When played sequentially (one right after another) or in unison (as in a chord), there are two ways to diagram the way a combination of notes will sound, melodically and harmonically.
Intervals, are the sequential semitone (1/2 step) distance between notes. However, this is not a good way to describe the way two notes will sound together, because two notes one interval apart played in unison (min2), sounds in resonance. It is the one of the most dissonant sounds a combination of two notes can make, or the sound heard when two strings that are slightly out of tune.
Harmonic Intervals, measure sound using perfect relationships, equal to seven semitones higher or lower in pitch. They describe the sound created between notes better than semitone intervals because notes played one harmonic interval apart sound in consonance or repose. Study the above to understand the difference between melodic and harmonic intervals.
The harmonics give us a way to analyze sound using a phenomenon of nature that describes the way two or more wavelengths of sound, sound together. By placing all the harmonic intervals next to each other, we can finally see the harmonic relationships of all the notes.