Objective audibility of tones in noise (ISO/PAS 20065)

A steady tone embedded in broadband noise stands out when it rises audibly above the noise that would otherwise mask it, the objective precondition for the tonal penalties applied in noise assessment. ISO/PAS 20065:2016 is the engineering method that quantifies this audibility: from a narrow-band FFT spectrum it derives, for every prominent tone, the audibility ΔL: how many decibels the tone level exceeds the masking threshold of the surrounding noise. It is the detailed method that ISO 1996-2:2017 defers to (the simpler Annex C route lives in environmental measurement); the mean audibility ΔL it produces feeds the ISO 1996-2 tonal adjustment Kt.

Show the code for this figure

import matplotlib.pyplot as plt
from phonometry import psychoacoustics

# ISO/PAS 20065 Annex E combustion-engine spectrum 1: nine tones (fT, LT, LS)
# from a narrow-band spectrum with line spacing 2.7 Hz
fT = [118.4, 137.3, 158.8, 314.9, 433.4, 592.2, 629.8, 643.3, 1582.7]
LT = [64.56, 67.96, 68.63, 68.50, 73.17, 78.31, 75.00, 79.75, 71.07]
LS = [48.91, 49.22, 50.50, 52.85, 58.29, 59.53, 59.71, 61.98, 54.16]

res = psychoacoustics.assess_tones(fT, LT, LS, 2.7)
print(round(res.decisive_audibility, 2), res.decisive_frequency)  # 5.01 137.3
res.plot()   # per-tone audibility bars, decisive tone highlighted
plt.show()

1. The critical band about the tone

Each tone of frequency fT is evaluated inside a critical band whose width is (Formula 2)

$$\Delta f_c=25.0+75.0{\left(1.0+1.4{\left(\frac{f_T}{1000}\right)}^2\right)}^{0.69}\mathrm{Hz}.$$

With a geometric placement of the corner frequencies about the tone (Formulae 3–5), √(f₁·f₂) = fT and f₂ − f₁ = Δfc, so f₁ = −Δfc/2 + √(Δfc² + 4·fT²)/2 and f₂ = f₁ + Δfc.

from phonometry import psychoacoustics

print(round(psychoacoustics.critical_bandwidth_engineering(137.3), 2))   # 101.36 Hz
f1, f2 = psychoacoustics.critical_band_corners(137.3)
print(round(f1, 2), round(f2, 2))                            # 95.67 197.04

2. Audibility of a tone

The mean narrow-band level LS of the masking noise (Formula 6, an iterative energy average of the lines in the critical band) and the tone level LT (Formula 8, the energy sum of the tonal lines) are derived from the narrow-band spectrum; mean_narrowband_level and tone_level do this directly (see §4). The critical-band level of the masking noise spreads LS over the critical bandwidth (Formula 12), the masking index accounts for the ear (Formula 13) and the audibility is their difference (Formula 14):

$$L_G=L_S+10\lg \frac{\Delta f_c}{\Delta f},a_v=-2-\lg [1+(\frac{f}{502}{)}^{2.5}],\Delta L=L_T-L_G-a_v.$$

A supplied tone is audible when ΔL > 0. Δf is the line spacing (frequency resolution); the energy sums over K > 1 lines carry a window correction of 10·lg(Δf/Δfe) (−1.76 dB for the recommended Hanning window, Δfe = 1.5·Δf, Formula (8)), while a single-line tone (K = 1) takes its level unchanged (Formula (7), no bandwidth correction).

from phonometry import psychoacoustics

# ISO/PAS 20065 Annex E, tone at 137.3 Hz (Δf = 2.7 Hz):
#   LS = 49.22 dB (Formula 6), LT = 67.96 dB (Formula 8).
print(round(psychoacoustics.tone_audibility(67.96, 49.22, 137.3, 2.7), 2))   # 5.01 dB
print(round(psychoacoustics.masking_index(137.3), 2))                        # -2.02 dB

3. Decisive and mean audibility

The decisive audibility of one narrow-band spectrum is the largest tone audibility in it (clause 5.3.8). Over J staggered spectra the mean audibility is their energy mean (Formula 20); a spectrum in which no tone is found contributes ΔLj = −10 dB (Formula 21). assess_tones applies the whole chain to a spectrum’s tones and reports the decisive tone.

How the decisive band is selected. The method does not scan a fixed set of bands: each detected tone defines its own critical band (§1), the audibility is evaluated tone by tone, and, after Step 3 has merged audible same-band tones into FG groups rated at their most audible member (§5), the decisive audibility is simply the largest ΔL left standing (Step 4). The “decisive band” is therefore the critical band centred on whichever tone or group wins, and it is free to move from spectrum to spectrum as the source runs through its operating states; the energy mean of Formula 20 then lets the loudest (most audible) spectra dominate the reported value, which is deliberate: a tone that is clearly audible part of the time is not excused by intervals in which it disappears.

from phonometry import psychoacoustics

# Annex E combustion-engine spectrum 1: nine tones (fT, LT, LS), Δf = 2.7 Hz.
fT = [118.4, 137.3, 158.8, 314.9, 433.4, 592.2, 629.8, 643.3, 1582.7]
LT = [64.56, 67.96, 68.63, 68.50, 73.17, 78.31, 75.00, 79.75, 71.07]
LS = [48.91, 49.22, 50.50, 52.85, 58.29, 59.53, 59.71, 61.98, 54.16]
res = psychoacoustics.assess_tones(fT, LT, LS, 2.7)
print(round(res.decisive_audibility, 2), res.decisive_frequency)  # 5.01 137.3

# Mean audibility of the five measured spectra (Table E.3 decisive values):
print(round(psychoacoustics.mean_audibility([9.18, 6.04, 7.46, 2.67, 7.17]), 2))  # 6.98 dB

res.plot(view="levels")   # tone levels above their critical-band masking noise

The same assessment reads two ways. The audibility bars at the top of this page answer “how far above the masking threshold is each tone”; the levels view answers “what did the analyser see”, which is the view an assessment report has to defend:

Show the code for this figure

import matplotlib.pyplot as plt
from phonometry import psychoacoustics

# The Annex E combustion-engine spectrum of the snippet above.
fT = [118.4, 137.3, 158.8, 314.9, 433.4, 592.2, 629.8, 643.3, 1582.7]
LT = [64.56, 67.96, 68.63, 68.50, 73.17, 78.31, 75.00, 79.75, 71.07]
LS = [48.91, 49.22, 50.50, 52.85, 58.29, 59.53, 59.71, 61.98, 54.16]
res = psychoacoustics.assess_tones(fT, LT, LS, 2.7)

# One line: the levels view, the same one the .report() fiche embeds.
res.plot(view="levels")
plt.show()

# The default view is the per-tone audibility instead:
res.plot()      # or res.plot(view="audibility")
plt.show()

Reading it left to right: each horizontal segment is the critical-band masking-noise level LG drawn across the band it applies to, each marker is the tone level LT, and the gap between them, less the masking index, is the audibility. A tone whose marker sits below its segment is masked, and no amount of level justifies a penalty for it.

3.1 Extended uncertainty of the audibility

Clause 5.4 attaches a 90 % bilateral extended uncertainty U to every audibility, and clause 6 makes it mandatory whenever fewer than 12 spectra have been averaged. assess_tones computes it per tone (res.extended_uncertainties), audibility_uncertainty evaluates it straight from the spectrum lines, and mean_audibility_uncertainty propagates it to the energy-averaged audibility of a spectrum set (Annex E: U = 2.80 dB for the 137.3 Hz tone against the printed 2,79).

How to read U. A 90 % bilateral interval leaves 5 % in each tail, so ΔL − U is a one-sided 95 % statement: when the whole interval ΔL ± U sits above 0 dB the tone is audible with at least 95 % confidence, and when the interval straddles zero the verdict is not statistically secured; the remedy is more spectra, since U shrinks with the number averaged (which is exactly why clause 6 makes reporting it mandatory below 12 spectra). The same logic guards the downstream penalty: ISO 1996-2:2017 Annex J converts the mean audibility into the tonal adjustment Kt in 1 dB steps (Table J.1: Kt = 0 for ΔL ≤ 0, up to Kt = 6 dB for ΔL > 12 dB, or the coarser 0/3/6 dB ladder of its note), so an uncertainty that spans a table boundary propagates straight into a 1–3 dB question mark on the rating level. Quoting ΔL ± U alongside Kt shows whether the adjustment is robust or hinges on one borderline spectrum.

4. From the narrow-band spectrum

Given the FFT lines of the critical band about a tone, mean_narrowband_level runs the iterative Formula 6 procedure (energy average, dropping any line more than 6 dB above the running LS, until stable within ±0.005 dB or fewer than five lines remain each side, Annex D) and tone_level sums the tonal lines contiguous with the peak (above both LS + 6 dB and L_peak − 10 dB). The mean always carries the −1.76 dB Hanning bandwidth correction; the tone level carries it only when the run spans more than one line (Formulae (7)/(8)).

from phonometry import psychoacoustics

# Annex E Table E.1: the 38 lines of the 137.3 Hz critical band (Δf = 2.7 Hz).
freqs = [96.9, 99.6, 102.3, 105.0, 107.7, 110.4, 113.0, 115.7, 118.4, 121.1,
         123.8, 126.5, 129.2, 131.9, 134.6, 137.3, 140.0, 142.7, 145.3, 148.0,
         150.7, 153.4, 156.1, 158.8, 161.5, 164.2, 166.9, 169.6, 172.3, 175.0,
         177.6, 180.3, 183.0, 185.7, 188.4, 191.1, 193.8, 196.5]
levels = [49.40, 50.68, 50.09, 53.37, 44.47, 50.91, 51.41, 59.40, 64.54, 57.57,
          51.02, 50.76, 59.93, 62.94, 58.49, 65.87, 62.66, 50.25, 51.32, 52.30,
          52.58, 53.15, 67.04, 67.27, 57.40, 57.17, 52.56, 51.39, 52.49, 47.68,
          51.26, 49.03, 61.42, 59.52, 48.43, 50.84, 48.20, 55.95]

ls = psychoacoustics.mean_narrowband_level(levels, freqs, 137.3)
lt = psychoacoustics.tone_level(levels, freqs, 137.3, ls)
print(round(ls, 2), round(lt, 2))                         # 49.22 67.96
print(round(psychoacoustics.tone_audibility(lt, ls, 137.3, 2.7), 2))   # 5.01 dB

5. Whole-spectrum detection

analyze_spectrum runs the full front-end over a spectrum (mean narrow-band level per line, peak detection (Clause 5.3.8 Step 1, a tone cannot sit on a slope), tone level, the distinctness test (Clause 5.3.4: bandwidth ≤ 26·(1 + 0.001·fT) Hz and edge steepness ≥ 24 dB), and audibility) and returns the distinct, audible tones. It then applies Step 3: audible tones sharing a critical band have their tone levels energy-summed (Formula 17, shared lines counted once, via combined_tone_level) into a combined “FG” entry rated at the most audible member, unless the exactly-two-tones-below-1000-Hz exception of §5.1 keeps them separate. The result’s group_sizes tells individual tones (1) from FG entries (N ≥ 2), and the decisive audibility (Step 4) is the maximum over all entries.

from phonometry import psychoacoustics

# Annex E Table E.1: the 38 lines of the 137.3 Hz critical band (Δf = 2.7 Hz).
freqs = [96.9, 99.6, 102.3, 105.0, 107.7, 110.4, 113.0, 115.7, 118.4, 121.1,
         123.8, 126.5, 129.2, 131.9, 134.6, 137.3, 140.0, 142.7, 145.3, 148.0,
         150.7, 153.4, 156.1, 158.8, 161.5, 164.2, 166.9, 169.6, 172.3, 175.0,
         177.6, 180.3, 183.0, 185.7, 188.4, 191.1, 193.8, 196.5]
levels = [49.40, 50.68, 50.09, 53.37, 44.47, 50.91, 51.41, 59.40, 64.54, 57.57,
          51.02, 50.76, 59.93, 62.94, 58.49, 65.87, 62.66, 50.25, 51.32, 52.30,
          52.58, 53.15, 67.04, 67.27, 57.40, 57.17, 52.56, 51.39, 52.49, 47.68,
          51.26, 49.03, 61.42, 59.52, 48.43, 50.84, 48.20, 55.95]

# Same Table E.1 spectrum as above.
res = psychoacoustics.analyze_spectrum(levels, freqs, 2.7)
singles = res.group_sizes == 1
print([round(f, 1) for f in res.tone_frequencies[singles]])  # [118.4, 137.3, 158.8]

# Step 3 already combined the three same-band tones into an FG entry:
fg = res.group_sizes > 1
print(int(res.group_sizes[fg][0]), round(float(res.tone_levels[fg][0]), 2))  # 3 72.15

# The same Formula 17 combination, called directly (LS from Table E.2):
lt_fg = psychoacoustics.combined_tone_level(levels, freqs, [118.4, 137.3, 158.8],
                               [48.91, 49.22, 50.50])
print(round(lt_fg, 2))                                # 72.15

res.plot()   # the detected entries, FG groups included, as audibility bars

Reproducing a decisive audibility exactly needs the complete narrow-band spectrum: Table E.1 is truncated to the 137.3 Hz critical band, so the 158.8 Hz tone’s mean narrow-band level is under-estimated from it (the algorithm itself matches the parent standard DIN 45681:2005-03 reference program). The peak detection and FG combination are verified against the Annex E worked example (the three tone frequencies and LT = 72.15 dB).

5.1 Two tones below 1000 Hz

When exactly two tones share a critical band and both lie below 1000 Hz, the ear can still tell them apart (so they are rated separately rather than FG-combined) if their frequency difference |fT1 − fT2| (Formula 18) exceeds

fD = 21·10^(1.2·|lg(fT/212)|^1.8)  Hz     (Formula 19, 88 Hz < fT < 1000 Hz)

evaluated at the more prominent tone fT (the larger audibility ΔL). The threshold bottoms out at 21 Hz at fT = 212 Hz and grows on either side. two_tone_separation_frequency gives fD; resolve_tones_separately applies the decision.

from phonometry import psychoacoustics

psychoacoustics.two_tone_separation_frequency(212.0)             # 21.0 Hz (minimum)
psychoacoustics.resolve_tones_separately(200.0, 260.0, 3.0, 2.0) # True  → rate separately
psychoacoustics.resolve_tones_separately(118.4, 137.3, 4.0, 5.0) # False → combine (Δf < fD)

Note

No ISO/PAS 20065 worked example exercises this branch; the Annex E band groups three tones, so the “exactly two tones” rule never fires there. The formula and decision are implemented clean-room from the text and verified against the DIN 45681:2005-03 Annex J reference program (fD = 21 * 10 ^ (1.2 * Abs(Log(fT / 212) / Log(10)) ^ 1.8)). Reassuringly, evaluated at the Annex E tones the threshold (≈ 24 Hz at 137.3 Hz) keeps them combined, consistent with that example’s FG grouping.

6. Tonal assessment report (.report())

ToneAudibilityResult.report(path) renders a one-page PDF fiche laid out like a tonal-assessment report of an environmental-noise laboratory, following the ISO 1996-2:2017 Annex J engineering method: a standard-basis line, an optional metadata header block (source/situation, client, measurement position, instrumentation and date, with the analysis line spacing Δf read from the result), a full-width table of the key quantities for every detected tone (tone frequency f_T, entry type, tone level L_pt, critical-band masking-noise level L_pn, critical bandwidth Δf_c and the audibility ΔL_ta) above the level-versus-frequency analysis plot with the tones and their critical-band masking noise marked, the boxed decisive audibility ΔL_ta together with the derived tonal adjustment K (Table J.1), an optional PASS/FAIL verdict row and a prominence note, and a footer with the fixed disclaimer.

It uses the same ReportMetadata container and rendering engine as the ISO 532-1 loudness fiche; a supplied requirement is read as the maximum acceptable decisive audibility ΔL_ta in dB (a quieter tone passes). Rendering needs reportlab (pip install phonometry[report]); only engine="reportlab" is supported. The fiche renders in English by default; pass language="es" for a Spanish fiche (translated fixed strings and a comma decimal separator), e.g. res.report("tone_fiche_es.pdf", language="es").

from phonometry import psychoacoustics, ReportMetadata

# The Annex E combustion-engine spectrum from §4/§5 (analyze_spectrum).
res = psychoacoustics.analyze_spectrum(levels, freqs, 2.7)
res.report(
    "tone_fiche.pdf",
    metadata=ReportMetadata(
        specimen="Combustion engine, steady operation",
        measurement_standard="ISO 1996-2",
        laboratory="Phonometry Reference Laboratory",
        requirement=6.0,             # maximum acceptable ΔL_ta (dB)
    ),
)                                    # decisive ΔL_ta (dB) and K (dB, Table J.1)

The example fiche is regenerated with make reports and kept rendered in the repository; click the preview to open the PDF.

One-page tonal-assessment fiche: a metadata header, a per-tone table of the tone level Lpt, the critical-band masking-noise level Lpn, the critical bandwidth and the audibility, the level-versus-frequency analysis plot with the tones and their masking noise marked, the boxed decisive ΔL_ta = 9.1 dB with the tonal adjustment K = 5 dB (ISO 1996-2:2017 Table J.1) and a FAIL verdict against a 6 dB audibility limit.

Download the report (PDF)

Tonal audibility fiche (ToneAudibilityResult.report), decisive ΔL_ta in dB with the tonal adjustment K.

What this guide covers

Covered. The ISO/PAS 20065:2016 engineering method in full: the critical bandwidth Δfc and corner frequencies (Formulae 2 to 5), the mean narrow-band level LS and tone level LT (Formulae 6 and 8, Annex D), the critical-band masking level LG, the masking index av and the audibility ΔL (Formulae 12 to 14), peak detection and the distinctness test (clauses 5.3.8 and 5.3.4), the multi-tone FG combination (Formula 17) and the two-tones-below-1000-Hz exception (Formulae 18/19), the decisive and energy-mean audibility (Formula 20) and the clause 5.4/6 extended uncertainty U, all through psychoacoustics.analyze_spectrum and assess_tones. ISO 1996-2:2017 Annex J is covered as far as it maps the mean audibility to the tonal adjustment Kt (Table J.1).

Not covered. Clause 5.3.2 requires the narrow-band spectrum to be A-weighted per IEC 61672-1 before analysis; this module is weighting agnostic and does not apply that weighting, so a caller must A-weight the spectrum first. The distinctness edge-steepness test follows the DIN 45681:2005-03 reading (its executable reference program), not the asymmetric formulas printed in the ISO/PAS 20065 text, which contradict it (see docs/ERRATA.md). Building the narrow-band FFT spectrum itself from a raw time-domain recording is not part of this module: every function here takes an already-computed spectrum (levels and frequencies).

See also

• API reference: psychoacoustics.tone_audibility.

References

• Deutsches Institut für Normung. (2005). Akustik — Bestimmung der Tonhaltigkeit von Geräuschen und Ermittlung eines Tonzuschlages für die Beurteilung von Geräuschimmissionen (DIN 45681:2005-03). The parent standard the −1.76 dB Hanning bandwidth correction, the iterative masking-level procedure and the detection/combination logic are confirmed against (its Annex J reference program).
• International Organization for Standardization. (2016). Acoustics — Objective method for assessing the audibility of tones in noise — Engineering method (ISO/PAS 20065:2016). The implemented engineering method: every formula on this page follows the 2016 PAS edition. The critical bandwidth Δfc (Formula 2) and its corner frequencies (Formulae 3–5), the critical-band level LG (Formula 12), the masking index av (Formula 13), the audibility ΔL = LT − LG − av (Formula 14) and the energy-mean mean audibility (Formula 20); the mean narrow-band level LS (Formula 6, iterative Annex D) and tone level LT (Formula 8) come from the critical-band spectrum, and analyze_spectrum adds peak detection (Clause 5.3.8) with the distinctness criteria (Clause 5.3.4), the multi-tone FG combination (Formula 17) and the separate evaluation of two tones below 1000 Hz (Formulae 18/19). Conformance is anchored on the Annex E combustion-engine worked example (Tables E.1/E.2/E.3). Withdrawn, superseded by ISO/TS 20065:2022 (https://www.iso.org/standard/81518.html).
• International Organization for Standardization. (2017). Acoustics — Description, measurement and assessment of environmental noise — Part 2: Determination of sound pressure levels (ISO 1996-2:2017). The environmental-noise standard this method serves: its Annex J adopts the engineering method and maps the mean audibility to the tonal adjustment Kt (Table J.1) discussed in §3.1.