I’m not expert in typology, but I wonder if it’s possible for a language to ban t+S sequences but have a phoneme č.
Does anyone know of an example of one such language?
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English is a prime example, but that is a matter of analysis. One problem with the question as asked is that it mixes alphabets, contrasting tS (IPA tʃ) and č. The letter č is not IPA, so to normalize the letters into IPA, the question would presumably be about the cluster t plus ʃ versus the single segment t͡ʃ. Polish czysta and trzysta, sometimes transcribed as [t͡ʃɨsta] versus [tʃɨsta], is an example of a contrast.
English on the other hand does not have a contrast. The question then is whether the thing in "church" should be analyzed as a cluster, or as a single segment. It is generally believed that it should be treated as a single segment, on the grounds that treating it as a cluster would complicate the rules for constructing syllable onsets. Stop-plus-fricative onsets are generally not found, and treating "ch" as a cluster would require a complex additional rule, which can be avoided be treating "ch" as a single consonant.
To be fair, we have to also see if there are any complications introduced by the single consonant analysis. "ch" does not combine with any consonant in the onset, so *chmee, *chrowl. This is similar to the restriction on ʃ, as long as you exclude (German and) Yiddish words like shmatte and Schweppes. However, there is one well-attested and unavoidable ʃC cluster, ʃr as in shrimp, shrove etc. One approach to this is to say that only /s/ can appear as a fricative in an onset fricative+C cluster, and treat [ʃ] in [ʃr] onsets as /sr/ (note the gap that there are no words like *[srɪmp]). If you accept the argument thus far, then it's simply a matter of also excluding the segment /t͡ʃ/ from onset clusters. (I am delaying the problem of "Toronto" being pronounced [t͡ʃranə] in said city, and "tree" being something like [t͡ʃri] in various parts of North America). The claim would be, specifically that [-anterior] coronals cannot appear in the onset with another consonant.
A large problem with this account is that it depends on there being an independent principle outlawing preconsonantal alveopalatal sibilants in the onset. While etymologically speaking all ʃC clusters derive from another language, that doesn't make them "not part of English". It is extremely unlikely that anyone pronounces the name Schwartz as [swɔrts], and Schwarts / sworts is probably a minimal pair for [ʃ,s] for people who know the word "sworts". So actually, we do have ʃC clusters.
This problem might be avoided by restating the relevant onset restriction: [-anterior] coronal stops are prohibited, but there is no ban on fricatives at the place of articulation. (Note that affricates qua single consonant are classically treated phonologically as stops). The problem is that the motivation for treating "ch" as a single segment is that treating it as a cluster requires adding otherwise unnecessary exceptional rules (exceptionally allowing t+ʃ). Alas, we need an otherwise unnecessary exclusion, to rule out [t͡ʃC].
The set of problematic onset clusters would be [t͡ʃw] and [t͡ʃr]. Stop plus obstruent is generally banned, likewise stop plus nasal, which leaves stop plus glide or liquid. There is also a ban on coronal stop plus [l], and you don't get [j] after alveopalatals... reducing the possibilities to [t͡ʃw] and [t͡ʃr]. As for [t͡ʃr], some speakers do pronounce initial "tr" something like that, so for them, it is possible that "tree" is /t͡ʃri/, and there is no gap. I say [tri], so for me there would be an unexplained gap. There is no independent rule that would rule out [t͡ʃw].
The point here is that invoking surface co-occurrence as the diagnostic for phonological analysis is a double-edged sword. In lieu of a Polish-type argument from contrast, the question is, how do you establish that a certain apparent CC sequence is a single consonant, versus a cluster? Analogous issues arise in the treatment of "prenasalized consonants" (NC clusters), "glottalized consonants (C+ʔ clusters), "rounded / palatalized consonants" (Cj and Cw clusters). Indeed, there are treatments of "sp, st" where these are "prespirantized stops" (Fujimura's CD model). Obviously, nothing can be concluded in a complete theoretical vacuum, but I find that it's extremely difficult to find nearly-universal agreement on the theoretical primitives pertaining to the problem of onset cluster vs. complex segment analysis.
There are many languages that wouldn't permit a sequence like /t/ + /ʃ/ because it involves either coda /t/ (in the case of heterosyllabic /t.ʃ/ or a tautosyllabic stop-fricative complex coda /tʃ./) or a stop-fricative onset cluster, and it's not that unusual for a language to ban both of these things, while still allowing affricates like /t͡ʃ/ to occur as onsets.
(As user6726's answer pointed out, it's actually a bit difficult to establish based on other parts of the phonology that a language bans stop-fricative onset clusters, but onsets like [tʃ] are conventionally analyzed as affricate phonemes rather than as stop-fricative clusters unless there is clear evidence that they are two phonemes rather than one.)
For example, Japanese has [tɕ] as a possible onset, which can be analyzed as an affricate phoneme /č/ (I think there are other possible more abstract analyses, like an allophone of /t/ or /tj/, but my impression is that this is typically only considered to be the diachronic or morphological source of /č/). Japanese doesn't have coda /t/, nor any contrasting sequence of /t/ + a sibilant fricative. It does have a distinct phoneme /t/, as well as something that is probably a phoneme /š/ (as with /č/, the analysis is a bit complicated because /š/ originated from conditioned palatalization of /s/, and there are still some neutralizations and morphological alternations between Japanese š and s that could be understood as indicating that š is "underlyingly" something like /s/ or /sj/).
Standard Chinese has [ts], [tʂ], [tɕ], [tsʰ], [tʂʰ], [tɕʰ] as onsets, but doesn't have coda /t/ or any (other) stop + fricative sequences as onsets, so these are analyzed as affricate phonemes. Chinese does have /t/, /tʰ/ and /s/, /ʂ/, /ɕ/ phonemes.
Nick Nicholas brought up the example of Tsakonian in a comment.
Another example I thought of since posting this answer is standard Italian: it has č, t and š as phonemes, but no tš clusters, because coda consonants and consonant clusters are fairly limited in the native vocabulary of standard Italian. There is geminate čč. It is phonetically realized not with a double release like [tʃtʃ], but with greater duration of the pre-release portion, something like [tːʃ], but Italian speakers seem to perceive this as phonologically involving coda /č/ rather than coda /t/, and may transcribe it phonologically as /t͡ʃt͡ʃ/. The consonant /š/ behaves a bit interestingly as it is one of the "intrinsic geminate" consonants that always occurs as a geminate when it is in intervocalic position.
I think word-final /t/ can occur in Italian in loanwords from English, but I don't know too much about the way this is typically realized phonetically, and I'm not sure whether loanwords should be included in a discussion of this kind of phonotactic "ban".