Frequency Ratio to Cents Converter – Online Microtonal Tool

A cent is a logarithmic unit of measure for musical intervals. One octave is divided into 1200 cents, so each semitone in 12-tone equal temperament equals exactly 100 cents. Cents allow precise comparison of intervals across different tuning systems. The formula to convert a frequency ratio r to cents is: cents = 1200 × log₂(r) = 1200 × ln(r) / ln(2). This tool performs this conversion instantly, making it invaluable for microtonal composers, instrument builders, and tuning theorists.

Use the Ratio → Cents tab. Enter the numerator and denominator of your frequency ratio (e.g., 3 and 2 for a perfect fifth, ratio 3:2 = 1.5). The tool instantly computes the equivalent cents value using the formula cents = 1200 × log₂(ratio). For example, 3:2 yields 701.955 cents. You can also enter decimal values by using 1 as the denominator and the decimal as the numerator.

Just Intonation (JI) uses frequency ratios based on small whole numbers (like 3:2, 5:4) derived from the harmonic series. These intervals are acoustically pure but can create tuning inconsistencies across keys. Equal Temperament (12-EDO) divides the octave into 12 equal semitones of 100 cents each, making all keys equally usable but slightly compromising the purity of intervals. For instance, a JI perfect fifth (3:2) is 701.955 cents, while an equal-tempered fifth is exactly 700 cents — a difference of about 2 cents that trained ears can perceive.

When a frequency ratio exceeds 2:1 (or cents exceed 1200), the interval spans more than one octave. Octave decomposition separates the interval into whole octaves plus the remaining interval within a single octave. For example, a ratio of 3:1 = 1901.955 cents decomposes into 1 octave (1200¢) + 701.955¢ (a perfect fifth). This helps musicians understand compound intervals and simplifies microtonal analysis by focusing on the octave-reduced form.

This tool uses JavaScript's double-precision floating-point arithmetic, providing accuracy to about 15 significant digits. Cents are displayed to 3 decimal places (0.001 cents), which is far more precise than human pitch discrimination (typically around 5-6 cents for trained listeners). For all practical microtonal composition, tuning, and analysis purposes, this precision is more than sufficient.

Yes! Negative cents represent descending intervals. For example, −1200 cents equals a descending octave (frequency ratio 1:2 = 0.5), and −701.955 cents equals a descending perfect fifth (ratio 2:3 ≈ 0.6667). The tool handles negative values correctly, including octave decomposition for negative intervals.

This converter is essential for: microtonal composers designing scales with specific frequency ratios; instrument makers calculating fret or hole placements; electronic musicians programming synthesizer pitch bends; tuning theorists comparing historical temperaments; sound designers creating precise harmonic relationships; and educators teaching the mathematics of musical intervals. It bridges the gap between acoustic physics and musical practice.