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Sorry if this question sounds a bit basic. I haven’t had a solid grounding in phonology/phonetics yet so I am a bit confused about these concepts.

We’re trying to build a model which studies certain properties of tonal languages. I understand that a vowel can be generally represented by four formant frequencies. However, I’m now trying to add tone into the representation, and I’m a bit confused by the relation between formant frequencies and pitch frequency.

In the following picture from Praat which shows four tones of Chinese, it seems that the numbers in black on the left are for formant frequencies (represented with red lines), while the numbers in blue on the right are for pitch frequency (represented with blue lines).

Four tones of Chinese

I've read the paper (de Boer 2000) which represents vowels with two dimensions: f1 frequency and a weighted result of f2-f4 frequencies. Now we're trying to add the tone property into a similar model. My main concern is, whether pitch frequency is completely independent of formant frequencies. If so, I might be able to just add it as extra one or two dimensions which are supposed to be independent of the already existing dimensions for formant frequencies (Though the scale of such dimensions would still be a concern). If not, then the task of adding tone into the model might be considerably more challenging.

Somebody has suggested that pitch is only related to the base frequency/base f0, and thus shouldn't interfere with the two dimensions already existing in de Boer's model. Does that make sense?

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Yes, F0 (the fundamental frequency) is the acoustic correlate of pitch (which is a perceptual concept). The fundamental frequency F0 is also the first harmonic H1 of the sound. If F0 is 100 Hz, the second harmonic H2 would be at 200 Hz, the third H3 at 300 Hz, the fourth H4 at 400 Hz, and so on. Vowel formants are located at different harmonics depending on the shape of the vocal tract (so F1 could be H3 in one vowel, but H4 in another). So, knowing the fundamental frequency F0 is not enough to know the formants of a vowel. If you want to know more about fundamental frequency and harmonics, I suggest:

Hayward, Katrina. 2013. Experimental phonetics: An introduction.

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There is a limited sense in which F0 (which is the acoustic property perceived as pitch) and F1-F5 are not independent, which is that if you have a tiny larynx (high F0), given the nature of human anatomy you will not have a really long tube (which determines formant frequencies). Another dependency is that resonance frequency can always be meaningfully computed, but F0 cannot. You can calculate formants from fricatives and stop bursts and in general from any aperiodic sound, but computing fundamental frequency presupposes a semi-periodic source. Thus anything with pitch has a formant, but the converse is not true.

Formants are derived from LPC analysis, which gives you the abstract resonance properties of the vocal tract, and coupled with an "impulse" (something like the "pure glottal wave") can reconstruct the speech waveform. A formant frequency is independent of the fundamental to the point that a formant frequency can be between the multiples of the fundamental (e.g. can be at 500 Hz when the fundamental is 300 Hz).

So you not only can add F0 as an independent dimension, you have to.

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