Chapter 2 Source, Filter and the Singer’s Formant

Two parts of one sound

The source–filter theory separates the voice into two parts (Fant 1960). The source is the buzz made at the glottis: a train of air pulses whose spectrum contains the fundamental frequency and all its harmonics, falling in strength towards high frequencies. The filter is the vocal tract, the air passage from the glottis to the lips, which resonates at certain frequencies, the formants, and so strengthens the harmonics that lie near them. Pitch is set mainly by the source; vowel colour and much of what we call tone are set by the filter.

The tract as a tube

The simplest model of the tract is a uniform tube about 17.5 cm long, closed at the glottis and open at the lips. Such a tube resonates when its length is an odd number of quarter wavelengths.

Calculation

$F_n = (2n-1)\,c/(4L)$ for three tract lengths.

# A uniform tube closed at the glottis and open at the lips resonates at odd
# quarter wavelengths: F_n = (2n − 1) · c / (4L).
c = 350.0                       # m/s, speed of sound in warm humid air
for L_cm in (17.5, 15.0, 19.0):
    F = [(2 * n - 1) * c / (4 * L_cm / 100) for n in range(1, 6)]
    print(f"L = {L_cm:4.1f} cm: " + "  ".join(f"F{n}={f:5.0f}" for n, f in enumerate(F, 1)))
# L = 17.5 cm: F1=  500  F2= 1500  F3= 2500  F4= 3500  F5= 4500
# L = 15.0 cm: F1=  583  F2= 1750  F3= 2917  F4= 4083  F5= 5250
# L = 19.0 cm: F1=  461  F2= 1382  F3= 2303  F4= 3224  F5= 4145

A 17.5 cm tube gives formants at 500, 1500, 2500, 3500 and 4500 Hz, close to an adult male’s neutral vowel. Shorter tracts, such as those of women and children, shift every formant up; lowering the larynx lengthens the tract and shifts them down. Vowels are made by changing the tube’s shape with the tongue, jaw and lips, which moves the first two formants in particular.

The singer’s formant

Evidence

In the singing of classically trained male singers, and to a lesser extent of altos, Sundberg found a prominent peak in the spectrum near 3 kHz, which he called the singing formant, now usually the singer’s formant. It arises because the third, fourth and fifth formants are drawn close together, so their resonances reinforce one another. Sundberg’s original model attributed the cluster to the larynx tube, the short narrow passage just above the vocal folds (the epilarynx), acting as a resonator of its own when its outlet is much narrower than the pharynx above it, about one sixth of its cross-section (Sundberg 1974). Later magnetic-resonance measurements found a smaller ratio, about 4.5 to 1, and showed that the fourth and fifth formants depend strongly, though not only, on the larynx tube (Sundberg 1995).

The energy near 3 kHz matters because the long-term spectrum of an orchestra is far weaker there than at its peak, several hundred hertz lower, and because the ear is most sensitive in that region (Sundberg 1987). A voice with a strong singer’s formant carries over loud accompaniment without being louder overall.

Calculation

A tract of 46 short cylindrical sections (epilarynx 2.7 cm, pharynx 8 cm, mouth 6.8 cm) with small losses. The program finds the formants from the transfer function and reports how far apart the third and fifth are, and how strong the strongest peak between 2.5 and 3.5 kHz is relative to the strongest overall, for three ratios of pharynx area to epilarynx area.

import cmath, math

C, RHO = 35000.0, 1.14e-3          # cm/s, g/cm³

def gain(f, sections, loss=0.008):
    """|U_lips / U_glottis| for tube sections (length cm, area cm²) from glottis to lips,
    with the lips treated as an open end (pressure zero)."""
    k = 2 * math.pi * f / C * (1 - 1j * loss)
    A, B, Cm, D = 1, 0, 0, 1
    for L, S in sections:
        Z = RHO * C / S
        a, b, c_, d = cmath.cos(k * L), 1j * Z * cmath.sin(k * L), 1j * cmath.sin(k * L) / Z, cmath.cos(k * L)
        A, B, Cm, D = A * a + B * c_, A * b + B * d, Cm * a + D * c_, Cm * b + D * d
    return 1 / abs(D)

def tract(epi_area, pharynx_area):
    # 2.7 cm epilaryngeal tube, 8 cm pharynx, 6.8 cm mouth: 17.5 cm in all
    return [(0.3, epi_area)] * 9 + [(0.4, pharynx_area)] * 20 + [(0.4, 3.0)] * 17

freqs = list(range(100, 5000, 5))
for name, epi, ph in (("speech-like", 2.0, 3.0), ("narrow epilarynx", 0.8, 3.0),
                      ("narrow epilarynx + wide pharynx", 0.7, 4.5)):
    g = [gain(f, tract(epi, ph)) for f in freqs]
    peaks = [freqs[i] for i in range(1, len(g) - 1) if g[i] > g[i - 1] and g[i] > g[i + 1]]
    band = max(g[i] for i, f in enumerate(freqs) if 2500 <= f <= 3500)
    print(f"{name:32s} ratio {ph / epi:4.1f}:1  formants " + " ".join(f"{p:4d}" for p in peaks[:5])
          + f"  | F5−F3 {peaks[4] - peaks[2]:4d} Hz, 2.5–3.5 kHz peak {20 * math.log10(band / max(g)):+5.1f} dB re strongest")
# speech-like                      ratio  1.5:1  formants  525 1560 2545 3480 4435  | F5−F3 1890 Hz, 2.5–3.5 kHz peak -11.3 dB re strongest
# narrow epilarynx                 ratio  3.8:1  formants  565 1670 2670 3425 4305  | F5−F3 1635 Hz, 2.5–3.5 kHz peak  -6.8 dB re strongest
# narrow epilarynx + wide pharynx  ratio  6.4:1  formants  500 1770 2720 3355 4300  | F5−F3 1580 Hz, 2.5–3.5 kHz peak  -5.4 dB re strongest

The tract transfer function of the simple tube model, for a speech-like tract and for a narrow epilarynx inside a wide pharynx. In the second, the third and fourth formants move together inside the 2.5–3.5 kHz band and the band as a whole is about 6 dB stronger.

Even this crude model shows the principle. Narrowing the epilarynx and widening the pharynx moves the third and fourth formants towards each other and raises the gain in the 2.5–3.5 kHz band by about 6 dB, without any extra pressure from the lungs. The cluster is not as tight as in real singers, whose tracts have shapes the simple tube does not capture, but the direction is right: the effect comes from geometry, not from force.

Note

“Ring” or “ping” near 3 kHz is not the only route to a carrying voice. Sopranos singing high notes rely mainly on tuning the first formant to the fundamental, and many styles of popular singing use a microphone and different resonance strategies. A stronger source, with sharper glottal closure, also adds energy at high frequencies. The singer’s formant is one well-understood mechanism, not a universal standard.

Hypothesis · not established

Not established. In untrained singers, an instruction directed to an external sound image (for example, sending the sound to a point across the room) increases the relative level near 3 kHz more than an instruction directed to the throat or larynx. The hypothesis predicts that the singing power ratio (the level of the strongest harmonic between 2 and 4 kHz relative to that between 0 and 2 kHz) of a sustained vowel at the same pitch and loudness is higher after the external instruction than after the internal one. It would be refuted if, in a randomised within-singer comparison with loudness controlled, the ratio did not differ by more than the measurement noise.

References

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

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

Sundberg, Johan. 1987. The Science of the Singing Voice. Northern Illinois University Press.

Sundberg, Johan. 1995. “The Singer’s Formant Revisited.” STL-QPSR (Speech Transmission Laboratory Quarterly Progress and Status Report, KTH) 36 (2–3): 83–96.

Fan, Y. (2026). The Biomechanics of Phonation and Attentional Shift: Myoelastic-Aerodynamic Theory, the Singer's Formant, Auditory-Motor Feedback and External Focus of Attention [发声的生物力学与注意力转移] (Version 1.0.0). 5D Library. https://doi.org/10.5281/zenodo.23719625 · Licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0).