Source: https://jmrplens.github.io/phonometry/guides/sound-quality/ # 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](https://jmrplens.github.io/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).

Show the code for this figure

```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() ```

```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) ```

Show the code for this figure

```python 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. ```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 ```

Show the code for this figure

```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() ```

### `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 ```

Show the code for this figure

```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() ```

### `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](https://jmrplens.github.io/phonometry/reference/api/psychoacoustics/psychoacoustic-annoyance/); this Clause 9 metric is the normative Sottek-model counterpart. See [Prominent Discrete Tones](https://jmrplens.github.io/phonometry/guides/tone-prominence/) for the ECMA-418-1 TNR/PR prominence verdicts, [Speech Transmission Index](https://jmrplens.github.io/phonometry/guides/speech-transmission/) for STI/STIPA, and [Theory](https://jmrplens.github.io/phonometry/reference/theory/perception/) for the underlying math. ## 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 psychoacoustics of the sensations quantified on this page: the high-frequency emphasis behind sharpness and the fast-modulation percept behind roughness. ## Standards DIN 45692:2009, *Messtechnische Simulation der Hörempfindung Schärfe*: sharpness in acum (clause 6 weighting, Annex B von Bismarck and Aures variants, Table A.2 targets). ECMA-418-2:2025, *Psychoacoustic metrics for ITT equipment — Part 2 (methods for describing human perception based on the Sottek Hearing Model)*: 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).