Sound Quality Metrics

Two sounds of equal loudness can still differ in how sharp, how tonal, how rough or how strongly fluctuating they are. This page covers the sound-quality metrics that complement loudness: sharpness (DIN 45692) and the ECMA-418-2 tonality, roughness and fluctuation strength of the Sottek Hearing Model. Loudness itself, including the ECMA-418-2 loudness that shares the same auditory front-end, lives in Loudness.

Sharpness in acum (DIN 45692)

Two sounds can be equally loud yet one feels “sharper” (hissy, metallic) because its loudness sits higher on the Bark scale. Sharpness is the g(z)-weighted first moment of the specific loudness pattern:

$$S=k\frac{{\int }_0^{24}{N}^{\prime }(z)g(z)zdz}{{\int }_0^{24}{N}^{\prime }(z)dz}\text{acum}$$

with $g(z)=1$ up to 15.8 Bark and rising exponentially beyond, and $k$ normalized so the reference sound (critical-band-wide noise at 1 kHz, 60 dB) is exactly 1.00 acum (DIN 45692 clause 6; the derived $k=0.108$ sits inside the normative window 0.105–0.115).

Show the code for this figure

import matplotlib.pyplot as plt
import numpy as np

# DIN 45692 sharpness weighting g(z): Eq. (1) plus the informative Annex B variants
z = np.arange(1, 241) * 0.1                     # Bark bins, 0.1 .. 24.0
g_din = np.where(z > 15.8, 0.15 * np.exp(0.42 * (z - 15.8)) + 0.85, 1.0)
g_bis = np.where(z > 15.0, 0.2 * np.exp(0.308 * (z - 15.0)) + 0.8, 1.0)
n = 4.0                                         # Aures depends on the total loudness (sone)
g_aures = 0.078 * np.exp(0.171 * z) / z * (n / np.log(n * 0.05 + 1.0))

fig, ax = plt.subplots()
ax.semilogy(z, g_din, label="DIN 45692 g(z)")
ax.semilogy(z, g_bis, "--", label="von Bismarck (Annex B)")
ax.semilogy(z, g_aures, "-.", label="Aures (Annex B, N = 4 sone)")
ax.axvline(15.8, linestyle=":", color="0.5")    # DIN knee: g rises beyond 15.8 Bark
ax.set(xlabel="Critical-band rate z [Bark]", ylabel="Weighting g(z)")
ax.grid(True, which="both", alpha=0.3)
ax.legend()
plt.show()

import numpy as np
from phonometry import psychoacoustics

# A raw recording plus its calibration so the guide runs standalone
fs = 48000
x = 0.2 * np.sin(2 * np.pi * 1000 * np.arange(fs) / fs)   # any recording (digital units)
sens = 1.0                                                # calibration_factor to pascals

s = psychoacoustics.sharpness_din(x, fs, calibration_factor=sens)      # acum
s_aures = psychoacoustics.sharpness_din(x, fs, method="aures")          # Annex B variant

CI verifies the Table A.2 target values (0.38 acum at 250 Hz up to 2.82 acum at 4 kHz) within the standard’s 5 % / 0.05 acum tolerance.

Tonality (ECMA-418-2)

A tonal component (a whistle, a fan’s blade-passing tone) stands out even at low level. ECMA-418-2 quantifies it from the autocorrelation function (ACF) of each band’s rectified signal: a periodic (tonal) component keeps a high ACF at nonzero lag, and the tonal-to-noise loudness ratio drives the specific tonality T′(z). The single value T is in tu_HMS, calibrated so a 1 kHz/40 dB tone is ≈ 1 tu_HMS; the result also tracks the tonal frequency f_ton per band.

import numpy as np
from phonometry import psychoacoustics

fs = 48000
t = np.arange(int(1.2 * fs)) / fs
x = np.sqrt(2) * 2e-5 * 10 ** (40 / 20) * np.sin(2 * np.pi * 1000 * t)

res = psychoacoustics.tonality_ecma(x, fs, field="free")
peak = int(np.argmax(res.specific_tonality))
print(f"T = {res.tonality:.3f} tu_HMS")                    # 1.000 tu_HMS
print(f"f_ton = {res.tonal_frequencies[peak]:.0f} Hz")     # 999 Hz

res.plot()   # average specific tonality T'(z) + time-dependent T(l)

Show the code for this figure

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

# The calibration anchor: a 1 kHz tone at 40 dB SPL is about 1 tu_HMS.
fs = 48000
t = np.arange(int(1.2 * fs)) / fs
x = np.sqrt(2) * 2e-5 * 10 ** (40 / 20) * np.sin(2 * np.pi * 1000 * t)
res = psychoacoustics.tonality_ecma(x, fs, field="free")

# One line: passing an axes draws the specific-tonality panel alone.
fig, ax = plt.subplots()
res.plot(ax=ax)
plt.show()

# Or draw T'(z) by hand against the critical-band-rate scale:
fig, ax = plt.subplots()
ax.fill_between(res.bark, res.specific_tonality, alpha=0.3, color="#d62728")
ax.plot(res.bark, res.specific_tonality, color="#d62728")
ax.set_xlabel("Critical-band rate z [Bark_HMS]")
ax.set_ylabel("Specific tonality T' [tu_HMS]")
plt.show()

The concentration is what distinguishes a tonal sound from a broadband one of the same loudness: the autocorrelation stage finds a periodic component in one band and almost nothing in the others. A fan with a blade-passing tone and a harmonic series shows several such peaks; a hiss shows a flat, low pattern. Calling .plot() without an axes adds the time-dependent T(l) panel, which is where an intermittent tone shows up.

tonality_ecma() parameters

Parameter	Type	Units	Range / default	Notes
signal_in	1D array	Pa	non-empty	Calibrated pressure signal
fs	float	Hz	> 0	Resampled to 48 kHz internally if needed
field	str	—	'free' (default) / 'diffuse'	Outer/middle-ear filter
f_low	float, optional	Hz	default None	Lower edge of a user band for the T(l) search
f_high	float, optional	Hz	default None	Upper edge of the user band

Returns an EcmaTonality: tonality (T, tu_HMS), specific_tonality (T′(z), 53 bands), bark, centre_frequencies, tonal_frequencies (f_ton,z), time, tonality_vs_time (T(l)), tonal_frequency_vs_time, field.

Roughness (ECMA-418-2): new capability

Roughness is the harsh, buzzing sensation of fast amplitude modulation (roughly 20–300 Hz, peaking near 70 Hz): the quality of a diesel idle or a distorted loudspeaker. It is a new metric for phonometry. ECMA-418-2 extracts each band’s envelope, weights its modulation spectrum by modulation rate and depth, and correlates the modulation across bands; the result R is in asper. The reference sound (1 kHz carrier, 100 % amplitude-modulated at 70 Hz, overall level 60 dB SPL) is defined as 1 asper; this clean-room implementation returns 0.9999 asper with the tabulated calibration constant c_R (Formula 104) used without reverse-fitting to the target.

import numpy as np
from phonometry import psychoacoustics

fs = 48000
t = np.arange(int(2.0 * fs)) / fs
x = (1.0 + np.cos(2 * np.pi * 70 * t)) * np.sin(2 * np.pi * 1000 * t)
x *= 2e-5 * 10 ** (60 / 20) / np.sqrt(np.mean(x**2))   # overall 60 dB SPL

res = psychoacoustics.roughness_ecma(x, fs, field="free")
print(f"R = {res.roughness:.4f} asper")   # 0.9999 asper (reference: 1 asper)

res.plot()   # time-dependent roughness R(l50) + specific-roughness heatmap

Show the code for this figure

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

fs = 48000
t = np.arange(int(2.0 * fs)) / fs
amp = np.sqrt(2) * 2e-5 * 10 ** (60 / 20)

# A pure tone (tonal, smooth) vs a 70 Hz amplitude-modulated tone (rough),
# both normalized to an overall level of 60 dB SPL:
tone = amp * np.sin(2 * np.pi * 1000 * t)
rough = (1.0 + np.cos(2 * np.pi * 70 * t)) * np.sin(2 * np.pi * 1000 * t)
rough *= 2e-5 * 10 ** (60 / 20) / np.sqrt(np.mean(rough**2))

scores = {
    "Pure tone": (psychoacoustics.tonality_ecma(tone, fs).tonality, psychoacoustics.roughness_ecma(tone, fs).roughness),
    "70 Hz AM tone": (psychoacoustics.tonality_ecma(rough, fs).tonality, psychoacoustics.roughness_ecma(rough, fs).roughness),
}
labels = list(scores)
tonal = [scores[k][0] for k in labels]
rough_v = [scores[k][1] for k in labels]
xpos = np.arange(len(labels))
fig, ax = plt.subplots()
ax.bar(xpos - 0.2, tonal, 0.4, label="Tonality [tu_HMS]")
ax.bar(xpos + 0.2, rough_v, 0.4, label="Roughness [asper]")
ax.set_xticks(xpos)
ax.set_xticklabels(labels)
ax.legend()
plt.show()

roughness_ecma() parameters

Parameter	Type	Units	Range / default	Notes
signal_in	1D array	Pa	non-empty	Calibrated pressure signal
fs	float	Hz	> 0	Resampled to 48 kHz internally if needed
field	str	—	'free' (default) / 'diffuse'	Outer/middle-ear filter

Returns an EcmaRoughness: roughness (R, asper, the 90th percentile of R(l50)), specific_roughness (R′(z), 53 bands), bark, centre_frequencies, time, roughness_vs_time (R(l50)), specific_roughness_vs_time ((n_times, 53) array), field.

Fluctuation strength (ECMA-418-2): new capability

Fluctuation strength is the slow, wobbling sensation of amplitude or frequency modulation below about 20 Hz: a siren, beating tones, speech at syllable rate. It is the slow counterpart of roughness: the same hearing model splits envelope modulation into a slow band-pass peaking near 4 Hz (fluctuation strength, in vacil_HMS) and a fast one peaking near 70 Hz (roughness). ECMA-418-2 Clause 9 analyses each band’s envelope with High-resolution Spectral Analysis (HSA), a least-squares fit of window-kernel spectral line pairs that resolves modulation rates far below the DFT bin width, using envelope-dependent analysis windows that skip quieter periods, then weights the dominant harmonic complex and scales it with an HSA-based specific loudness. The reference sound (1 kHz carrier, 100 % amplitude-modulated at 4 Hz, overall level 60 dB SPL) is defined as 1 vacil_HMS; this clean-room implementation converges to 0.9958 vacil_HMS by 12 s with the tabulated calibration constant c_F (Formula 163) used without reverse-fitting to the target (the 8 s example below prints 0.9957). A signal whose single value F exceeds 0.2 vacil_HMS has a prominent fluctuation strength (Clause 9.2).

import numpy as np
from phonometry import psychoacoustics

fs = 48000
t = np.arange(int(8.0 * fs)) / fs
x = (1.0 + np.cos(2 * np.pi * 4 * t)) * np.sin(2 * np.pi * 1000 * t)
x *= 2e-5 * 10 ** (60 / 20) / np.sqrt(np.mean(x**2))   # overall 60 dB SPL

res = psychoacoustics.fluctuation_strength_ecma(x, fs, field="free")
print(f"F = {res.fluctuation_strength:.4f} vacil_HMS")   # 0.9957 vacil_HMS (reference: 1 vacil_HMS)

res.plot()   # time-dependent F(l50) + specific-fluctuation-strength heatmap

Show the code for this figure

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

fs = 48000
t = np.arange(int(3.0 * fs)) / fs
carrier = np.sin(2 * np.pi * 1000 * t)

def am_tone(fmod):
    # 100 % AM at an overall level of 60 dB SPL (the Clause 7/9 convention)
    x = (1.0 + np.sin(2 * np.pi * fmod * t)) * carrier
    return x * 2e-5 * 10 ** (60 / 20) / np.sqrt(np.mean(x**2))

fm_slow = [0.5, 1, 2, 4, 8, 16, 32]
fm_fast = [20, 40, 70, 100, 150, 200]
f_vals = [psychoacoustics.fluctuation_strength_ecma(am_tone(fm), fs).fluctuation_strength
          for fm in fm_slow]
r_vals = [psychoacoustics.roughness_ecma(am_tone(fm), fs).roughness for fm in fm_fast]

fig, ax = plt.subplots()
ax.semilogx(fm_slow, f_vals, "o-", label="Fluctuation strength F [vacil_HMS]")
ax.semilogx(fm_fast, r_vals, "s-", label="Roughness R [asper]")
ax.set(xlabel="Modulation frequency [Hz]", ylabel="F [vacil_HMS] / R [asper]")
ax.legend()
plt.show()

fluctuation_strength_ecma() parameters

Parameter	Type	Units	Range / default	Notes
signal_in	1D array	Pa	non-empty	Calibrated pressure signal
fs	float	Hz	> 0	Resampled to 48 kHz internally if needed
field	str	—	'free' (default) / 'diffuse'	Outer/middle-ear filter

Returns an EcmaFluctuationStrength: fluctuation_strength (F, vacil_HMS, the 90th percentile of F(l50)), specific_fluctuation_strength (F′(z), 53 bands), bark, centre_frequencies, time, fluctuation_strength_vs_time (F(l50)), specific_fluctuation_strength_vs_time ((n_times, 53) array), field.

The Fastl & Zwicker fluctuation-strength models (closed form for AM broadband noise and the Osses 2016 signal model) live in Psychoacoustic Annoyance; this Clause 9 metric is the normative Sottek-model counterpart.

See Prominent Discrete Tones for the ECMA-418-1 TNR/PR prominence verdicts, Speech Transmission Index for STI/STIPA, and Theory for the underlying math.

What this guide covers

Covered. DIN 45692:2009 through sharpness_din(): the clause 6 weighting over the ISO 532-1 specific-loudness pattern, plus the informative Annex B von Bismarck and Aures variants, with the Table A.2 targets checked in CI. ECMA-418-2:2025 on the shared clause 5 auditory front-end: the clause 6.2 tonality output stages (tonality_ecma()), the clause 7 roughness chain (roughness_ecma()) and the clause 9 HSA fluctuation strength (fluctuation_strength_ecma()). Each uses the standard’s tabulated calibration constant, not a reverse fit to the reference sound.

Not covered. The three ECMA-418-2 entry points are monaural: the binaural quadratic-mean combinations of Formula 112 (clause 7.1.11) and Formula 170 (clause 9.1.15) are not implemented, so analyse each channel separately. The optional entropy weighting of clause 7.1.6 needs an external rotational-speed signal and is left out, as is the ±0.25 % adjustment of $c_F$ that footnote 47 permits. The ECMA-418-2 loudness that shares this front-end lives in Loudness, and the Fastl and Zwicker fluctuation-strength models in Psychoacoustic Annoyance.

See also

• API reference: psychoacoustics.sharpness, psychoacoustics.tonality_ecma, psychoacoustics.roughness_ecma and psychoacoustics.fluctuation_strength_ecma.

References

• Deutsches Institut für Normung. (2009). Messtechnische Simulation der Hörempfindung Schärfe (DIN 45692:2009). Sharpness in acum: the clause 6 weighting, the Annex B von Bismarck and Aures variants and the Table A.2 targets.
• Ecma International. (2025). Psychoacoustic metrics for ITT equipment — Part 2 (methods for describing human perception based on the Sottek Hearing Model) (ECMA-418-2:2025). The Sottek Hearing Model tonality (tu_HMS, clause 6), roughness (asper, clause 7) and fluctuation strength (vacil_HMS, clause 9, the HSA-based envelope analysis).
• Fastl, H., & Zwicker, E. (2007). Psychoacoustics: Facts and models (3rd ed.). Springer. https://doi.org/10.1007/978-3-540-68888-4The psychoacoustics of the sensations quantified on this page: the high-frequency emphasis behind sharpness and the fast-modulation percept behind roughness.