The point of the example in the question in the title is that, to my knowledge, there are no minimal pairs that contrast [ð] and [θ] in English, yet, if someone pronounced a word with those sounds correctly we would think it sounds wrong or at least think they must be using a different dialect than us. It is not free variation, and it is not phonologically conditioned.

I believe 'x' and 'ng' also behave this way in English, although in these cases we are dealing with two segments, always with 'x', and sometimes with 'ng'. Actually it is not about the orthography, so I should probably talk about [ks]/[gz] and [ŋ]/[ŋg]. You can correct me if I am wrong, but I don't know of any words that contrast only on these sounds. In this case there my be some phonological motivation for [ks] vs [gz]. I haven't thought it through. I know in the case of 'ng' there is some morphological motivation, but I think that would still qualify for the phenomena I am asking about.

I am studying a language which has prenasalized many words that begin with voiced consonants, but some words do not get prenasalized. I don't believe there are any minimal pairs to be found where one the only difference is prenasalization at the beginning of the word.

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    Contrary to your supposition, I believe there may be minimal pairs with [ð] and [θ] in English, they are just very rare for historical reasons to do with regular devoicing of word-final sibilants in OE, hence /bɑːθ/ and /beɪð/ in RP (not quite a minimal pair because of the vowel thing). I will see if I can think of a proper one. – OmarL Jun 13 '19 at 7:42
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    Ah yes. Apparently in some varieties of English (which the wiktionary lists as chiefly American), <swath> may be pronounced /swɔθ/ for "a broad sweep or expanse" or /swɔð/ for "some kind of bandage". – OmarL Jun 13 '19 at 7:46
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    There are thigh and thy. Loath and loathe are also a minimal pair for many people, though homophonous for others. – Nardog Jun 13 '19 at 8:22
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    For /ks/ v. /gz/: ex vs eggs in some dialects.I think your assumption that the contrasts you list aren't phonemic due to lack of minimal pairs may be mistaken – Mark Beadles Jun 13 '19 at 12:47
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    @curiousdannii the question is not about which phonological feature contrasts those two sounds, but whether there is a term for a presumably phonemic distinction that has no minimal pairs to strengthen the case for its existence, yet it's clearly also not allophones in free variation or predictably conditioned by the environment. I don't know if there is such a term but personally I think minimal pairs should be seen as an aid to show phonemicity, not as a thing that absolutely has to be there or else sounds are just allophones. – LjL Jun 13 '19 at 14:30

First, there are no terms that everyone agrees with. The thing you are talking about is phonemes, but where there are no minimal pairs that utterly prove phonemic status. Two sounds are "phonemes" if their distribution is not rule-governed. If we get rid of the few minimal pairs, the two fricatives still contrast. Some people might maintain that you need minimal pairs to utterly prove contrast, but lack of MPs has never been sufficient to prove contrast.

So there is no standard term for contrast without minimal pairs.

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Quick note, there are two minimal pairs for [θ] versus [ð] in English: ether ~ either, and thigh ~ thy.

But even if there aren't minimal pairs, a distinction can be phonemic if it is unpredictable. The distribution of [ŋ] versus [ŋg] can be predicted quite reliably:

  • [ŋ] is used at the end of a morpheme (sing, sing-er)
  • [ŋg] is used in the middle of a morpheme (finger, anger)

But the distribution of [θ] versus [ð] can't be. The best you can do is list all of the words with [ð] in them, and say "everything else has [θ]". Which is strong evidence that the two are separate phonemes, even if the words ether and thy drop out of the language entirely.

EDIT: Colin Fine has pointed out a few more minimal pairs: mouth (n) ~ mouth (v), sooth ~ soothe, and wreath ~ wreathe. Many thanks!

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    I think the logic that dictates minimal pairs cannot be an absolute requirement for attributing phonemic status to a pair of sounds can be seen by doing a mental experiment where minimal pairs exist, but then the words that form them stop being current, essentially disappearing from the language: thy is still understood and used in some very limited contexts, but let's say it disappears completely from any speaker's knowledge and that the ether/either pair doesn't exist... surely the accidental fact that an example showing contrast is now gone cannot, on its own, change phonemic status. – LjL Jun 13 '19 at 18:40
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    Also mouth (n) vs mouth (v); wreath vs wreathe, and sooth vs soothe. – Colin Fine Jun 13 '19 at 19:15
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    Also 'backs' and 'bags' for minimal pair /ks/ vs gz/. – Mitch Jun 13 '19 at 19:17
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    @Draconis Oh, right. I was just trying to relate to the pairs the OP gave. – Mitch Jun 13 '19 at 19:22
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    What about "longer"? That segment is [ŋg] but would be [ŋ] according to your bullet points. Also consider that nouns end in [θ] always, a fairly reliable prediction from what I can tell – OmarL Jun 14 '19 at 12:25

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