Source: https://jmrplens.github.io/phonometry/perception/psychoacoustics/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 lives in [Loudness](https://jmrplens.github.io/phonometry/perception/psychoacoustics/loudness/); the ECMA-418-2 loudness that shares the same auditory front-end lives in [Advanced loudness](https://jmrplens.github.io/phonometry/perception/psychoacoustics/advanced-loudness/). The four metrics form one family: a single calibrated signal splits across two auditory front ends, and every branch is anchored to a reference sound whose normative target is exactly 1. The diagram maps the family with the values the library actually computes for each reference (rounded results such as 0.9999 asper and 0.9957 vacil_HMS sit against that target of 1). Block diagram of the sound-quality metric family: a calibrated signal in pascals feeds two auditory front ends, the Zwicker specific loudness pattern over 24 Bark and the ECMA-418-2 Sottek Hearing Model with 53 auditory bands; the first drives the DIN 45692 sharpness whose critical-band-wide noise reference at 1 kHz and 60 dB reads 1.00 acum, the second drives the tonality, roughness and fluctuation strength, whose references, a 1 kHz tone at 40 dB and 1 kHz carriers fully amplitude-modulated at 70 Hz and 4 Hz at 60 dB, read 1.000 tu_HMS, 0.9999 asper and 0.9957 vacil_HMS; a closing note states that N5, sharpness, roughness and fluctuation strength feed the Fastl and Zwicker psychoacoustic annoyance ## 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). 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
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 $\approx 1$ tu_HMS; the result also tracks the tonal frequency $f_\text{ton}$ 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) ``` ECMA-418-2 average specific tonality over the 53 Bark_HMS bands for a 1 kHz tone at 40 dB SPL: the tonality is concentrated in a single peak at the tone's critical band around 9 Bark_HMS and falls to nearly zero elsewhere, integrating to T = 1.00 tu_HMS
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` ($f_{\text{ton},z}$), `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 ``` 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
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 $R(l_{50})$), `specific_roughness` ($R'(z)$, 53 bands), `bark`, `centre_frequencies`, `time`, `roughness_vs_time` ($R(l_{50})$), `specific_roughness_vs_time` ((n_times, 53) array), `field`. All three ECMA-418-2 entry points on this page 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 ear channel separately. The optional entropy weighting of clause 7.1.6, which needs an external rotational-speed signal, is also left out. ## 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 and without the ±0.25 % adjustment that footnote 47 permits (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 ``` 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
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 $F(l_{50})$), `specific_fluctuation_strength` ($F'(z)$, 53 bands), `bark`, `centre_frequencies`, `time`, `fluctuation_strength_vs_time` ($F(l_{50})$), `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/perception/psychoacoustics/psychoacoustic-annoyance/); this Clause 9 metric is the normative Sottek-model counterpart. See [Prominent Discrete Tones](https://jmrplens.github.io/phonometry/perception/psychoacoustics/tone-prominence/) for the ECMA-418-1 TNR/PR prominence verdicts, [Speech Transmission Index](https://jmrplens.github.io/phonometry/perception/speech/speech-transmission/) for STI/STIPA, and [Theory](https://jmrplens.github.io/phonometry/reference/theory/perception/) for the underlying math. ## See also - API reference: [`psychoacoustics.quality.sharpness`](https://jmrplens.github.io/phonometry/reference/api/psychoacoustics/sharpness/), [`psychoacoustics.quality.tonality_ecma`](https://jmrplens.github.io/phonometry/reference/api/psychoacoustics/tonality-ecma/), [`psychoacoustics.quality.roughness_ecma`](https://jmrplens.github.io/phonometry/reference/api/psychoacoustics/roughness-ecma/) and [`psychoacoustics.quality.fluctuation_strength_ecma`](https://jmrplens.github.io/phonometry/reference/api/psychoacoustics/fluctuation-strength-ecma/). - Theory: [Advanced loudness models and sound quality](https://jmrplens.github.io/phonometry/reference/theory/perception/#advanced-loudness-models--sound-quality): the specific-loudness pattern the sharpness, roughness and fluctuation-strength metrics are all built on. ## 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).