Music intervals explained: names, semitones and inversions

An interval is the distance between two notes, and its name has two parts: a number, found by counting letter names from the lower note to the upper one inclusively, and a quality, found by counting the semitones. C up to E is a 3rd because it spans three letters (C, D, E), and it is a major 3rd because it covers four semitones; C up to E♭ is still a 3rd, but with three semitones it is a minor 3rd. The interval calculator names any interval from two spelled notes and plays them.

The number: counting letters

Count the lower note as one and keep counting letters up to the upper note. C to D is a 2nd, C to G a 5th (C, D, E, F, G), C to the next C an octave, which is an 8th. Accidentals do not change the number: C to G♯ and C to G♭ are both 5ths, because both reach the letter G. The same note twice is a unison. This letter count is why the theory names can look odd at first, but it is what keeps spelling consistent.

The quality: perfect, major, minor

Intervals fall into two families. Unisons, 4ths, 5ths and octaves are perfect. Seconds, 3rds, 6ths and 7ths are major or minor, with minor one semitone smaller than major. Either family can be stretched or squeezed further: one semitone larger than perfect or major is augmented, and one smaller than perfect or minor is diminished. Perfect intervals stay perfect when inverted, and they sound especially stable and blended.

Intervals within an octave

Here are the standard intervals above C, by semitones:

• 0: perfect unison (C to C)
• 1: minor 2nd (C to D♭)
• 2: major 2nd (C to D)
• 3: minor 3rd (C to E♭)
• 4: major 3rd (C to E)
• 5: perfect 4th (C to F)
• 6: augmented 4th (C to F♯) or diminished 5th (C to G♭), the tritone
• 7: perfect 5th (C to G)
• 8: minor 6th (C to A♭)
• 9: major 6th (C to A)
• 10: minor 7th (C to B♭)
• 11: major 7th (C to B)
• 12: perfect octave (C to C)

These twelve are the raw material of chords: a major chord is a major 3rd plus a minor 3rd stacked on top, which together span a perfect 5th, as How to identify a chord from its notes shows.

Why spelling matters

On a piano, E♭ and D♯ are the same key, so C to E♭ and C to D♯ both cover three semitones. But C to E♭ spans three letters and is a minor 3rd, while C to D♯ spans two and is an augmented 2nd. The sound is identical in equal temperament; the name tells you the job the note is doing. In C minor the E♭ is the third of the chord, while a D♯ would be a chromatic note heading up to E. That is why the interval calculator asks for spelled notes rather than piano keys, and why it can name rarer intervals like a doubly augmented 4th.

Inversions and compound intervals

Flip an interval by moving the lower note up an octave and you get its inversion. The numbers always add up to nine and the quality swaps: a major 3rd (C to E) inverts to a minor 6th (E to C), a perfect 4th to a perfect 5th, augmented to diminished. Perfect stays perfect. Intervals larger than an octave are compound: a 9th is an octave plus a 2nd, a 10th an octave plus a 3rd. They keep the quality of the simple interval inside them, so C to the D an octave and a step above is a major 9th, the note added in a C9 chord.

Equal temperament and pure ratios

Each interval also has a frequency ratio. In twelve-tone equal temperament every semitone multiplies the frequency by the twelfth root of 2, about 1.0595, so each semitone is 100 cents and an octave is exactly 2:1. Simple whole-number ratios, called just intonation, are the acoustically pure versions: a just perfect 5th is 3:2, about 702 cents, almost identical to the piano's 700; a just major 3rd is 5:4, about 386 cents, noticeably narrower than the piano's 400. Singers and string players often drift towards the pure versions when they hold a chord. The interval calculator shows both. To hear intervals rather than calculate them, use the ear trainer, check notes on the virtual piano, and read Why A is 440 Hz: how concert pitch was standardised for the frequencies behind them, which the note frequency calculator gives for every note.