shiqu, and the (3) national high varieties SC, as well as the corresponding t-statistics are illustrated by clustered heatmaps in Fig. 6. Within thirty years, all of the sub-dialects experienced a decrease in systematic distances to the Shiqu variety (Fig. 6, b1, except for onsets and tones). Conversely, some sub-dialects’ diverged from the SC variety, while others converged (Fig. 6, c1). Nonetheless, the OSA distances of the local sub-dialects to the SC variety decreased consistently (Fig. 6, c2, except for finals and tones), whereas changes in their OSA distances from the Shiqu variety varied (Fig. 6, b2).

Fig. 6: Clustered heatmaps displaying the changes in systematic distance (sys_distm) and Optimal String Alignment (OSA) distances in the Shanghai dataset.
figure 6

a1, a2 Changes in mean sys_dist (a1) and OSA (a2) between statistical centroid in the Shanghai local dialectal varieties (labelled on the right). b1, b2 Changes in mean sys_dist (b1) and OSA (b2) between the Shanghai Shiqu (central urban) dialect and the other Shanghai sub-dialects (labelled on the right). c1, c2 Changes in mean sys_dist (c1) and OSA (c2) between Standard Chinese (SC) and Shanghai local dialectal varieties (labelled on the right).The phonemic units of syllables (Syl), segmental combinations (Seg), onset consonants (Ons), rhymes (Rhy), vowels (Vow), final consonants (Fin), and tone classes (Ton) are labelled at the bottom of each plot, ordered according to the result of clustering. The heatmaps are colour-coded so that blue indicates decreasing distances and red indicates increasing distances, with higher saturation indicating greater changes. Significance codes and directions of paired t-tests comparing by-sub-dialect mean old and new sys_dist/OSA are annotated at the top of each plot (red↑for increasing, blue ↓for decreasing).

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Systematic distances analysis also showed a decreased distance of the Shiqu variety to SC (Fig. 6, c1, 7th row, except for tones), in contrast to the divergence revealed by OSA distances (Fig. 6, c2, 1st row). See also the modelling of weighted cosine systematicity data in Figs. 7 and 8.

Fig. 7: Modelling of the weighted cosine systematicity (sys_cos_w) between Shanghai Shiqu and the other local dialectal varieties.
figure 7

Butterfly-scatter-violin-box plots are used to show the sys_cos_w values (multiplied by 1000 and natural log-transformed) for different time-slices (old and new) and mapping directions (left: Shiqu→SH local, right: SH local→Shiqu). Each translucent point on the scatter plots represents a single mapping, with its horizontal coordinate representing the sys_cos_w value (the vertical coordinate is jittered for visualisation purposes). The coloured violin shapes indicate the probability density of the sys_cos_w values, while the box plots represent the range, quartiles, median and odd values, annotated with means (yellow for Shiqu→SH loca and red for SH local→Shiqu). Based on the LME results, significance codes are marked on conditions with positive effects, with main effects annotated on the grey cross lines at the centre, and interaction effects in the panel related to the interaction term. Blue shades are added to cells representing positive main effects for the new varieties, and red shades are added to cells representing positive main effects for the new varieties. In addition, blue shades on the right corner of cells represent positive interaction effects for the mapping SH local→Shiqu, and red shades represent negative interaction effects. The subplots are vertically arranged according to the pairs of linguistic varieties, and horizontally according to the phonemic units.

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Fig. 8: Modelling of the weighted cosine systematicity (sys_cos_w) between Standard Chinese (SC) and the Shanghai local dialectal varieties.
figure 8

The butterfly-scatter-violin-box plots show the value of sys_cos_w (multiplied by 1000 and natural log-transformed) for each time slice (old, middle, and new) and mapping direction (SC to SH local, and SH local to SC). Each translucent point on the scatter plot represents one mapping, with its horizontal coordinate representing the sys_cos_w value (the vertical coordinate is jittered). The coloured violin shapes indicate the probability density of the sys_cos_w values, while the box plots represent the range, quartiles, median and odd values, with means indicated in yellow (for SC to SH local) and red (for SH local to SC). Based on the Linear Mixed Effects results, significance codes are marked on conditions with positive effects, main effects annotated on the grey cross lines at the centre, and interaction effects in the panel relating to the interaction term. Blue shades are added to cells representing positive main effects for the new varieties, and red shades for positive main effects for the old varieties. Additionally, blue shades in the right corner of cells signify positive interaction effects for the mapping SH local to SC, and red shades for negative interaction effects. The subplots are arranged vertically according to the pairs of linguistic varieties, and horizontally according to the phonemic units.

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Furthermore, Fig. 9 reveals that the systematic distances between the sub-dialects and the Shiqu high variety are correlated with geographical distances and commuting times: for the old pronunciations, geographic distance and biking time are stronger correlative factors, whereas public-transportation-time is more prominent for the new pronunciations. In general, the correlation coefficients increased.

Fig. 9: The Correlation between real-world and systematic distances in the Shanghai dataset.
figure 9

a1–d7 The vertical coordinates represent scaled mean sys_dist between old (dashed lines) and new (solid lines) Shanghai Shiqu (urban centre) varieties and the other Shanghai local dialectal varieties; the horizontal coordinates represent scaled geographic distance (a1–a7), scaled walking time (in minutes, b1–b7), scaled biking time (in minutes, c1–c7), or scaled public-transportation time (in minutes, d1–d7) between city centre and these locations. The panels are arranged vertically according to phonemic units: syllables (Syl: a1, b1, c1, d1), segmental combinations (Seg: a2, b2, c2, d2), onset consonants (Ons: a3, b3, c3, d3), rhymes (Rhy: a4, b4, c4, d4), vowels (Vow: a5, b5, c5, d5), final consonants (Fin: a6, b6, c6, d6), and tone classes (Ton: a7, b7, c7, d7). Both Pearson and Spearman correlations are annotated in right bottom legends. Triangles on the plots indicate the real-world distance measurements tested that show the highest correlation with the old varieties (blue for Pearson, light-blue for Spearman), and with the new varieties (red for Pearson, fuchsia for Spearman). e Geographic locations where the Shanghai dialectal data were collected, the new varieties are marked with coloured points, the old varieties marked with coloured circles, administrative boundaries marked in grey, and hydrographic objects marked in light-blue (according to You 2013; Baidu Maps 2022a; National Geomatics Center of China 2022). f, g, h MDS plots created using walking time (Walking, f), biking time (Biking, g), and public-transportation time (PT, h) across the sampling sites (according to Baidu Maps 2022b), labelled with the corresponding dialectal varieties. Chongming and Buzhen cannot be accessed exclusively by walking or cycling.

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Comparing changes in systematic (Figs. 5 and 9) and OSA distances (Fig. 4a, b) in MDS spaces shows that new local varieties are mostly on or near the connecting lines between SC (national high variety, which locates much farther away) and old versions in the OSA space, but not in the sys_dist space.

Regression effects on the change of systematicity

Regarding individual word pairs, it appears that they have drastically different directions of co-evolutionary changes. Nonetheless, a word-wise underlying pattern can be seen in both datasets: the Regression Effect over time (Galton, 1879; Senn, 2011) . As shown in Fig. 10, word pairs with high sys_cos_w scores in an earlier time slice tend to have lower or less increased scores in subsequent time-slices. Conversely, those with low sys_cos_w scores earlier tend to have higher or less reduced systematicity later on. This underlying pattern is evident despite the overall converging trend seen in the Shanghai dataset and testified in both datasets.

Fig. 10: Weighted cosine systematicity (sys_cos_w) and its word-wise changes.
figure 10

The horizontal coordinate of each point in each subplot represents the lemma’s original sys_cos_w, and the vertical coordinate represents its subsequent change, as calculated by subtracting the original sys_cos_w from the latter sys_cos_w. The zero line of change (grey solid line) divides each plot into two parts. Mappings considered from different directions are colour-coded, as annotated with legends within the subplots. Fitting lines are plotted within each subplot (solid lines for the mappings left variety→right variety, dashed lines for the mappings right variety→left variety). a For changes of English-German sys_cos_w, vertically arranged according to phonemic units—consonants in general (Con gen), initial consonants (Con ini), inter-syllabic consonants (Con int), final consonant (Con fin), vowels in general (Vow gen), initial vowels (Vow ini), inter-syllabic vowel (Vow int), and final vowels (Vow fin), horizontally arranged according to the original and later time-slices (left: old to modern, right: modern to recent). b For changes of sys_cos_w between SC and Shanghai local dialectal varieties, vertically arranged according to pairs of linguistic varieties, horizontally arranged according to phonemic units—syllables (Syl), segmental combinations (Seg), onset consonants (Ons), rhymes (Rhy), vowels (Vow), final consonants (Fin), and tone classes (Ton). c For changes of sys_cos_w between the Shanghai Shiqu (urban centre) variety and the other Shanghai local dialectal varieties, arranged in a similar way as in b.

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Discussion

Hypotheses tested

This study investigates how social ecology affects systematic correspondence in co-evolving languages, providing insights into general sound change mechanisms derived from a pool of synchronic variability (Ohala, 1989). On the one hand, we examined two socially independent yet related languages, English and German, which have maintained a balanced relation in systematic correspondence over centuries of diverging sound changes. On the other hand, the local dialectal varieties in Shanghai have systematically converged toward the regional and national high variety within decades, while still maintaining their distinct lexical pronunciations. The findings suggest that the systematicity of lexical relations between co-evolving languages is not doomed to increase or decrease. Therefore, neither the attrition hypothesis nor the integration hypothesis is adequately supported.

Instead, evidence exists to support the self-regulated adaptation theory. Given the variations in linguistic ecologies between these two cases, these findings indicate that having a stable structure with standardised higher linguistic varieties as “prototypes” (Dixon, 1997) or “superstrata” may have a significant impact on aligning lower forms of language with their corresponding higher forms. On the other hand, when two populations with their own separate standards come into contact without a shared prototype, this contact alone may not be enough to produce a similar effect. These findings support the notion that linguistic ecology plays a crucial role in shaping the development of languages co-evolution.

Furthermore, it should be noted that there is evidence of correspondence-based (Fig. 6, b1, c1) and similarity-based (Fig. 6, b2, c2) convergence at work. These two types of convergence work together in a complementary manner. Additionally, there is the phenomenon of segmental convergence and tonal divergence counteracting each other (Fig. 6, b1). Moreover, there is the Regression Effect (Galton, 1879; Senn, 2011), whereby word pairs that had a higher degree of correspondence in the past tend to have lower or less increased correspondence in the present (and vice versa for word pairs with a lower degree of systematicity). All these findings support the self-regulated adaptation hypothesis and the more general self-organisation theory (Green et al., 2008).

Additionally, the change in systematic correspondence is influenced by external factors such as the strength of contact, language status, and geographical distances, consistent with Labov’s (1963) theory of social motivations for sound change, Mufwene’s (2001) ecological theory for language contact, and Dixon’s (1997) punctuated equilibrium model.

Unveiling the historical transformations of specific languages

The current findings align with previous studies in historical linguistics and dialectology, providing further insight into the specific languages under investigation: German, English, and Chinese dialects.

Historical linguists can uncover significant historical linguistic events by examining the systematic distances in the MDS spaces. For example, the divergence of English and German vowels, as well as the larger systematic distances in vowels between Old English and Modern English compared to Old High German and Modern German (Fig. 3, b1–b4) indicate that English vowel evolution is less regular than German vowel evolution during this stage. This finding may represent the influence of the Great Vowel Shift (1350–1700) in English (Jespersen, 1909). This shift involved conditional sound changes and lexical diffusion (Wang, 1969), contributing to the misalignment of pronunciations we observe in modern English and German spellings.

Conversely, when examining the evolution of final consonants (Fig. 3, d4), we find that the systematic distances between Old High German and Modern German are larger than those between Old English and Modern English. This suggests that English final consonants evolve more regularly than German final consonants at this stage, possibly influenced by the drop of English infinitive verb suffices (although see Szmrecsanyi, 2012) and the complex changes of German verb conjunctions, such as the alternation between strong and weak inflections (Bailey, 1997).

On the other hand, the observation that the local dialectal varieties in Shanghai have systematically converged toward the national standard aligns with previous examples in dialectology, which demonstrate that certain sound categories of the central urban Shanghainese varieties have become more in sync with Standard Chinese (Qian, 2003; Chen, 2019). Interestingly, we not only statistically consolidated the previous findings on the sub-dialects in Shanghai, but also observed a previously ignored phenomenon, namely that the standardised national high variety (SC) and the regional high variety (Shiqu, not standardised but strongly associated with higher social status) influence the low varieties with different biases on two complementary mechanisms: one based on pronunciation similarity, the other based on systematic correspondence. Furthermore, it is worth emphasising that lexical tones in these Chinese dialects, being suprasegmental phonemes, exhibit distinct co-evolution patterns as opposed to segmental phonemes. Moreover, certain local linguistic varieties further highlight this distinctive nature of tones, as they appear to be accompanied by onsets or finals. This correlation between onsets/finals and the historical shift of tones in Chinese has been identified in previous renown studies (Pan and Zhang, 2015; Karlgren, 1915–1916).

In addition, the present findings provide additional illustrations for two commonly applied rules that are relevant across different areas. Firstly, the value of sys_cos_w for each cross-linguistic mapping rule is found to be inversely proportional to its ranking, aligning with Zipf’s Rule (Chao and Zipf, 1949). Secondly, word pairs that had stronger correspondence in the past tend to exhibit lower or less significant increase in correspondence in the present (and vice versa for pairs with lower systematicity), which aligns with the Regression Effect (Senn, 2011).

Limitations

It is important to consider several limitations of our method.

Firstly, due to the unavailability of aligned lexical frequency information and lexical stress of Old English and Old High German, as well as the lack of certainty in the syllabic boundaries of English and German, we were unable to incorporate them into our method, which may have implications for the accuracy and completeness of our analysis.

Secondly, although our sample size was the largest to our knowledge, the possibility of enlarging it should be explored in future studies to enhance the statistical power and reliability of our findings.

Moreover, as we only utilised the two available datasets, questions regarding their representativeness naturally arise. While we made efforts to include two diverse and large datasets, we acknowledge that further datasets could provide a more comprehensive perspective.

To ensure the applicability of our theories to a broader range of historical linguistics and dialectology, it would be valuable to apply the proposed measurement, weighted cosine systematicity, to datasets from a broader range of linguistic ecology and to include a larger variety of time-slices. For instance, datasets from Sprachbund, where systematic correspondences are formed purely through contact rather than shared origins, e.g., the Balkan languages, can provide valuable insights. Similarly, datasets from adstratum languages, such as two standard languages are used in parallel in a country, e.g., modern French and Dutch in Belgium, can offer significant contributions. Additionally, exploring dialects within one country that do not explicitly denote related vocabularies with the same set of ideographic symbols, e.g., the Dutch, Norwegian, or English dialect (Trudgill, 1986), can also provide valuable information. By studying these diverse datasets, we may enhance our understanding of historical linguistics and dialectology in a more comprehensive manner.

Lastly, it is crucial to consider the potential biases that could arise from our approach to handling pronunciation variants. While we made a conscious effort to minimise assumptions about speakers’ knowledge, variations in the decision to include certain lexical variants during the original data collection process may still have influenced the results. It is important to acknowledge that these variations could introduce biases and potential limitations that might impact the overall findings and interpretations of our study.

By examining and acknowledging these limitations, we may ensure a more accurate and robust evaluation of our methodology.

Conclusion

This study has focused on two important scenarios. Firstly, we have examined the co-evolution of two closely related national languages with equal social status by analysing the English-German dataset. Secondly, we have investigated the co-evolution of non-literal local sub-dialects alongside a regional and national high variety using the Shanghai dataset. By utilising weighted cosine systematicity, a vector-based measurement, we have been able to explore the quantitative impact of linguistic ecology on language co-evolution and have tested various co-evolutionary theories. This study provides valuable quantitative insights into the historical transformations of specific languages, which can be generalised to a broader scope of historical linguistics and dialectology. Overall, it sheds light on the mechanisms and patterns of language evolution, contributing to a deeper understanding of the complex and dynamic nature of languages over time.