Octave & Decade Frequency Spans

Formula

Description

Octaves and decades are logarithmic measures of frequency span used extensively in filter design, audio engineering, and Bode plot analysis. An octave is a doubling of frequency (ratio of 2:1), while a decade is a 10:1 ratio. Filter rolloff is typically specified as dB/octave or dB/decade: a first-order filter rolls off at 6 dB/octave (20 dB/decade), and each additional order adds another 6 dB/octave. The forward result returns the span in octaves; divide by 3.322 to get decades.

Variables

• f₁ — Lower frequency (Hz)
• f₂ — Upper frequency (Hz)

Practical Notes

Conversions: 1 decade = 3.322 octaves, 1 octave = 0.301 decades. The audio range (20 Hz to 20 kHz) spans about 10 octaves or 3 decades. A Bode plot typically covers 3-5 decades. Common filter specs: 2nd-order Butterworth = 12 dB/octave = 40 dB/decade. 4th-order = 24 dB/octave = 80 dB/decade.