Interval to Cents Calculator

Musical intervals can be measured in cents, a unit that provides precise information about the size of an interval. This calculator converts musical intervals to cents, helping musicians, composers, and music theorists work with exact interval sizes.

What are cents in music?

The cent is a logarithmic unit of measurement used in music to describe the size of intervals. It provides a more precise way to measure intervals than the traditional equal temperament system, which divides the octave into 12 equal semitones.

Key fact: One cent is defined as 1/100th of a semitone, and one semitone is 100 cents. An octave is 1200 cents because it consists of 12 semitones.

Cents are particularly useful in microtonal music, where intervals smaller than a semitone are common. They allow musicians to specify exact interval sizes, which is important for tuning instruments and composing music.

Why use cents instead of semitones?

While semitones provide a useful way to describe intervals in equal temperament, cents offer several advantages:

They provide a more precise measurement of interval size
They allow for the description of intervals smaller than a semitone
They are useful in microtonal music and tuning systems
They help in the precise tuning of instruments

For example, a minor third in 12-tone equal temperament is approximately 300 cents, while a major third is about 400 cents. The exact cent value can vary depending on the tuning system used.

How to convert intervals to cents

Converting musical intervals to cents involves understanding the logarithmic relationship between frequency ratios and interval sizes. The formula for converting an interval to cents is:

cents = 1200 × log₂(f₂/f₁)

Where:

f₂ is the frequency of the higher note
f₁ is the frequency of the lower note
log₂ is the logarithm base 2

This formula works because the human ear perceives pitch differences logarithmically. The factor of 1200 ensures that an octave (which has a frequency ratio of 2:1) equals exactly 1200 cents.

Step-by-step conversion process

Determine the frequencies of the two notes in the interval
Calculate the ratio of the higher frequency to the lower frequency (f₂/f₁)
Take the base-2 logarithm of this ratio
Multiply the result by 1200 to get the interval size in cents

For example, to convert a perfect fifth (frequency ratio of 3:2) to cents:

cents = 1200 × log₂(3/2) ≈ 1200 × 0.58496 ≈ 702 cents

This means a perfect fifth is approximately 702 cents.

Examples and applications

Here are some common musical intervals and their cent values in equal temperament:

Interval	Frequency Ratio	Cents
Minor second	16/15	100
Major second	9/8	200
Minor third	6/5	300
Major third	5/4	400
Perfect fourth	4/3	500
Tritone	7/5	600
Perfect fifth	3/2	700
Minor sixth	8/5	800
Major sixth	5/3	900
Minor seventh	9/5	1000
Major seventh	15/8	1100

These cent values are useful for:

Tuning instruments to specific intervals
Composing music with precise interval sizes
Understanding the differences between tuning systems
Analyzing the harmonic structure of music

Microtonal music applications

In microtonal music, which uses intervals smaller than a semitone, cents are essential. For example:

A quartertone (1/4 of a semitone) is 25 cents
A third of a semitone is 33.33 cents
A half semitone is 50 cents

These precise measurements allow composers to create music with intervals that are not possible in traditional 12-tone equal temperament.

FAQ

What is the difference between cents and semitones?: Semitones are a traditional unit of measurement in equal temperament music, where an octave is divided into 12 equal semitones. Cents provide a more precise measurement, with 100 cents equal to one semitone. This allows for the description of intervals smaller than a semitone.
Why is an octave 1200 cents?: An octave has a frequency ratio of 2:1. Using the cent formula, log₂(2) equals 1, and multiplying by 1200 gives 1200 cents. This makes the cent system consistent with the octave as the fundamental interval in music.
Can I use this calculator for microtonal music?: Yes, this calculator is particularly useful for microtonal music, as it allows you to specify intervals smaller than a semitone. You can input the exact frequencies of the notes to get their cent value.
How accurate are the cent values in this calculator?: The calculator uses precise mathematical calculations to determine cent values. The results are accurate to several decimal places, providing a reliable reference for musical intervals.
Can I use this calculator for tuning instruments?: Yes, the cent values provided by this calculator can be used to tune instruments to specific intervals. By comparing the calculated cent values with the desired interval size, you can adjust the instrument's tuning accordingly.