Two-Tube Approximations of the Vocal Tract

Simplified models using two uniform tubes connected in series provide powerful insights into how vocal tract shape affects formant frequencies. These models explain why different vowels have characteristic formant patterns and how articulatory changes shift formant frequencies.

The [a] Vowel: Pharyngeal Constriction

Figure 6.9: (a) Two-tube approximation for an [a] vowel, (b) standing-wave pressure pattern for F₁, and (c) standing-wave pressure pattern for F₂. Dashed lines are for a uniform tube of equal length.

Assume that the [a] shape can be approximated by two uniform tubes:

• Narrow tube: Representing the pharynx
• Wider tube: Representing the mouth

The 90° bend in the vocal tract is eliminated for simplicity.

First Formant (F₁) Effects

The standing wave pressure pattern for F₁ shows:

1. Reduced pressure in mouth: The wider mouth tube has lower acoustic pressure than the pharynx
2. Virtual zero shifts left: The point where pressure crosses zero moves toward the pharynx
3. Acoustic length decreased: The tube acts shorter than its physical length
4. F₁ increases: Shorter acoustic length → higher frequency

Result: F₁ is raised from the neutral position (500 Hz → ~700-900 Hz)

Second Formant (F₂) Effects

The situation for F₂ is different:

1. Pharynx dominates: Overall pressures are much higher in the narrow pharynx
2. Pressure node shifts right: The zero-crossing point moves toward the tube junction
3. Acoustic length increased: The pharynx tube acts longer than its physical length
4. F₂ decreases: Longer acoustic length → lower frequency

Result: F₂ is lowered from the neutral position (1500 Hz → ~1000-1300 Hz)

Spectral Compacting

The combined raising of F₁ and lowering of F₂ for the [a] vowel creates a compact spectrum—acoustic energy is concentrated in a narrow frequency range. F₁ and F₂ move toward each other, almost becoming the same formant.

Physical interpretation: Pharyngeal narrowing of the vocal tract results in F₁ and F₂ moving toward each other. In the limit, if the mouth tube had infinite area, only the pharynx would remain as a resonator, and the two formants would indeed become one.

The [i] Vowel: Oral Constriction

Figure 6.10: (a) Two-tube approximation for an [i] vowel, (b) standing-wave pressure pattern for F₁, and (c) standing-wave pressure pattern for F₂. Dashed lines are for a uniform tube of equal length.

For the [i] vowel:

• Wide pharynx: Back of vocal tract expanded
• Narrow mouth: Constriction at palate (tongue raised and fronted)

First Formant (F₁) Effects

The first formant pressure pattern shows:

1. Nearly constant in pharynx: Pressure flat across the back cavity
2. Virtual zero far to right: Acoustic zero extends well beyond the physical tube
3. Acoustic length increased: Tube acts longer than physical length
4. F₁ decreases: Longer acoustic length → lower frequency

Result: F₁ is lowered from neutral (500 Hz → ~250-350 Hz)

Second Formant (F₂) Effects

For F₂, the mouth tube dominates:

1. Reduced pharynx pressure: Overall reduction in back cavity pressure
2. Pressure zero shifts toward interface: Node moves left toward the constriction
3. Mouth supports half wavelength: Narrow mouth cavity has two nearly-open ends
4. F₂ increases: Shorter effective length → higher frequency

Result: F₂ is raised from neutral (1500 Hz → ~2200-2500 Hz)

Spectral Diffusion

The combined lowering of F₁ and raising of F₂ creates a diffuse spectrum—acoustic energy is spread over both low and high frequencies. The formants move apart.

Physical interpretation: Mouth narrowing diffuses the vowel spectrum, spreading F₁ and F₂ apart. The key is the opposing manner in which the F₁ and F₂ pressure patterns change from neutral. F₁ pressures become large wherever F₂ pressures are small, and vice versa.

Acoustic Principles Revealed

These two-tube models reveal fundamental acoustic principles:

Principle 1: Constriction Location

• Pharyngeal constriction → F₁ and F₂ move together (compacting)
• Oral constriction → F₁ and F₂ move apart (diffusion)
• Velar constriction → Intermediate effect (requires three-tube model)

Principle 2: Pressure Distribution

• High pressure at constrictions - Opposite to Bernoulli (aerodynamic) pressure
• Low pressure at expansions - Including the open mouth
• Pressure maxima shift - Determine effective acoustic length

Principle 3: Formant Sensitivity

• F₁ sensitive to pharynx/jaw opening - Controlled primarily by jaw height
• F₂ sensitive to tongue position - Controlled by tongue front-back placement
• Different formants dominated by different cavities - Allows independent control

Quantifying the Effects

For a pharyngeal constriction ([a] vowel):

• Area ratio (mouth/pharynx): ~6:1
• F₁ shift: +200 to +400 Hz
• F₂ shift: -200 to -300 Hz
• Net spectral compacting: ~400-700 Hz

For an oral constriction ([i] vowel):

• Area ratio (pharynx/mouth): ~6:1
• F₁ shift: -150 to -200 Hz
• F₂ shift: +700 to +1000 Hz
• Net spectral diffusion: ~850-1200 Hz

Pedagogical Applications

Understanding these models helps explain:

1. Why jaw opening raises pitch of vowels - Increases F₁ by shortening acoustic length
2. Why tongue fronting brightens vowels - Increases F₂ through oral constriction
3. Why [a] has focused energy - Compact formants create concentrated spectrum
4. Why [i] sounds bright - High F₂ places energy at high frequencies

Limitations of Two-Tube Models

While powerful, two-tube approximations have limitations:

• Cannot model velar constrictions accurately
• Ignore lip rounding effects (need extended tube)
• Assume uniform areas within each tube
• Neglect nasal coupling and side branches

Despite these limitations, two-tube models capture the essential acoustic principles governing formant frequency patterns.

Summary

Two-tube approximations of the vocal tract provide elegant explanations for formant frequency patterns. Pharyngeal constrictions (as in [a]) raise F₁ and lower F₂, creating spectral compacting. Oral constrictions (as in [i]) lower F₁ and raise F₂, creating spectral diffusion. These models reveal that acoustic pressure increases at constrictions and that different formants are dominated by different vocal tract cavities, allowing for independent articulatory control of formant frequencies.

Key Takeaways

• ✅ Two-tube models use one tube for pharynx and one for mouth cavity
• ✅ Pharyngeal constriction raises F₁ and lowers F₂ (spectral compacting)
• ✅ Oral constriction lowers F₁ and raises F₂ (spectral diffusion)
• ✅ Acoustic pressure is high at constrictions, opposite to aerodynamic pressure
• ✅ Pressure node positions determine effective acoustic length of cavities
• ✅ Different formants are dominated by different portions of the vocal tract

Related Topics

• Three-Tube Approximations
• Articulatory and Acoustic Descriptions
• Rules for Modifying Vowels
• Acoustic Impedance and Reflection

Further Reading

1. Fant, G. (1960). Acoustic theory of speech production. The Hague: Mouton.
2. Stevens, K. N. (1998). Acoustic phonetics. MIT Press.
3. Titze, I. R. (2000). Principles of voice production (2nd ed.). National Center for Voice and Speech.