A Frequency Ratio to Cents Calculator is a specialized tool for musicians, music theorists, and acousticians that translates the mathematical relationship between two musical pitches into a standard, perceptually uniform unit of measurement called the "cent." In music, the interval between two notes is defined by the ratio of their frequencies. This calculator takes that frequency ratio and converts it into cents, where 100 cents is equal to one equal-tempered semitone (the distance between two adjacent keys on a piano). This conversion is essential for analyzing tuning systems, comparing different musical intervals, and for the precise calibration of electronic instruments, as it provides a standardized way to talk about pitch differences.
formula of Frequency Ratio To Cents Calculator
The formula to convert a frequency ratio into cents uses a base-2 logarithm, which is perfectly suited for musical octaves.
Cents = 1200 * log₂(Frequency Ratio)
Where:
- Cents: The final value of the musical interval.
- 1200: This constant is used because there are 12 semitones in an octave, and each semitone is defined as 100 cents (12 * 100 = 1200).
- log₂: The base-2 logarithm function. This function answers the question: "To what power must I raise the number 2 to get my frequency ratio?"
- Frequency Ratio: The ratio of the higher frequency to the lower frequency (f₂ / f₁).
Common Musical Interval Ratios and Their Cent Values
This table shows the simple integer frequency ratios for several key intervals in Just Intonation (a tuning system based on these pure ratios) and their corresponding values in cents.
| Musical Interval | Just Intonation Ratio | Value in Cents (approx.) |
| Unison | 1/1 | 0 cents |
| Minor Third | 6/5 | 316 cents |
| Major Third | 5/4 | 386 cents |
| Perfect Fourth | 4/3 | 498 cents |
| Perfect Fifth | 3/2 | 702 cents |
| Major Sixth | 5/3 | 884 cents |
| Octave | 2/1 | 1200 cents |
Example of Frequency Ratio To Cents Calculator
Let's calculate the cent value for the musical interval of a perfect fifth, which has a pure frequency ratio of 3/2.
Step 1: Determine the frequency ratio.
Frequency Ratio = 3 / 2 = 1.5
Step 2: Apply the formula.
Cents = 1200 * log₂(Frequency Ratio)
Cents = 1200 * log₂(1.5)
Step 3: Calculate the base-2 logarithm.
Using a scientific calculator, we find that log₂(1.5) is approximately 0.58496.
Step 4: Complete the calculation.
Cents = 1200 * 0.58496
Cents ≈ 701.95 cents
Therefore, the musical interval of a perfect fifth is approximately 702 cents.
Most Common FAQs
A cent is a logarithmic unit of measure used for musical intervals. The modern Western tuning system, called equal temperament, divides an octave into 12 equal semitones (like the distance from one piano key to the next). The cent system further divides each of those semitones into 100 equal parts. This provides a very fine, standardized scale for measuring and comparing pitch differences.
The base-2 logarithm is used because the fundamental musical interval, the octave, corresponds to a doubling of frequency (a frequency ratio of 2/1). The log₂ of 2 is exactly 1. By using a base-2 logarithm, the formula is naturally scaled to the structure of the musical octave, making the subsequent multiplication by 1200 (for the 1200 cents in an octave) a direct and meaningful conversion.
Just Intonation is a tuning system that uses simple whole number frequency ratios (like 3/2 for a perfect fifth) to create pure, consonant-sounding intervals. Equal Temperament is a tuning system that slightly adjusts these pure intervals so that all 12 semitones in an octave are exactly the same size (exactly 100 cents). While equal temperament's intervals are not perfectly pure, it allows music to be played in any key without retuning the instrument, which is why it has become the standard for most Western music.