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Suppose that there is a race of humanoid beings whose mouth parts are able to make a wide, continuous range of sounds (as opposed to humans who I'm considering to have a medium-sized discrete range of sounds). Each sound in these creature's range sounds similar but distinctly different from each "neighboring sound". There are four different transitions that this race can use to transform one sound into another (by moving the tongue up, down, left, or right) and each transform can be done in varying levels (e.g. x1, x2, x0.5, x$\sqrt{2}$, x$\pi$ etc). As these creatures have a "smooth" range of sounds, they have developed a "smooth," "contextual" vocabulary, where each word is derived from the prior word through the eight possible transform combinations and almost infinitude of amounts. The initial concept is indicated through gesture (i.e. pointing or equivalent) and this gesture is typically sustained throughout the conversation.
Words in this language consist of one vowel as previously mentioned, followed by a single consonant that describes the type of word (noun, verb, adjective, or adverb) and the case, gender, number, and/or tense of the word. Communication often occurs simultaneously under ordinary circumstances with the listener merely repeating back what the speaker was saying. Also this species can breath simultaneously while talking but must pause approximately every minute or so to rest their tongue.

My question is how would such a species think, and would their thoughts be translatable to human language or even human thought?

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    $\begingroup$ You might need to expand on why you consider that to be "continuous." To me, it sounds very discrete, with 16 symbols (4 tongue movements, 4 levels), and a highly agglutinative language. I would argue that the human mouth is substantially more expressive than what you describe the alien to be. We see this expression when conveying emotions with our language. The words themselves are very discrete, but the emotion behind them is enormously fluid. $\endgroup$ – Cort Ammon - Reinstate Monica Dec 9 '17 at 23:00
  • $\begingroup$ @CortAmmon 4 tongue movements at almost infinite varying levels, hence I used eg and etc, but I will edit to make it clearer $\endgroup$ – tox123 Dec 9 '17 at 23:15
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    $\begingroup$ @tox123 How does that significantly differ from vowel production in humans? $\endgroup$ – sphennings Dec 9 '17 at 23:22
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    $\begingroup$ Would you have a problem with me providing an answer which is "this is no different from humans?" Our language is actually pretty incredible! $\endgroup$ – Cort Ammon - Reinstate Monica Dec 9 '17 at 23:27
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    $\begingroup$ No reason for such a race of people, who speak until their tongues tire, to develop any different from a race of ventriloquists or overtone singers: youtu.be/vC9Qh709gas . $\endgroup$ – user43711 Dec 10 '17 at 20:38
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Most thinking that doesn't involve a piece of paper is associative

No matter how different the vocalizations are, it is always possible for two species of equivalent intelligence to remind one another of the shared experience. This is because we can always reduce ourselves to drawing pictures or using hand gestures. Over time, the two species can recognize the vocalization associated with that experience. There is only a problem if one species can exprerience something the other species cannot (e.g., one can fly, the other can't), in which case the vocalizations can't be interpreted as there is no common association.

Most thinking that uses paper is symbolic

Paper is limited in how much information it can retain. This forces civilizations to develop "languages" that can express meaningful chunks of information using discrete symbols. Mathematics are the ultimate example of this problem. A symbol can be as simple as summation or as complex as integration... but they're just symbols.

And this particular point is incredibly important. As a civilization advances and must better express itself either mathematically or informationally, its vocal language will eventually conform to its written language. That's because, eventually, vocalizations for "addition" and "integration" must exist, and that means definitions, and that further reduces language to basic, discrete components.

What thinking involving paper isn't symbolic is probably art

This is where vocalization and thinking would be difficult to express to an alien society. What is art? Heck, most humans would have trouble answering that question. Whole books are written in an effort to describe what a single image (worth at lest 1,000 words) "means." Art, or how a species emotionally perceives their experiences, is the one thing that would be difficult to translate no matter the nature of the vocalizations.

Conclusion

If the social, technological, and intellectual capabilities of the aliens were similar to our own, there would be no difficulty expressing scientific and technical ideas; little difficulty expressing common experiences (such as politics, snowboarding, stubbing your toe, and how much it hurts to be hit by a truck); more difficulty communicating philosophy and religion; and we'd have the devil of a time understanding why the beggars are crying (it's not my tie, honestly!).

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The human voice is already able to process an infinite range of sounds. Witness the yoddle. It is not the ability to make sounds that has caused human speech to become discrete, it is the necessity for storing the speech.

In order to store speech so it can be associated with a word and higher level abstract meaning in the mind, vocalizations have to be able to be categorized and compartmentalized. We use discrete phonemes because we can represent them in our brains as discrete phonemes, and we can write them on paper as words made out of discrete elements (letters).

Human speech, in fact, uses continuously changing 'sliding' pitch in phonemes such as "buh" and kuh'. In the lexicon of speech, they are referred to as pre-voiced explosive consonants. In fact, we do not understand speech by its pitch, but by the interplay between four levels of pitch spoken at the same time, in the same phoneme. That is why we can understand high pitched female voices and low-pitched male voices equally. We recognize phonemes based on whether the pitch goes up or down relative to the other three levels as the phoneme is spoken.

So it is not the mind that responds to the species ability to make continuously varying sounds, it is that our speech is limited to specific sounds as a result of the requirements of our minds.

So putting the horse in front of the cart, you are asking us to come up with some form of thinking that is continuous in nature, and does not function on the basis of discrete elements and logical connections. THEN we should come up with a form of communication that such a mind would use.

In fact, you are referring to the human 'right brain', which stores information holistically in pictures. The 'right brain' (I use quotes because it is not physiologically the right brain, but conceptually in popular thought 'right brain thinking') has no speech. The closest it comes is music and melody conferring feeling and emotion, but not concrete substance and logical reasoning. Try communication in pictures instead of words, and you have some concept of the scope. I suppose the closest thing to formal holistic continuous communication we have on earth besides non-verbal music would be true (not finger spelling) sign language with hand gestures representing thoughts, and not actual words. Think mimes and charades. Notice these forms of communication do not require vocalizations at all. They are seen, not heard. They communicate in pictures.

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