In this chapter, we have explained Rayleigh's rule, that constriction at a standing-wave antinode lowers its frequency, vice versa at nodes rather than antinodes, and vice versa for widening rather than constriction. Viewing the resonating vocal tract as a half-open, ideal acoustic tube which normally (in the sense of ``on average'') has a uniform cross-section, we applied this rule to relate many of the central observed facts of acoustic phonetics to particular (and correct) tube configurations, and to map them directly to articulatory configurations. Through this model we related the acoustic and articulatory effects of vowel height and frontness, rhoticity and laterality, and several effects of increasing constriction in the transitions from vowels to various consonants. We have shown that F1 measures mouth opening, F2 measures relative constriction in the middle third of the vocal tract (i.e., front-back tongue body position), and F3 is lowered by very local, articulatorily peculiar constrictions at nodes of the 5/4 wavelength standing wave. Also, bilabial closure, constricting the open-end antinode of each formant, lowers all formant frequencies; and front-velar closure, close to an F2 node and an F3 antinode, raises F2 and lowers F3 towards each other. We explained the heretofore mysterious, central fact about stop-vowel coarticulation, the so-called ``locus'' effect for coronals: a constriction, located between the open-end antinode and the palatal node of F2, shifts the sum of antinode and node constrictions towards, but not all the way to, a balance, so that relatively high F2 falls and relatively low F2 rises. Finally, in a small speech production experiment, we verified the theory's prediction that the F2 locus for dentals should be lower in frequency than for more retracted coronals, since the location of constriction adds a greater component to the F2 antinode sum for dentals than more posteriorly articulated coronal stops.
In this derivation of formant structure from the shape of the resonating vocal tract, we have explained many of the most interesting, most complicated, most well-documented patterns in phonetics.
This quite old theory has interesting implications for current phonetic and phonological theory. For example, in important recent work on vowel articulation and phonology,2.14 Wood and his colleagues have challenged the traditional basic vowel dimensions, height and backness, arguing that the model of Stevens and House (1955), taken up also in Fant (1960), is both phonetically and phonologically superior. Their non-traditional account of vowel phonology, claims that ``the vocal tract is narrowed at one of four locations: along the hard palate for [i-], and [y-9#9]-like vowels, along the soft palate for [u-U] and -like vowels, in the upper pharynx for [o-] and -like vowels, and in the lower pharynx for [æ-a]-like vowels.'' They claim that ``one cannot deduce articulation by translating F1 into `height' and F2 into `backness' ''(Pettersson & Wood, 1987). However, this is a rather good summary of what we have just shown can be done. They themselves refer repeatedly to the tuning of the shape of the vocal tract in relation to the sensitive nodes and antinodes of the standing waves. It was their mention of the idea, in fact, that led me to pursue the theory of nodes and antinodes.2.15
We have seen that the effects of the shape of the vocal tract on formant frequencies match very closely with the traditional vowel features: openness and changes in F1 are directly related, and F2 is a measure of the combined effects of tongue-body frontness and lip aperture. Thus their complicated articulatory theory of place-of-articulation for vowels seems unnecessary from an acoustic point of view.
Such a theory also seems unnecessary from a phonological point of view. In an economical and accountable description of phonetic/phonological dimensions, there should be a consistent interpretation of the dimensions in both articulation and acoustics. Articulatory dimensions should map uniquely onto acoustic dimensions, and vice versa. For example, different degrees of articulatory openness should translate unambigously onto some acoustic dimension (F1), and indeed they do. It would be undesirable if different points on the articulatory dimension should map onto the same point on the acoustic dimension, so that if there were 5 degrees of height, for example, the third and fifth had the same phonetic value. Changes in an articulatory dimension should be realized by similar corresponding changes in the acoustic dimension, no matter where on the scale. Doubling back is undesirable.
This criterion eliminates many possible articulatory-phonetic dimensions from consideration as phonological dimensions. The theory of node-antinode constrictions implies that constriction at a node is acoustically equivalent to widening at an antinode, though the articulatory configurations are different. It implies that ``there is an essential many-to-one mapping from articulation to acoustics.''2.16 Only those articulatory dimensions which do have an unambiguous acoustic interpretation are acceptable, according to this criterion.
The theory that vowels have a place of articulation which varies from the palate back to the low pharynx has just this kind of undesirable property. As the place of constriction moves back in the vocal tract from the palate to the glottis, the acoustic effect on F2 is first to lower it (while moving the constriction towards the F2 antinode in the upper pharynx), and then to raise F2 (moving past the antinode, towards the node at the glottis). So the mapping from the articulatory dimension to the acoustic dimension actually changes direction: moving the constriction back can have opposite acoustic effects. Thus the place-of-articulation theory of vowels not only is overly complicated from the point of view of predicting acoustic structure, but also has an inconsistent mapping from articulatory to acoustic dimensions.
The theory presented here of the articulatory-acoustic mapping amounts to a reaffirmation of the traditional phonetic/phonological dimensions of openness and frontness (or height and backness; these are opposite sides of the same coin). However, some differences between the traditional articulatory dimensions and these dimensions which are simultaneously articulatory and acoustic should be noted.
``Openness'' is a property of the acoustic tube formed by the vocal tract, not a tongue feature or a jaw feature alone, but a feature that derives from both tongue and jaw raising at the same time. The gradual widening of the tube at its open end along with gradual narrowing at its closed end are what constitute openness. It is not uniquely a lingual dimension nor a jaw dimension, but a dimension that measures their joint effect on the shape of the tube.
In several of the linguistically distinctive phonetic patterns discussed in this chapter, it was found that unrelated articulations had the same acoustic effect. The most striking confirmation of this was the predicted and observed pattern of three places of articulation for : labial, post-palatal, and low pharyngeal. Constrictions at each of these places have the same acoustic effect, that of lowering F3 by constricting at each of the velocity antinodes of the third standing wave in the vocal-tract.
Similarly, the raising of F3 sometimes found for [l] is consistent with a double apical/velar articulation of [l], where the constrictions are at points in the vocal tract where  has the greatest widening: at nodes of the F3 standing wave.
Other cases support the same conclusion. For example, the usual co-occurrence of lip rounding with tongue-body backing has the joint effect of lowering F2, by constricting at both of the antinodes of the second standing wave. The articulatorily unrelated constrictions have the same acoustic effect. It almost seems that the articulation is designed to make the acoustic effect; it is put together out of separate pieces, as it were. It is no coincidence that the lips are often spread in the production of the extreme front vowel, [i], since widening of the tube at the F2 antinode that is located at the lips has the same acoustic effect as constricting the tube at the F2 node at the palate. Extreme tongue-body frontness has the same effect on F2 as lip-spreading. Again, separate articulations are joined for a similar effect.
Between Jakobson, Fant and Halle (1954) and Chomsky and Halle (1968), the phonetic substance of phonological features was shifted in the theory from acoustics to articulation. This chapter might seem to suggest a great leap backwards, at least in the analysis of vowel sounds.2.17 Here we find that the simplest dimensions for vowels are acoustic, not articulatory, while quite complicated simultaneous articulations are found to occur whose best explanation seems to be the acoustic effect that they produce. However, rather than hopping back and forth over the articulatory/acoustic fence, let us instead simply recognize that the mapping from articulation to acoustics is causal, direct and simple. And if the patterns of constriction of the tube formed by the vocal tract are symmetric with respect to the nodes and antinodes of the standing waves in the tube, then the mapping is invertible as well.
Thus the measurements of the the first two formant frequencies, F1 and F2, that constitute the bulk of the phonetic data of this thesis, are to be taken as not just acoustic measurements, but phonetic measurements, which directly reflect articulatory dimensions of vocal-tract openness and tongue-body frontness/lip aperture. They are not mere acoustic epiphenomena, that are articulatorily uninterpretable. They are objective and rather precise measures of the fundamental, continuous phonetic dimensions of vowel quality, which are directly realized both in articulatory patterns and in acoustic patterns.