---
title: "Objective audibility of tones in noise (ISO/PAS 20065)"
description: "The ISO/PAS 20065:2016 engineering method for the audibility ΔL of a tone above the masking threshold: the critical band about the tone (Δfc, Formula 2), the critical-band masking level LG = LS + 10 lg(Δfc/Δf) (Formula 12), the masking index av = −2 − lg[1 + (f/502)²·⁵] (Formula 13), the audibility ΔL = LT − LG − av (Formula 14), and the decisive and energy-mean mean audibility (Formula 20)."
references:
  - type: standard
    organization: "International Organization for Standardization"
    year: 2016
    title: "Acoustics — Objective method for assessing the audibility of tones in noise — Engineering method"
    designation: "ISO/PAS 20065:2016"
    url: "https://www.iso.org/standard/66941.html"
    note: "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 Withdrawn, superseded by ISO/TS 20065:2022 (https://www.iso.org/standard/81518.html)."
  - type: standard
    organization: "International Organization for Standardization"
    year: 2017
    title: "Acoustics — Description, measurement and assessment of environmental noise — Part 2: Determination of sound pressure levels"
    designation: "ISO 1996-2:2017"
    url: "https://www.iso.org/standard/59766.html"
    note: "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."
  - type: standard
    organization: "Deutsches Institut für Normung"
    year: 2005
    title: "Akustik — Bestimmung der Tonhaltigkeit von Geräuschen und Ermittlung eines Tonzuschlages für die Beurteilung von Geräuschimmissionen"
    designation: "DIN 45681:2005-03"
    url: "https://www.dinmedia.de/en/standard/din-45681/76463232"
    note: "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)."
---

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

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](/phonometry/guides/levels/)); the mean
audibility `ΔL` it produces feeds the ISO 1996-2 tonal adjustment `Kt`.

<ThemeImage
  src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/tone_audibility.svg"
  alt="Per-tone audibility ΔL of the nine tones of the ISO/PAS 20065 Annex E combustion-engine spectrum, with the decisive tone at 137.3 Hz highlighted and the ΔL = 0 dB threshold marked"
  width="82%"
  loading="eager"
/>

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

```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()
```

</details>

## 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
```

### 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)).

```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
```

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

```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)
```

:::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`<sub>T</sub>, entry type, tone level `L`<sub>pt</sub>, critical-band
masking-noise level `L`<sub>pn</sub>, critical bandwidth `Δf`<sub>c</sub> and the
audibility `ΔL`<sub>ta</sub>) above the level-versus-frequency analysis plot with
the tones and their critical-band masking noise marked, the boxed decisive
audibility `ΔL`<sub>ta</sub> 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](/phonometry/guides/loudness/#iso-532-1-report-report);
a supplied `requirement` is read as the maximum acceptable decisive audibility
`ΔL`<sub>ta</sub> 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 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.

<ReportPreview
	name="iso1996_tone_audibility_example"
	title="ISO 1996-2 tonal audibility example report (PDF)"
	description="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."
	caption="Tonal audibility fiche (ToneAudibilityResult.report), decisive ΔL_ta in dB with the tonal adjustment K."
/>

## See also

- API reference: [`psychoacoustics.tone_audibility`](/phonometry/reference/api/psychoacoustics/tone-audibility/).
