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Humans can speak making precise movements of our tongues, teeth and vocal chords.

I want a sentient creature whose carnivorous mouth-parts are completely unsuitable for speech to be able to communicate fluently with others of its species.

• A trumpet can certainly make different notes by vibrating the lips against the mouthpiece whilst blowing.




• A manatee can finely control its flatulence as an alternative to a swim bladder.
  https://youtu.be/ktLwBHsA_vk




• Humans have been known to be able to inhale both air and water through their anus.

The great early 20th Century French flatulist, Le Petomane, was able
to do this, ... By sucking in large quantities of air [through his anus], he was able to
perform lengthy shows on stage, and could imitate musical instruments,
farm animals, and bird songs, whistle melodies, and play the ocarina.
https://megafart.wordpress.com/2012/02/18/is-it-possible-to-inhale-suck-in-air-via-ones-anal-opening/

EDIT - On following this up, I fear the claims made above of what he could do were greatly exaggerated. The main point was that he could draw air in at will which is something most people can't.

Question

Could an alien anus-breather evolve to communicate entirely by taking in and expelling wind? Would it have to evolve extra appendages to do so?

I'm talking about a complex language similar to that of humans. Not just alarm calls.

Totally doable. Some animals on Earth communicate by making noises with their third eye:

It's a little boy's dream: talking through farting. But it may be a part of everyday life for herring. A new study finds that the fish make noise by squeezing air bubbles out of their backsides. Researchers believe the noise could be used for communication.

(...)

The team noticed that herring often release bubbles from their anuses when scared or during ascent or descent. They reported this curious finding earlier this year in Aquatic Living Resources.

To get an even closer look, Ben Wilson of Simon Frasier University in British Columbia, Canada, set up tanks in the lab. Wilson's group videotaped the fish and correlated the bubble blowing with the sounds they recorded, which they termed fast repetitive ticks (FRTs). Wilson's group found that the herring need to gulp air at the surface for continued FRTs; when the researchers blocked the herring's access to air, the fish FRTed for a little while then stopped. The team also noted that the herring made this particular noise just as darkness was falling, when they gather. This suggests that the FRTing has a social function, Wilson and colleagues report online 5 November in Biology Letters.

Although related fish have been caught farting nondescriptively, these new herring FRTs are distinctive. "They're really cool signals," says acoustic biologist Michael Fine of Virginia Commonwealth University, Richmond, because they're streams of distinct pulses.

Your aliens could communicate by making pulses of different length and pitch. Pulse length can be controlled by regulating the amount of expelled gas; pulse pitch may be controlled by using more or less rectal length for the fart, or by controlling the anal aperture.

As for how to get the air in there, use a mechanism similar to aerophagia.

While not impossible, the biggest problems with anus speaking is that the colon is designed to be filled with solid waste and you'd basically be unable to speak without purging first. A second issue is that if you are picturing a similar placement for the anus to that you find in most earth animals, you'd also have trouble speaking while moving since you have all those large gluteal muscles around it that flex and extend as you walk. If you think about human speech, it is not so much the faculty of our mouth to eat that makes it good for speech, but the fact that it ties into our respiratory system. Our respiratory system is always clear by design, and it is already very specialized at inhaling and exhaling large volumes of air for which there is not really an evolutionary pressure to cause that adaptation in the anus.

While it would certainly make for an entertaining plot point, and is not entirely unreasonable that this might happen, I can think of several other ways an organism would probably have a stronger evolutionary reason to develop communication than via the anus.