---
title: "Sound Quality Metrics"
description: "Sharpness in acum (DIN 45692) and the ECMA-418-2 Sottek Hearing Model tonality (tu_HMS), roughness (asper) and fluctuation strength (vacil_HMS)."
references:
  - type: book
    authors: ["Fastl, H.", "Zwicker, E."]
    year: 2007
    title: "Psychoacoustics: Facts and models"
    edition: "3rd ed."
    publisher: "Springer"
    doi: "10.1007/978-3-540-68888-4"
    note: "The psychoacoustics of the sensations quantified on this page: the high-frequency emphasis behind sharpness and the fast-modulation percept behind roughness."
  - type: standard
    organization: "Deutsches Institut für Normung"
    year: 2009
    title: "Messtechnische Simulation der Hörempfindung Schärfe"
    designation: "DIN 45692:2009"
    url: "https://www.dinmedia.de/en/standard/din-45692/117635111"
    note: "Sharpness in acum: the clause 6 weighting, the Annex B von Bismarck and Aures variants and the Table A.2 targets."
  - type: standard
    organization: "Ecma International"
    year: 2025
    title: "Psychoacoustic metrics for ITT equipment — Part 2 (methods for describing human perception based on the Sottek Hearing Model)"
    designation: "ECMA-418-2:2025"
    url: "https://ecma-international.org/publications-and-standards/standards/ecma-418/"
    note: "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)."
---

import ThemeImage from '../../../components/ThemeImage.astro';

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](/phonometry/guides/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:

$$
S = k\,\frac{\int_0^{24} N'(z)\, g(z)\, z\ dz}{\int_0^{24} N'(z)\ dz}\ \text{acum}
$$

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).

<ThemeImage
  src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sharpness_weighting.svg"
  alt="DIN 45692 sharpness weighting g(z) against critical-band rate on a log axis, comparing the DIN, von Bismarck and Aures curves with the 15.8 and 15 Bark knees marked"
  width="80%"
  loading="eager"
/>

<details>
<summary>Show the code for this figure</summary>

```python
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()
```

</details>

```python
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.

```python
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)
```

### `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.

```python
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
```

<ThemeImage
  src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/tonality_roughness_demo.svg"
  alt="ECMA-418-2 sound-quality demo: a tonal sound scores high tonality and near-zero roughness, while a 70 Hz amplitude-modulated sound scores high roughness and low tonality"
  width="80%"
/>

<details>
<summary>Show the code for this figure</summary>

```python
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()
```

</details>

### `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).

```python
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
```

<ThemeImage
  src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/hms_modulation_bandpass.svg"
  alt="ECMA-418-2 slow vs fast modulation perception: fluctuation strength forms a band-pass over modulation frequency peaking near 4 to 6 Hz while roughness of the same 1 kHz amplitude-modulated tones peaks near 70 Hz"
  width="80%"
/>

<details>
<summary>Show the code for this figure</summary>

```python
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()
```

</details>

### `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](/phonometry/guides/psychoacoustic-annoyance/);
this Clause 9 metric is the normative Sottek-model counterpart.

See [Prominent Discrete Tones](/phonometry/guides/tone-prominence/) for the
ECMA-418-1 TNR/PR prominence verdicts,
[Speech Transmission Index](/phonometry/guides/speech-transmission/) for
STI/STIPA, and [Theory](/phonometry/reference/theory/perception/) for the underlying math.

## See also

- API reference: [`psychoacoustics.sharpness`](/phonometry/reference/api/psychoacoustics/sharpness/), [`psychoacoustics.tonality_ecma`](/phonometry/reference/api/psychoacoustics/tonality-ecma/), [`psychoacoustics.roughness_ecma`](/phonometry/reference/api/psychoacoustics/roughness-ecma/) and [`psychoacoustics.fluctuation_strength_ecma`](/phonometry/reference/api/psychoacoustics/fluctuation-strength-ecma/).
