The music interval calculator does two related jobs. Given two notes, it names the interval between them with its correct quality and number — major third, augmented fourth, minor seventh — and reports the semitone count, the size in cents, the twelve-tone equal temperament frequency ratio and the interval's inversion. Given one note and an interval, it works in the other direction and spells the note you land on.
Arb Digital publishes free calculators across music, maths and everyday work. This page uses twelve-tone equal temperament throughout, with a default reference of A4 = 440 Hz, and both of those choices are stated openly because they change the numbers even though they never change the interval names.
What This Calculator Does
In naming mode you pick two notes, each as a letter, an accidental and an octave in scientific pitch notation, and the tool reports the interval. In build mode you pick one note plus an interval number, a quality and a direction, and it reports the note that lies there. Both modes show the semitone count, cents, ratio and inversion for the interval involved.
Crucially, it names intervals by spelling, not by sound. That is what distinguishes a real interval calculator from a semitone counter. C to F♯ and C to G♭ sound identical on a piano and are both six semitones, but the first is an augmented fourth and the second a diminished fifth, and a piece of music treats them as different things heading in different directions.
This is a distinct job from converting notes into frequencies. Our note frequency converter maps a note name to hertz and back, reporting cents deviation and MIDI note numbers; it does not name the relationship between two notes. If you want a scale rather than an interval, the music scale generator builds those, and the music transposer shifts a whole passage.
How to Use It
- Choose a mode. Naming uses the two note rows; building uses the first note row plus the interval row.
- Set each note as letter, accidental and octave. Octave numbers follow scientific pitch notation, where middle C is C4 and the number increments at C rather than at A — so B3 is a semitone below C4.
- In build mode, pick a number, a quality and a direction. Perfect goes with unisons, fourths, fifths and octaves; major and minor go with seconds, thirds, sixths and sevenths. Invalid pairings are reported rather than guessed at.
- Adjust the reference pitch if you need to. It moves the frequencies shown and leaves every interval name, semitone count and ratio untouched.
- Read the inversion. It is often the quicker route to an answer, because a descending interval is usually easier to think about as the inversion of an ascending one.
The Formula / How It's Calculated
Interval naming is two independent counts that must agree. The number comes from letter names only: count the letters inclusively, so C to E is C-D-E, a third, regardless of any accidentals. The quality comes from the semitone distance compared with what that number would be in its plain form.
Concretely, each letter has a natural semitone value within the octave — C 0, D 2, E 4, F 5, G 7, A 9, B 11 — and each accidental shifts it by ±1 or ±2. The plain sizes are: unison 0, second 2, third 4, fourth 5, fifth 7, sixth 9, seventh 11, octave 12. Unisons, fourths, fifths and octaves are perfect at their plain size; seconds, thirds, sixths and sevenths are major. From there, one semitone narrower than major is minor, two narrower is diminished; one narrower than perfect is diminished; and one wider than either perfect or major is augmented.
Worked example. C4 to E4: the letters C-D-E give a third, and the semitone distance is 4 − 0 = 4, which is exactly the plain size of a third, so it is a major third — 4 semitones, 400 cents, ratio 2^(4/12) = 1.259921. Its inversion is a minor sixth. C4 sits at 261.626 Hz and E4 at 329.628 Hz with A4 = 440.
Second example, showing spelling at work. C4 to F♯4: letters C-D-E-F give a fourth, and the distance is 6 semitones against a plain fourth of 5, so it is an augmented fourth. C4 to G♭4: letters C-D-E-F-G give a fifth, and the distance is again 6 against a plain fifth of 7, so it is a diminished fifth. Identical sound, different name, because the number came from the letters.
The Temperament This Page Uses, and Why It Matters
All ratios and cents figures here are twelve-tone equal temperament, in which the octave is divided into twelve identical semitones. Each semitone multiplies frequency by the twelfth root of two, roughly 1.059463, and each is defined as exactly 100 cents, so an octave is exactly 1,200 cents. That definition is what makes the cents column trivially equal to semitones × 100 on this page.
Equal temperament is a compromise, and it is worth knowing how large the compromise is. Georgia State University's HyperPhysics notes in its treatment of musical scales and intervals that the ear responds to frequency ratios rather than differences, and that consonant intervals correspond to small whole-number ratios such as 2:1 for the octave and 3:2 for the fifth. Its page on equal temperament compares that system against just and Pythagorean tuning.
Run the numbers and the picture is clear. A just perfect fifth at 3:2 is 1,200 × log₂(1.5) = 701.955 cents, while the equal-tempered fifth is exactly 700 — narrow by about two cents, which almost nobody hears. A just major third at 5:4 is 386.31 cents against an equal-tempered 400, which is nearly fourteen cents sharp and is audible as a slow beating in a sustained chord. That is why equal-tempered thirds sound restless to ears trained on choral or string tuning, and why ensembles without fixed pitch drift toward just intervals on their own.
Why the Reference Pitch Changes Nothing About Intervals
The A4 field moves every frequency the page reports, and moves none of the interval mathematics. That is the defining property of a ratio-based system: an interval is a multiplication, so it is the same multiplication wherever you start. A major third above A4 = 440 gives 554.365 Hz; a major third above A4 = 415 gives 522.87 Hz; both are 1.259921 times their starting note, and both are a major third.
A4 = 440 Hz is the widely used standard reference and is what this page defaults to. Alternatives exist in practice — a number of orchestras tune slightly higher, and historical performance often uses a lower reference such as 415 Hz for baroque repertoire, which is close to a semitone below modern pitch. None of that affects a single interval name. If a piece of software disagrees with this page about a frequency but agrees about the interval, the reference pitch is the difference.
Inversion, and the Shortcut It Gives You
Invert a simple interval — move the lower note up an octave — and two things happen predictably. The numbers sum to nine: a third inverts to a sixth, a second to a seventh, a fourth to a fifth. And the qualities swap: major becomes minor, minor becomes major, augmented becomes diminished, diminished becomes augmented, while perfect stays perfect.
That gives a genuinely useful shortcut. Descending intervals are harder to identify by ear and on paper than ascending ones, so instead of counting down a minor sixth from C, count up its inversion — a major third — and then drop an octave. It also explains why the tritone is special: six semitones inverts to six semitones, so an augmented fourth inverts to a diminished fifth and the interval is symmetrical about the octave. It is the only one that is.
Compound Intervals and When to Reduce Them
Intervals larger than an octave are compound, and they are named by continuing the count: a ninth, a tenth, an eleventh, a thirteenth. A major tenth is a major third plus an octave, and to reduce a compound interval to its simple form you subtract seven from the number rather than eight, because both endpoints are counted inclusively — a tenth reduces to a third, not a second.
Whether to reduce depends on what you are doing. Harmonic analysis usually reduces, since a tenth functions as a third in a chord. Voicing and orchestration usually do not, because a tenth sounds open where a third sounds close, and that difference is the entire point of writing it that way. This calculator reports the number as entered and gives you the semitone count, so both readings are available.
Compound intervals also connect directly to the overtone series, where the intervals between successive partials shrink as you go up. The harmonic series calculator covers that relationship, and the music duration calculator handles the time side of the same repertoire when you need running times rather than pitches.
Arb Digital maintains a large library of free tools covering music, maths, measurement and everyday planning. Have a look, or get in touch.
Browse Free Tools Contact Arb DigitalCommon Mistakes to Avoid
- Counting semitones and stopping there — six semitones is an augmented fourth or a diminished fifth depending on spelling, and the two are not interchangeable in a key.
- Forgetting that interval numbers count inclusively — C to E is a third because C, D and E are three letters, not because there are two steps between them.
- Pairing perfect with a third or a sixth — perfect applies only to unisons, fourths, fifths and octaves; everything else takes major or minor.
- Getting the octave boundary wrong — scientific pitch notation changes number at C, so B3 to C4 is a semitone while A3 to B3 is a whole tone within one octave number.
- Reducing a compound interval by eight — subtract seven, so a tenth reduces to a third and a ninth to a second.
Related Free Tools From Arb Digital
Convert note names to frequencies with the note frequency converter, build scales with the music scale generator, and shift a passage to another key with the music transposer. The harmonic series calculator covers overtones and their intervals, while the music duration calculator and BPM calculator handle tempo and running time. Browse the free online tools hub for everything else.
Frequently Asked Questions
The sound is the same in equal temperament — six semitones — but the spelling differs. An augmented fourth spans four letter names, such as C to F sharp; a diminished fifth spans five, such as C to G flat. They resolve in different directions in tonal music, which is why the distinction is kept.
Because perfect is reserved for unisons, fourths, fifths and octaves, which are the intervals whose plain forms correspond to the simplest frequency ratios. Seconds, thirds, sixths and sevenths come in major and minor forms instead, and can be augmented or diminished from there.
Subtract the number from nine and swap the quality. A major third inverts to a minor sixth, a minor seventh to a major second, an augmented fourth to a diminished fifth. Perfect intervals stay perfect, and the tritone inverts to itself because six semitones is exactly half an octave.
No. An interval is a frequency ratio, so it is the same multiplication wherever it starts. Moving A4 from 440 Hz to 415 Hz shifts every absolute frequency but leaves every interval name, semitone count, cents figure and ratio exactly as it was.
Exactly 100 in twelve-tone equal temperament, because the octave is defined as 1,200 cents and is divided into twelve equal parts. Each semitone multiplies frequency by the twelfth root of two, which is approximately 1.059463.
Because it is about fourteen cents wider than a just major third. The 5:4 ratio measures 386.31 cents while the equal-tempered third is exactly 400, and that difference is audible as beating in a sustained chord. The equal-tempered fifth, by contrast, is only about two cents from just.
One larger than an octave, named by continuing the count — a ninth, tenth, eleventh or thirteenth. To reduce it to its simple form, subtract seven rather than eight, because interval numbers count both endpoints. A major tenth is a major third plus an octave.
All ratios and cents figures on this page are twelve-tone equal temperament with a default reference of A4 = 440 Hz. Other temperaments and reference pitches give different frequencies and different ratios, though the interval names are unchanged.
