Source: https://jmrplens.github.io/phonometry/guides/tone-audibility/ # 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. (Whether a tone is *annoying* is a separate, downstream rating judgement.) It is the detailed method that **ISO 1996-2:2017** defers to (the simpler Annex C route lives in [environmental measurement](https://jmrplens.github.io/phonometry/reference/api/levels/levels/)); the mean audibility `ΔL` it produces feeds the ISO 1996-2 tonal adjustment `Kt`.

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```python 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) $$ \Delta f_c = 25.0 + 75.0\left(1.0 + 1.4\left(\tfrac{f_T}{1000}\right)^{2}\right)^{0.69}\ \mathrm{Hz}. $$ 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`. ```python 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): $$ L_G = L_S + 10\lg\!\frac{\Delta f_c}{\Delta f}, \qquad a_v = -2 - \lg\!\Big[1 + \big(\tfrac{f}{502}\big)^{2.5}\Big], \qquad \Delta L = L_T - L_G - a_v . $$ 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). ```python 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. ```python 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:

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```python 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. The Annex E example reproduces the printed Table E.2 column (for the 137.3 Hz tone, `U = 2.80 dB` 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)). ```python 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. ```python 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 above 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 ```text 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. ```python 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) ``` > **No numeric oracle.** 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. ## 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 `fT`, entry type, tone level `Lpt`, critical-band masking-noise level `Lpn`, critical bandwidth `Δfc` and the audibility `ΔLta`) above the level-versus-frequency analysis plot with the tones and their critical-band masking noise marked, the boxed decisive audibility `ΔLta` 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 (documented under [Field insulation](https://jmrplens.github.io/phonometry/guides/insulation-field/#report-metadata-reportmetadata)) and rendering engine as the [ISO 532-1 loudness fiche](https://jmrplens.github.io/phonometry/guides/loudness/#iso-532-1-report-report); a supplied `requirement` is read as the maximum acceptable decisive audibility `ΔLta` 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")`. ```python from phonometry import psychoacoustics, ReportMetadata # The Annex E combustion-engine spectrum (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, regenerated with `make reports`, is kept rendered in the repository; click the preview to open the PDF. [](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iso1996_tone_audibility_example.pdf) *Tonal audibility fiche (`ToneAudibilityResult.report`), decisive ΔL_ta in dB with the tonal adjustment K.* ## References - International Organization for Standardization. (2016). *Acoustics — Objective method for assessing the audibility of tones in noise — Engineering method* (ISO/PAS 20065:2016; withdrawn, superseded by [ISO/TS 20065:2022](https://www.iso.org/standard/81518.html)). [iso.org catalogue](https://www.iso.org/standard/66941.html). The implemented engineering method: every formula on this page, from the critical band to the mean audibility and its uncertainty, follows the 2016 PAS edition. - International Organization for Standardization. (2017). *Acoustics — Description, measurement and assessment of environmental noise — Part 2: Determination of sound pressure levels* (ISO 1996-2:2017). [iso.org catalogue](https://www.iso.org/standard/59766.html). 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. ## Standards ISO/PAS 20065:2016, *Acoustics — Objective method for assessing the audibility of tones in noise — Engineering method*: 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) are computed from the critical-band spectrum, and `analyze_spectrum` adds peak detection (Clause 5.3.8) with the distinctness criteria (Clause 5.3.4) and the multi-tone `FG` combination (Formula 17), plus the separate evaluation of two tones below 1000 Hz (Formulae 18/19). The −1.76 dB Hanning bandwidth correction, the iterative masking-level procedure and the detection/combination logic are confirmed against the parent standard **DIN 45681:2005-03** (its Annex J reference program). Conformance is anchored on the Annex E combustion-engine worked example (Tables E.1/E.2/E.3): `LS` and `LT` from the spectrum, tone detection and the `FG` combined level, the per-tone audibility, the masking index and the mean audibility of the five spectra.