Chapter 6. The Physics of the Voice: Harmonics, Resonance and Standing Waves

The three stages are about attention. This chapter is about the instrument the attention is directed at. It collects the physics that the earlier chapters used, gives the formulas, and shows the calculations in code, so that the vocabulary of singing (“resonance”, “placement”, “ring”) can be checked against what is actually happening in the air.

Source and filter

Evidence

The standard description of the voice is the source–filter theory set out by Gunnar Fant: the vibrating vocal folds produce a source sound rich in harmonics, and the vocal tract, the tube of air from the folds to the lips, filters that sound, strengthening some frequencies and weakening others (Fant 1960). The peaks of the filter are called formants. The source decides the pitch; the filter decides the vowel and much of the colour.

The vocal folds do not vibrate in a pure sine wave. Each cycle they close sharply, and a sharp event contains many frequencies. A sung note at fundamental frequency $f_0$ therefore contains the harmonics $f_k = k \, f_0, \qquad k = 1, 2, 3, \dots$ At $f_0 = 220\ \text{Hz}$ (the A below middle C) these are 220, 440, 660, 880 Hz and so on, each weaker than the last on average.

The vocal tract as a quarter-wave resonator

Calculation

For a rough first model, treat the vocal tract as a uniform tube about $L = 17\ \text{cm}$ long, closed at the glottis (where the folds are nearly closed) and open at the lips. Such a tube resonates when its length holds an odd number of quarter wavelengths: $L = (2n-1)\frac{\lambda}{4} \quad\Longrightarrow\quad F_n = \frac{(2n-1)\,c}{4L},$ where $c \approx 350\ \text{m/s}$ is the speed of sound in the warm, humid air of the tract. Pressure is greatest at the closed end and zero at the open end, so each resonance is a standing wave with a pressure node at the lips:

c = 350.0      # speed of sound in warm, humid air in the vocal tract, m/s
L = 0.17       # vocal tract length, glottis to lips, m

for n in (1, 2, 3):
    F = (2 * n - 1) * c / (4 * L)
    wavelength = c / F
    # pressure nodes (zero pressure) measured from the glottis
    nodes = [L - m * wavelength / 2 for m in range(n)]
    print(f"F{n} = {F:6.0f} Hz, wavelength {wavelength*100:5.1f} cm, "
          f"pressure nodes at {sorted(round(x*100, 1) for x in nodes)} cm")
# F1 =    515 Hz, wavelength  68.0 cm, pressure nodes at [17.0] cm
# F2 =   1544 Hz, wavelength  22.7 cm, pressure nodes at [5.7, 17.0] cm
# F3 =   2574 Hz, wavelength  13.6 cm, pressure nodes at [3.4, 10.2, 17.0] cm

These numbers, about 500, 1500 and 2500 Hz, are close to the measured formants of the neutral vowel (the “uh” in “about”) in adult male speakers. Changing the shape of the tube with the tongue, jaw and lips moves the formants, and that is what makes different vowels.

The first three standing waves of pressure in a uniform 17 cm tube closed at the vocal folds and open at the lips. Gold dots mark pressure nodes.

This is where the standing waves of the voice are: in the air of the vocal tract, between the vocal folds and the lips. They are not in the skull or the sinuses (Chapter 4). What singers feel as buzzing in the face is vibration carried from this air column and from the larynx into bone and tissue.

Harmonics meet formants

Harmonics of a note at 220 Hz (gold) under the resonance curve of a uniform 17 cm vocal tract (blue). Harmonics that fall near a formant are strengthened.

The figure shows the two halves of the source–filter theory together. The harmonics are fixed by the pitch; the formants are fixed by the shape of the tract. Whichever harmonics happen to fall near a formant come out strong.

Calculation

Which harmonic sits nearest each formant depends on the pitch:

formants = {"F1": 515, "F2": 1544, "F3": 2574}
for f0 in (110, 220, 440, 660):
    harmonics = [f0 * k for k in range(1, 30)]
    near = {name: min(harmonics, key=lambda h: abs(h - F)) for name, F in formants.items()}
    print(f0, {name: (h, h // f0) for name, h in near.items()})
# 110 {'F1': (550, 5), 'F2': (1540, 14), 'F3': (2530, 23)}
# 220 {'F1': (440, 2), 'F2': (1540, 7), 'F3': (2640, 12)}
# 440 {'F1': (440, 1), 'F2': (1760, 4), 'F3': (2640, 6)}
# 660 {'F1': (660, 1), 'F2': (1320, 2), 'F3': (2640, 4)}

Each pair is (harmonic frequency, harmonic number). At a low male pitch, many harmonics lie under each formant, so the formants are well “sampled” and the vowel is clear. At 660 Hz the fundamental is already above the first formant of this tube, so no harmonic can be strengthened by it unless the singer changes the tract.

Evidence

Singers do change the tract. Elodie Joliveau, John Smith and Joe Wolfe measured the vocal tract resonances of sopranos and found that, once the fundamental rose above the first resonance, the singers raised the first resonance along with the pitch so that it stayed close to the fundamental, between about 600 and 1000 Hz (Joliveau et al. 2004). This “resonance tuning” increases loudness for the same effort, and it is one reason why vowels on very high notes are hard to tell apart.

The singer’s formant

Evidence

Trained classical male singers, and many altos, show a strong peak in their spectrum between about 2.5 and 3 kHz that speech lacks. Johan Sundberg showed that this “singer’s formant” arises when the third, fourth and fifth formants are brought close together, which he attributed to a widened lower pharynx and a lowered larynx (Sundberg 1974). Because an orchestra’s sound is strongest at much lower frequencies, a voice with energy around 3 kHz can be heard over it without being louder overall.

This is the acoustic answer to the problem in Chapter 3. Being heard is mainly a matter of where the voice’s energy lies, not how much effort goes into it. A ringing voice is efficient; a pushed one is not.

What the physics says about the three stages

Note

The physics does not prove the method; it makes its claims more precise.

Stage One depends on the fact that the singer’s own blend of air- and bone-conducted sound differs from what any listener hears (Chapter 2). The filtering is different, not the timing.
Stage Two uses the vibration felt in the head as an object of attention. The physics says that vibration comes from the air column and the larynx, so it carries real information about the vowel and the pitch, even though it is not where the sound “resonates”.
Stage Three concerns the pitch, that is, the fundamental $f_0$ and with it the whole harmonic series. The formant structure is a separate matter, governed by the shape of the tract. A singer can hold a steady inner $f_0$ while the vowel changes.

Summary

The voice is a source (the vocal folds, producing harmonics at multiples of $f_0$ ) filtered by the vocal tract (whose resonances are the formants).
A uniform 17 cm tube closed at one end has resonances near 515, 1544 and 2574 Hz; these are standing waves in the air between the folds and the lips.
Harmonics that fall near formants are strengthened; at high pitches, sopranos move the first resonance to follow the fundamental.
The singer’s formant near 3 kHz lets a voice carry over an orchestra without extra loudness.
The physics supports the method’s account of where vibration comes from and why effort is an inefficient way to be heard; it does not show that the method works.

References

Fant, Gunnar. 1960. Acoustic Theory of Speech Production. Mouton.

Joliveau, Elodie, John Smith, and Joe Wolfe. 2004. “Vocal Tract Resonances in Singing: The Soprano Voice.” Journal of the Acoustical Society of America 116 (4): 2434–39. https://doi.org/10.1121/1.1791717.

Sundberg, Johan. 1974. “Articulatory Interpretation of the ‘Singing Formant’.” Journal of the Acoustical Society of America 55 (4): 838–44.

Fan, Y. (2026). Turning the Hearing Inward: A Bioacoustic Study of the Ergen Yuantong Listening Method [反闻闻自性] (Version 1.0.0). 5D Library. https://doi.org/10.5281/zenodo.23666557 · Licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0).