Semitone Calculator

Enter two frequencies in hertz. The calculator finds the interval with k = 12 × log₂(f2/f1), plus cents and the ratio f2/f1. 440 and 880 Hz give exactly 12 semitones (an octave). 440 and 660 Hz, a just fifth 3:2, give about 7.02, not 7. The nearest equal-tempered interval name comes from that same pair.

Inputs

Enter two frequencies in hertz. Cents, the ratio, and the interval name are results from that pair.

Result

Enter both frequencies. The result shows up here.

Raw quotient: frequency ratio. Cents from a tuner: cents converter. A hertz shift: transposition. Start from A4: note frequency.

How it works

440 Hz and 880 Hz → 12.00
f1440
f2880
semitones12.00

In equal temperament the semitone count between two tones is twelve times the base-2 log of f2/f1. An octave, ratio 2, gives k = 12. A unison, f2 = f1, gives 0. Two octaves, 110 Hz to 440 Hz, give 24. That is the same grid that multiplies by 2^(k/12) on the transposition page.

A just fifth 3:2 does not land on 7. 660/440 = 1.5, and 12 × log2(1.5) ≈ 7.02 semitones, about 702 cents. The 12-TET fifth is 2^(7/12) and about 659.26 Hz from A4. The pair 440 and 659.26 returns 7.00. The gap of about 2 cents is what lets twelve fifths close the octaves.

When f2 sits below f1, k is negative. 880 Hz down to 440 Hz is minus 12, not a form mistake. 440 and 415.30 Hz give −1.00, a semitone down. Both frequencies must be positive: a log of zero or a negative has no place here.

Cents are semitones times 100. The cents result uses one decimal place, and the semitone result uses two. 440 and 466.16 Hz is 1.00 semitone, one hundred cents. This is not a microphone tuner. The calculator uses two entered numbers.

Enter f1, then f2, and click Calculate. A comma and a period both work in either field. If you want the raw quotient without a logarithm, open the frequency ratio calculator. If a tuner already gave cents, convert them on the cents calculator.

To go the other way, take k and a base on transposition: 440 Hz and k = 7 give 659.26 Hz. Or start from A4 on note frequency when the base should stay 440.

The calculator also shows the ratio f2/f1. At 440 Hz and 880 Hz that ratio is 2.00000. The interval name follows the semitones rounded to the nearest equal-tempered step. Twelve semitones is an octave, and seven semitones is a tempered fifth. When the second frequency is lower, the result stays negative and the direction is down.

How to use

  1. Enter f1 in hertz, for example 440.
  2. Enter f2, for example 880 for an octave or 660 for a just fifth.
  3. Click Calculate. 440 and 880 give 12.00 semitones and 1200 cents. 440 and 660 give 7.02 and about 702 cents.
  4. When f2 is lower, the sign is a minus. 880 to 440 is −12.00.
  5. For the quotient without semitones, open frequency ratio. To shift by this k, open transposition.

Formula

k = 12 × log2(f2 / f1)

Cents = k × 100. Ratio = f2 / f1.

Both frequencies are greater than zero. Negative k is correct when f2 sits below f1. The name is the nearest equal-tempered step after rounding k. A cent is one hundredth of that semitone. Alexander John Ellis defined that unit in 1885.

Hz and cents between two pitches

Here k = 12 × log2(f2/f1). 440 and 880 are exactly 12 semitones. 440 and 660, a just fifth 3:2, is 7.02, not 7. Cents = k × 100. Ratio = f2/f1.

Hz
The f1 and f2 pair in hertz. 440 and 880 give k = 12. When f2 sits under f1, k is negative. Both inputs must be positive.
Cents
k × 100 in the result next to the semitones. 12 semitones is 1200 cents. 7.02 from 440/660 is about 702, not a clean tuner 700.
ratio
f2 / f1. At 440 Hz and 880 Hz it is 2.00000. At 440 Hz and 660 Hz it is 1.50000. The quotient without a logarithm is on the frequency-ratio page.
direction
The sign of k. When f2 is higher, the direction is up. When f2 is lower, the direction is down and k is negative. Equal frequencies are the same pitch.

Real-life examples

Example 1

440 Hz and 880 Hz -> 12.00 semitones.

Example 2

440 Hz and 660 Hz -> 7.02 semitones, not 7.

Example 3

220 Hz and 440 Hz -> 12.00 semitones.

Example 4

440 Hz and 440 Hz -> 0.00, the same pitch.

Example 5

440 Hz and 466.16 Hz -> 1.00 semitone.

Example 6

440 Hz and 415.30 Hz -> -1.00 semitone.

Example 7

440 Hz and 659.26 Hz -> 7.00 semitones.

Example 8

110 Hz and 440 Hz -> 24.00 semitones.

Ways to use this calculator

  • You check how many semitones split two analyzer peaks before you draw the interval on a keyboard.
  • You compare a just fifth 3:2 (7.02) with a tempered fifth (7.00 at 659.26 Hz).
  • You flip direction: 880 to 440 confirms minus 12 before you lower an oscillator by an octave.

Related calculators

Frequency quotient f2/f1
f2 / f1 from a hertz pair
Cents to semitones converter
Semitones from cents, 100 cents = 1 semitone
Frequency transposition
New hertz after a shift of semitones
Note hertz from A4 = 440 Hz
Note Hz in equal temperament, A4 = 440
Beat time from BPM
Quarter-note ms from BPM tempo
Bar length from tempo
Seconds of one bar from BPM
Note time from BPM and denominator
Note milliseconds from BPM and denominator
Track time from bars
Seconds and mm:ss from bars and BPM

All music calculators

Frequently asked questions

How many semitones sit between 440 and 880 Hz?

A clean 12.00, or 1200 cents, and the ratio is 2. That is an octave up. 220 to 440 also gives 12.00.

Why are 440 and 660 Hz 7.02, not a flat 7?

660/440 is a just fifth 3:2. In 12-TET a fifth is 2^(7/12), about 659.26 Hz. The gap is about 2 cents.

What is the result at 440 and 659.26 Hz?

7.00 semitones and 700 cents. That is the grid fifth, not 3:2.

What happens when the second frequency is lower?

k comes out negative. 880 Hz to 440 Hz is minus 12. 440 and 415.30 Hz is −1.00.

How many semitones between 440 and 466.16 Hz?

1.00, one hundred cents. 466.16 Hz is A4 plus one semitone in 12-TET.

What is 110 Hz to 440 Hz?

24.00 semitones, two octaves. 440/110 = 4, and 12 × log2(4) = 24.

Do 440 and 440 give zero?

Yes, 0.00 semitones and 0 cents. A unison, the log of 1.

Where is the raw quotient without a log?

On the frequency-ratio page. There 660/440 stays 1.5000.

How do I turn these semitones into new Hz?

Take k and a base on transposition. 440 and 7 give 659.26 Hz.

Does a comma in 466.16 work?

Yes. 466,16 and 466.16 are the same f2. Against 440 the result is 1.00.

Where does the interval name come from?

The calculator rounds the semitone count to the nearest equal-tempered step and gives that step a name. Twelve semitones is an octave, and seven semitones is a tempered fifth. When the result does not land on a grid step, the name sits next to an offset in cents. Up or down follows the sign of the result.

How do I calculate semitones between two frequencies?

Use k = 12 × log₂(f2/f1). At 440 Hz and 880 Hz: 12 × log₂(2) = 12.00. Cents are k × 100, and the ratio is f2/f1.

How many semitones sit between 220 Hz and 440 Hz?

Exactly 12.00 — one octave up, the same as 440 to 880. The ratio is 2, and cents are 1200.

How many cents is 440 Hz → 442 Hz?

k = 12 × log₂(442/440) ≈ 0.0785 semitone, about 7.85 cents. That is a small concert-pitch drift, not a full semitone.

How is this different from the cents-to-semitones converter?

Here you enter two hertz values and get k from a logarithm. The cents converter divides ready tuner cents by 100 and does not ask for f1 or f2.

How is this different from frequency transposition?

Transposition takes one f1 and a semitone count k, then finds new f2 = f1 × 2^(k/12). Here both frequencies are inputs, and k is the result.

Why can the result be a fraction instead of a whole number?

Because a Hz pair rarely lands exactly on the 12-TET grid. 440 and 660 give ≈ 7.02. The interval name uses the nearest step after rounding k.

· Hubert Ligocki

At 440 Hz and 880 Hz the result is 12.00 semitones, 1200 cents, and a ratio of 2. A just fifth, 440 Hz and 660 Hz, is 7.02. New hertz: transposition.

Knowledge sources

The interval on this page is a step of twelve-tone equal temperament. The cent is Alexander John Ellis’s unit from 1885. The 440 Hz example follows A440 concert pitch.

Page updated November 3, 2026.