I am learning a Tigrinya for the last couple of months, and find it difficult to grasp and produce the central vowels of the language (see picture). I want to know if there are other languages which have this three way distinction regarding backness, the only other language I found was Norwegian.
If you're looking for an list of languages, you're probably going to have to do some research yourself. I looked at WALS, but it doesn't seem to have a chapter specifically related to backness. But you could have a look at languages with many vowel phonemes The more vowels, the more likely that there's a 3-way backness distinction.
If you're looking for familiar vowel sounds to help you pronounce the Tigrinya vowels or other central vowels, I suggest you find the vowels on Wikipedia, which usually has a list of languages with a specific vowel. Scanning through that list might give you an idea of how the sound should be pronounced. E.g. start with close central rounded vowel and jump to other central vowels using the IPA vowel chart.
In general, vowel backness (as well as all the other attributes) is not the property of a certain language. It's because of the nature of the human vocal tract.
This way, each vowel sound produced has a certain backness (as well as the other features).
However, languages use different mappings of phones to phonemes. In other words, each point on a vowel diagram is not a point, but rather an area. Everything that falls into that area is mapped to a certain phoneme.
There are plenty of languages that do not distinct backness (on IPA vowel diagram, it looks like an area occupying the whole width of the diagram).
For example:
Polish "mysz" (mouse) [mɘ̟ʂ]. Different speakers may actually say [mɪʂ] or [mʊʂ], but a listener would map it to an existing phoneme, ɘ̟.
German "bitte" (please) [ˈbɪ-tə]
Greek "ακακία" (acacia) [ɐ-kɐ-ˈci.-ɐ]
There is yet another consideration. Although some languages lack cardinal central vowels, those vowel appear in a spoken language due to various phonetic phenomenons (of course, individual for every language). For instance, it's a common effect of centralization/reduction of unstressed vowels in Russian:
"молоко" (milk) [mə-ɫʌ-'kɔ]
Note that this word contains three "o", the last one in stressed, and two unstressed ones are reduced in different ways.
"этот" (this) [ˈɛ-tət]
Note that ɛ (front mid) in this word, regardless of stress, is often pronounced ɜ (central mid).
The realisation of Tigrinya vowels, especially the 'sixth-row' vowel (marked as central in your table) differs and depends on the environment. Please do contact me though if you're still learning Tigrinya, let's share resources for this under-documented languages. cheers
