Psychoacoustics, Equal Loudness Contours & Frequency Selectivity

Psychoacoustics, Equal Loudness Contours & Frequency Selectivity

Psychoacoustics investigates the psychological and physiological relationship between physical acoustic parameters (sound pressure, frequency, spectrum, duration) and subjective perceptual attributes (loudness, pitch, timbre, spatial location). A doctoral grasp of psychoacoustic mechanisms is vital for configuring advanced digital signal processing in hearing instruments, interpreting auditory brainstem evaluations, and identifying central auditory processing deficits.

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Absolute Sensitivity: MAP vs. MAF

Human hearing sensitivity varies systematically across frequency. Historically, psychophysicists quantified absolute audibility thresholds using two fundamental paradigms:

Minimum Audible Pressure (MAP)

• Paradigm: Absolute threshold measured monaurally using an earphone (supra-aural or insert) with sound pressure determined at or near the tympanic membrane via a calibrated probe tube (or in a standard acoustic coupler).
• Acoustic Characteristics: MAP measurements exclude outer ear transformations (pinna diffraction, torso reflection, and natural concha resonance). They are also elevated at low frequencies by physiological noise generated within the occluded ear canal (vascular flow, breathing, muscle contraction) and the occlusion effect.

Minimum Audible Field (MAF)

• Paradigm: Absolute threshold measured binaurally in an anechoic free field with calibrated loudspeakers positioned at $0^\circ$ azimuth at a distance of $1\,\text{meter}$ from the listener's head.
• Sensitivity Advantage: MAF thresholds are consistently $6\,\text{dB}$ to $10\,\text{dB}$ lower (more sensitive) than MAP thresholds across the audiometric spectrum:
  • Binaural Summation: Listening with two ears yields an automatic physiological threshold improvement of approximately $3\,\text{dB}$ at near-threshold levels.
  • Outer Ear Acoustic Resonance & Torso Diffraction: The acoustic filtering of the head, torso, pinna flange, and unobstructed external auditory meatus provides an acoustic gain of $+15$ to $+20\,\text{dB}$ peak in the $2000 - 4000\,\text{Hz}$ region ($f_r \approx 2700\,\text{Hz}$ for an average $2.5\,\text{cm}$ adult canal acting as a quarter-wavelength resonator).
  • Coupler Discrepancies: Earphone couplers alter physiological impedance and lack real-ear high-frequency resonant transfer functions.

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Equal Loudness Contours: The Phon and Sone Scales

Sound pressure level ($\text{dB SPL}$) does not scale linearly with human perceived loudness. Because the ear exhibits non-uniform sensitivity across frequency, tones of equal physical intensity are perceived with vastly different subjective loudness.

Equal Loudness Contours (Fletcher-Munson & ISO 226)

In 1933, Fletcher and Munson established equal loudness contours (modernized by Robinson and Dadson, and currently standardized in ISO 226:2023). Subjects adjusted the level of comparison tones across frequencies until they matched the subjective loudness of a reference tone at $1000\,\text{Hz}$.

• The Phon Scale: The unit of loudness level. By international definition, the loudness level of any sound in phons is numerically equal to the $\text{dB SPL}$ of an equally loud $1000\,\text{Hz}$ pure tone: $\text{Loudness Level (phons)} = \text{dB SPL of equally loud } 1000\,\text{Hz tone}$ For example, if a $100\,\text{Hz}$ tone must be presented at $60\,\text{dB SPL}$ to sound equally loud to a $1000\,\text{Hz}$ tone at $40\,\text{dB SPL}$, the $100\,\text{Hz}$ tone has a loudness level of $40\,\text{phons}$.
• Contour Characteristics:
  • At low levels ($20-40\,\text{phons}$), the curves bow steeply upward below $500\,\text{Hz}$ and above $6000\,\text{Hz}$, reflecting the ear's poor sensitivity at the frequency extremes.
  • A prominent trough occurs between $2000\,\text{Hz}$ and $4000\,\text{Hz}$, reflecting the acoustic resonance of the external ear canal and middle ear ossicular chain.
  • Contour Flattening at High Intensities: As intensity increases (e.g., $90-100\,\text{phons}$), the equal loudness contours flatten substantially. At high intensities, human hearing is nearly equally sensitive across low, middle, and high frequencies. This explains why bass notes grow disproportionately louder as overall volume is increased (the basis for loudness compensation / "bass boost" circuits at low listening levels).

The Sone Scale: Ratio Loudness

While phons establish whether two tones are equally loud, they do not quantify how much louder one sound is compared to another. Stanley Smith Stevens developed the sone scale as a true psychophysical ratio scale:

• Reference: $1\,\text{sone}$ is defined as the loudness of a $1000\,\text{Hz}$ tone presented at $40\,\text{dB SPL}$ (i.e., $40\,\text{phons}$).
• Stevens' Power Law: Perceived loudness follows a power function of sound pressure: $L = k \cdot P^{0.6}$ In terms of decibels, for levels above $40\,\text{phons}$, a $10\,\text{phon}$ increase corresponds to a doubling of perceived loudness:
  • $40\,\text{phons} = 1\,\text{sone}$
  • $50\,\text{phons} = 2\,\text{sones}$ (twice as loud as $40\,\text{phons}$)
  • $60\,\text{phons} = 4\,\text{sones}$ (four times as loud as $40\,\text{phons}$)
  • $70\,\text{phons} = 8\,\text{sones}$ (eight times as loud as $40\,\text{phons}$)
  • $30\,\text{phons} \approx 0.5\,\text{sones}$ (half as loud as $40\,\text{phons}$)

Clinical Relevance: In sensorineural hearing loss with recruitment (abnormal loudness growth secondary to outer hair cell loss), loudness grows at an accelerated rate. A patient with recruitment may perceive a $5\,\text{dB}$ physical increment above threshold as a doubling or tripling of subjective loudness.

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Pitch Perception & Frequency Selectivity

Pitch is the perceptual correlate of acoustic frequency. While frequency is an objective physical rate of oscillation, pitch represents an internal psychological construct.

The Mel Scale

The mel scale (Stevens, Volkmann, & Newman, 1937) is a psychophysical scale where equal numerical intervals correspond to equal perceived pitch intervals:

• Reference: A $1000\,\text{Hz}$ pure tone presented $40\,\text{dB}$ above threshold ($40\,\text{dB SL}$) is assigned a pitch of $1000\,\text{mels}$.
• Non-Linearity: Pitch does not double when frequency doubles. Doubling perceived pitch from $1000\,\text{mels}$ to $2000\,\text{mels}$ requires raising the physical frequency from $1000\,\text{Hz}$ to approximately $3000\,\text{Hz}$. Below $1000\,\text{Hz}$, the relationship is approximately linear; above $1000\,\text{Hz}$, pitch grows exponentially slower than frequency.

Theories of Pitch Perception

Two complementary neurophysiological theories explain pitch encoding:

1. Place Theory (Tonotopic / Traveling Wave Theory): Proposed by Georg von Békésy. The basilar membrane varies systematically in mechanical properties from base (narrow, light, stiff) to apex (wide, heavy, compliant). High-frequency tones produce peak displacement at the base; low-frequency tones travel the length of the cochlea to displace the apex. Pitch is encoded by which auditory nerve fibers are activated along the tonotopic array. Place theory accounts for pitch discrimination across all frequencies, particularly above $4000\,\text{Hz}$.
2. Temporal Theory (Phase-Locking & Volley Principle): Formulated by Ernest Wever. Auditory nerve fibers discharge action potentials phase-locked to a specific point in the stimulus cycle. Because single auditory neurons have absolute and relative refractory periods limiting maximum firing rates to approximately $1000\,\text{spikes/s}$, the Volley Principle posits that groups of neurons fire in coordinated, staggered bursts. Together, the ensemble matches stimulus periodicities up to approximately $4000 - 5000\,\text{Hz}$. Above $5000\,\text{Hz}$, phase-locking degrades completely, and the auditory system relies exclusively on place coding.

Critical Bands & Auditory Filters

Harvey Fletcher (1940) discovered that the human cochlea behaves as a bank of overlapping band-pass filters:

• Critical Bandwidth: In a tone-in-noise detection experiment, Fletcher held tone level constant and progressively widened the bandwidth of centered masking noise. Tone detection threshold increased linearly with noise bandwidth up to a specific inflection point—the critical band. Once noise bandwidth exceeded the critical band, adding more noise produced zero additional masking because energy outside the filter was rejected by the cochlea.
• The Bark Scale (Zwicker): Divides the audible spectrum into 24 critical bands. At low frequencies ($< 500\,\text{Hz}$), critical bandwidth remains roughly constant at approximately $100\,\text{Hz}$. Above $500\,\text{Hz}$, bandwidth widens proportionally to frequency (approximately $15 - 20\%$ of center frequency).
• Equivalent Rectangular Bandwidth (ERB): Developed by Glasberg and Moore (1990) using notched-noise masking. The auditory filter shape is quantified as an ideal rectangular filter passing equal total power: $\text{ERB}(f) = 24.7(4.37f + 1)$ where $f$ is center frequency in $\text{kHz}$. At $1000\,\text{Hz}$ ($f = 1.0$), $\text{ERB} = 24.7(4.37[1] + 1) \approx 132.6\,\text{Hz}$.
• Psychophysical Tuning Curves (PTCs): Measure the masker level needed to just mask a fixed, low-level probe tone across various masker frequencies. Normal PTCs feature a razor-sharp tip at the probe frequency and a steep high-frequency slope. In cochlear hearing loss, damage to outer hair cells impairs the active electromotile cochlear amplifier (prestin motors), causing PTC tips to elevate and broaden, degrading frequency resolution and speech discrimination in noise.

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Acoustic Masking: Simultaneous vs. Temporal Paradigms

Masking is the process by which the threshold of audibility for one sound (the probe/signal) is elevated by the presence of another sound (the masker).

1. Simultaneous Masking

Occurs when probe and masker are presented concurrently.

• Upward Spread of Masking: A low-frequency masker elevates the threshold of a higher-frequency probe much more effectively than a high-frequency masker elevates the threshold of a lower-frequency probe. Mechanism: The basilar membrane traveling wave travels from base to apex. Low-frequency energy must travel past the high-frequency basal region, exciting basal hair cells on its way, whereas high-frequency energy dissipates rapidly near the base without reaching apical low-frequency hair cells.

2. Temporal (Non-Simultaneous) Masking

Masking that occurs when signal and masker do not overlap in time:

• Forward Masking (Post-stimulatory Masking): The masker precedes the probe signal. The threshold of the probe is elevated for up to $100 - 200\,\text{ms}$ following masker offset. Mechanisms: Short-term neural adaptation / synaptic depletion at the inner hair cell-auditory nerve junction, persistent basilar membrane ringing, and post-inhibitory neural suppression in the brainstem. Forward masking decreases exponentially as the silent delay between masker and probe increases.
• Backward Masking (Pre-stimulatory Masking): The probe signal precedes the masker. The threshold of the probe is elevated even though it finishes before the masker begins, effective for silent intervals up to $20 - 50\,\text{ms}$. Mechanisms: A purely central neural processing phenomenon. The intense, following masker initiates a high-velocity neural discharge in large-diameter, rapidly conducting fibers that overtakes and interferes with the weaker, slower neural volley elicited by the preceding low-level probe within the brainstem and cortex.

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Binaural Hearing Advantages & Spatial Localization

Binaural auditory processing confers critical real-world advantages over monaural hearing.

Binaural Advantages

1. Binaural Summation:
   • At threshold, binaural sensitivity improves by approximately $3\,\text{dB}$ compared to monaural listening.
   • At suprathreshold conversational levels ($40 - 80\,\text{dB SL}$), central binaural summation improves perceived loudness by $5 - 6\,\text{dB}$, allowing lower hearing aid gain settings and reducing acoustic feedback risks.
2. Binaural Squelch: The central auditory nervous system (specifically the superior olivary complex) exploits interaural differences in phase and level to perceptually segregate a target speech stream from spatially disparate background noise, conferring a $2 - 3\,\text{dB}$ Signal-to-Noise Ratio (SNR) advantage.
3. Binaural Release from Masking / Masking Level Difference (MLD):
   • MLD measures the improvement in signal detection threshold when the phase relationship of the signal or the masker is reversed between ears.
   • Homophasic Conditions (baseline): $S_0 N_0$ (signal in phase, noise in phase) or $S_\pi N_\pi$ (both signal and noise $180^\circ$ out of phase).
   • Antiphasic Conditions (release from masking): $S_\pi N_0$ (signal $180^\circ$ out of phase, noise in phase) or $S_0 N_\pi$ (signal in phase, noise $180^\circ$ out of phase).
   • In normal listeners, shifting from $S_0 N_0$ to $S_\pi N_0$ at $500\,\text{Hz}$ produces a massive $12 - 15\,\text{dB}$ release from masking (the tone can be heard $15\,\text{dB}$ softer simply by phase-inverting it in one ear).
   • Diagnostic Value: MLD is maximal at low frequencies ($250 - 500\,\text{Hz}$) and drops to $< 3\,\text{dB}$ above $1000\,\text{Hz}$. A reduced MLD ($< 7\,\text{dB}$ at $500\,\text{Hz}$) indicates lower brainstem pathology involving the trapezoid body and superior olivary complex (e.g., multiple sclerosis, pontine gliomas, or central auditory processing disorders).

Duplex Theory of Sound Localization (Lord Rayleigh)

Horizontal sound localization relies on two complementary acoustic cues:

• Vertical Localization & Cone of Confusion: Sounds originating from equidistant points along a cone radiating from the ear canal create identical ITDs and ILDs (the "cone of confusion"). Ambiguities in front-back and vertical plane localization are resolved by spectral pinna cues (pinna notches created by the convolutions of the concha and helix in the $4000 - 10,000\,\text{Hz}$ range, characterized as Head-Related Transfer Functions / HRTFs).