Source: https://jmrplens.github.io/phonometry/reference/api/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.
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 `N5` (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/guides/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`:
$$
PA = N_5\left(1 + \sqrt{w_S^2 + w_{FR}^2}\right), \qquad
w_S = (S - 1.75)\,0.25\,\lg(N_5 + 10)\ \ (S > 1.75\ \mathrm{acum}), \qquad
w_{FR} = \frac{2.18}{N_5^{0.4}}\,(0.4\,F + 0.6\,R).
$$
`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:
```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 `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.
```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
> `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](https://jmrplens.github.io/phonometry/guides/sound-quality/).
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 `fmod`:
$$
F = \frac{5.8\,(1.25\,m - 0.25)\,[0.05\,(L/\mathrm{dB}) - 1]}
{(f_{mod}/5\,\mathrm{Hz})^2 + (4\,\mathrm{Hz}/f_{mod}) + 1.5}\ \ \mathrm{vacil}.
$$
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.
```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)
```
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
> `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.
## 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 = N5·(1 + √(wS² + wFR²))` with the
sharpness weighting `wS` and roughness/fluctuation weighting `wFR`
(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/guides/loudness/) and
[Sound Quality Metrics](https://jmrplens.github.io/phonometry/guides/sound-quality/)).
