A brief recap: it is the epilarynx

Chapter 2 of Volume 10 used a model of tube sections to explain the origin of the singer’s formant. Here we only recap the conclusions and correct one statement in the framework.

The framework’s “pharynx to ventricle-exit area ratio above 6:1” therefore needs two corrections. First, the structure that matters is the epilarynx (larynx tube), not the laryngeal ventricle, which is a small recess between the vocal folds and the ventricular folds, a different structure. Second, the direction and size of the ratio: in the 1974 model the exit of the larynx tube had about 1/6 of the pharyngeal area (the pharynx about six times as large), and the 1995 MRI measurements gave only about 4.5 times, not “more than 6:1”. Chapter 2 showed that the same narrow epilarynx also increases the inertance of the tract: it affects both the “filter” (clustering of formants) and the “source” (source–filter coupling).

The singer’s formant lies roughly in the 2.5–3 kHz region, varying with voice type, lower in low voices and higher in high ones (Sundberg 1977). This region coincides with the resonance of the outer ear.

“A signal-to-noise gain of 10–15 dB”: first, what it means

The framework states that the singer’s formant raises a singer’s “signal-to-noise ratio against the orchestra by 10–15 dB”. On checking, no study was found that measured this figure directly; it looks like an estimate derived from “the formant cluster rises by so many decibels”. Rather than copy it, we can lay the accounting open: under what assumptions would one get this figure?

We do a transparent calculation with long-term average spectra in one-third-octave bands. The orchestral spectrum peaks at 450 Hz, falls by 6 dB per octave below and by $S_o$ dB per octave above (three values: 6, 8 and 10). The voice starts from a “speech-like” spectrum (peak at 500 Hz, falling 8 dB per octave above), to which a “formant cluster” $H$ dB high is added near 2.8 kHz; the overall voice level is then reset to its original value, since the singer’s formant needs no extra total energy. The orchestra as a whole is 10 dB louder than the singer. The outer-ear gain is the same for both.

# Singer-to-orchestra ratio by band under explicit, illustrative assumptions (1/3-oct
# LTAS): orchestra peaks at 450 Hz (-6 dB/oct below, -So above); a "speech-like" voice
# peaks at 500 Hz (-6 / -8 dB/oct); the singer adds an H dB formant-cluster bump at 2.8
# kHz (Gaussian, 0.35 oct s.d.), rescaled to the SAME overall level; the orchestra is 10
# dB louder overall. An outer-ear gain (about +15 dB near 2.7 kHz) applies to both
# sources.
import math

BANDS = [round(1000 * 2 ** (k / 3)) for k in range(-9, 10)]          # 125 Hz ... 8 kHz
# the 2.5 and 3.2 kHz bands
SF = [BANDS.index(f) for f in BANDS if 2400 <= f <= 3300]

def ltas(f, peak, below, above):
    o = math.log2(f / peak)
    return below * o if o < 0 else -above * o

total = lambda levels: 10 * math.log10(sum(10 ** (v / 10) for v in levels))
ear = lambda f: 15 * math.exp(-0.5 * (math.log2(f / 2700) / 0.6) ** 2)

def spectra(h, so=8.0, d=10.0):
    orch = [ltas(f, 450, 6, so) for f in BANDS]
    voice = [ltas(f, 500, 6, 8) + h * math.exp(-0.5 * (math.log2(f / 2800) / 0.35) ** 2)
             for f in BANDS]
    shift = total(orch) - d - total(voice)
    return orch, [v + shift for v in voice]

def band_snr(h, so=8.0):
    orch, voice = spectra(h, so)
    heard = lambda s: total([s[i] + ear(BANDS[i]) for i in SF])
    i5 = BANDS.index(500)
    return heard(voice) - heard(orch), voice[i5] - orch[i5]

if __name__ == "__main__":
    print("orchestra 10 dB louder overall; SNR = singer minus orchestra in the same band")
    print("bump H | 2.5-3.2 kHz SNR, So = 6 / 8 / 10 dB/oct | gain vs H = 0 | 500 Hz SNR")
    base = band_snr(0.0)[0]
    for h in (0.0, 6.0, 10.0, 15.0):
        r = [band_snr(h, so) for so in (6, 8, 10)]
        print(f" {h:4.0f}  |   {r[0][0]:+5.1f} / {r[1][0]:+5.1f} / {r[2][0]:+5.1f} dB"
              f"   |   {r[1][0] - base:+5.1f} dB    |  {r[1][1]:+5.1f} dB")
    orch, voice = spectra(10.0)
    lev = total(voice) - total(orch)
    print(f"check: with H = 10 the singer's overall level is {lev:+.1f} dB re orchestra"
          " (by construction)")
    print(f"outer-ear gain at 2.5 / 3.2 kHz: +{ear(2520):.1f} / +{ear(3175):.1f} dB"
          " to BOTH sources")
# orchestra 10 dB louder overall; SNR = singer minus orchestra in the same band
# bump H | 2.5-3.2 kHz SNR, So = 6 / 8 / 10 dB/oct | gain vs H = 0 | 500 Hz SNR
#     0  |   -13.5 /  -9.0 /  -4.2 dB   |    +0.0 dB    |   -9.0 dB
#     6  |    -8.2 /  -3.7 /  +1.1 dB   |    +5.3 dB    |   -9.1 dB
#    10  |    -4.8 /  -0.2 /  +4.6 dB   |    +8.8 dB    |   -9.2 dB
#    15  |    -0.6 /  +3.9 /  +8.7 dB   |   +12.9 dB    |   -9.5 dB
# check: with H = 10 the singer's overall level is -10.0 dB re orchestra (by construction)
# outer-ear gain at 2.5 / 3.2 kHz: +14.8 / +13.9 dB to BOTH sources

The table shows three things.

First, the gain in band signal-to-noise ratio is set almost entirely by the height of the formant cluster: a 6 dB cluster gains 5.3 dB, a 10 dB cluster 8.8 dB, a 15 dB cluster 12.9 dB (a little less than $H$ because the peak of the cluster does not sit exactly at the centres of the two bands), whatever the slope of the orchestral spectrum. So “a gain of 10–15 dB” amounts to saying “the singer’s formant cluster stands about 11–17 dB above a speech-like spectrum”, a figure that has to be measured in real recordings and cannot be taken as known. For reference, in the tube model of Volume 10, narrowing the epilarynx and widening the pharynx strengthened the 2.5–3.5 kHz band by about 6 dB, which corresponds to a gain of about 5 dB in this table.

Second, the absolute signal-to-noise ratio, whether the voice actually rises above the orchestra in this band, depends strongly on how fast the orchestra falls off at high frequencies: with the same 10 dB cluster the band ratio is still −4.8 dB if the orchestra falls 6 dB per octave and +4.6 dB if it falls 10 dB per octave. The same singer can be in a completely different position with different orchestration or dynamics.

Third, at 500 Hz the ratio stays near −9 dB throughout: in the orchestra’s strongest band the singer’s formant does not help. The outer ear adds about 14–15 dB at 2.5 and 3.2 kHz, but it amplifies voice and orchestra equally and does not change the band ratio; what it does is make that band more prominent in loudness.

Chapter summary

The singer’s formant arises from a clustering of the third to fifth formants near 3 kHz; what produces it is the area mismatch between a narrow epilarynx and a wider pharynx (about 1:6 in the 1974 model, about 4.5:1 in the 1995 MRI measurements), not the laryngeal ventricle. Its frequency coincides with the resonance of the outer ear, while orchestral energy is concentrated in the few hundred hertz range. Under stated spectral assumptions, the gain in signal-to-noise ratio in the 2.5–3.2 kHz bands is about the number of decibels by which the formant cluster exceeds a speech-like spectrum (about 8.8 dB for a 10 dB cluster), and the absolute ratio depends on orchestration; “a gain of 10–15 dB” is not an established measurement. The next chapter turns to the nonlinearity of the source itself: registers, the two-mass model and bifurcations.