I'm making a song-synthesizing software, so I built some models about human speech, and I'm testing them. But it turns out there is an obstacle. I seem to have misunderstood how vowels are rounded.
I modeled the mouth to be controlled by two factors: Compression (abbr. COM) and protrusion (abbr. PRO). These play a role when synthesizing rounded vowels, and are represented by a number from 0 to 1. The more COM is, the more the mouth closes vertically. The more PRO is, the more the mouth closes horizontally.
I recorded the "extreme" vowels, a.k.a. [a], [ɶ], [i], [y], [ɑ], [ɒ], [ɯ], and [u]. I've thought three factors, namely height (abbr. C/O; takes a number from -1 to +1), backness (abbr. B/F; takes a number from -1 to +1), and PRO, were sufficient to model them. But it failed, for apparently there were a fourth factor: COM.
So let me summarize how I'm going to refine my model. I set [ə] to be the "neutral" vowel where C/O, B/F, COM, and PRO are all zero. The followings are apparent:
For open vowels, the jaw opens (along with the mouth), and C/O becomes negative.
For closed vowels, the dorsum lifts, and C/O becomes positive.
The jaw-opening and the dorsum-lifting never occur simultaneously. (It's a rather awkward position)
For unrounded vowels, both COM and PRO are zero.
For [a], C/O = -1; B/F = -1; COM = 0; PRO = 0.
For [e], C/O = +⅓; B/F = -1; COM = 0; PRO = 0.
For [i], C/O = +1; B/F = -1; COM = 0; PRO = 0.
But rounded vowels? This is where I have a confusion. As far as I've observed my own pronunciation:
For [u], C/O = +1; B/F = +1; COM = 1; PRO = 1.
For [ɶ], C/O = -1; B/F = -1; COM = 0; PRO = 1.
For [y], C/O = +1; B/F = -1; COM = 1; PRO = 0...?
Despite Wikipedia states that compression and protrusion are two different types of rounding, I had to model like this, because eventually I would model consonants as well. It seems feasible to model approximants like this:
For [β], C/O = 0; B/F = 0; COM = 1; PRO = 0.
For [βʷ], C/O = 0; B/F = 0; COM = 1; PRO = 1.
For [ʁ], C/O = 0; B/F = +1; COM = 0; PRO = 0.
For [ʁ̹], C/O = 0; B/F = +1; COM = 0; PRO = 1.
For [β͡ʁ], C/O = 0; B/F = +1; COM = 1; PRO = 0.
For [ʁʷ], C/O = 0; B/F = +1; COM = 1; PRO = 1.
So far, is my model, or is the analysis above, consistent with usual phonetics?