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Since that the F0 i.e. the pitch is the first harmonic and all formants are the i-th harmonics, is it possible that the formants of a vowel in a high tone are higher than those in a lower tone?

For example, imagin a language which has a L tone and a H tone, and there is a vowel in the L tone satisfies:

F0=100Hz, F1=800Hz, F2=1500Hz

Then, if that vowel in the H tone(assume that F0=200Hz) satisfies:

F0=200Hz, F1=1600Hz, F2=3000Hz

If so, should I use the value F1/F0, F2/F0 etc. instead of F1, F2 etc. to eliminate such influence?

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  • I think those are two different vowels, and the only real answer is 'it's much more complicated than that'. If you have Praat (free download) you can experiment with this, but I can say from experience that you can't define a vowel by the ratio of its formant frequencies - the absolute values count for at least something.
    – user23078
    Mar 24, 2019 at 16:30

2 Answers 2

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F0 does not affect formant frequency, for a slightly obscure reason. When you look at a spectral cross section, you see the amplitudes of individual harmonics (determined via a Fourier transform). The peaks occur pretty much at multiples of the fundamental. The harmonic with the highest amplitude in the area where you expect a formant is not necessarily at the formant's frequency: the actual formant's peak is usually between two harmonics.

A formant frequency is computed using LPC analysis, usually, which gives you a general picture of the resonances in the vocal tract, independent of fundamental frequency. The underlying theory is known as the source-filter theory, where a given glottal source (pitch and voice quality characteristics) is "shaped" by the LPC coefficients to give an actual waveform.

In other words, the mathematics of formant analysis guarantees that pitch and formant frequency are independent. However: the situation you describe in the formulas does not correspond to a real situation, an you will not find a vowel whose formant's are doubled when pitch is doubled. 3000 Hz is not a realistic F2, nor is 1600 a realistic F1

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  • Are the 'formants' calculated by Praat really the resonances, though, or are they the peaks in the spectrum - i.e. those harmonics that are near enough to a resonance to be amplified by it?
    – user23078
    Mar 25, 2019 at 16:26
  • Least really computed formant's from LPC analysis, not peak picking from Spectra. You can get a spectral slice, ctrl-l if I recall correctly
    – user6726
    Mar 25, 2019 at 20:23
  • I am a little confused, because the Praat manual speaks only of "resonances (formants)" They link papers for the many LPC methods they implement, of which "autocorrelate" follows Markel and Gray's "Linear Prediction of Speech", which is in gbooks at least, and seems like a good entry.
    – vectory
    Mar 25, 2019 at 20:23
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    I got it. What we called as formants are the properties of the vocal trace i.e. the resonator, which can be predicted by the LPC anlysis. However harmonics are the properties of the output sounds, which can be derivated directly through Fouier Transformation. They are in fact two different things, right?
    – C.K.
    Mar 26, 2019 at 8:34
  • @c.k. yes, exactly.
    – user6726
    Mar 26, 2019 at 22:40
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I'm the OP and after four years I'm sure I have enough knowledge to answer the question in terms of acoustics as well as signal processing.

My original question is a misunderstanding of the formant frequency. The key is, the formant frequencies of a specific vowel from the same person is approximately fixed (if we assume the same tongue shape), that is, the formant frequency is a fixed value but NOT a nth order harmonic.

To extend the explanation, a little more acoustical analysis is needed. A certain tongue shape in a given vocal tract can be approximated by a certain vocal tract area function. Given the vocal tract area function, we can solve the wave equation and then we know that in which frequencies the vocal tract maximizes the wave's amplitude. That's the formant frequency values. That is why formant frequency is a fixed value if the tongue shape is given.

Now let's consider the source. In an ideal condition, the wave contains and only contains waves whose frequencies are integer multiples of the fundamental frequency. Let's say a vowel has F1=800Hz F2=1550Hz, and a source with F0=200Hz. Since the source contains only waves with frequencies 200Hz, 400Hz, ..., 800Hz, ..., 1400Hz, 1600Hz, ... Therefore, the amplitude of the 800Hz wave is gained locally maximally, and since there's no 1550Hz wave to be enhanced, the 1400Hz and 1600Hz wave are enhanced maximally(the formant frequencies are only finite locally maximal peaks and the waves with frequencies near the formant frequencies are also enhanced by the vocal tract in the real cases), but maybe larger on the 1600Hz wave.

When LPC algorithm is implemented on this output, we may believe the formants of the vowel are 800Hz and 1550Hz, or we might say the F2 value is between 1400Hz and 1600Hz, but we can not know its specific value, though in most cases, we also don't want to know that.

Anyway, in terms of source-filter theory, source and vocal tract shape are independent. F0 is source-dependent, and formant frequencies are vocal-tract-shape-dependent. That's all.

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    There was a recent paper by Whalen et al in JASA on the problem of computing formants vs. resonances which actually supports the idea that F0 does influence formant computations. The crucial point that they make is that the acoustic measurement "formant" is not the same thing as the physical property "resonance", and that formants can be surprisingly far off from resonances.
    – user6726
    Aug 27 at 4:50
  • Thanks for telling! I went through the paper and realized that's true. The main influence of F0 on formant measurement lies on mainly F1, especially when F0 is high but F1 is low. The key point is, the amplitude of the 2nd or 3rd harmonics is so large (and also gained if the bandwidth of the F1 is not so narrow) that even when the harmonics nearest to the F1 frequency (maybe the 4th, say) is gained, it's still less than the 2nd or 3rd harmonics, and that causes the artifact.
    – C.K.
    Aug 27 at 7:56

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