Logarithmic Frequency Perception

How We Perceive Frequency: The Logarithmic Nature of Hearing

1. How Our Hearing Perceives Frequencies

The human ear does not perceive pitch and frequency linearly, but logarithmically. This means that we do not experience frequency changes as equal absolute steps, but rather as **ratios** (relative differences).

For example:

• The jump from **440 Hz (A4)** to **880 Hz (A5)**—a one-octave increase—is perceived as being the same size as the jump from **220 Hz (A3)** to **440 Hz (A4)**, even though the absolute frequency difference in the first case is 440 Hz, and in the second case only 220 Hz.
• Why? Because both are a **doubling** of frequency—a ratio of 2:1—and our ears perceive this ratio, not the absolute number.

2. Linear vs. Logarithmic Perception

If our ears worked linearly, then a constant frequency increase would result in equal perceived pitch steps. For instance:

• From **220 Hz to 320 Hz** is a 100 Hz increase.
• From **320 Hz to 420 Hz** is also a 100 Hz increase.

If our hearing were linear, we would perceive both jumps as equal. But we don’t. Instead, we hear the jump from **220 Hz to 320 Hz** as *larger* than the jump from **320 Hz to 420 Hz**. Why?

Because the perceived change depends on the **ratio**, not the absolute difference:

• 320 / 220 ≈ **1.45**
• 420 / 320 ≈ **1.31**

Our hearing interprets the **larger ratio** (1.45) as a bigger step in pitch than the smaller ratio (1.31), even though both differences are 100 Hz.

3. Equal Step Sizes in Musical Scales

In Western music, we want each step between two notes in a scale to *sound* equally spaced. Since our perception is logarithmic, this means that the actual frequencies must increase **exponentially**.

That’s why the interval between two adjacent notes (e.g., a semitone) corresponds to a **constant multiplication factor**, not a fixed frequency increment.

The 12th Root of 2

• One octave spans 12 semitones and doubles in frequency.
• Therefore, each semitone increases the frequency by a factor of:

$2^{1/12}\approx 1.059463$

So:

• Starting at 440 Hz, the next semitone is 440 × 1.059463 ≈ **466.16 Hz**, and so on.

If frequencies were spaced linearly (e.g., every note 100 Hz apart), the steps would **not** sound equal. Higher notes would sound increasingly farther apart, and lower notes would seem compressed—a very unnatural musical experience.

4. Summary

• Logarithmic perception: Our ears perceive frequency differences **relatively**, not absolutely.
• Exponential frequency steps: To produce equal steps in pitch, frequencies must increase by a **constant ratio**.
• Practical example: A jump from **220 Hz to 440 Hz** sounds equal to a jump from **440 Hz to 880 Hz**, even though the second is a much larger absolute increase.

This logarithmic model is the foundation of the **equal temperament tuning system** used in modern music—and it matches how we naturally hear and understand pitch.