Psychoacoustic annoyance and fluctuation strength

How annoying a sound is depends on more than how loud it is. Fastl & Zwicker, Psychoacoustics: Facts and Models, combine four psychoacoustic sensations (loudness, sharpness, roughness and fluctuation strength) into a single psychoacoustic annoyance PA, a scalar that grows with loudness and is lifted further when the sound is sharp, rough or slowly fluctuating. This page covers the exact PA model (Eqs 16.2–16.4), the fluctuation strength it consumes, both the closed form for amplitude-modulated broadband noise (Eq. 10.2) and the Osses et al. (2016) signal model, and the signal convenience that derives all four sensations from a recording.

Show the code for this figure

import matplotlib.pyplot as plt
import numpy as np
from phonometry import psychoacoustics

n5 = np.linspace(4.0, 60.0, 200)
profiles = [
    ("Baseline: S = 1.75 acum, F = R = 0", 1.75, 0.0, 0.0),
    ("Sharp: S = 3.5 acum", 3.5, 0.0, 0.0),
    ("Rough + fluctuating: F = 1.2 vacil, R = 0.7 asper", 2.0, 1.2, 0.7),
]
fig, ax = plt.subplots()
for label, s, f, r in profiles:
    pa = [psychoacoustics.psychoacoustic_annoyance(v, s, f, r).annoyance for v in n5]
    ax.plot(n5, pa, label=label)

ex = psychoacoustics.psychoacoustic_annoyance(30.0, 2.0, 0.5, 0.3)
ax.plot([30.0], [ex.annoyance], "o", label=f"Worked example (PA = {ex.annoyance:.2f})")
ax.set_xlabel("Percentile loudness N5 [sone]")
ax.set_ylabel("Psychoacoustic annoyance PA")
ax.legend()
plt.show()

1. The four sensations

Psychoacoustic annoyance rests on four hearing sensations, each with its own model and unit in the library:

• Loudness: the percentile loudness N5 (the loudness exceeded 5 % of the time), in sone, from the ISO 532-1 Zwicker time-varying model (loudness_zwicker, see Loudness).
• Sharpness S, in acum: the spectral balance towards high frequencies, DIN 45692 (sharpness_din).
• Roughness R, in asper: the harshness of fast (~70 Hz) amplitude modulation, ECMA-418-2 Sottek model (roughness_ecma).
• Fluctuation strength F, in vacil: the sensation of slow (~4 Hz) loudness fluctuation (§3 below).

2. Psychoacoustic annoyance (Eqs 16.2–16.4)

The exact model (Fastl & Zwicker Eq. 16.2; origin Widmann 1992) scales N5 by a factor that grows with the sharpness weighting wS and the combined roughness/fluctuation weighting wFR:

$$PA=N_5\left(1+\sqrt{w_S^2+w_{FR}^2}\right),w_S=(S-1.75)0.25\lg (N_5+10)(S>1.75\mathrm{acum}),w_{FR}=\frac{2.18}{N_5^{0.4}}(0.4F+0.6R).$$

wS is zero for S ≤ 1.75 acum (sharpness only adds annoyance above that threshold); wFR weights roughness more heavily than fluctuation strength (0.6 vs 0.4). psychoacoustic_annoyance takes the four quantities directly and returns the annoyance together with the two intermediate weightings:

from phonometry import psychoacoustics

res = psychoacoustics.psychoacoustic_annoyance(30.0, 2.0, 0.5, 0.3)   # N5, S, F, R
print(round(res.annoyance, 4))   # 37.0478
print(round(res.w_s, 4), round(res.w_fr, 4))   # 0.1001 0.2125

res.plot()   # PA beside its wS and wFR weightings (needs matplotlib)

The figure above sweeps PA against N5 for three profiles: a neutral baseline (S = 1.75 acum, F = R = 0, so PA = N5), a sharp sound and a rough-and-fluctuating sound, both lifted above the baseline, with the worked example marked.

2.1 From a signal (engineering estimate)

psychoacoustic_annoyance_from_signal is a convenience that derives all four sensations from a calibrated pressure signal and combines them: N5 from the ISO 532-1 Zwicker time-varying loudness, S from DIN 45692 sharpness, R from the ECMA-418-2 Sottek roughness and F from the fluctuation-strength signal model.

from phonometry import psychoacoustics

res = psychoacoustics.psychoacoustic_annoyance_from_signal(x, fs, field="free")
print(res.annoyance, res.n5, res.sharpness, res.roughness, res.fluctuation_strength)

res.plot()   # the same PA / wS / wFR view, now from the four derived sensations

Caution

Model-mixing caveat. This composite mixes model families: Zwicker-family N5 and S, Sottek R and Osses F. The PA model was calibrated with Zwicker-family sensations, so treat the signal convenience as an engineering estimate. For exact, reproducible results, compute the four quantities with whichever models you trust and pass them to psychoacoustic_annoyance directly (the four-argument function is the exact model).

3. Fluctuation strength

Fluctuation strength F (vacil) quantifies the perception of slow loudness fluctuation. Like roughness it is a band-pass sensation of the modulation frequency, but it peaks about an order of magnitude lower, at fmod ≈ 4 Hz rather than the ~70 Hz roughness peak. By definition, a 1 kHz tone at 60 dB, 100 % amplitude-modulated at 4 Hz, produces 1 vacil. This page covers the Fastl & Zwicker models; the normative Sottek-model fluctuation strength of ECMA-418-2 Clause 9 lives in Sound Quality Metrics.

Show the code for this figure

import matplotlib.pyplot as plt
import numpy as np
from phonometry import psychoacoustics

# Exact closed form (Eq. 10.2), AM broadband noise at 60 dB, 100 % modulation:
fmod = np.logspace(np.log10(0.5), np.log10(32.0), 240)
f_bbn = [psychoacoustics.fluctuation_strength_am_noise(60.0, 1.0, fm) for fm in fmod]

# Osses 2016 signal model on a 1 kHz / 70 dB AM tone over the same sweep:
fs = 48000
t = np.arange(int(2.0 * fs)) / fs
carrier = np.sin(2 * np.pi * 1000 * t)
fm_tone = [1.0, 2.0, 4.0, 8.0, 16.0, 32.0]
f_tone = []
for fm in fm_tone:
    am = (1.0 + np.sin(2 * np.pi * fm * t)) * carrier
    am = am / np.sqrt(np.mean(am ** 2)) * 2e-5 * 10 ** (70 / 20)
    f_tone.append(psychoacoustics.fluctuation_strength(am, float(fs)).fluctuation_strength)

fig, ax = plt.subplots()
ax.semilogx(fmod, f_bbn, label="AM broadband noise (closed form)")
ax2 = ax.twinx()
ax2.plot(fm_tone, f_tone, "s--", color="tab:green", label="AM tone (signal model)")
ax.axvline(4.0, ls="--", color="0.4")
ax.set_xlabel("Modulation frequency f_mod [Hz]")
ax.set_ylabel("Fluctuation strength F [vacil]")
h1, l1 = ax.get_legend_handles_labels()
h2, l2 = ax2.get_legend_handles_labels()
ax.legend(h1 + h2, l1 + l2, loc="upper right")
plt.show()

3.1 Closed form for AM broadband noise (Eq. 10.2)

For sinusoidally amplitude-modulated broadband noise, Fastl & Zwicker give a closed form (Eq. 10.2) in modulation factor m, level L and modulation frequency fmod:

$$F=\frac{5.8(1.25m-0.25)[0.05(L/\mathrm{dB})-1]}{(f_{mod}/5\mathrm{Hz}{)}^2+(4\mathrm{Hz}/f_{mod})+1.5}\mathrm{vacil}.$$

The denominator is the 4 Hz band-pass: it bottoms out near fmod ≈ 3.7 Hz and rises on either side. The result is clamped at 0 (the sensation vanishes below ~20 dB or m < 0.2). This exact form is the value to quote for AM broadband noise.

from phonometry import psychoacoustics

print(round(psychoacoustics.fluctuation_strength_am_noise(60.0, 1.0, 4.0), 4))   # 3.6943 vacil

3.2 The Osses 2016 signal model

fluctuation_strength implements the Osses et al. (2016) signal model: it builds an excitation pattern over 47 auditory filters, extracts the ~4 Hz envelope modulation per band, weights and combines it, and returns the overall F (vacil) plus the specific fluctuation strength over the Bark axis and the time-dependent trace.

from phonometry import psychoacoustics

res = psychoacoustics.fluctuation_strength(x, fs)
print(res.fluctuation_strength)   # vacil
res.plot()   # specific fluctuation strength F′(z) over the Bark axis (needs matplotlib)

Show the code for this figure

import matplotlib.pyplot as plt
import numpy as np
from phonometry import psychoacoustics

# The reference-like stimulus: a 1 kHz tone at 70 dB SPL, 100 % amplitude
# modulated at 4 Hz, where the sensation peaks.
fs = 48000
t = np.arange(int(2.0 * fs)) / fs
am = (1.0 + np.sin(2 * np.pi * 4.0 * t)) * np.sin(2 * np.pi * 1000 * t)
am = am / np.sqrt(np.mean(am ** 2)) * 2e-5 * 10 ** (70 / 20)
res = psychoacoustics.fluctuation_strength(am, float(fs))
print(round(res.fluctuation_strength, 2))    # 1.09 vacil

# One line: the specific fluctuation strength over the Bark axis.
res.plot()
plt.show()

# Or draw f'(z) by hand from the arrays the result carries:
fig, ax = plt.subplots()
ax.fill_between(res.bark_axis, res.specific, alpha=0.3)
ax.plot(res.bark_axis, res.specific)
ax.set_xlabel("Critical-band rate z [Bark]")
ax.set_ylabel("Specific fluctuation strength [vacil/Bark]")
plt.show()

The sensation stays where the modulated energy is, in the critical bands around the carrier, which is exactly why the closed form of §3.1 and the signal model diverge for broadband modulated noise: there the modulation is spread over the whole Bark axis and the band-by-band model accumulates more of it than the single closed form allows.

Note

No numeric standard. Fluctuation strength has no ISO/IEC standard. This is a clean-room implementation from the Osses 2016 paper, calibrated so that the 1 kHz / 60 dB / m = 1 / 4 Hz AM tone reads 1.00 vacil by construction, and cross-checked against the Osses 2016 Table 1 literature values and the open SQAT reference (used only as a numeric oracle). Over the 70 dB AM-tone sweep fmod ∈ {1, 2, 4, 8, 16, 32} Hz it gives [0.40, 0.79, 1.09, 1.05, 0.17, 0.09] vacil against the literature [0.39, 0.84, 1.25, 1.30, 0.36, 0.06] (Pearson r = 0.98, correct 4 Hz peak, within ~2.1×). FM-tone accuracy is explicitly not pursued. For AM broadband noise the signal model overshoots the absolute level (it spreads the modulated energy across bands); quote the closed form fluctuation_strength_am_noise (§3.1) for that stimulus.

What this guide covers

Covered. Fastl & Zwicker’s Psychoacoustics: Facts and Models gives the exact PA model, Eqs 16.2-16.4 (origin Widmann 1992), run by psychoacoustic_annoyance. It also gives the closed form for AM broadband noise, Eq. 10.2, run by fluctuation_strength_am_noise. The Osses et al. (2016) signal model for fluctuation strength runs as fluctuation_strength, cross-checked against its Table 1 literature values and the SQAT reference. psychoacoustic_annoyance_from_signal derives the four inputs from a signal (ISO 532-1 Zwicker N5, DIN 45692 S, ECMA-418-2 R) and combines them with the exact model.

Not covered. The Osses 2016 model’s accuracy for FM tones is explicitly not pursued. Only AM stimuli are validated here. The front-end also departs from the paper’s exact framing (2 s frames, 90 % overlap, absolute-threshold gating). As a result, a steady 1 kHz tone reads about 0.09 vacil instead of 0, and steady broadband noise reads about 0.4 vacil. For AM broadband noise, quote fluctuation_strength_am_noise (§3.1) instead of fluctuation_strength, which overshoots that stimulus. The normative Sottek-model fluctuation strength of ECMA-418-2 Clause 9 is not implemented on this page: it lives in Sound Quality Metrics.

See also

• API reference: psychoacoustics.psychoacoustic_annoyance and psychoacoustics.fluctuation_strength.

References

• Fastl, H., & Zwicker, E. (2007). Psychoacoustics: Facts and models (3rd ed.). Springer. https://doi.org/10.1007/978-3-540-68888-4The source of the psychoacoustic-annoyance model of section 2 (Eqs 16.2-16.4, chapter 16; origin Widmann 1992) and of the closed-form fluctuation strength for AM broadband noise of section 3.1 (Eq. 10.2, chapter 10).
• Felix Greco, G., Merino-Martínez, R., Osses, A., & Lotinga, M. J. B. (2025). SQAT: A sound quality analysis toolbox for MATLAB. GitHub. https://doi.org/10.5281/zenodo.7934709The open MATLAB reference (open-source software) used as the numeric oracle for the fluctuation-strength cross-checks on this page.
• Osses Vecchi, A., García León, R., & Kohlrausch, A. (2016). Modelling the sensation of fluctuation strength. Proceedings of Meetings on Acoustics, 28, 050005. https://doi.org/10.1121/2.0000410The fluctuation-strength signal model implemented in section 3.2 (presented at ICA 2016; clean-room, with no numeric standard, calibrated to 1 vacil at the reference AM tone), including the Table 1 literature values used as its cross-check.