An idea from XKCD

[zompist]

Oh, responding to the original cartoon: it's not necessary, so far as I know, to posit formants that are complex numbers.

But suppose there's some situation where a force is defined as F = F₀ cos ωt. And of course we have to do calculus on these, which is a bear.

But a complex number F₀e^iωt happens to be equivalent to F₀ cos ωt + i F₀ sin ωt. And calculus on exponentials is easy. So we could represent that force as a complex number, do the calculus, and just ignore the imaginary parts.

You may know, or have guessed, that this isn't made up; it's something actually done in physics— these forces come up in explaining oscillatory motion. See chapter 23 of Feynman's lectures in physics.

[Richard W]

The thing to complexify is the frequency (to within a factor) omega, which gives one oscillatory effects that tail off with time, which is actually natural and potentially controllable, thought the achieved shapes are unlikely to be neat exponentials. A more natural extra axis would be one for frequency modulation, often (always?) seen in diphthongs.

[alice]

The problem with this idea is what about central vowels — if rounded front vowels and unrounded back vowels are simply more central than their unrounded front and rounded back counterparts, how do we handle distinctions such as beween [ɨ] and [ʉ]?

Does anybody actually know what the formants are for [ɨ] and [ʉ]? According to the Internet, they're a secret.

[bradrn]

When I refer to Labov here, I'm mostly referring to Principles of Linguistic Change: Internal Factors (1994). It's big and dry, but it's packed with data. He refers to laxness to solve a couple of old puzzles in historical linguistics (for English and some other languages). To over-summarize, tense vowels can be seen as the outer perimeter of the triangle, and lax vowels a track just inside that area, and vowels can move along either track, neatly avoiding each other. (The puzzles came in that earlier studies had vowels in about the same place at the same time, so it was not at all clear how they could re-separate later.)

Sounds interesting; I’ll have to read it sometime!

As for whether it’s needed, I’d say it really is. As I said, no-one seems to agree on the exact meaning of any of the symbols, and that’s a real problem in narrow (and even broad) phonetic transcription.

Isn't there general agreement that [i] and [u] are as high and [a] as low as you can manage, while [i] is as front as possible and [u] as back as possible? I realize I'm using articulatory terms, but I do so because that's where the constraints lie. So far as I know a higher-F2 [i] could be generated, just not by a human.

I entirely agree with this. As a matter of fact, so does everyone else — I’ll note that [a i u] are pretty much the only vowels where there is widespread agreement on quality, most probably because they do in fact have the straightforward articulatory definitions you give.

(Then again, even [a] is a bit controversial — the IPA makes it a specifically front vowel, which is a questionable choice.)

We could define [e] and [ɛ] at even steps between [i] and [a], but it's an empirical question whether they would sound like they're at even steps. And it's yet another empirical question whether those points correlate well with the median [e/ɛ] heard in languages with four level divisions.

Yes, I’d be very interested in a study of this sort! Though Ladefoged (again) notes that the [e/ɛ] distinction can itself be quite different between languages, so it’s an interesting point whether taking the average is valid in the first place. (He mentions Italian and Yoruba specifically, with F1/F2 charts showing the former has fairly evenly spaced vowels, while the latter has [e o] closer to [i u].)

The thing to complexify is the frequency (to within a factor) omega, which gives one oscillatory effects that tail off with time, which is actually natural and potentially controllable, thought the achieved shapes are unlikely to be neat exponentials. A more natural extra axis would be one for frequency modulation, often (always?) seen in diphthongs.

What would be even better is to have a nice representation for both the F1 and F2 axes. Alas, complexification of this doesn’t give any obvious improvement, algebraically speaking, though there might be some clever trick with Fourier series to get things working. (I’m taking Quantum Control this semester, so it’s entirely probable that I’ll be learning the relevant abstractions soon. After all, both that and this reduce down to sinusoidals within sinusoidals, more or less…)

[Travis B.]

One thing I found curious was that on an international Zoom call yesterday with quite a few Germans on it, the Germans' /æ/ I heard as my /ɛ/, even though my [ɛ] is /æ/ and I presume they were realizing /æ/ as [ɛ] which would imply that I would hear their /æ/ as my /æ/ despite it not being [æ]. Is StG /ɛ/, which I presume they were substituting English /æ/ with, slightly centralized or something, such that it would fall in the vowel space for my /ɛ/? (My own [ɛ] for /æ/ contrasts with my [ɛ̠]~[ɜ] for /ɛ/ primarily on the basis of frontness, with the former being fully fronted while the latter is somewhat centralized.)

[Travis B.]

Yes, I’d be very interested in a study of this sort! Though Ladefoged (again) notes that the [e/ɛ] distinction can itself be quite different between languages, so it’s an interesting point whether taking the average is valid in the first place. (He mentions Italian and Yoruba specifically, with F1/F2 charts showing the former has fairly evenly spaced vowels, while the latter has [e o] closer to [i u].)

StG in particular is notorious for having a very high [eː], such that many native English-speakers have trouble telling apart StG /eː/ and /iː/.

[bradrn]

One thing I found curious was that on an international Zoom call yesterday with quite a few Germans on it, the Germans' /æ/ I heard as my /ɛ/, even though my [ɛ] is /æ/ and I presume they were realizing /æ/ as [ɛ] which would imply that I would hear their /æ/ as my /æ/ despite it not being [æ]. Is StG /ɛ/, which I presume they were substituting English /æ/ with, slightly centralized or something, such that it would fall in the vowel space for my /ɛ/? (My own [ɛ] for /æ/ contrasts with my [ɛ̠]~[ɜ] for /ɛ/ primarily on the basis of frontness, with the former being fully fronted while the latter is somewhat centralized.)

I wouldn’t know, but I do experience this regularly from the other side — lots of people seem to think my /æ/ is /e/. (‘But why is your name Bread?’, I often get asked.)

[Travis B.]

One thing I found curious was that on an international Zoom call yesterday with quite a few Germans on it, the Germans' /æ/ I heard as my /ɛ/, even though my [ɛ] is /æ/ and I presume they were realizing /æ/ as [ɛ] which would imply that I would hear their /æ/ as my /æ/ despite it not being [æ]. Is StG /ɛ/, which I presume they were substituting English /æ/ with, slightly centralized or something, such that it would fall in the vowel space for my /ɛ/? (My own [ɛ] for /æ/ contrasts with my [ɛ̠]~[ɜ] for /ɛ/ primarily on the basis of frontness, with the former being fully fronted while the latter is somewhat centralized.)

I wouldn’t know, but I do experience this regularly from the other side — lots of people seem to think my /æ/ is /e/. (‘But why is your name Bread?’, I often get asked.)

One thing I find is that there is surprisingly little confusion between /æ/ and /ɛ/ between different NAE varieties even considering the NCVS. Other Inland North-speakers can still tell apart non-Inland North /æ/ and /ɛ/ without fail, and non-Inland North-speakers can likewise still tell apart Inland North /æ/ and /ɛ/ without fail. My suspicion is that non-Inland North [ɛ] is already weakly centralized, while non-Inland North [æ] is not unless it is diphthongized (e.g. centralizing diphthongs for /æ/ before nasals are not limited to Inland North dialects*), so non-Inland North /æ/ and /ɛ/ can be recognized as like Inland North /æ/ and /ɛ/ respectively despite their being nominally quite different.

* Classically /æ/ in Inland North dialects is a centering diphthong in all cases, but I find that in the dialect here it is only a diphthong before nasals, and otherwise it is simply raised without diphthongization.