Scale Degrees
Music's number system — each note named by its role relative to the root
When one note is heard as the root, every other note can be described in relation to it. The most useful way is to label each note by its interval above the root — these are the scale degrees, music's number system. Like the intervals, they are named against the major scale as the reference: 5 is the perfect 5th, the fifth note of the major scale.
Any melody or harmony can be written in scale degrees, and doing so captures how the music is actually heard better than naming absolute notes does. In modern equal-temperament music the specific root matters much less than the relationships between notes: a major scale resembles a major scale on any other root far more than it resembles any minor scale. The same set of notes can even sound like any of the major modes depending on which one is the root — which is exactly why a root, and degrees measured from it, is such a useful idea.
Notes outside the major scale are named by sharpening or flattening its degrees: a minor 3rd is the ♭3, the bluesy 7th the ♭7 — so all seven numbers still appear, each once. Smaller notesets are numbered the same way against the major scale, which is why a triad's middle note is the 3rd, not the 2nd: it sits a third above the root.
Solfège is the same system made singable — a syllable per degree, for carrying the numbers onto melodies by voice.
Hearing one note as the root and the rest in relation to it feels abstract at first, but each degree's character grows familiar with practice — and, with it, the ability to pick the degrees out in real songs. The degrees also reinforce one another: a firmer feel for one sharpens the rest, so they are learned as a network rather than one at a time.