Octave & Decade Frequency Spans
octaves = log2(f2/f1), decades = log10(f2/f1)
Quick answer
This calculator solves octaves = log₂(f₂/f₁), decades = log₁₀(f₂/f₁) for octaves. With Lower Frequency 1 kHz, Upper Frequency 2.2 kHz, the result is 1.138.
Calculator
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.
Symbols and units
| Symbol | Meaning | Default unit | Accepted units |
|---|---|---|---|
| f1 | Lower Frequency | Hz | Hz, kHz, MHz, GHz |
| f2 | Upper Frequency | Hz | Hz, kHz, MHz, GHz |
Step by step
- 1. What you know: Lower Frequency 1 kHz, Upper Frequency 2.2 kHz
- 2. The relation: octaves = log₂(f₂/f₁), decades = log₁₀(f₂/f₁)
- 3. Substitute: f1 = 1 kHz, f2 = 2.2 kHz
- 4. Result: octaves = 1.138
Worked examples
Minimum range
Given: Lower Frequency 5 Hz, Upper Frequency 11 Hz
Result: octaves = 1.138
Low range
Given: Lower Frequency 50 Hz, Upper Frequency 110 Hz
Result: octaves = 1.138
Typical
Given: Lower Frequency 1 kHz, Upper Frequency 2.2 kHz
Result: octaves = 1.138
High range
Given: Lower Frequency 20 kHz, Upper Frequency 44 kHz
Result: octaves = 1.138
Maximum range
Given: Lower Frequency 0.2 MHz, Upper Frequency 0.44 MHz
Result: octaves = 1.138
How octaves changes with Lower Frequency
| Lower Frequency (input) | octaves (result) |
|---|---|
| 100 Hz | 4.459 |
| 0.25 kHz | 3.138 |
| 0.5 kHz | 2.138 |
| 0.75 kHz | 1.553 |
| 1 kHz | 1.138 |
| 1.5 kHz | 0.5525 |
| 2 kHz | 0.1375 |
| 4 kHz | -0.8625 |
| 7 kHz | -1.67 |
| 10 kHz | -2.184 |
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<p><a href="https://vbatpower.com/formulas/octave-decade/">Octave & Decade Frequency Spans</a> Calculator by vbatpower.com</p>Frequently asked questions
How do you calculate Octave & Decade Frequency Spans?
Use octaves = log₂(f₂/f₁), decades = log₁₀(f₂/f₁). With Lower Frequency 1 kHz, Upper Frequency 2.2 kHz, octaves works out to 1.138.
How do you calculate octave vs decade?
Rearrange for octaves: with Lower Frequency 50 Hz, Upper Frequency 110 Hz, octaves is 1.138.
How do you calculate pi filter db/octave?
Rearrange for octaves: with Lower Frequency 1 kHz, Upper Frequency 2.2 kHz, octaves is 1.138.
What is the formula for Octave & Decade Frequency Spans?
The formula is octaves = log₂(f₂/f₁), decades = log₁₀(f₂/f₁). Octaves and decades are logarithmic measures of frequency span used extensively in filter design, audio engineering, and Bode plot analysis.
Related Concepts
All Signal & dB formulas →Need more features?
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