Source: https://jmrplens.github.io/phonometry/perception/psychoacoustics/psychoacoustic-annoyance/ # Psychoacoustic annoyance and fluctuation strength (Fastl & Zwicker) 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. Psychoacoustic annoyance PA against percentile loudness N5 for three sensation profiles: a neutral baseline where PA equals N5, a sharp sound and a rough-and-fluctuating sound both lifted above the baseline, with the worked example PA = 37.05 marked
Show the code for this figure ```python 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 $N_5$ (the loudness exceeded 5 % of the time), in **sone**, from the ISO 532-1 Zwicker time-varying model (`loudness_zwicker`, see [Loudness](https://jmrplens.github.io/phonometry/perception/psychoacoustics/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 $N_5$ by a factor that grows with the sharpness weighting $w_S$ and the combined roughness/fluctuation weighting $w_{FR}$: $$ PA = N_5\left(1 + \sqrt{w_S^2 + w_{FR}^2}\right), \qquad w_S = (S - 1.75)\,0.25\,\log_{10}(N_5 + 10)\ \ (S > 1.75\ \mathrm{acum}), \qquad w_{FR} = \frac{2.18}{N_5^{0.4}}\,(0.4\,F + 0.6\,R). $$ $w_S$ is zero for $S \le 1.75\ \text{acum}$ (sharpness only adds annoyance above that threshold); $w_{FR}$ 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: ```python 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 $N_5$ for three profiles: a neutral baseline ($S = 1.75\ \text{acum}$, $F = R = 0$, so $PA = N_5$), a sharp sound and a rough-and-fluctuating sound, both lifted above the baseline, with the worked example marked. The model is small enough to draw whole: two weightings and one combination. The diagram traces the worked example through it, sensation by sensation. Block diagram of the Fastl and Zwicker psychoacoustic annoyance model: the four sensations of the worked example, sharpness of 2.0 acum, percentile loudness N5 of 30 sone, fluctuation strength of 0.5 vacil and roughness of 0.3 asper, feed two weightings, the sharpness weighting wS of 0.1001 that is zero below 1.75 acum and the roughness-and-fluctuation weighting wFR of 0.2125 that weighs roughness 0.6 against 0.4, and combine into PA equal to N5 times one plus the root of wS squared plus wFR squared, giving 37.05; a closing note recalls that a neutral sound sits on the baseline PA equal to N5 ### 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: $N_5$ 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. ```python 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 ``` > **Model-mixing caveat.** This composite mixes model families: Zwicker-family > $N_5$ 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 $f_\mathrm{mod} \approx 4\ \text{Hz}$ 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](https://jmrplens.github.io/phonometry/perception/psychoacoustics/sound-quality/). Fluctuation strength versus modulation frequency on a log axis: the closed-form AM broadband-noise curve and the AM-tone signal-model sweep both show a band-pass characteristic peaking at 4 Hz
Show the code for this figure ```python 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 $f_\mathrm{mod}$: $$ F = \frac{5.8\,(1.25\,m - 0.25)\,[0.05\,(L/\mathrm{dB}) - 1]} {(f_\mathrm{mod}/5\,\mathrm{Hz})^2 + (4\,\mathrm{Hz}/f_\mathrm{mod}) + 1.5}\ \ \mathrm{vacil}. $$ The denominator is the 4 Hz band-pass: it bottoms out near $f_\mathrm{mod} \approx 3.7\ \text{Hz}$ 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. ```python 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. ```python 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) ``` Specific fluctuation strength over the Bark axis for a 1 kHz tone at 70 dB SPL fully amplitude-modulated at 4 Hz: the sensation is concentrated in the critical bands around the carrier near 8 Bark and integrates to about 1.1 vacil
Show the code for this figure ```python 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 section 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. > **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 > $f_\mathrm{mod} \in \{1, 2, 4, 8, 16, 32\}\ \text{Hz}$ 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; only AM stimuli are validated. The front-end also > departs from the paper's exact framing (the paper's 2 s frames with 90 % > overlap and absolute-threshold screening become 50 % overlap, a Hann window > and relative floors here), so perfectly steady signals do not read exactly > zero: a steady 1 kHz tone comes out near 0.09 vacil and steady broadband > noise about 0.15-0.19 vacil, partly the physical level fluctuation of the > noise itself. 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. ## See also - [Loudness](https://jmrplens.github.io/phonometry/perception/psychoacoustics/loudness/): the ISO 532-1 percentile loudness $N_5$ this model takes as its first argument, and the acquisition rules the recording has to satisfy before any of it applies. - [Sound Quality Metrics](https://jmrplens.github.io/phonometry/perception/psychoacoustics/sound-quality/): the DIN 45692 sharpness $S$, the ECMA-418-2 roughness $R$ and the normative vacil_HMS fluctuation strength of the same sensation this page models in vacil. - [Advanced Loudness](https://jmrplens.github.io/phonometry/perception/psychoacoustics/advanced-loudness/): the alternative loudness models, and why mixing families makes the composite an engineering estimate. - [Environmental Levels](https://jmrplens.github.io/phonometry/environment/assessment/environmental-levels/): community annoyance, the other thing the word means, and the rating levels it is assessed with. - Theory: [Advanced loudness models and sound quality](https://jmrplens.github.io/phonometry/reference/theory/perception/#advanced-loudness-models--sound-quality): the loudness and sound-quality metrics the annoyance model combines, derived one at a time. - API reference: [`psychoacoustics.quality.annoyance`](https://jmrplens.github.io/phonometry/reference/api/psychoacoustics/annoyance/) and [`psychoacoustics.quality.fluctuation_strength`](https://jmrplens.github.io/phonometry/reference/api/psychoacoustics/fluctuation-strength/). ## References - Fastl, H., & Zwicker, E. (2007). *Psychoacoustics: Facts and models* (3rd ed.). Springer. [doi:10.1007/978-3-540-68888-4](https://doi.org/10.1007/978-3-540-68888-4). The source of the psychoacoustic-annoyance model of section 2 (Eqs 16.2–16.4, chapter 16) and of the closed-form fluctuation strength for AM broadband noise of §3.1 (Eq. 10.2, chapter 10). - Osses Vecchi, A., García León, R., & Kohlrausch, A. (2016). Modelling the sensation of fluctuation strength. *Proceedings of Meetings on Acoustics*, 28, 050005. [doi:10.1121/2.0000410](https://doi.org/10.1121/2.0000410). The fluctuation-strength signal model implemented in §3.2, including the Table 1 literature values used as its cross-check. - Felix Greco, G., Merino-Martínez, R., Osses, A., & Lotinga, M. J. B. (2025). *SQAT: a sound quality analysis toolbox for MATLAB* (open-source software). [github.com/ggrecow/SQAT](https://github.com/ggrecow/SQAT), [doi:10.5281/zenodo.7934709](https://doi.org/10.5281/zenodo.7934709). The open MATLAB reference used as the numeric oracle for the fluctuation-strength cross-checks on this page. ## Standards Fastl & Zwicker (2007, 3rd ed.), *Psychoacoustics: Facts and Models* (Springer): psychoacoustic annoyance $PA = N_5(1 + \sqrt{w_S^2 + w_{FR}^2})$ with the sharpness weighting $w_S$ and roughness/fluctuation weighting $w_{FR}$ (Eqs 16.2–16.4; origin Widmann 1992), and the closed form for the fluctuation strength of amplitude-modulated broadband noise (Eq. 10.2). The fluctuation- strength signal model follows Osses García & Kohlrausch (2016), *Modelling the sensation of fluctuation strength* (ICA 2016), clean-room and with no numeric standard: calibrated to 1 vacil at the reference AM tone and cross-checked against the paper's Table 1 literature trends and the open SQAT reference oracle. The four sensations reuse the library's ISO 532-1 loudness, DIN 45692 sharpness and ECMA-418-2 roughness (see [Loudness](https://jmrplens.github.io/phonometry/perception/psychoacoustics/loudness/) and [Sound Quality Metrics](https://jmrplens.github.io/phonometry/perception/psychoacoustics/sound-quality/)).