A solution to the mystery of the sub-harmonic series via a linear model of the cochlea

This paper proposes a linear model of the cochlea as vibrating strings to demonstrate the emergence of the sub-harmonic series, thereby explaining the historical hypothesis regarding minor chord consonance and offering new insights into the generation of combination tones through energy nonlinearity.

The Gist

Imagine your ear isn't just a microphone that records sound; it's more like a grand piano with thousands of tiny, invisible strings stretched inside a fluid-filled tube. This is the cochlea.

For centuries, musicians and scientists have been puzzled by a musical mystery: Why do certain chords sound "happy" (consonant) while others sound "sad" or "tense"?

• The Happy Chord (Major Triad): We know why a C-Major chord (C-E-G) sounds happy. If you play a low C, your ear naturally hears a "ghostly" series of higher notes (harmonics) that perfectly match the C-Major chord. It's like the note C is shouting, "I am the root of this family!"
• The Sad Chord (Minor Triad): But what about a C-Minor chord (C-E♭-G)? The "ghostly" notes of a low C don't naturally form a minor chord. In fact, for 500 years, musicians hypothesized that the ear must be hearing a "sub-harmonic" series (notes lower than the root) to explain why minor chords sound the way they do. But physics said, "Impossible! You can't have frequencies lower than the one you are playing."

The New Discovery
This paper by Boscain, Ma, Prandi, and Turco solves the mystery using a simple, elegant idea. They didn't need to invent complex, messy physics. They just looked at how the ear measures sound.

Here is the explanation in everyday language:

1. The Ear as a Set of Guitar Strings

Imagine the inside of your ear is lined with thousands of tiny guitar strings.

• The strings near the entrance are short and tight (they hear high pitches).
• The strings near the back are long and loose (they hear low pitches).

When a sound comes in, it makes these strings vibrate.

2. The "Energy" Hypothesis

The authors propose a simple rule: The brain doesn't listen to the vibration itself; it listens to the energy stored in the strings.

Think of it like this:

• If you pluck a guitar string, it wiggles back and forth.
• The vibration is the wiggle.
• The energy is how hard the string is working. It's like the "heat" or "intensity" of the vibration.

3. The Magic of the "Sub-Harmonic" (The Ghost Notes)

Here is where the magic happens. The authors found that when a string vibrates, it doesn't just vibrate at the main frequency. It has "modes" of vibration, like a rope that can wiggle once, twice, or three times at the same time.

• The Scenario: You play a single note, C4 (262 Hz).
• The Reaction: The string tuned to C4 vibrates happily.
• The Twist: But because the brain measures Energy, it also notices that the string tuned to F2 (a much lower note, 87 Hz) is vibrating in its third mode to match the C4.
• The Result: Even though you only played C4, the "energy" in the F2 string lights up. The brain thinks, "Hey, I'm detecting an F2!"

Because the brain sees energy at F2, F3, F4, etc., it reconstructs a Sub-Harmonic Series. This series naturally forms a Minor Chord.

The Analogy:
Imagine you are shining a flashlight (the sound) on a wall.

• The Old View: You only see the spot of light directly hit by the beam.
• The New View: The wall is made of special mirrors. When the light hits one spot, it reflects in a pattern that creates other spots of light on the wall that weren't directly hit. The brain sees these reflected spots (the sub-harmonics) and thinks, "Oh, a minor chord is being played!"

4. Solving the "Tartini's Third" Mystery

There is another old mystery: If you play two notes at once (say, a high C and a G), you sometimes hear a third, lower note (a low C) that isn't actually there. This is called Tartini's Third.

Usually, scientists say this happens because the ear is "non-linear" (it distorts sound like a broken speaker).

The Paper's Solution:
The authors show you don't need a broken speaker. You just need the Energy rule again.

• When two sounds hit the strings, the energy of the strings interacts.
• Mathematically, when you square the vibration (to get energy), the two frequencies mix together.
• This mixing naturally creates a new frequency: The difference between the two notes.
• So, if you play 300Hz and 400Hz, the energy calculation naturally highlights the 100Hz difference. The brain hears the "ghost" note.

Why This Matters

This paper is beautiful because it simplifies a complex problem.

1. It's Linear: The ear's mechanical parts (the strings) can be perfectly simple and linear (like a spring).
2. It's Non-Linear in Perception: The "magic" happens because the brain calculates Energy (which is a squared, non-linear math operation).

The Takeaway:
We don't need to assume the ear is a chaotic, broken machine to explain why we hear minor chords or phantom notes. We just need to realize that the ear is a sophisticated energy detector. By measuring the intensity of the vibrations rather than just the movement, our brain naturally constructs the "sub-harmonic" world that makes music sound so rich, deep, and emotional.

In short: The music isn't just in the sound waves; it's in the energy they leave behind.

Technical Summary

1. Problem Statement

The paper addresses two long-standing mysteries in psychoacoustics and music theory:

• The Sub-harmonic Series: Historically hypothesized by Zarlino (16th century) to explain the consonance of the minor chord, the sub-harmonic series consists of frequencies $F/n$ (where $F$ is the fundamental). However, mathematically, a standard Fourier decomposition of a harmonic sound contains no frequencies below the fundamental. The question remains: how does the auditory system perceive these "undertones" if they are not physically present in the sound wave?
• Tartini's Third Sound (Combination Tone): When two tones $F_1$ and $F_2$ are played simultaneously, listeners often perceive a third tone at frequency $F_2 - F_1$. While traditionally attributed to nonlinearities in the inner ear (specifically outer hair cells), the paper investigates whether this can be explained within a linear framework.

The core challenge is to explain the emergence of these phenomena using a linear model of the cochlea, challenging the prevailing view that such effects strictly require nonlinear mechanical amplification.

2. Methodology

The authors propose a simplified mathematical model of the cochlea based on the following assumptions:

• Physical Model (Linear): The basilar membrane is modeled as a collection of non-interacting, damped vibrating strings. Each string corresponds to a specific location $x$ along the cochlea (from base to apex) and has a specific resonance frequency. The dynamics are governed by the linear damped wave equation:
  $$\rho(x) u_{tt} = T(x) u_{zz} - \gamma(x) u_t + c(x)F(t)$$
  where $u$ is displacement, $\rho$ is linear mass, $T$ is tension, $\gamma$ is damping, and $F(t)$ is the external sound force.
• Key Hypothesis (The Information Signal): The authors introduce a crucial hypothesis regarding what information is transmitted from the cochlea to the auditory cortex. Instead of transmitting the displacement $u(t)$ or velocity, they hypothesize that the signal is the total energy $E(x, t)$ stored in each string (kinetic + potential energy).
  $$E(x, t) = \int_0^{\ell(x)} \left( \frac{1}{2}\rho(x) u_t^2 + \frac{1}{2}T(x) u_z^2 \right) dz$$
  This introduces a nonlinearity at the level of signal processing (energy is quadratic in displacement), even though the mechanical system itself is linear.
• Oscillation Modes: Unlike models that only consider the fundamental mode of vibration, this model accounts for all oscillation modes (harmonics) of the vibrating strings. A string with fundamental resonance $\xi$ also has modes at $2\xi, 3\xi, \dots$.

3. Key Contributions and Theoretical Derivations

A. Emergence of the Sub-harmonic Series

The authors demonstrate that when a harmonic sound of frequency $F$ (containing harmonics $F, 2F, 3F, \dots$) excites the cochlea:

1. Resonance Matching: The string with fundamental resonance $F$ vibrates in its fundamental mode.
2. Higher Mode Activation: Crucially, the string with fundamental resonance $F/2$ is excited in its second mode (which vibrates at $2 \times (F/2) = F$). Similarly, the string with resonance $F/3$ vibrates in its third mode.
3. Energy Transmission: Since the transmitted signal is the energy of the string, the auditory cortex receives a strong signal from strings located at positions corresponding to $F, F/2, F/3, \dots$.
4. Result: The energy distribution $E(x, t)$ exhibits peaks not only at the harmonic frequencies but also at the sub-harmonic frequencies ($F/n$).
   • Note: Due to the symmetry of the forcing function (assuming uniform force across the string), even sub-harmonics ($F/2, F/4$) are mathematically suppressed in this specific model, leaving only odd sub-harmonics ($F, F/3, F/5$). This is significant because the first five odd sub-harmonics of a root note generate an inversion of a minor triad, providing a physical explanation for the consonance of the minor chord.

B. Explanation of Tartini's Third Sound

When two frequencies $F_1$ and $F_2$ are input:

1. The displacement $u(t)$ is a linear sum of the responses to $F_1$ and $F_2$.
2. However, the energy $E(t) \propto u(t)^2$ contains cross-terms (interference).
3. Expanding the square of the sum of two sinusoids generates terms with frequencies: $2F_1, 2F_2, F_1+F_2$, and $|F_2 - F_1|$.
4. Result: The quadratic nature of the energy function naturally produces the combination tone $F_2 - F_1$ within the energy signal sent to the cortex, without requiring nonlinear mechanics in the basilar membrane itself.

4. Results

• Numerical Simulations: The authors performed simulations using parameters derived from Nobili et al. (2003) and modified parameters to fit the human audible range (20Hz–20kHz).
• Sub-harmonic Peaks: The simulations of the energy function $E(\xi, t)$ clearly show peaks at the fundamental frequency and at sub-harmonic frequencies ($F/3, F/5$, etc.), confirming the theoretical prediction.
• Minor Chord Consonance: The model successfully generates the specific frequency ratios associated with the minor triad (e.g., for a root C, the sub-harmonics produce F, A♭, C), supporting Zarlino's historical hypothesis.
• Combination Tones: The energy spectrum of two simultaneous tones explicitly shows the difference frequency component, validating the mechanism for Tartini's third sound.

5. Significance and Conclusion

• Paradigm Shift: The paper challenges the necessity of complex nonlinear mechanical models (like Hopf bifurcations in outer hair cells) to explain sub-harmonics and combination tones. It suggests these phenomena can arise from a linear mechanical system coupled with a quadratic (energy-based) readout mechanism.
• Historical Validation: It provides a rigorous mathematical justification for hypotheses made by Zarlino and contemporaries in the 16th and 17th centuries regarding the "undertone series."
• Computational Efficiency: By keeping the mechanical model linear, the mapping from input sound to cortical signal can be written in closed form, making the model computationally efficient and analytically tractable compared to fully nonlinear active models.
• Future Directions: The authors suggest extending the model to include interactions between strings and feedback loops (where the energy signal modulates the endolymph), which could further refine the prediction of combination tones.

In summary, the paper resolves the "mystery" of the sub-harmonic series by proposing that the auditory system perceives the energy distribution across the basilar membrane's higher vibration modes, a mechanism that naturally generates both the minor chord's undertones and combination tones through simple quadratic nonlinearity in signal processing.