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In 'A course in Phonetics' P. Ladefoged writes:

If we consider vowels to be specifiable in terms of three dimensions, this implies that the cardinal vowels fall on a plane in this three-dimensional space, as shown in Figure 9.4.

The picture: Figure 9.4 from the book

But this vowels are not fall on a single common plane! If this plane existed, it would contain at least a, ɑ and ɔ by definition and would be something like this (we can construct a plane by three points):

enter image description here

(Sorry for inaccurate picture, but the main idea is shown)

As you can see, there is no way to place all of this vowels on the same plane. (I can assign numeric coordinates to vowels and prove it in a more accurate way by calculations if it's needed).

Am I wrong? Is P. Ladefoged wrong?


UPDATE: Mathematical proof for @GregLee

Statement: There is no plane that contains all of this eight vowels at the same time.

Proof: This plane, if it existed, would contain lines a-ɛ and ɑ-ɔ, but this lines are skew, so there is no such plane. Why are they skew? Because

  1. They do not cross each other, because

1.1. If they crossed, it would be either common point of line segments [a-ɛ] and [ɑ-ɔ], either of rays [ɛ, upwards) (part of line a-ɛ) and [ɔ, upwards) (part of line ɑ-ɔ), either of rays [a, downwards) (part of line a-ɛ) and [ɑ, downwards) (part of line ɑ-ɔ) because of same height intervals of elements of this pairs.

1.2. Line segments [a-ɛ] and [ɑ-ɔ] do not cross

1.3. Ray [ɛ, upwards) (part of line a-ɛ) does not cross with ray [ɔ, upwards) (part of line ɑ-ɔ), because

1.3.1. Roundness of ɛ is less than roundness of ɔ

1.3.2. Roundness decreases upwards in ray [ɛ, upwards) (part of line a-ɛ) and increases upwards in ray [ɔ, upwards) (part of line ɑ-ɔ)

1.4. Ray [a, downwards) (part of line a-ɛ) does not cross with ray [ɑ, downwards) (part of line ɑ-ɔ), because

1.4.1. Roundness of a is less or equals in comparison with roundness of ɑ

1.4.2. Roundness increases downwards in ray [a, downwards) (part of line a-ɛ) and decreases downwards in ray [ɑ, upwards) (part of line ɑ-ɔ)

  1. They do not lay in the same plane, because lines a-ɛ and ɑ-ɔ have derivatives of roundness with respect of height with different signs.
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  • It's an abstract plane, on the centroid of the saggital section. But in fact such diagrams are only intended to be two-dimensional; details like lip rounding, nasalization, pharyngealization, etc. use independent articulators, and therefore don't really appear on that trapezoidal tongue placement diagram independently. Apr 7 '15 at 20:16
  • @johnlawlerinexile, there is no plane in geometrical sense of the word that contains all of this vowels at the same time, isn't it? Apr 7 '15 at 20:21
  • Your figure does not seem to be accurately drawn. The line connecting ɔ and ɑ should slant toward the back of the trapezoidal solid, but you've drawn it as though it is parallel with the faces of the solid. You can't prove anything by drawing a sloppy figure.
    – Greg Lee
    Apr 8 '15 at 4:05
  • @GregLee, I updated post with some nongraphical math Apr 8 '15 at 7:25
  • A plane can contain non-parallel lines, so from the fact that lines are not parallel, it does not follow that they are not co-planar.
    – Greg Lee
    Apr 8 '15 at 14:51
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I think we have to conclude that one of the authors mis-spoke or mis-drew. A plane can be defined off of the positions of [i u ɑ], but not passing through all of the cardinal vowels where they are in the chart. I think with some small adjustment on the position of the vowels in the middle, you can get them all on the [i u ɑ] plane. Perhaps the thing to do is compose the graph numerically based on the [i u ɑ] plane and then see whether the other vowels can be forced into acceptable positions on that plane.

Edit: something like this (I'm terrible at drawing-magic).enter image description here

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  • Roundness increases throughout i-a and ɑ-u, so there is no way to construct the desired plane. Apr 7 '15 at 21:20
  • I don't understand. Define the plane with the values (1,0,1), (0,1,1), (.5 ,1,0). Why can't you locate 5 points on that plane which are approximately where he drew [e o ɛ ɔ a]?
    – user6726
    Apr 7 '15 at 21:46
  • "Why can't you locate... [e o ɛ ɔ a]" - because line "e-ɛ-a" and line "ɔ-o" are not parallel (height of points of the first line negatively correlates with roundness; in case of the second line - positively). Apr 7 '15 at 21:59
  • Comment on EDIT: Your picture suggests that "a" is much higher than "ɑ" (But as it seems to me it's wrong: e.g. en.wikipedia.org/wiki/…) Apr 7 '15 at 22:33
  • Well, my vowel positions are crude, and the plane extends downward from the dashed line, I just put it on the line to indicate that I'm not straying from the plane. But the question of whether the resulting planar object is "close enough" is legitimate. It's not obvious what real-world object the "vowel space" is supposed to be, see Ladefoged 1975 et seq.
    – user6726
    Apr 7 '15 at 22:39

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