Geometric Mean vs. Arithmetic Mean in Bandpass Filters

[JJBladester]

Why is the geometric mean used to define the center frequency of a bandpass filter instead of the arithmetic mean?

I read in this book that

1. All the lowpass elements yield LC pairs that resonate at ω = 1.
2. Any point of the lowpass response is transformed into a pair of points of the bandpass filter. The frequencies of the pair of points are reciprocals. This means that, after frequency scaling, we can write

$$f_{0}=\sqrt{f_{1}f_{2}}$$

where f₁ and f₂ are the scaled frequncies of the transforms of a single LP point, and f₀ was scaled from ω = 1. This effect tells us that the bandpass filter has geometric-mean symmetry.

I get that you need to use the geometric mean when multiplying (scaling) elements but I'm still having a hard time seeing why we can't just use the arithmetic mean in the case of defining the bandpass filter center frequency.

 

[NascentOxygen]

I'll try a qualitative justification. If you plot the separate responses (dB vs. frequency) on a logarithmic frequency scale (the x-axis), the centre of the response does indeed seem to be about midway. But I said this is when plotted on a logarithmic scale, so using the arithmetic mean would miss the mark by a long way.

To throw in some figures. Suppose we have one corner frequency at 1Hz, and another at 100Hz. On a log paper plot, midway between these values corresponds to 10Hz and that's about where the response peaks. The response certainly does not peak near 50Hz.