There are two main scales formulas: major and minor.
Each is built on the concept of “steps,” which are either half or whole. A half-step is the distance between a white key and the black key next to it on the piano. Or the distance of one fret on a guitar. The first two notes of the jaws theme are a half-step.
And two halves make a whole; Add two half-steps together and you get a whole-step. Here’s what a half- and whole-step sound like:
Side note: half-steps can be called semitones, whole-steps can be called whole tones.
And, of course, we can stack these steps on top of one another. I tell my guitar students to imagine a ladder. Some steps in the later are 6 inches from the next, some are 12 inches. Stack a bunch of those steps together and you can climb to the next story of a building.
This is the idea behind a scale; steps are organized in such a way that they arrive one octave up. “An octave” refers to a note twelve semitones above another. It’s important to know that an octave is the repetition of the same note. C an octave up is C, right? It’s just a higher C.
With that in mind, here’s a major scale formula:
So what does that all mean? Here’s the way I notate it. A “^” means a half-step:
1 2 3^4 5 6 7^1
Notice, first of all, that I repeat “1” once the scale hits the octave. This is because it’s not a new note, remember? A “C” played an octave above another “C” is still a “C.”
Alright, so that formula shows you that there are two half-steps in the major scale. One is between 3 and 4, and one between 7 and 1.
On a piano, if you play a scale starting on C, you’ll notice the half steps line up with the places where no black keys are present. Two white keys without a black key in between them on a piano are a half-step apart. Or on an open guitar string, you would play the frets: 0-2-4-5-7-9-11-12.
TL;DR: Scales are built by stacking half-steps and whole-steps. The major scale is made of five whole-steps, and two half-steps: between 3 and 4, and between 7 and 1. It can be written like this, where “^” means half-step: 1 2 3^4 5 6 7^1.