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If you plotted an audio equalizer on a standard linear ruler from 20 Hz to 20,000 Hz, the entire bass range would be crushed into a microscopic 1-millimeter sliver on the left edge.
To give equal visual weight to every musical octave, audio engineers and exam boards use a logarithmic frequency axis paired with a linear decibel grid.
Why does human hearing require this special mathematical grid in the first place?
The Logarithmic Frequency Axis (X-Axis)
The human ear detects sounds from 20 Hz to 20 kHz across ten octaves, where each octave represents a doubling of frequency (20Β Hzβ40Β Hzβ80Β Hz).
In 1860, psychologist Gustav Fechner formulated the Weber-Fechner Law, proving that human pitch and loudness perception scales proportionally with ratios rather than arithmetic addition.
πInteractive diagram
How do we divide this axis into clear, standardized sections on an exam paper?
Spacing Decades and Intermediate Gridlines
A decade is a ten-fold multiplication of frequency (10Β Hzβ100Β Hzβ1Β kHzβ10Β kHz). Each decade takes up an equal width on the page.
The geometric distance of any frequency f within a decade spanning fstartβ to fendβ=10β
fstartβ is proportional to log10β(fstartβfβ).
For example, the distance from 100Β Hz to 200Β Hz covers log10β(2)β30.1% of the decade's width, while the step from 800Β Hz to 900Β Hz occupies only log10β(9/8)β5.1%.
This causes intermediate division lines to compress toward the upper end of each decade. For example, moving from 100Β Hz to 200Β Hz occupies log10β(2)β30.1% of the decade width, while 800Β Hz to 900Β Hz takes up only log10β(9/8)β5.1%.