Question: I was trying to transcribe my cat's noises into IPA. For example, her wanting to be petted is like [njæː] or [njaː]. But purring doesn't seem to fit any existing notation, except maybe [ʜ]. Has anyone ever mapped a cat's phonetic inventory? Their vocal tracts don't seem to be much different than ours.

Answer: 
Sure, why not? The main problem you will run into is establishing contrastiveness. I strongly recommend reading ch. 4 of the Handbook of the IPA. You can hear two different cats purring at the [wiki page](https://en.wikipedia.org/wiki/Purr), so the question is whether those are two different purr phonemes (technically known as "purreme") or one with a big variation in realization? You could transcribe it as [ʔ̆ɨ̥̃̆ʔ̆ɨ̥̃̆ʔ̆ɨ̥̃̆ʔ̆ɨ̥̃̆ʔ̆ɨ̥̃̆ʔ̆ɨ̥̃̆], though you might have to explain that the breve implies a duration of about 5 msc. Apart from the inconvenience of writing all this junk, there is a deeper scientific question of whether it's wrong to imply that a purr is really made up of lots of tiny segments. So this is where, in dealing with language, somebody would propose a new symbol for this newly discovered sound. You could then propose [ﮛ] to represent this sound, but I doubt it would get approved.

With real language, the IPA symbol [a] represents a wide range of physical sounds, not a precise single physical sound (that doesn't even mean anything). At a certain point, an apparent "a" is far enough back that it sounds like the range of sounds represented as [ɑ], and then you can argue that writing "a" is an error and instead "ɑ" should be used. But that's only possible because there are languages that contrast [a] and [ɑ], so we know that [a] and [ɑ] are different things. The [a] of Arabic and the [a] of English (not US English) are different, but that doesn't motivate inventing new vowel letters: because, IPA is not a system for reducing arbitrary acoustic waveforms to combinations of letters.
Aside from @user6726's answer this is my personal view:
Phonetic transcription is based on human's articulation (as human's produce for this case you can say for example the /k/ sound is palatal but do cats use hard palate to produce sounds?!) Imagine a hypothetical creature that makes a sound like modem's `(actually old dial-up modems)` sound. we would transcribe that sound in a way that our mind hears it like it's language. for better understanding you can have this in mind that in languages like Persian the cat's meow is transcribed as `/miu:/` (not sure if I transcribed it correctly but it's long /u/) having said that you first have to see how the cat articulates this sound (you could use MRI or cameras) then you have to see how the cat produces it and then you can assign a symbol for it.

**BUT** there is another assumption in question: Can we use IPA for transcribing cat's sounds? YES. but it might and will cause a lot of confusions. Do cat's use the same manner of articulation as humans do?

**Further Reading:**

* Remmers, J.E. and H. Gautier. Neural and mechanical mechanisms of feline purring. Respiration Physiology, v. 16, December 1972: 351- 361.
* Holub, Joan. Why do cats meow? New York, Dial Books for Young Readers, c2001. 46 p. (juvenile).
* Frazer-Sissom, Dawn E., D.A. Rice, and G. Peters. How cats purr. Journal of zoology: proceedings Zoological Society of London, v. 223, January 1991: 67-78..
* De Lanerolle, Nihal C. and Frederick F. Lang. Functional neural pathways for vocalization in the domestic cat. In Physiological control of mammalian vocalization. Edited by John D. Newman. Plenum Press, New York, London, c1988: 21-41