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# Loudness

Level metrics tell you how much *sound pressure* there is; loudness tells you
how loud a listener actually *perceives* it. This page covers the three
loudness model families phonometry ships: the Zwicker method (ISO 532-1), the
Moore-Glasberg methods (ISO 532-2/3) and the Sottek Hearing Model loudness
(ECMA-418-2), plus the equal-loudness contours of pure tones (ISO 226).
Sharpness, tonality and roughness live in
[Sound Quality Metrics](sound-quality.md); speech metrics in
[Speech Transmission Index](speech-transmission.md) and
[Speech Intelligibility Index](speech-intelligibility.md).

## Loudness in sones (ISO 532-1, Zwicker)

Decibels compress perception: 10 dB more reads as *twice as loud*, and two
sounds with the same dB(A) can differ audibly depending on how their energy
spreads over the ear's **critical bands**. The Zwicker method models the
hearing chain explicitly (outer/middle-ear transmission, critical-band
analysis on the 24 Bark scale, level-dependent masking slopes) and outputs
**loudness N in sones**, a ratio scale: 4 sones is twice as loud as 2 sones.
By definition a 1 kHz tone at 40 dB SPL is 1 sone, and every +10 phon
doubles the sone value.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_zwicker_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_zwicker.svg" alt="ISO 532-1 Zwicker loudness chain: 28 one-third-octave band levels, transmission and lower-critical-band grouping, core loudness of the 20 critical bands, specific loudness over Bark, integrated into total loudness N in sones and loudness level in phons" width="78%"></picture>

The animation below shows that integration at work: as the band level of a
1 kHz narrowband sound steps up, the specific-loudness pattern N'(z) grows
along the Bark axis and the area under it is the total loudness in sones.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_specific_loudness_dark.gif"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_specific_loudness.gif" alt="Animation: the specific-loudness pattern of a 1 kHz narrowband sound builds along the Bark axis as the band level steps from 45 to 85 dB, and the area under the pattern integrates to the total loudness in sones" width="640" height="360" loading="lazy"></picture>

[Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_specific_loudness.webm)

```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
levels_28 = np.full(28, 60.0)                             # 28 one-third-octave band levels (dB)

# From a raw recording: calibration_factor scales digital units to Pa
res = psychoacoustics.loudness_zwicker(x, fs, field="free", calibration_factor=sens)
print(f"N = {res.loudness:.1f} sone  ({res.loudness_level:.0f} phon)")   # 13.1 sone (77 phon)

# Time-varying signals: percentile loudness N5 is the reporting standard
res = psychoacoustics.loudness_zwicker(x, fs)          # stationary=False (default)
print(f"{res.n5:.1f} {res.n10:.1f} {res.loudness:.1f}")   # 13.1 13.1 13.1 — N5, N10, Nmax

# From 28 one-third-octave band levels (25 Hz .. 12.5 kHz)
res = psychoacoustics.loudness_zwicker_from_spectrum(levels_28, field="diffuse")

res.plot()   # N'(z) over the Bark scale — the specific-loudness pattern (needs matplotlib)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/loudness_pattern_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/loudness_pattern.svg" alt="Specific loudness patterns over the Bark scale for a 1 kHz narrowband sound and a broadband sound of equal band level" width="80%"></picture>

*Same band level, very different loudness: energy spread over many critical
bands (red) sums to far more sones than the same level concentrated in one
band (blue). The area under N'(z) is the total loudness.*

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

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import psychoacoustics

levels_28 = np.full(28, 60.0)                             # 28 one-third-octave band levels (dB)
# From 28 one-third-octave band levels (25 Hz .. 12.5 kHz)
res = psychoacoustics.loudness_zwicker_from_spectrum(levels_28, field="diffuse")

# One line — the specific-loudness pattern N'(z) straight from the result:
res.plot()
plt.show()

# Or reproduce the figure by hand — two patterns of equal band level (60 dB),
# energy spread over many critical bands vs concentrated in the 1 kHz band:
narrow = psychoacoustics.loudness_zwicker_from_spectrum(np.r_[np.full(16, -60.0), 60.0, np.full(11, -60.0)])
broad = psychoacoustics.loudness_zwicker_from_spectrum(np.full(28, 60.0))
z = np.arange(1, narrow.specific.size + 1) * 0.1          # Bark axis
fig, ax = plt.subplots()
for r, color, label in [
    (broad, "#ff7f0e", f"Broadband  N = {broad.loudness:.1f} sone"),
    (narrow, "#1f77b4", f"1 kHz narrowband  N = {narrow.loudness:.1f} sone"),
]:
    ax.fill_between(z, r.specific, color=color, alpha=0.3)
    ax.plot(z, r.specific, color=color, label=label)
ax.set_xlabel("Critical-band rate z [Bark]")
ax.set_ylabel("Specific loudness N' [sone/Bark]")
ax.legend()
plt.show()
```

</details>

The implementation is a clean-room port of the standard's **normative
reference program** (Annex A.4): all twelve data tables are digit-exact and
the full Annex B validation set runs in CI: the stationary test case
reproduces the published value to every printed digit, and the tone-pulse
N(t) traces stay inside the standard's per-sample 5 % tolerance band.

### `loudness_zwicker()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `x` | 1D array | Pa (after calibration) | ≥ 8 ms at 48 kHz | Resampled internally to 48 kHz if needed |
| `fs` | int | Hz | > 0 | |
| `field` | str | — | `'free'` (default) / `'diffuse'` | Sound-field correction (Table A.5) |
| `stationary` | bool | — | default `False` | `True`: single N from the averaged spectrum |
| `calibration_factor` | float | Pa per digital unit | default `1.0` | From `sensitivity()` |

Returns a `ZwickerLoudness` dataclass: `loudness` (N, sones), `loudness_level`
(phon), `specific` (N′(z), 240 bins of 0.1 Bark), and for time-varying runs
`n5`, `n10`, `time`, `loudness_vs_time` (500 Hz trace).

### ISO 532-1 report (`.report()`)

`ZwickerLoudness.report(path)` renders a one-page PDF fiche laid out like an
accredited loudness report: a standard-basis line, an optional metadata header
block, a compact metrics table (total loudness *N*, loudness level *L*<sub>N</sub>,
and the *N*<sub>5</sub>/*N*<sub>10</sub> percentiles for a time-varying result)
beside the specific-loudness pattern *N*′(z) (the result's own `.plot()`), the
boxed `N = X sone (LN = Y phon)` single number, an optional verdict row and a
footer with the fixed disclaimer. It uses the same `ReportMetadata` container
(documented under [Field insulation](insulation-field.md#report-metadata-reportmetadata))
and rendering engine as the ISO 717 insulation fiche; a supplied `requirement`
is read as the maximum permitted loudness in sone (a lower loudness 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("loudness_fiche_es.pdf", language="es")`.

```python
from phonometry import psychoacoustics, ReportMetadata

res = psychoacoustics.loudness_zwicker_from_spectrum(levels_28, field="free")
res.report(
    "loudness_fiche.pdf",
    metadata=ReportMetadata(
        specimen="Household appliance, steady operating noise",
        measurement_standard="ISO 532-1 method 1",
        laboratory="Phonometry Reference Laboratory",
        requirement=12.0,             # maximum permitted loudness (sone)
    ),
)                                     # N (sone) and LN (phon)
```

The example fiche, regenerated with `make reports`, is kept rendered in the
repository. Click the preview to open the PDF:

[![ISO 532-1 loudness example report: metadata header, a metrics table with total loudness N and loudness level LN, the specific-loudness pattern over Bark, the boxed N = 8.2 sone (LN = 70.4 phon) single number and a PASS verdict against a 12 sone limit](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iso532_loudness_example.webp)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iso532_loudness_example.pdf)

*Zwicker loudness fiche (`ZwickerLoudness.report`), N in sone with LN in phon.*

## Loudness level of pure tones (ISO 226:2023)

The normal equal-loudness-level contours relate the SPL of a pure tone to its
perceived *loudness level* in phons (the SPL of an equally loud 1 kHz tone).
`equal_loudness_contour(phon)` evaluates ISO 226:2023 Formula (1) at the 29
preferred third-octave frequencies of Table 1, `loudness_level(spl, frequency)`
is the exact inverse (Formula 2), and `hearing_threshold()` returns the
threshold-of-hearing column. `equal_loudness_contours(phons)` bundles a whole
family of contours with the threshold into a plottable `EqualLoudnessContours`
result:

```python
from phonometry import psychoacoustics

freqs, spl = psychoacoustics.equal_loudness_contour(40.0)   # the classic 40-phon contour
phon = psychoacoustics.loudness_level(73.0, 63.0)           # 73 dB @ 63 Hz -> 40 phon

# The whole family (20-90 phon by default) plus the hearing threshold:
res = psychoacoustics.equal_loudness_contours()
res.plot()   # the iconic ISO 226 chart (needs matplotlib)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/equal_loudness_contours_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/equal_loudness_contours.svg" alt="ISO 226:2023 normal equal-loudness-level contours from 20 to 90 phon with the hearing threshold curve" width="80%"></picture>

ISO 226:2023 defines the contours from 20 to 90 phon; above 80 phon the formula is valid only up to 4 kHz, so the 90 phon contour stops there and no higher contours are defined.

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

```python
import matplotlib.pyplot as plt
from phonometry import psychoacoustics

# One line — the contour family straight from the result:
res = psychoacoustics.equal_loudness_contours()
res.plot()
plt.show()

# Or reproduce the figure by hand — ISO 226:2023 Formula (1) at the 29
# preferred frequencies of Table 1, one contour per loudness level:
fig, ax = plt.subplots()
for phon in [20, 40, 60, 80, 90]:
    freqs, spl = psychoacoustics.equal_loudness_contour(float(phon))
    ax.semilogx(freqs, spl, color="C0")
    ax.annotate(f"{phon} phon", xy=(1000, phon + 1), fontsize=9)
ft, tf = psychoacoustics.hearing_threshold()
ax.semilogx(ft, tf, "--", color="C1", label="Hearing threshold $T_f$")
ax.set(xlabel="Frequency [Hz]", ylabel="Sound pressure level [dB re 20 µPa]")
ax.grid(True, which="both", alpha=0.3)
ax.legend()
plt.show()
```

</details>

Validity per clause 4.1: 20-90 phon (80 phon above 4 kHz); the implementation
is verified against the Annex B tables in CI. Note this is the loudness of
*pure tones*; the loudness of arbitrary signals in sones is what the ISO 532
models on this page compute.

## Advanced loudness models

ISO 532-1 above is one of the **three** loudness model families phonometry
ships (four methods in the table below). This
section adds the **Moore-Glasberg** loudness of ISO 532-2/532-3 and the
**Sottek Hearing Model** loudness of ECMA-418-2:2025, whose shared auditory
front-end also powers the tonality and roughness metrics of
[Sound Quality Metrics](sound-quality.md).

### Choosing a loudness model

| Model | Standard | Stationary / time-varying | Output | When to use |
| :--- | :--- | :--- | :--- | :--- |
| Zwicker | ISO 532-1:2017 | both | sone | Reference method; one-third-octave input; fast and widely cited |
| Moore-Glasberg | ISO 532-2:2017 | stationary | sone | roex excitation pattern; better for tones and explicit binaural summation |
| Moore-Glasberg-Schlittenlacher | ISO 532-3:2023 | time-varying | sone (STL/LTL) | Time-varying loudness with short-/long-term traces and the peak N_max |
| Sottek (Hearing Model) | ECMA-418-2:2025 | time-varying | sone_HMS | Shares one auditory front-end with the ECMA tonality and roughness metrics |

All four methods are anchored so a **1 kHz tone at 40 dB SPL is ≈ 1 sone**; the values
are not interchangeable digit-for-digit because the models differ in their
auditory filters and their loudness summation.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/loudness_models_comparison_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/loudness_models_comparison.svg" alt="Loudness of a 1 kHz tone as a function of level for the Zwicker, Moore-Glasberg and Sottek models, all passing through 1 sone at 40 dB SPL" width="80%"></picture>

*The three models agree at the 1 sone / 40 dB anchor and diverge with level:
Zwicker doubles the sone value every +10 phon, while the Sottek model grows
more slowly (about 1.65× per 10 dB), an intrinsic difference between the
auditory summations, not a calibration error.*

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

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import psychoacoustics

# 1 kHz tone, 20..80 dB SPL: all three models pass through 1 sone at 40 dB
fs = 48000
t = np.arange(fs) / fs
levels = np.arange(20.0, 81.0, 10.0)
zw, mg, ec = [], [], []
for spl in levels:
    x = np.sqrt(2) * 2e-5 * 10 ** (spl / 20) * np.sin(2 * np.pi * 1000 * t)
    zw.append(psychoacoustics.loudness_zwicker(x, fs, stationary=True).loudness)
    mg.append(
        psychoacoustics.loudness_moore_glasberg_from_spectrum([(1000.0, float(spl))]).loudness
    )
    ec.append(psychoacoustics.loudness_ecma(x, fs).loudness)

fig, ax = plt.subplots()
ax.plot(levels, zw, "o-", label="Zwicker (ISO 532-1)")
ax.plot(levels, mg, "s--", label="Moore-Glasberg (ISO 532-2)")
ax.plot(levels, ec, "^-.", label="Sottek (ECMA-418-2)")
ax.plot(40.0, 1.0, "o", color="k", markerfacecolor="none", markersize=10)   # the shared anchor
ax.set(xlabel="Sound pressure level [dB SPL]", ylabel="Total loudness N [sone]")
ax.legend()
plt.show()
```

</details>

### Moore-Glasberg loudness (ISO 532-2)

Where Zwicker uses fixed critical bands on the Bark scale, Moore-Glasberg builds
an **excitation pattern** with level-dependent rounded-exponential (roex)
auditory filters on the ERB-number ("Cam") scale, then applies a compressive
excitation → specific-loudness transform with C = 0.0617 sone/Cam
(ISO 532-2:2017, Formula 7) and a binaural-inhibition stage. It reproduces the
tone and broadband cases of Annex B to a percent or two and, unlike ISO 532-1,
models binaural summation explicitly.

```python
import numpy as np
from phonometry import psychoacoustics

# The definitional anchor: one 1 kHz sinusoidal component at 40 dB SPL,
# free field, binaural -> 1 sone / 40 phon by construction of the sone.
res = psychoacoustics.loudness_moore_glasberg_from_spectrum([(1000.0, 40.0)], field="free")
print(f"N = {res.loudness:.3f} sone  ({res.loudness_level:.1f} phon)")   # 1.000 sone (40.0 phon)

# From a calibrated recording: the narrowband (FFT) line spectrum is formed
# (power-preserving normalization) and fed to the exact sinusoidal-component
# method (ISO 532-2 clauses 5.2/5.4).
fs = 48000
x = np.sqrt(2) * 2e-5 * 10 ** (40 / 20) * np.sin(2 * np.pi * 1000 * np.arange(fs) / fs)
res = psychoacoustics.loudness_moore_glasberg(x, fs, field="free", presentation="binaural")

res.plot()   # specific loudness N'(i) over the ERB-number (Cam) scale
```

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

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import psychoacoustics

# From a calibrated recording: the narrowband (FFT) line spectrum is formed
# (power-preserving normalization) and fed to the exact sinusoidal-component
# method (ISO 532-2 clauses 5.2/5.4).
fs = 48000
x = np.sqrt(2) * 2e-5 * 10 ** (40 / 20) * np.sin(2 * np.pi * 1000 * np.arange(fs) / fs)
res = psychoacoustics.loudness_moore_glasberg(x, fs, field="free", presentation="binaural")

# One line — the specific-loudness pattern N'(i) straight from the result:
res.plot()
plt.show()

# Or draw it by hand from the ERB-number grid the result already carries:
fig, ax = plt.subplots()
ax.fill_between(res.erb_number, res.specific, alpha=0.3)
ax.plot(res.erb_number, res.specific)
ax.set_xlabel("ERB number [Cam]")
ax.set_ylabel("Specific loudness N' [sone/Cam]")
plt.show()
```

</details>

#### `loudness_moore_glasberg()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `x` | 1D array | Pa | non-empty | Calibrated pressure signal (signal wrapper) |
| `components` | list of `(f, L)` | Hz, dB SPL | — | `_from_spectrum`: discrete sinusoidal components |
| `band_levels` | 29-vector | dB SPL | 25 Hz .. 16 kHz | `_from_third_octave` input (IEC 61260-1 bands) |
| `fs` | int | Hz | > 0 | Signal wrapper only |
| `field` | str | — | `'free'` (default) / `'diffuse'` / `'eardrum'` | Outer-ear transfer |
| `presentation` | str | — | `'binaural'` (default) / `'diotic'` / `'monaural'` | Binaural summation |

Returns a `MooreGlasbergLoudness`: `loudness` (N, sone), `loudness_level`
(phon), `specific` (N′(i), 372 bins of 0.1 Cam), `erb_number`,
`centre_frequencies`, `field`, `presentation`.

### Time-varying loudness (ISO 532-3)

ISO 532-3 wraps the same excitation / specific-loudness model in a running
multi-resolution spectral analysis (six parallel FFTs, updated every 1 ms) and
two cascaded temporal integrators: the fast **short-term loudness** S′(t) and
the slower **long-term loudness** S″(t). The peak long-term loudness N_max
predicts the loudness of sounds up to about 5 s.

```python
import numpy as np
from phonometry import psychoacoustics

fs = 32000
t = np.arange(int(1.3 * fs)) / fs
x = np.sqrt(2) * 2e-5 * 10 ** (40 / 20) * np.sin(2 * np.pi * 1000 * t)

res = psychoacoustics.loudness_moore_glasberg_time(x, fs, field="free")
print(f"N_max = {res.n_max:.3f} sone  ({res.loudness_level_max:.0f} phon)")   # 1.000 sone (40 phon)
print(f"long-term loudness exceeded 5% of the time: {res.percentiles[5.0]:.3f} sone")   # 0.999 sone

res.plot()   # short-term S'(t) and long-term S''(t) loudness vs time
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/moore_glasberg_time_loudness_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/moore_glasberg_time_loudness.svg" alt="Short-term and long-term Moore-Glasberg loudness traces for a tone burst, showing the fast attack of the short-term loudness and the slower release of the long-term loudness" width="80%"></picture>

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

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import psychoacoustics

fs = 32000
t = np.arange(int(1.3 * fs)) / fs
x = np.sqrt(2) * 2e-5 * 10 ** (40 / 20) * np.sin(2 * np.pi * 1000 * t)
res = psychoacoustics.loudness_moore_glasberg_time(x, fs, field="free")

# The result carries both traces on a 1 ms time axis:
res.plot()
plt.show()

# Or plot them directly to see the fast STL vs the slow LTL:
fig, ax = plt.subplots()
ax.plot(res.time, res.short_term_loudness, label="Short-term S'(t)")
ax.plot(res.time, res.long_term_loudness, label="Long-term S''(t)")
ax.set_xlabel("Time [s]")
ax.set_ylabel("Loudness [sone]")
ax.legend()
plt.show()
```

</details>

#### `loudness_moore_glasberg_time()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `signal` | 1D or `(n, 2)` array | Pa | non-empty | Mono = diotic; two columns = left/right ears |
| `fs` | int | Hz | > 0 | |
| `field` | str | — | `'free'` (default) / `'diffuse'` / `'eardrum'` | Outer-ear transfer |
| `presentation` | str | — | `'binaural'` (default) / `'diotic'` / `'monaural'` | Binaural summation |
| `percentiles` | sequence | percent | default `(1, 5, 10, 50, 90, 95)` | Exceeded long-term loudness levels |

Returns a `MooreGlasbergTimeVaryingLoudness`: `time` (1 ms grid),
`short_term_loudness` / `long_term_loudness` (sone), their `_level` in phon,
`n_max`, `loudness_level_max`, a `percentiles` dict, `field`, `presentation`.

### Sottek Hearing Model loudness (ECMA-418-2)

ECMA-418-2:2025 specifies a single auditory front-end (outer/middle-ear
filtering, a 53-band gammatone-like filter bank on the Bark_HMS scale
with z = 0.5 .. 26.5, half-wave rectification, block RMS and a compressive
nonlinearity, Formula 23) that is **shared** by its loudness, tonality and
roughness metrics. The loudness N is reported in **sone_HMS**, and the same
1 kHz/40 dB anchor calibrates the front-end (our clean-room value 0.984,
with the full Clause 6.2.3 band averaging; the residual's origin is
documented in the module docstring).

```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.loudness_ecma(x, fs, field="free")
print(f"N = {res.loudness:.3f} sone_HMS")   # 0.984 sone_HMS
print(res.specific_loudness.shape)          # (53,) average specific loudness N'(z)

res.plot()   # average specific loudness N'(z) + time-dependent N(l) at 187.5 Hz
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sottek_specific_loudness_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sottek_specific_loudness.svg" alt="Sottek Hearing Model average specific loudness N'(z) over the 53 Bark_HMS bands for a 1 kHz tone, peaking at the tone's critical band" width="80%"></picture>

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

```python
import matplotlib.pyplot as plt
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.loudness_ecma(x, fs, field="free")

# The result carries the average specific loudness over the 53 Bark_HMS bands:
res.plot()
plt.show()

# Or draw N'(z) by hand against the critical-band-rate scale:
fig, ax = plt.subplots()
ax.fill_between(res.bark, res.specific_loudness, alpha=0.3)
ax.plot(res.bark, res.specific_loudness)
ax.set_xlabel("Critical-band rate z [Bark_HMS]")
ax.set_ylabel("Specific loudness N' [sone_HMS/Bark_HMS]")
plt.show()
```

</details>

#### `loudness_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 (Clause 5.1.1) |
| `field` | str | — | `'free'` (default) / `'diffuse'` | Outer/middle-ear filter (Clause 5.1.3) |

Returns an `EcmaLoudness`: `loudness` (N, sone_HMS), `specific_loudness`
(N′(z), 53 bands), `bark`, `centre_frequencies`, `time`, `loudness_vs_time`
(N(l) at 187.5 Hz), `field`.

## 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 critical-band and masking psychoacoustics behind the Zwicker model and
  the loudness sensation the newer models refine.
- Fletcher, H., & Munson, W. A. (1933). Loudness, its definition, measurement
  and calculation. *The Journal of the Acoustical Society of America*, 5(2),
  82-108. [doi:10.1121/1.1915637](https://doi.org/10.1121/1.1915637).
  The original equal-loudness measurements behind the loudness-level concept
  of the pure-tone section.
- International Organization for Standardization. (2023). *Acoustics —
  Normal equal-loudness-level contours* (ISO 226:2023).
  [iso.org catalogue](https://www.iso.org/standard/83117.html).
  The contour model and Table 1 parameters behind the pure-tone loudness
  levels.

## Standards

ISO 532-1:2017, *Acoustics — Methods for calculating
loudness — Part 1: Zwicker method* — stationary and time-varying loudness in
sones from the normative Annex A.4 reference program, with the N5/N10
percentile loudness, validated against the Annex B set. ISO 532-2:2017,
*... Part 2: Moore-Glasberg method*: stationary loudness from roex excitation
patterns on the ERB-number scale, with explicit binaural summation.
ISO 532-3:2023, *... Part 3: Moore-Glasberg-Schlittenlacher method*:
time-varying short-term and long-term loudness and the peak N_max.
ISO 226:2023, *Acoustics — Normal equal-loudness-level contours* — the
contours (Formula 1), the loudness level of pure tones (Formula 2) and the
hearing threshold. 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 loudness (sone_HMS).

## See also

- [Sound Quality Metrics](sound-quality.md): sharpness,
  tonality and roughness, the other half of the sound-quality story.
- [Psychoacoustic annoyance and fluctuation strength](psychoacoustic-annoyance.md):
  the Zwicker and Fastl model that consumes the percentile loudness N5.
- [Theory](theory-perception.md): the equations behind the loudness models.
- API reference: [`psychoacoustics.loudness_zwicker`](https://jmrplens.github.io/phonometry/reference/api/psychoacoustics/loudness-zwicker/), [`psychoacoustics.loudness_moore_glasberg`](https://jmrplens.github.io/phonometry/reference/api/psychoacoustics/loudness-moore-glasberg/) and [`psychoacoustics.loudness_contours`](https://jmrplens.github.io/phonometry/reference/api/psychoacoustics/loudness-contours/).
