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I'm new in speech-processing and there are some notions confused me.

Through Fourier Transform, we can get spectra of specific sound signals, the harmonic frequencies can be shown as the peaks in the spectrum. So are those harmonic frequencis i.e. the peak frequencies equivalent to the formant frequencies? Or what's the relationship between harmonic frequencies and formant frequencies?

BTW, I have anothor question, that is, why the harmonic frequencies are integer multiples of the fundamental frequency? What if I synthesize a sound with a 440Hz wave and a 441Hz wave? Will the fundamental frequency of that sound be 1Hz!?

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why the harmonic frequencies are integer multiples of the fundamental frequency?

Physics!

Pretty much all of the physical processes that create vibrations at a base frequency, also create vibrations at integer multiples of that base frequency. This includes the way the human vocal chords work.

You can artificially synthesize a sound with any frequencies you like, but you'll find it sounds somewhat "unnatural" unless you add some extra frequencies at integer multiples of the fundamental.

Or what's the relationship between harmonic frequencies and formant frequencies?

Imagine for a moment that the human vocal tract didn't exist. In other words, you just had a voice box, and nothing above that to alter the sound.

Based on the physics of the vocal chords, this would produce a "sawtooth wave", very much like the sound of a bowed string. If you Fourier-transform such a wave, the nth harmonic's amplitude is roughly proportional to 1/n.

But with the vocal tract in place, that's not what we see. The geometry of the vocal tract imposes an "envelope" on the wave, damping some parts and amplifying others. The peaks of the envelope are what we call the formants.

diagram of envelope superimposed on harmonics

In this diagram, the blue line is a Fourier-transformed speech sample, with each peak being a harmonic. The red line is a reconstruction of the envelope, a polynomial that transforms a sawtooth wave into this particular speech sound—where the red line is high, it's amplifying the harmonics, and where it's low, it's damping them. And the green bars on top/blue bars on the bottom mark the formants.

(The diagram is from my own work, released under CC-BY; the units on the horizontal axis are Hz, and the units on the vertical axis don't really have any physical meaning.)

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    "The peaks of the envelope are what we call the formants" – I wish someone had explained that to me in these terms. If you have more material published I want to read it :) – melissa_boiko Aug 2 '19 at 9:20
  • @melboiko It is not that a formant, his explanation is very very simplified. – amegnunsen Aug 2 '19 at 10:27
  • Thank you very much! I got it. Only sounds which satisfy that all the harmonics are integer multiples of fundamental frequencies can be perceived as "natrual", "harmonic" or "musical", am I right? Even though I just put two or three coprime frequencies(but not infinite frequencies) together, the synthesized sound will also sound "inharmonic"? – C.K. Aug 2 '19 at 15:17
  • @C.K. To a certain extent, yes. Like Amegnunsen says, this is a very simplified explanation—sounds with multiple sets of harmonics can also sound "natural", such as a chord on a violin, but anything without any harmonics at all will tend to sound "artificial" (put a couple sine waves together and you'll hear it). – Draconis Aug 2 '19 at 16:07
  • @melboiko Thanks, that's high praise! Like Amegnunsen said, there are a lot of details I glossed over, but thinking of the sound as a source (sawtooth wave) plus a filter (envelope) was what gave me an intuition for formants in the first place. – Draconis Aug 2 '19 at 16:09

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