There is a global downtrend or declination in a sentence (e.g., Gussenhoven, 2004). Articulatorily, declination is arguably caused by a decrease in subglottal pressure over time (Lieberman, 1967; Collier, 1975; Pierrehumbert 1980; Gelfer et al., 1983). Beside declination, lexical tones and tonal interactions also cause downtrend in F0 contours, e.g., downstep lowers the following H tones. Downstep has long been discussed in African languages (Yoruba (Niger-Congo): Ward, 1952; cf. Courtenay, 1971; Luo (Nilotic): Tucker and Creider, 1975; Twi (Akan): Stewart, 1965; Genzel and Kügler, 2011; Kügler 2017; Tswana (Southern Bantu): Zerbian and Kügler, 2015, 2021; among many others). In the African linguistic tradition, downstep is distinguished from downdrift (Stewart, 1965; Hombert, 1974; see Hyman and Leben, 2017 for an overview), terrace (Courtenay, 1971), or automatic and non-automatic downstep (see detailed discussion in Connell, 2001; Rialland and Somé, 2011; Leben, 2014). Strictly speaking, downstep refers to a new register or ceiling established for subsequent H tones after a L tone (Snider, 1990; Snider and van der Hulst, 1993; Connell, 2017; cf. Akumbu, 2019). The differentiation between downstep and downdrift, or automatic and non-automatic downstep concerns the fact that in several African languages, both an overtly realized L tone and a floating L tone functions as the trigger of the lowering process. A floating L tone triggers non-automatic downstep, whereas a phonetically realized L tone triggers automatic downstep (Hyman and Leben, 2017). Phonetically, no difference is found between these two types of downstep (e.g., Genzel and Kügler, 2011; Kügler 2017 for Akan). Since there is no floating L tone in Mandarin Chinese, we do not need to differentiate them. We here take the broad definition of downstep as the lowering of F0 after a L tone, following Shih (1988); Xu (1999); Laniran and Clements (2003), and Genzel (2013) among many others, see (1).

Downstep: In a HLH tone sequence, the second H is realized with lowered F0 compared to the first H, due to the L tone (sequential-comparison). The size of downstep can be paradigmatically calculated as the difference of F0-maximum in the H tones after a L tone and in the corresponding H tones of an all-H tone phrase (cross-comparison).

In some West-African languages, downstep initiates a new pitch register to which subsequent tones are scaled, phonologically termed as register tones (e.g., Snider, 1998) or register features (e.g., Akumbu, 2019). In Mandarin, there is no study directly concerning the effect of downstep as setting up a new register tone or register line. We here introduce three studies, which suggest that downstep in Mandarin does not seem to set up a new register tone. First, it showed that several H tones after a L are lowered in F0 as compared to all H-tone sentences, then the pitch gradually reaches the target in the all-H sentence toward the end of the sentence (see Figure 4, pp. 66 in Xu, 1999). Second, Gu and Lee (2009) found that the lowering effect in the H tones is greater when the preceding L tone is lower. Third, Wang and Xu (2011) used sentence with HLHL…HL and LHLH…LH tone sequences, the sentence-medial H tones reach roughly the same height as the corresponding H tones in an all-H sentence, which is explained as the balance between pre-Low raising and downstep. Thus, downstep seems to be a tonal feature with gradual change in pitch. A terracing pattern of H tones in the LH sequence—as found in the West-African pattern—does not seem to exist in Mandarin Chinese.

What lacks in previous studies is that how downstep interacts with other informative functions, e.g., prosodic boundary and focus. The first question relates to the domain of downstep. The domain of downstp appears to vary across languages. In Kishamba, morpheme boundaries act as a trigger of downstep (Odden, 1986). In Tswana (Southern Bantu), downstep occurs between prosodic words within a phonological phrase, whereas phonological phrase boundaries block downstep (Zerbian and Kügler, 2015, 2021). In Yoruba, downstep applies across all boundaries within a breath group, which could roughly be interpreted as an intonation phrase (Courtenay, 1971). In Japanese, only an accented word (H*L) within a Major Phrase (MaP) triggers downstep (Pierrehumbert and Beckman, 1988; Selkirk and Tateishi, 1991). The downstep effect in Mandarin as reported in Xu (1999) showed that a phrase boundary does not seem to block downstep, though no systematic data on this issue was provided.

As for the interaction of downstep and focus, we here introduce two studies. Ishihara (2007) studied downstep systematically with sentences in the structure as N1 + N2 + N3 + VP (N and VP are abbreviations of noun and verb phrase respectively). It showed that downstep between N2 and N3 is only partially reset, when N3 is focused and when the syntactic boundary is stronger between N2 and N3. It indicated that downstep is weakened by a strong phrase boundary, and a focused H tone after the L tone. Xu (1999) has shown similar results in Mandarin that the size of downstep seems to be reduced when the H tone after the L tone is focused.

It has been also found that downstep can be canceled in yes/no questions in Hausa (Lindau, 1986), meaning that final F0 raising may counter-balance the downstep effect. In Mandarin, however, it does not seem to be the case as shown in Xu (1999). It requires more systematic analysis on whether sentence final F0 raising interferes with the downstep effect.

As mentioned above, another term easy to be confused with downstep is declination, which refers to the F0 downtrend from the beginning through the end of an utterance. We can see that the crucial difference between declination and downstep is its scope. While declination is a gradual lowering of F0 within an intonation phrase, downstep is a local lowering of F0. Declination has been found in both non-tonal languages (‘t Hart & Cohen, 1973; Maeda, 1976; Cooper and Sorensen, 1977; Pierrehumbert, 1979; Sorensen and Cooper, 1980; Cohen et al., 1982; Umeda, 1982; Ladd 1988) and tonal languages (Cantonese: Zhang, 2017; Ge and Li, 2018; Chinese: Xu, 1999; Shih, 2000; Shih and Lu, 2010). Some researchers argue that declination is a fundamental effect in human speech due to a drop in subglottal air pressure (Lieberman, 1967; Collier, 1975; Pierrehumbert, 1979; Gelfer et al., 1983; Gussenhoven, 2004). However, other researchers stated that declination is a combined effect from different functions, e.g., sentence stress and terminal fall (Lieberman and Tseng, 1980; Xu, 1999; Liu and Xu, 2005), topic initial F0 raising (Umeda, 1982; Wang and Xu, 2011) and discourse structure (Hirschberg and Pierrehumbert, 1986; Nakajima and Allen, 1993; Sluijter and Terken, 1993). Downstep and pre-low bouncing, as introduced earlier, also contribute to the overall declination (Liberman and Pierrehumbert, 1984; Pierrehumbert and Beckman, 1988; Shih, 1988; Xu, 1999). Shih (2000) used sentences with the tone sequence of LRH…HN (L, R, H and N stands for low, rising, high and neutral tone respectively), and found that the H tones show declination in the way that the lowering slope is steeper in shorter sentences, after taking apart focus and final lowering. In Shih and Lu (2010) the intonation of an all H tone digital string (338–811-3783) drops from 300 Hz to almost 100 Hz. Similarly, Yuan and Liberman (2014) found that shorter utterances have steeper declination in both the top line and the baseline, after excluding the initial rising and final lowering effects. They are in favor of the idea that declination is linguistically controlled, but not just a by-product of the physics and physiology of talking. It is possible that the declination in the previous three studies still involves some other unknown effects which are hidden by the regression model. In the current study, we calculated declination syllable-by-syllable, as the F0 drop between two adjacent H tones.

A L tone could also cause F0 raising after it, especially when the following syllables carry the neutral tone, termed as post-low-bouncing (Mandarin and Cantonese: Chao, 1968; Lin and Yan, 1980; Shih, 1988; Chen and Xu, 2006; Gu and Lee, 2009; cf. Prom-on et al., 2012). As discussed in Prom-on et al. (2012), post-low-bouncing has been considered mostly as an articulatory phenomenon, limited to the first neutral tone after the low tone. They emphasized that post-low F0 bouncing is different from a carryover effect, although it occurs between tones. The carryover effect shows in the way that the initial F0 of a syllable is heavily assimilated to the final F0 of the preceding tone, but over the course of the current syllable, F0 gradually approaches its own tonal target. To account for such assimilatory effect, Xu and Wang (2001) proposed the Target Approximation model, which represents the production of successive tones as a process of asymptotically approaching each tonal target within the time interval of the respective syllable, starting from the offset F0 of the preceding syllable. Post-low-bouncing, instead, is the process that pitch increases first then drops back to the underlying target. They discussed the possible physical mechanism behind the low-bouncing effect and suggested a balance-perturbation hypothesis. In simple words, after producing a very low F0, the extrinsic laryngeal muscles, especially the sternohyoids (Ohala, 1972; Atkinson, 1978), stop contracting and thus temporarily tip the balance between the two antagonistic forces maintained by the intrinsic laryngeal muscles, resulting in a sudden increase of the vocal fold tension (Prom-on et al., 2012, pp. 422). It still requires articulatory studies to verify the balance-perturbation hypothesis. From pitch analysis, one way to test it is to vary syllable duration in the L tone. A longer L tone may reduce post-low-bouncing as it gives more time for the muscles releasing the force. We hence predict that post-low-bouncing is weakened if the L tone is at a phrase boundary, as the L tone is with final lengthening.

There has been extensive research on how focus is realized prosodically in many languages (for an overview see Kügler and Calhoun, 2020). In Mandarin, focus is realized by increasing the pitch range, intensity, duration and articulatory fullness of the focused word, and reducing the F₀ and intensity of the following words (post-focus-compression, PFC), while leaving the pre-focus words largely unchanged (Xu, 1999; Chen and Gussenhoven, 2008; Wang and Xu, 2011). Although Mandarin is tonal, its prosodic focus pattern is very similar to English (Cooper et al., 1985; de Jong, 1995; Xu and Xu, 2005), German (Féry and Kügler, 2008) and many other Indo-European languages (Xu et al., 2012). A recent study found that PFC can go across a relative strong prosodic boundary in Mandarin (e.g., a boundary between to clauses), indicating that phrasing does not interfere with post-focus constituents (Wang et al., 2018b). In other words, focus and phrasing are largely encoded in parallel in intonation, though focus may cause prosodic boundaries in some languages (e.g., Kügler and Calhoun, 2020).

Prosodic boundaries are generally indicated by different phonetic cues such as pre-boundary lengthening, silent pause, F0 reset, phonological boundary tones and changes in voice quality (for detailed discussion, see Wang et al., 2018b). In Mandarin Chinese, boundary strength is realized with gradient means rather than categorical ones, differentiated mainly in pre-boundary lengthening and optional silent pause, but not F0 (Xu and Wang, 2009; Wang et al., 2018b). Although pitch reset has been found at a strong boundary (Dutch: de Pijper and Sandeman, 1994; Swerts, 1997; English: Ladd, 1988), F0 plays a limited role to distinguish boundary strength in Mandarin Chinese when tones and focus are carefully controlled (Xu and Wang, 2009; Wang et al., 2018b). Minimum F0 is lowered at a strong boundary with a silent pause for about 200 ms, but not at a phrase boundary within a sentence (Wang et al., 2018b).

It has been found in many languages that pre-boundary syllables are longer than non-final syllables (e.g., English: Byrd, 2000; Finnish: Nakai et al., 2009; Dutch: Swerts and Geluykens, 1994), and articulatory gestures have slower velocity (Krivokapic and Byrd 2012). The prolonged syllable might give rise to fully realized phonetic targets (Lindblom, 1990; DiCanio et al., 2021), e.g., tones in Mandarin (see Figure 4 in Wang et al., 2018b, p. 36). Phrase-final tones carry both lexical tone and post-lexical tone, e.g., a pitch accent and a boundary tone (Arvaniti and Fletcher, 2020). On the other hand, phrase-final position may also be the locus of glottalization (Huffman, 2005), devoicing (Wagner, 2002), and a gradual decay in intensity and F0 (Gussenhoven, 2004; Ladd, 2008; cf. DiCanio et al., 2021).

Relating to the current study, we aim to find out whether a phonological phrase boundary reduces or even blocks downstep and post-low-bouncing effect, assuming that the pre-boundary syllable carrying a L tone is longer and hence the tone is fully realized with pitch raising toward the end of the syllable, since fall-rise is the citation form of the L tone in Mandarin. Another possibility is that pitch goes lower when the L tone is longer, thus makes a greater downstep effect.

After introducing the studies on the linguistic meaning of tone and intonation, we here would like to go back to articulatory studies on pitch control. It will help us to understand downstep and post-low-bouncing, since down to the bottom of the questions raised above, it is all about how the muscles, bones, vocal folds and brain cooperate to realize the pitch targets. The observed pitch contours reflect both linguistic meanings and articulatory constrains.

Yuan and Liberman (2014) discussed articulatory studies on how F0 is controlled. We here just briefly cite some most relevant studies. F0 is determined by the stiffness and effective mass of the vocal folds and the subglottal air pressure (Murry, 1971; Hollien, 1974, 1983; Baer, 1979; Titze, 1988; Stevens, 2000; Zhang, 2016). Intrinsic laryngeal muscles, especially the cricothyroid muscle (CT), are the main contributor to the adjustment of the stiffness and effective mass of the vocal folds. The contraction of CT raises F0; the relaxation of CT, along with the activity of other laryngeal muscles, lowers F0 (Collier, 1975; Atkinson, 1978). Extrinsic laryngeal muscles, which suspend and support the larynx, can also change the states of the vocal folds through vertical larynx movement (Ohala 1972; Honda 1995; Hirose 1997), and F0 falls as the larynx moves down.

F0 lowering is not only accompanied by larynx lowering, relating to extrinsic laryngeal muscles (Honda, 1995; Hirose, 1997) but also involves the joint supraglottal action (Lindqvist-Gauffin, 1969, 1972; cf. Lindblom, 2009). In Mandarin, the basic role of larynx height in the execution of tone is complicated by the relationship of larynx height to the state of the larynx: constriction of the supra-glottal laryngeal structures is facilitated by raising the larynx (Edmondson and Esling, 2006) and inhibited by lowering the larynx (Moisik et al., 2014; Moisik and Esling, 2014). Moisik et al. (2014) shows that a L tone target can be reached either by lowering the larynx, or by combining the raise of larynx height and laryngeal constriction, which may lead to creakiness in the low tone. They show that producing the H tone requires any tone involving lowering in pitch is easily becoming creaky, especially the L tone (Kuang, 2017, 2018).

Ladefoged (1973, p. 75) suggested that the creaky voice phonation mechanism is that “because the arytenoid cartilages move forward as they come together; the vocal cords tend to be less stretched in creaky voiced sounds; they are therefore likely to vibrate at a lower frequency. But the coming together of the arytenoids and the movements of the thyroid cartilage that stretch the vocal cords are independent laryngeal gestures, so that it is quite possible for creaky voiced sounds to occur on any pitch.” Creaky voice in Mandarin L tone exhibits various laryngealization properties in acoustic waveforms, including aperiodicity, period doubling, or low-frequency pulse-like vibratory patterns (Gerratt and Kreiman, 2001; Keating et al., 2015). In Mandarin, creaky voice relates to the low target in pitch that the L tones are less creaky when the pitch range is raised, but creakier when the pitch range is lowered (Kuang, 2017, 2018). In previous studies on downstep and post-low-bouncing, creaky voice is usually not taken into account. A consequence of this discussion leads to the question whether creakiness causes downstep or not.

The main goal of the current study is to understand the property of downstep in Mandarin. The second goal is to provide some analysis on how post-low-bouncing interacts with boundary strength. These will lead us to better understand how intonation is shaped by both informative functions and articulatory constrains. The research questions and hypotheses are summarized as the following.

The carrier sentences contained only H tones, except for a neutral tone at sentence-final position. Two target words were embedded in the middle of the carrier sentence, one consisted of a LH word (named as syllable X and Y) triggering downstep and post-low-bouncing, and the other one consisted of a HH word, serving as the reference. The two sentences of each item were read in varied contexts eliciting 4 different focus, and 2 boundary conditions.

Three variables were independently manipulated in this experiment, that is, tone of syllable X (either H or L tone), boundary strength between syllable X and Y (syllable boundary or phrase boundary) and focus type (wide focus (WF), focus on syllable X (XF), on syllable Y (YF) and in sentence final position (ZF)). One set of the sentences in the condition of syllable and phrase boundary were provided in (1a) and (1b). Each sentence was with the syntactic structure as S-V1-O1-V2-O2, and the target words (syllables X and Y) were put in the O1 and V2 position, respectively. Here, by comparing the F0 of syllable Y and that of the following H tones between the two sentences (LH and HH), we can calculate the effect size and scope of downstep. The statistical analysis will then test for how many syllables after the L tone the downstep effect lasts, with consideration of boundary and focus conditions.

For the two boundary conditions, a monosyllabic homophone of the target syllable Y was used to construct sentences with different syntactic boundaries. Based on the assumption of the syntax-phonology interface, prosodic boundaries, in particular in this experimental setting, are the result of matching syntactic constituents onto prosodic constituents (Selkirk, 2011). Thus, in the syllable boundary condition (1a), the HL_XH_Y was one word, whereas in the phrase boundary condition (1b), the HL_X was a word, and the following H_Y was an adverb, phrased together with the following words as a verb phrase (VP). Thus, prosodic boundary in condition (1a) was weaker than that in (1b), named as a syllable boundary (SylB) and a phrase boundary (PhrB) respectively. In example (1a), the HHH sequence (yin1ou1dou1樱欧兜, Ying1ou1 bag1) meant a bag printed with ying1ou1 (a make-up word for an exotic plant), whereas in (1b), ‘dou1’ in the HHH sequence (ying1ou1.dou1樱欧都, Ying1ou1 all1) was an adverb, meant “all” to modify the following verb “lingchu (take-out).” In this way, the two boundary conditions were clearly distinguished by using two different characters (兜 vs. 都, bag vs. all). It was the same construction for the HLH sequence, in which the HL tone word is ying1ou3 (樱藕Ying1ou3), which was also a make-up word for an exotic plant. Here, the contrast of syllable X, either being L or H toned, was straightforward by using the two different characters (藕vs 欧, ou3 vs. ou1). In this way, no specific explanation of the material was necessary for the speakers. They were easily able to read the sentences with different tones and phrasing conditions in a natural way.

Focus was elicited by varying a preceding background sentence, which required a correction of the corresponding word in the target sentence. Taken the HL_XH_Y sentence in the syllable boundary condition (see 1a), the four focus conditions are presented in (2). Here, the H tone (syllable Y) is critical to test the effects of downstep and post-low-bouncing, and their interaction with focus. Thus, syllable Y was manipulated as either post-focus (focus on syllable X), on-focus (focus on syllable Y) or pre-focus (focus on syllable Z). A wide focus condition served as the baseline. Similar contexts were constructed for the other sentences, see Appendix I for the whole sentence sets.

The background sentences of the four focus conditions for the sentence (1a) are as follows.

Wide focus: “ni3 ting1shuo1 le0 ma0?” (Have you heard about it?)

We constructed two sets of items. In total, 2 (tone of syllable X) × 2 (boundary between X and Y) × 4 (focus) × 2 (sets) × 3 (repetitions) × 8 (speaker) = 768 sentences were analyzed.

Eight native Mandarin speakers participated in the experiment at Minzu University of China (5 female and 3 male speakers), from the age of 20 to 28. They were born and brought up in Beijing, spoke no other Chinese dialects and reported no hearing or speaking impairments. They were paid with small amount of money for taking part in the experiment.

The subjects were recorded individually in the speech lab at Minzu University of China. They were asked to read aloud both the context and the target sentences at a normal speed and in a natural way. They sat before a computer monitor, on which the test sentences were displayed, using AudiRec, a custom-written recording program. To make the reading task a little easier for the speakers, the focused words were highlighted with color. A Shure 58 Microphone was placed about 10 cm in front of the speaker. All sentences were digitized directly into a Thinkpad computer and saved as WAV files. The sampling rate was 48 KHz and the sampling format was one channel 6-bit linear. Each speaker repeated the whole set of sentences 3 times in different random order, with about 5 minutes break between sessions. Before the formal recording, they read the sentences silently to get familiar with them, and to make sure that they understood the meaning. The total recording time was about an hour.

The target sentences were extracted and saved as separate WAV files. ProsodyPro (Xu, 2013) running under Praat (Boersma and Weenink, 2013–2022), was used to take F0 and duration of each syllable measurements from the target sentences, which were all segmented into syllables manually, and at the same time hand-checked vocal cycles markings generated for errors, such as double-marking and period skipping. ProsodyPro then generated syllable-by-syllable F0 contours that were either time-normalized or in the original time scale. At the same time, the script extracted various measurements, including maximum F0, minimum F0 and duration of each syllable. We could measure F0 at the offset of a syllable, however maximum F0 is toward the very end of the syllable (see Figure 5), it is highly probable that the two values are with very little difference. Maximum F0 is much more widely applied in previous studies (e.g., Xu, 1999; Genzel and Kügler, 2011; Prom-on et al., 2012). Thus, we choose maximum F0 to measure downstep effect.

The statistic tests were carried out in the R environment (R Core Team, 2016) by using lme4 package Version 1.1–18 (Bates, et al., 2016) to estimate the effect of the fixed factors (tone, boundary and focus) and the random factors (speaker and sentence set) on the acoustic parameters, e.g., maximum F0 and duration. Regression coefficients (bs), standard errors (SEs) and t-values (t = b/SE) are reported, taken t > 2.0 as reaching the significant level at p < 0.05 (Gelman and Hill, 2006). In the results, we reported the best-fit model according to the model comparisons with the lowest AIC and BIC. For the fixed factors, we took the model with interaction only when there was significant interaction.

Creaky L tone was visually identified by checking the spectrum and the WAV files. Zhang (2016) distinguished four types of creaky voice (see Figure 3 in that paper). We grouped all these types as creaky voice. Since F0 is the main concern in this paper, we here labeled the part with aperiodic pulses as creaky, see Figure 1. In this way, the part of the regular pulses was used to get the F0 values of the syllable. Most of the creaky L tone was similar to what Figure 1 shows.

First, we present intonation contours to show how focus is encoded in intonation. Figure 2 presents the HH and LH sentences in the condition of syllable boundary, with the 4 focus conditions overlaid in one figure. In each sentence, 10 time-normalized F0 points for each syllable were averaged across 48 observations (8 speakers × 2 sets × 3 repetitions).

We can see in Figure 2 that focus is realized as the tri-zone pattern as defined in Xu (1999) and repetitively found in many other studies (e.g., Wang et al., 2018b). Looking at the HH tone sentences, we can clearly see that the on-focus syllables show raised F0 and expanded pitch range; the post-focus words exhibit lowered and compressed pitch; while the pre-focus words are similar to the wide focus condition. It holds in the LH sentences as well, except that when the L tone word (e.g., ying1ou3) is focused (XF), the pre-low H is raised. And in the YF condition, on-focus F0 raising still applies in the H tone after the L tone. Thus, downstep does not override (or cancel) on-focus F0 raising. The sentences in the phrase boundary show a very similar pattern, which is not presented here for the interest of space. A phrase boundary does not block post-focus F0 compression (PFC), as likewise reported in Wang et al. (2018b). In general, it confirms that tonal interactions and phrasing do not change how focus is realized, though the amount of focal raising appears to differ between tone conditions.

Secondly, Figure 3 presents how boundary strength is encoded in intonation in the XF and YF conditions. No clear difference in F0 between the two boundary conditions can be seen here, in both the HH and LH (lower row) sentences. In WF and ZF conditions, the two boundary conditions do not show clear difference either, which is not presented here for the interest of space. It is in consistence with Wang et al. (2018b) that F0 plays a limited role on phrasing, especially on boundaries within a sentence. Importantly, when pre- and post-boundary syllables are under focus (the X and Y focus condition), there is still no clear sign of using F0 to mark boundary strength. Thus, focus in Mandarin does not seem to invulnerably insert a prosodic boundary.

The above graphic observations show that F0 variation is mainly triggered by focus and tone, but not by prosodic boundaries. We further analyzed the nature of the boundary and whether speakers distinguished the two boundary conditions phonetically. The following analysis of syllable duration (see section “Acoustic analysis on the interaction of focus and boundary,” Figure 4) confirms that boundary strength was encoded mainly in pre-boundary lengthening, but not F0.

From the graphic analysis (see Figures 2, 3), we can see that the intonation patterns of focus and phrasing are consistent with previous studies, e.g., Xu (1999) and Wang et al. (2018b). Since focus and boundary effects have already been extensively studied, statistical analysis on all the syllables is not presented here. Statistic test on syllable X is of particular interest as it interacts with downstep and post-low-bouncing. To better understand the interaction between boundary and focus on syllable X, we present the boxplot of maximum F0 and duration of syllable X in Figure 4.

Linear-mixed-models on maximum F0 and duration in syllable X were carried out in HH and LH sentences separately, with focus and boundary as two non-interactive fixed factors, while speaker and sentence set as random factors (see Table 1). Wide-focus and syllable-boundary were set as the baseline conditions. The LMM model was chosen to meet the criteria that (1) the model with presumed interaction did not show significant interactions, thus we took this model without interaction; and (2) it was with the lowest AIC and BIC while we tried different ways of setting the random effects.

As for focus effect on syllable X, together with the observations in Figure 4, the statistical analysis in Table 1 shows that focus significantly increases both maximum F0 (about 2.8 st) and duration (about 66 ms) of syllable X (see the line of XF in Table 1). In the Y focus condition, the 3 syllables HXY (H means the high tone before syllable X, e.g., ying1ou1dou1 ‘Yingou bag’) is possibly grouped as one prosodic word, thus syllable X is also with increased maximum F0 (about 1.4 st) and duration (about 24 ms; see the line of YF) in Table 1, which is in consistent with the findings in Chen (2006) on the durational domain of focus.

As for boundary effect on syllable X, the data in Table 1 (also see Figures 3, 4) show that boundary does not have any effect in maximum F0 (92.7 st vs. 92.6 st), but only in duration of syllable X (192 ms vs. 212 ms). No interaction was found between focus and boundary in the duration of syllable X, meaning that the pre-boundary lengthening applies to roughly the same degree in all the focus conditions (see Figure 4), which is in consistent with Wang et al. (2018b). The above results hold for both HH and LH sentences. It leads us to conclude that focus and tone do not interfere with pre-boundary lengthening. Thus, durational adjustment due to focus, boundary and tone is also largely encoded in parallel. We can then further test whether the lengthened L tone decreases or increases the level of downstep and post-low-bouncing in the following sections.

From Figure 3, we can see that maximum F0 in the L tone is actually the end point of the preceding H tone, which does not show any difference between the two boundary conditions (see Table 1 and Figure 4). Does the minimum F0 in the L tone differ between the boundary conditions? With similar LMM tests in the LH and HH sentences separately, taken boundary and focus as two fixed factors with interaction, and speaker as the random factor, the minimum F0 of syllable X in the LH sentence showed no difference in the two boundary conditions either (86.4 st on average in both conditions) (Estimate = −0.369, SE = 0.426, df = 352, t = −0.866, p = 0.387). However, there was an interaction between focus and phrasing, i.e., when the L tone is focused (XF), the minimum F0 in the phrase boundary condition is significantly lower than in the syllable boundary condition (Estimate = −1.402, SE = 0.598, df = 352, t = −2.344, p = 0.0196). In the other three focus conditions, no difference in minimum F0 was found between the two boundary conditions.

When we labeled the speech data, we noticed that most of the L tones were creaky, that was 84.1% and 74.6% in the phrase and syllable boundary, conditions respectively. It is possible that creakiness is an additional feature of a stronger boundary, when minimum F0 cannot go any lower at a phrase boundary (Kuang, 2017).

To summarize, (1) focus is reliably realized in a tri-zone pattern, i.e., pre-focus F0 is largely intact, on-focus F0 is raised and post-focus F0 is lowered and compressed; in addition, focus increases duration of the focused syllable; (2) boundary strength has very little effect on maximum or minimum F0, but mainly realized by pre-boundary lengthening, which is independent from focus and tone; (3) The L tone is more likely to be creaky when it is before a phrase boundary than a syllable boundary.

The analysis on focus and boundary in section “Graphic analysis on focus and phrasing” and section “Acoustic analysis on the interaction of focus and boundary” shows that the current experiment is in agreement with previous findings on these two effects (Xu, 1999; Wang et al., 2018b). It validates the following analysis on the interaction of these two functional variations with the tonal effects, i.e., downstep and post-low-bouncing. As introduced in the beginning of the results section, we here firstly report the cross-comparison on assessing the downstep effect adopted from Xu (1999) and Shih (2000) among many others, by comparing crossly between the HH and LH sentences (see Figure 5).

In the wide- and Z focus sentences, we can see in Figure 5 that F0 raises greatly in syllable Y in the LH sentence, which is the procedure of target approximation from a low starting point to the H target. As expected, F0 in syllable Y does not reach the height as the HH tone sentences in several H tones after the L tone, showing a clear downstep effect. We can also see that the downstep effect becomes weaker when the H tones are in a longer distance from the L tone. Five new findings are as below.

In summary, the graphic analysis of Figure 5 shows that: (1) The downstep effect is relatively robust in varied focus and boundary conditions. More specifically, downstep is not blocked by a phrase boundary, neither is it overridden by on-focus F0 raising or phrase boundary. (2) When the L tone is under focus, post-low bouncing is found in the following H tones, and seems to be weakened by a phrase boundary.

The main questions to be quantitatively analyzed are the size and the domain of downstep and post-low-bouncing effect, and their interactions with focus and phrase boundary.

Downstep is firstly analyzed by comparing the LH and the corresponding HH sentences in the WF, ZF and YF conditions. In the cross-comparison, the size of the downstep effect is calculated by the difference in maximum F0 between the H tones in the LH and HH sentence in syllable Y (syllable 7) and the following syllables (syllable 8 to 14). The post-low-bouncing effect is calculated in the X-focus condition in a similar way (see section “F0 analysis on post-low-bouncing effect”).

Figure 6 presents the size of downstep effect in the three focus and two boundary conditions. The mean values show how much F0 maximum is lowered in the LH sentence as compared to the HH sentence in the corresponding syllable. Paired-sample T tests were applied in each syllable to test whether the difference reached statistical significance at the level of p < 0.05, which is marked by a * in Figure 6.

To get an overall statistical analysis of the factors on the downstep effect, a LMM was applied, setting focus, boundary, and syllable as fixed factors with interactions presumed (WF, syllable boundary, and the 7^th syllable are set as the base-line condition), while speaker is the random factor (see Table 2). Putting it together with the t-test in Figure 6, the following findings are statistically supported: (1) The downstep effect in Y-focus condition is significantly smaller than that in wide-focus condition, while no difference is found between Z-focus and wide-focus condition. (2) The downstep effect decreases as the H tones are in longer distance from the L tone. (3) Unexpectedly, the downstep effect is greater in the phrase boundary condition than the syllable boundary condition, especially in the wide-focus conditions. It is probably because the L tone is with lower minimum F0 and with more creaky voice (see section “Acoustic analysis on the interaction of focus and boundary”), thus the following H tone is with a larger difference from the all-H reference, as compared to the syllable boundary condition. (4) In the Y-focus condition, the downstep effect interacts with focus and boundary, in the way that the downstep effect in the adjacent syllable of the L tone is greater in the syllable boundary condition than in the phrase boundary condition.

Another way to analyze downstep is the degree of F0 lowering after a L tone. In this way, downstep effect can be compared with declination, which was analyzed by calculating the difference of the maximum F0 in the adjacent H tones in all-H sentences. Firstly, we just analyzed the two H tones surrounding the L tone, that is the difference of maximum F0 between syllable 7 and 5 (Difsy7sy5), presented in Figure 7 as boxplots divided by focus conditions, with HH and LH sentences compared directly. Since the results already show that boundary has no effect on maximum F0 in syllable X (Figure 4), we here averaged the two boundary conditions in Figure 7.

To evaluate whether there is declination in all H tone sentence, we compared “Difsy7sy5” in the wide focus condition of the HH sentences with 0 in a one-sample t-test (t = −3.359, df = 107, p = 0.001). The 95% confidence interval is −0.3 to −0.07. With the same analysis, however, declination is not found in the Z-focus condition (t = −1.46, df = 105, n.s.). Thus, declination is to a much less degree and vulnerable to be cancelled by a final focus.

The LMM model on “Difsy7sy5,” with focus, boundary and tone as fixed factors and speaker as random factor, showed a main effect in tone and focus, but not in boundary (see Table 3). It further confirms that F0 plays a limited role on differentiating boundary degrees.

In general, by comparing HH with LH in Figure 7, we can see that the difference on the degree of F0 drop in the all-H tone sentence is significantly less than the downstep effect in XF, ZF and WF conditions (p < 0.05). The interaction between focus and tone is not found in XF condition. We can see that Difsy7sy5 is greater in LH than in the HH sentence. Besides, Difsy7sy5 in the HH sentence is much smaller in the XF than in the WF condition, which reflects post-focus-compression (PFC) in F0. The new finding here is that downstep still shows aside from PFC. It means that the downstep effect is not just the general downtrend of F0. Declination and downstep are presumably not from the same articulatory mechanism.

When focus is on the H tone after the L tone (YF), the pitch difference between the two H tones (syl5 and syl7) is greater in the LH than the HH sentences. This comes from pre-low-raising (Lee et al., 2021). Here, it also shows the pre-low-raising is independent of on-focus F0 raising.

Then, we further tested whether declination holds all along the sentence by comparing maximum F0 of each adjacent H tones, see Figure 8. We here only consider the wide-focus condition. The *** in the figure indicates that the F0 raise or drop between two adjacent H tones in all-H sentence is greater than 0 by on-sample t-test with p < 0.001, otherwise there is no difference between the two H tones. To put it in a simple way, the *** means that there is either F0 raising or declination in the current syllable. We can see that in the HH sentences, F0 goes up in the beginning of the sentence (increased 0.34 st), then drops gradually for about 3 syllables (decreased 0.25 st). However, between syllable 7 and 6 and between syllable 9 and 8, no significant difference is found in maximum F0. These two positions are phrase boundaries. It is possible that a phrase boundary cancels declination. Toward the end of the sentence, declination is absent as well. The last syllable is a neutral tone, which causes a sharp drop in F0. Thus, declination is with a very small pitch drop between two H tones, and can be easily cancelled due to topic, boundary, tone and other reasons.

If we look at the LH sentences, we can see that the H tones around the L tone causes much greater F0 change than the all-H sentence. Toward the end of the sentence, the adjacent H tones do not differ in F0, which is similar to the all-H sentence. It is in agreement with the cross-comparison of the downstep effect, that downstep gets weaker as the H tones are further from the L tone.

All-together, both the cross- and sequential-comparison show that downstep effect is in a much greater degree than declination. Downstep effect is robust, lasting for about 2–3 syllables. Downstep effect is not cancelled by focus or boundary, whereas declination can be cancelled by these two informative functions.

The very last question concerning downstep is whether it is caused by creaky voice, or whether creaky L tones cause greater downstep effect. The number of creaky L tone in different conditions is presented in Table 4. In line with previous studies, we also see that when the L tone is under focus and at a phrase boundary, it is more likely to be creaky. We do not go into detailed analysis on the acoustic parameters of the creaky L tone. Instead, we simply calculate the amount of creakiness in L tones to answer the question whether a creaky low tone causes greater downstep effect. In Figure 9, the maximum F0 of syllable Y is plotted against the duration of the creaky part in syllable X, with four focus conditions divided in different plots. When the creaky duration is 0, it means this is a normal L tone. Here we do not see any clear trend of a creaky L tone causes lower F0 in the following H tone, which is supported by the LMM model analysis with creaky, focus and gender as fixed factors and speaker as a random factor (lmer(maxF0syl7 ~ Creakylablel*focus*Gender+ (1|speaker), data = creaky)). The LMM shows significant effect in focus and gender, whereas creaky does not show any effect (Estimate = −0.2455, SE = 0.416, df = 347, t = −0.59, n.s.). Thus, creaky L tone is not the direct cause of downstep, but strengthens downstep. It then explains why downstep effect is greater after a phrase boundary (Figure 6).

The post-low-bouncing effect was calculated as the difference of maximum F0 in the H tones between the LH and HH sentences in the XF condition (the L tone is on-focus). As can be seen in Figures 7, 10, F0 maximum is lower in the syllable right after the L tone (syllable 7), that is because the F0 maximum of syllable 7 in the HH sentence is the offset of the previous H tone (the maximum F0 hence appears at the onset of the syllable 7 representing the transition from the focused H tone to a post-focally H tone). Post-low-bouncing shows at the end of syllable 7, which can be observed in the maximum F0 of syllable 8 and 9, then the pitch gradually drops back. The linear-mixed-model analysis was carried out with syllable and boundary as two fixed factors (with interaction), while speaker and set are random factors. In this statistical test, we only considered syllable 8 to 10, since no difference is shown between HH and LH sentences after the 10^th syllable (see Figure 10). The LMM analysis shows a significant effect in syllable (SE = 0.093, t = −5.087*), boundary (SE = 1.193, t = −3.018*) and the interaction (SE = 0.132, t = 2.899*). Thus, the following observations in Figures 5, 10 are statistically supported: (1) The post-low-bouncing effect gradually decreases in the syllables after the L tone; (2) A phrase boundary weakens post-low-bouncing effect, especially in the second H tone after the L tone.

This is actually a very complicated question, since focus in Mandarin involves both on-focus raising and post-focus-compression in F0 (Shih, 1988; Xu, 1999; Chen and Gussenhoven, 2008; Wang and Xu, 2011; Wang et al., 2018b), see Figure 2 in the current study. Besides, downstep refers to the relevant pitch height in the H tones, either as compared to all-H reference line (cross-comparison answering Q1-Q4), or as the F0 drop between the H tones before and after the L tone (sequential-comparison answering Q5). To make the question even more complicated, L tones usually become creaky (Q6). No previous study has considered the influence of creakiness on downstep (Q6). Thus, the first question is split to the following 6 sub-questions, aiming to fully understand the property of downstep, and to take apart declination and downstep. The results are interpretable and coherent to each other if we take the idea that downstep is mostly constrained by articulatory movement, instead of conveying linguistic meaning.

According to the balance-perturbation hypothesis (Prom-on et al., 2012), we predicted that post-low-bouncing is weakened if the L tone is at a phrase boundary. It is indeed the case, as shown in Figure 7. They hypothesized that after producing a very low F0, the extrinsic laryngeal muscles (e.g., sternohyoids) stop contracting to maintain the balance between the two antagonistic forces in the intrinsic laryngeal muscles. When the L tone is focused, the extra force may cause a sudden increase of the vocal fold tension, resulting in the raise in F0 in the following H tone. We here see that when the L tone is before a prosodic phrase boundary, it then probably gives a little more time to release the tension between the extrinsic and intrinsic laryngeal muscles. This would explain the difference in size of post-low bouncing found in our data. In line with Prom-on et al. (2012), we also found that post-low-bouncing occurs in H tones when the L tone is under focus. In their study, neutral tones after a L tone show post-low-bouncing. The reason might lie in the fact that post-focal words are weakened in intensity and compressed in F0. The weakened H tones at post-focal position might share some similar mechanism with weak articulatory movement in the neutral tones.

At last, we here briefly introduce some preliminary findings in the current study, relating to Moisik et al. (2014). They have found that low F0 tone targets in Mandarin can not only be reached by lowering the larynx, but also by combining the raise of larynx height and laryngeal constriction, which may lead to creakiness in the low tone. In the L tone, the amount of F0 lowering correlates with larynx lowering in male speakers (r = 0.73 and 0.86), while the female speaker uses larynx raising (r = 0.13; Figures 11-13, pp. 39 in their study). In our study, the minimum F0 of the low tone (X) is positively correlated to the maximum F0 of the following H tone (Y) in the male speakers (wide focus: y = −2 + 0.99x, r² = 0.66; X-focus condition: y = 9.4 + 0.87x, r² = 0.736), but not in the female speakers (wide focus: y = 65 + 0.27x, r² = 0.073; X-focus condition: y = 79 + 0.12x, r² = 0.01). To fully understand the anatomical process in downstep, articulatory studies considering gender difference are required.