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I'm wondering about the rate of occurence of complex tone contours like the Mandarin Chinese third tone, the falling-rising tone. By "complex" I mean that its contour isn't simply a rising, falling, or steady pitch (except of course under tone sandhi). I'm not necessarily talking about languages like Vietnamese, where tones are distinguished based on secondarily articulated characteristics, namely the breathy, glottal, and creaky-voiced tones.

My question: among languages with lexical tone--and more narrowly, languages with contour-based tones (so excluding, for instance, Yoruba)--how common are tones and tonal contrasts like this, as opposed to simpler ones? What are their similarities or differences between these languages?

I'd much appreciate any help in better defining and rewording this question, I know the topic of lexical tone is extremely complicated. I'm only using Mandarin Chinese as an example because that's the tonal language that I'm most familiar with.

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    However, the 3rd tone of Mandarin Chinese, as actually uttered in connected speech, is usually just a low tone: contour-wise it's roughly flat, maybe falling slightly. – Michaelyus Aug 3 '15 at 14:28
  • I think, there are no "complex" or "simple" tone contours for the native speakers of tonal languages. Each lexical tone is a tool; the only key is whether a native speaker is able to tell two tones apart. For instance, I learned Mandarin a while ago and then started learning Thai. Thai has no equivalent of Mandarin 3rd tone, but it has rising and high-rising tones that were difficult for me to distinguish and reproduce. However, I noticed that when I occasionally used Mandarin 3rd tone for Thai rising and Mandarin 2nd tone for Thai high-rising, the Thai speakers understood me perfectly. – bytebuster Aug 3 '15 at 15:34
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I understand your question as asking about tone-collocations on syllables which contain at least three pitch-height elements. I do not understand your basis for excluding Yoruba. One reason might be that underlyingly Yoruba has just H M L, and contours result from rules; but the other is that rise and fall in Yoruba clearly decompose into H and L on a single syllable. So it is unclear whether you are excluding languages with superficial contours, or those where contours are arguably not "atomic" (as Chinese is sometimes analyzed). I will assume that you are only excluding superficial contours. In which case, Mandarin 3rd tone can't be argued to be a complex (tri-tonal) contour. And finally, "tone" is often used to describe the tone profile of entire words so that HLH might be a pattern in a disyllabic word (high+rise), and could be called "a tone". Such an analysis would be irrelevant, under the understanding that you're looking for tone density within a single syllable. If these assumptions aren't what you had in mind, you should edit the question to clarify.

The main result is that tri-tonal contours are much less common than bi-tonal contours, in fact very few languages have non-superficial tritonal contours. In no language do you find tri-tonal contours without there being bi-tonal contours. The number of allowed tri-tonal contours is always less than the number of bi-tonal contours. Tritonal contours always seem to involve a change in pitch-movement direction (relevant in a language with 3+ levels), so LML or MHL might be possible but not LMH contrasting with LH.

However, if you suspend the consideration of non-superficiality and map phonetic pitch traces to "tones", tritonal contours become more common, possibly more common that bitonal contours, given a fine-enough grained analysis of surface pitch.

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I've seen the claim that when it comes to tonal contours, contour tonal languages "select from as many categories as possible" (Routledge Linguistics Encyclopedia). They posit the four basic tonal shapes as: level, rising, falling and convex/concave; naturally, standard Mandarin's four tones fit into one of each of these.

However, this is noticeably absent from standard descriptions of any Yue Chinese group (which includes Cantonese and Toisanese as well as the 10-tone Bobai dialect) that I'm aware of, as well as from the main Hakka varieties and from the Hokkien branch of the Minnan Chinese group. Both of these groups have more than four phonemic tones (although the Teochew branch of Minnan has a low falling-rising tone for the 阴去 category).

Many Mandarin varieties, not just standard 普通话 / 國語, have a concave tone. From Wikipedia's page on the four tones of Middle Chinese, Jinan in Shandong, Chengdu in Sichuan, and Nanjing in Jiangsu all share this feature with Beijing. The Nanchang variety of Gan and the Wenzhounese variety of Wu seem to also have concave tones (Wenzhounese having this longest of contours reliably paradoxically in the historical 入声 tones, which are the shortest tones in most of the rest of south China!).

Of those rarities that have both, Fuzhounese of the Min Dong family is probably the most well-known Chinese variety, with the 阴去 tone a concave one, whilst the 阳去 tone is convex. However, to my ear the concave tone is subject to a similar "smoothing" as in standard Mandarin, whilst the convex tone is somewhat more salient. The status of the tones in Suzhounese of the Wu family is subject to debate (as with all the tones of Wu really - tone sandhi has just become too complicated there!): the concave one is well-attested, but the convex one is allophonic to... something else (either low rising or low falling).

Leaving behind the Sino-Tibetan for the Oto-Manguean, Yoloxóchitl Mixtec has both too, although I'm not sure what other tones there are.

Just glancing at the literature for Vietnamese phonetics, the case with Vietnamese phonation and pitch-register is particularly interesting, as the "citation" concave tone "hỏi" is probably the one "tone" where the pitch contour is most important; the breathiness has been analysed has merely a phonetic realisation. However, it seems that it may be confused with huyền in Hanoi where the rising end of the hỏi contour is weakening. Southern Vietnamese on the other hand does not have a canonical concave tone: hỏi has merged with ngã to form a low-rising tone.

What generalisations can be drawn then? Common ancestry? Areal effects? Even support for the maximisation of the distribution of tone shapes is shaky. The only commonality would be that rising and falling tones always exist alongside the concave and convex.

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