Musical Pitch Cents and Frequency Ratio
Measures logarithmic microtonal pitch intervals where an octave (ratio 2:1) equals exactly 1200 cents and one semitone equals 100 cents.
\[ C = 1200 \cdot \log_2\left( \frac{f_2}{f_1} \right) = \frac{1200}{\log_{10}(2)} \cdot \log_{10}\left( \frac{f_2}{f_1} \right) \approx 3986.31 \cdot \log_{10}\left( \frac{f_2}{f_1} \right) \]
Plain Text Formula:
C = 1200 * log2(f2 / f1) = 3986.31 * log10(f2 / f1)
📐 Variable Definitions & Parameters
| Symbol | Variable Name | Standard Unit | Physical Definition |
|---|---|---|---|
| \[ C \] | Musical Interval | cents (¢) | Logarithmic microtonal distance (100 cents = 1 semitone) |
| \[ f_2 / f_1 \] | Frequency Ratio | ratio | Relative frequency ratio between upper and lower pitch |
| \[ 1200 \] | Octave Constant | cents/octave | Alexander Ellis scale definition constant |
💡 Worked Engineering Example
Problem Statement:
Find the difference in cents between a pure just-intonation fifth (ratio 3:2 = 1.500) and an equal-tempered fifth (700 cents).
Step-by-Step Solution:
- C_pure = 1200 * log2(1.5) = 1200 * 0.58496 = 701.955 cents
- Delta = 701.955 - 700.000 = +1.955 cents
🎯 Result: A pure fifth is approximately 2.0 cents wider than an equal-tempered fifth.