Numbering intervals

To number an interval you count how many note names you cover from the lower note to the upper one, counting both of them. From C to E you pass through C, D and E: three names, therefore a third. From C to G you count C-D-E-F-G, five names: a fifth.

Since you count including the two endpoints, the unison (the same note repeated) is a first, two neighbouring notes form a second, and so on up to the octave, which is an eighth: the same name seven degrees higher.

This count is done over note names or, on the staff, over consecutive lines and spaces; not over semitones. To fix the number, accidentals are also ignored: both C–E and C–Eb span three degrees and are both thirds, even though their size in semitones differs.

To size an interval reliably, however, what really counts is the semitones: from X to Y there are Z semitones, and that is what determines which interval it is. Counting note names on the staff only gives the correct quality when there are no accidentals, that is, in the key of C (C major): there each next name adds the expected number of semitones and number and quality line up on their own. When sharps or flats are present you have to go back to the semitones so as not to be fooled.

Think of it like the rungs of a physical ladder: the number of the interval only tells you how many rungs apart you are, counting the starting one and the destination. If you step on the first and the third rung, you have covered three rungs: a third. How many semitones those rungs are exactly is another question —the quality— that comes later.

That is why we say the number is only half of an interval's full name: it gives you the diatonic size (third, fifth…), and the quality (major, minor or perfect) adds the precision in semitones in the sibling nodes.