Linear and Logarithmic Frequency Scales

When we talk about Linear and Logarithmic frequency scales (not to be confused with fractional octave banding) we are really just talking about how frequencies are plotted on charts and graphs. On a linear frequency scale, let’s say every 100 Hertz (you can pick any number), occupies the same amount of space on the chart as every other. On an octave scale, each octave is the same width as every other, even though the linear frequency range for each band doubles as you ascend in frequency (125 Hz, 250 Hz, 500 Hz, 1 kHz, 2 kHz, 4 kHz…). On a logarithmic decade scale, each power of 10 Hz, (10, 100, 1000, 10,000) is the same width as every other. Logarithmic scales work the same way for any base, but the bases we use for log scales in Smaart are two and ten (octaves and decades).

Linear vs Logarithmic frequency scaling. Two views of the same comb filter on a linear and log scaled magnitude graph.

We most often look at frequency on octave or decade scales because these correlates better with our own logarithmic perceptions of sound. However, linear scales are very useful for some things as well, and sometimes correlate better with the underlying physics of sound and acoustics. Charting frequency on a linear scale can make comb filters and harmonic distortion products stand out more clearly since the lobes or peaks are linearly spaced. Another example might be the phase shift associated with a fixed delay, which becomes a straight-line slope on a linear frequency scale.

Note that when you look at FFT data from acoustical measurements or other noisy signals on a log frequency scale, the trace gets fuzzier-looking at higher frequencies. That doesn’t necessarily mean there is more noise in the HF. It is a natural consequence of packing more and more linearly-space FFT points into a smaller and smaller amount of chart space. That is one of the reasons for the MTW transfer function option, as noted earlier. Smoothing also helps to reduce visual noise in the HF in transfer function measurements, as does fractional octave banding for spectrum measurements.

MTW vs 16K FFT transfer function on a logarithmic frequency scale. The MTW uses larger time constants at low frequencies to improve LF resolution while smaller time constants at higher frequencies reduce visual “noise” due to excess resolution.