Interval Notation Calculator Music: What It Does

An interval notation calculator for music works out the distance between two notes in semitones and then names that interval (for example, major third, perfect fifth). You enter two note names or two frequencies, and it returns the interval name, the number of semitones, and often the frequency ratio. This page provides a simple calculator that converts a frequency ratio into cents and semitones, plus the background you need to understand musical intervals.

Formula

The core formula for converting a frequency ratio to cents is:

cents = 1200 × log₂(f₂ / f�)

where:

• fâ‚� = the lower frequency (Hz)
• fâ‚‚ = the higher frequency (Hz)
• logâ‚‚ = logarithm base 2

To get semitones, divide the cents by 100 (since one equal-tempered semitone = 100 cents). The frequency ratio is simply f₂ / f�.

Worked Example

Using the default values: f� = 440 Hz (A4) and f₂ = 523.25 Hz (C5, the note a minor third above A4 in equal temperament).

1. Calculate the ratio: 523.25 / 440 = 1.1892
2. Compute the base-2 logarithm: log₂(1.1892) ≈ 0.2500
3. Multiply by 1200 to get cents: 1200 × 0.2500 = 300.0 cents
4. Divide by 100 to get semitones: 300 / 100 = 3.00 semitones

The result is 300 cents, which is exactly 3 semitones – a minor third in equal temperament. The calculator rounds to 1 decimal for cents, 2 for semitones, and 4 for the ratio.

What Is a Musical Interval?

In music, an interval is the distance between two notes. It is measured by the ratio of their frequencies, not by a simple subtraction of hertz. For example, the difference between 440 Hz and 880 Hz is 440 Hz, but the interval is an octave because the ratio is 2:1. The same interval (an octave) exists between 220 Hz and 440 Hz, even though the hertz difference is only 220 Hz.

Intervals are named by their size in scale steps (e.g., second, third, sixth) and quality (perfect, major, minor, augmented, diminished). The most common system in modern music is equal temperament, which divides the octave into 12 equal semitones. Each semitone is exactly 100 cents, so an octave is 1200 cents.

How the Calculator Works

This calculator takes two frequencies and computes the interval in cents and semitones using the logarithmic formula above. It also shows the raw frequency ratio. The math is the same regardless of whether you enter frequencies in Hz, kHz, or any other unit – as long as both are in the same unit, the ratio works.

If you want to find the interval between two named notes (like C and E), you would first need to convert those notes to frequencies. Many online tools can do that, or you can use the standard A4 = 440 Hz reference and the formula for equal temperament: f = 440 × 2^(n/12), where n is the number of semitones above A4.

Common Mistakes and Misconceptions

• Using linear subtraction: Subtracting frequencies gives a hertz difference, not a musical interval. Always use a ratio.
• Forgetting to use the same unit: If one frequency is in Hz and the other in kHz, the ratio will be off by a factor of 1000. Convert first.
• Assuming all intervals are equal-tempered: The calculator assumes equal temperament. In just intonation, the same note names can have slightly different frequencies, and the cents values will differ.
• Confusing cents with semitones: 100 cents = 1 semitone. A whole tone is 200 cents, a perfect fifth is 700 cents (7 semitones).
• Rounding too early: If you round the ratio before taking the logarithm, you can introduce errors. Keep several decimal places until the end.

When to Use an Interval Calculator

Musicians, composers, and audio engineers use interval calculators in several situations:

• Tuning instruments: Checking if a note is sharp or flat relative to a reference.
• Transcribing music: Identifying intervals between notes in a recording.
• Sound design: Creating specific beat frequencies or harmonic ratios.
• Music theory study: Visualizing the size of intervals in cents.
• Electronic music production: Setting oscillator frequencies to create chords or melodies.

For example, if you want to set a detuned oscillator to be 10 cents sharp, you can calculate the required frequency: f₂ = f� × 2^(10/1200) ≈ 1.00579 × f�.

Limitations of the Calculator

This calculator only works with frequencies. It does not name intervals like “major third� or “perfect fifth� – that requires knowing the note names and the key. To get the interval name, you need to map the semitone distance to a scale degree and quality. For instance, 3 semitones is a minor third, 4 semitones is a major third, and 7 semitones is a perfect fifth.

Also, the calculator assumes equal temperament. If you are working with just intonation or other tuning systems, the cents values will not match the standard names. For example, a just major third has a ratio of 5:4, which is 386.3 cents, not 400 cents as in equal temperament.

Frequently Asked Questions

How do I convert a frequency ratio to cents?

Use the formula: cents = 1200 × log₂(ratio). For example, a ratio of 2:1 (octave) gives 1200 cents, and a ratio of 3:2 (perfect fifth) gives about 702 cents.

What is the difference between cents and semitones?

A cent is one hundredth of a semitone. So 100 cents = 1 semitone, 200 cents = 2 semitones, and so on. The calculator gives both, so you can use whichever is more convenient.

Can I use this calculator for any two frequencies?

Yes, as long as both frequencies are positive numbers in the same unit. It works for any audible or inaudible frequencies, though musical intervals are typically between 20 Hz and 20,000 Hz.

Why does the calculator show a ratio but not the interval name?

Interval names depend on the musical context (key, scale, note names). The calculator only computes the mathematical size. To get a name, you need to know the note names and the scale system.

Is this calculator accurate for microtonal music?

Yes, it gives precise cents values, which is exactly what microtonal musicians need. However, the standard interval names may not apply to microtonal intervals that fall between semitones.

What if I enter the frequencies in reverse order?

The calculator will give a negative value for cents and semitones, indicating a descending interval. The absolute value is the size of the interval, but the sign tells you the direction.

Related

• Final Exam Grade Calculator for High School
• Final Grade Calculator After Exam: What You Need
• Final Grade Calculator After Final: How It Works

in f1 | Frequency 1 | Hz | 440 | 0
in f2 | Frequency 2 | Hz | 523.25 | 0
out cents | Interval size | cents | 1200 * log2(f2 / f1) | 1
out semitones | Interval size | semitones | 12 * log2(f2 / f1) | 2
out ratio | Frequency ratio | - | f2 / f1 | 4