Fluctuation Strength Analysis

Fluctuation strength quantifies the perception of slow amplitude modulations in a sound. While roughness captures rapid modulations (15 – 300 Hz), fluctuation strength addresses modulation frequencies below approximately 20 Hz, with peak sensitivity at about 4 Hz. At these low modulation rates, the listener perceives a rhythmic "fluctuating" or "throbbing" character rather than a continuous rough texture.

Theory

Low-Rate Modulation Detection

The auditory system resolves slow envelope fluctuations as distinct temporal events. When the modulation rate is sufficiently low (below ~20 Hz), each modulation cycle is perceived individually, producing a sensation of temporal variation. As the rate increases beyond 20 Hz, the individual cycles merge into a continuous roughness percept.

Modulation Transfer Function

The fluctuation strength model applies a modulation transfer function that is band-pass in nature, peaking at 4 Hz. This function reflects the temporal resolution limit of the auditory system:

F = 0.008 · ∫₀^{24 Bark} ΔL(z) / ( f_mod / 4 + 4 / f_mod ) dz [vacil]

Here ΔL(z) represents the level variation within each critical band due to modulation, and f_mod is the modulation frequency in Hz. The denominator reaches its minimum at f_mod = 4 Hz, producing the maximum fluctuation strength.

Relationship to Roughness

Fluctuation strength and roughness are complementary metrics that together cover the full range of temporal modulation perception. The crossover region lies near 15 – 20 Hz, where both models produce low values. In practice, most real-world sounds exhibit modulation components in both regimes, and both metrics should be evaluated together.

Parameters

Parameter	Options	Description
Block Size	200 ms – 2000 ms	Longer blocks are needed compared to roughness to resolve modulations at 2 – 4 Hz
Overlap	0% – 75%	Overlap between consecutive blocks; higher overlap provides smoother time traces
Field Type	Free field / Diffuse field	Ear transfer function for critical-band analysis
Modulation Frequency Range	0.5 Hz – 20 Hz (fixed)	Effective modulation frequency range captured by the model

Output Quantities

Quantity	Unit	Description
F	vacil	Overall fluctuation strength
F(t)	vacil	Fluctuation strength vs Time trace

Fluctuation Strength vs Time

Time-varying fluctuation strength F(t) reveals how low-frequency modulation evolves over the course of a recording. Typical applications include:

Detecting idling-speed irregularities in combustion engines where cylinder misfires produce ~4 Hz modulation patterns
Evaluating HVAC duct pulsation that may produce a "breathing" or "surging" quality at low blower speeds
Assessing the rhythmic beat frequency when two closely spaced tones produce audible beating at 2 – 8 Hz

Hearing Model Fluctuation Strength (ECMA-418-2 Edition 4)

Two fluctuation strength models are available side by side. The classic model described above follows the Zwicker/Fastl formulation. In addition, Deepsona provides the hearing model variant defined in ECMA-418-2, Edition 4 (2025), in which the signal passes through the standardized auditory filter bank and an ear transfer function before the low-rate modulation is evaluated.

Aspect	Classic model	Hearing model (ECMA-418-2 Edition 4)
Basis	Critical-band level-variation model	Standardized hearing model with defined auditory filter bank
Unit	vacil	vacil; the band-resolved result is a density in vacil/Bark
Sound field	Free field / Diffuse field	Automatic (default) / Free field / Diffuse field — Automatic determines the sound field per channel from the equalization information stored with the recording
Frequency resolution	Overall value	53 half-Bark bands, 0.5 – 26.5 Bark
Typical use	Established comparisons and legacy targets	Reporting against a current standardized method; locating the fluctuating frequency region

The band-resolved hearing-model analysis is offered as three variants — band spectrum, vs Time and vs RPM — which share one single-value definition. They are documented on the Specific Fluctuation Strength (Hearing Model) page.

The classic and hearing-model results are not numerically interchangeable. Fix one model for a comparison series and state which one was used — including the edition and the sound field for the hearing model — in every report.

Fluctuation Strength vs RPM

Hearing-model fluctuation strength is also available as a vs RPM variant. Instead of a time axis, the result is plotted over the speed of rotation, which shows directly in which part of the operating range the sound becomes uneven or throbbing.

Applicable to run-up and coast-down measurements; the recording must contain speed information.
Speed step: 50 rpm by default; the slope (run-up or coast-down) is detected automatically unless it is set explicitly.
Result: one curve in vacil per sound-pressure channel over the speed axis, with the 90th percentile reported as the single value.

For the parameters shared with the other speed-resolved psychoacoustic analyses, see Psychoacoustics vs RPM.

Interpretation Guidelines

Reference value: 1 vacil is defined as the fluctuation strength of a 1 kHz tone at 60 dB SPL, 100% amplitude-modulated at 4 Hz.

< 0.1 vacil — Imperceptible fluctuation; steady sound character.
0.1 – 0.3 vacil — Slight fluctuation; may be subliminal but can contribute to perceived unsteadiness.
0.3 – 0.7 vacil — Moderate fluctuation; clearly perceptible throbbing or pulsing character.
> 0.7 vacil — Strong fluctuation; dominant temporal pattern, likely to cause annoyance if sustained.

Practical Application Notes

In automotive applications, fluctuation strength is relevant for idle quality assessment. Irregular combustion at idle produces modulation near 4 Hz, which maximizes the fluctuation strength and is perceived as engine "lumpiness."
Fluctuation strength is less commonly specified as a stand-alone target metric than loudness or sharpness. However, it is frequently used as an input to composite sound quality indices (e.g., Aures pleasantness model).
Due to the long analysis blocks required (200+ ms), temporal resolution of F(t) is inherently lower than for roughness. Rapid transients may be smoothed out.
Beat phenomena between two tonal components separated by a few Hz will produce high fluctuation strength. This is a common source of complaint in multi-compressor installations or multi-fan systems.

Standard References

E. Zwicker & H. Fastl, Psychoacoustics: Facts and Models, 3rd ed., Springer, 2007, Chapter 10: Fluctuation Strength
H. Fastl, "The Psychoacoustics of Sound-Quality Evaluation," Acustica / acta acustica, vol. 83, 1997
W. Aures, "Ein Berechnungsverfahren der Rauhigkeit," Acustica, vol. 58, 1985 (companion roughness model for comparison)
ECMA-418-2:2025 — Psychoacoustic metrics for ITT equipment — Part 2: Models based on human perception (Edition 4) — hearing model fluctuation strength