I'll just say it: the set of distinct vowel sounds humans generate, is a vector space; this implies the consonants are too. Which means that the set of human phonemes can be embedded in . . . Euclidean n-space. But someone here might object, so maybe that means it can't (??). Vectors of course, are mathematical, but in general have some physical basis (which among other things, could mean the vector spaces are "useful"). In order to have a vector space, you need a way to tell the difference between two vectors (i.e. a metric). You need the operation of vector addition to be defined, and you need a scalar field and scalar multiplication. There can also be tighter rules or axioms which must be satisfied, but addition and scalar multiplication are fundamental/foundational. Angles and distances can be abstract and not necessarily related to anything physical (ah, the "power" of mathematics! bwaaha hhaaa).
I suppose if you consider them as sounds, then they can be added as well as change in volume (scalar multiply). Whether n space will suffice for finite n is questionable.
The vowels a,e,i,o,u are all distinctly vocalisable. How do we define "addition" of two or more? In English, a has at least two variants: as in "cat" or "hat", and as in "bar", or "far". What does the addition of these two variants give, and does it correspond to a phoneme? Does the physical fact that a single human can't pronounce two distinct vowels at the same time have any significance here?
It's your claim that they can be represented as a vector space. Isn't it up to you to show how addition is defined? That suggests to me that a symbol like "a" is not representable as a distinct element in your vector space of vocalisations. It would have to be represented as several different elements. This discussion looks suspiciously like it might suffer the same problems as your claim that colors form a vector space.
Why is it up to me? What about all the people who've tried (perhaps succeeded) in representing vowels as a vector space in the last say, 300 years? What about music, is that a vector space? Does the difference between musical notes like G and B say, form a metric? This time think before you post a reply. Ok, what about the symbol A in music? It represents several distinct notes all separated by an octave or factor of 2, as do all the symbols for notes or tones. What about the thousands of other claims from thousands of other people? What about your obvious lack of understanding of what a vector space is?
arfa brane: What's this obsession of yours with vector spaces really all about? You do realise that mathematical models of physical reality are just that, do you not? Sounds aren't a vector space. Colors aren't a vector space. It is possible that some aspects of those things might be usefully given a mathematical representation by vector spaces, but don't mistake the map for the territory. It's your thread. It's your idea. Your claim. Who are you thinking of? You didn't mention any of those people in your opening post. No. Why would you think that it is? You tell me. Does it? If so, how? Look, it's fine if you just want to throw a random thought into the ether to see what comes back. I gave your thread some of my attention. If I'd given it no thought I wouldn't have bothered posting a reply. If you have nothing to add, I'll move on to something more interesting. That suggests that a musical "A" is not a unique element of the general space of musical notes, then, doesn't it? Perhaps this is why instrument makers and others refer to the As in different octaves using numbers, like A1, A2, A5, A7 etc. They will presumably become relevant when you mention them and make them relevant. Until then, they are abstractions in an unspecified ether. Sorry, you've lost me. What are you talking about?
Here's are a couple of questions for you, arfa brane: What's the identity element in your vector space of vowel sounds? What's the inverse of the vowel sound "a"?
Not sure if this helps as I not sure what the thread is about Apparently there is a number 2 version Please Register or Log in to view the hidden image!
It helps, at least by cheering up the reader. (The stretch labeled "USA English" does not sound like any USA English I have heard - are the other languages similarly warped, for some reason? (The Spanish sounds like the Spanish of some familiar region, to me, but I am not a native speaker). This thread has no coherent subject yet. The set of "distinct" vowel sounds humans generate varies according to the language of the humans generating the vowels - there are hundreds of different sets of that kind. With consonants included, there are thousands. The set of "distinct" vowel or consonant sounds a human perceives depends largely on the languages they learned to speak before they were nine years old. Additions or alterations acquired after that age usually involve focused and time-consuming effort, tend to be sparse and auxiliary and unstable. Again: there are at least hundreds, probably thousands, of different sets of that kind. Also, less significantly here but of interest: There are people who can generate two different and distinct vowel sounds simultaneously, by isolating overtone combinations from a single ground tone in different parts of the throat and mouth - commonly in singing, but I have heard its adepts talk in that mode as well. (This guy, for one, live into a microphone on stage at the Cedar Riverside Cultural Center in Minneapolis, MN, years ago: https://en.wikipedia.org/wiki/Kaigal-ool_Khovalyg . Vowels separate by timbre, mostly, and the music of that region roots in, is structured around, manipulations of timbre analogous to the manipulations of pitch more familiar to Western ears). And so forth.
Indeed. This was brought home to me when my son, who is bilingual in English and French, spoke to his mother in French about a family nonsense word he had learnt from me in English. In his French sentence he pronounced this word as a Frenchman would, even though when speaking to me he pronounced as I do. So he had (and has) two separate sets of sounds to draw on, one to speak English and one to speak French.
I guess it's about what vector spaces really are. And it's about what vector spaces are not. Kind of a dual thing . . .
That's not all; the set of distinct vowel sounds a male can make varies quite substantially from a female, or a child of either sex. That variation seems to span all languages and cultures; unsurprisingly researchers have tried to quantify it.
Here's a relevant point: artificial speech systems have to use a standard set of sounds. People who use speech translation technology are not being ripped off. How do you explain Hawking's speech generator; or how do you explain the small number of complaints (i.e. more or less no complaints) from people who listen to his lectures or speeches? Likewise people who buy a modern color TV generally don't bitch about the color realism too hard. Why, if colors and speech sounds are so variable? . . . what does someone who can pronounce two vowels at once do when their speech translator doesn't? Why can most people tell when they're looking at an artificially generated image that it isn't natural? Most people can recognise Hawking's artificial voice and tell it isn't natural, how come we can do this? Most of all, how many posters in this thread have actually read anything about the mathematical structure of music, or speech, or color perception? Without the observers who have the perceptions, what is it?
This is pretty crazy: Evan Parker and Sainkho Namtchylak "Extended techniques" with instruments is one thing, but when the instrument is the mouth*... And, of course, any sort of dental work--or simply losing a tooth--necessitates adaptation of said techniques. * well, mouth and throat
