Music theory formulas are a way to describe in words and symbols the tonal mix of a given chord or scale. We can use Numera, which labels the root of the system as a zero [the math works], or we can use the traditional system, which labels the root one. On the formula charts in this lesson, the Numera equivalents are given beneath the traditional chord tone name. The traditional system uses Major things as points of comparison. What follows is an explanation for the traditional system.
The basic premise of the traditional system is this: the Major chord and the Major scale provide templates as points of comparison. Major = normal. For chords, what is normal means what is true for a Major chord [R-3-5]. For scales, what is normal means what is true for the tones in a Major scale [R-2-3-4-5-6-7]. R = 1 = Root.
The formula for the Major chord is R, 3, 5, where R = Root, 3 = third (4 half steps away from root), and 5 = fifth (3 half steps away from 3rd, 7 half steps away from root). These relationships need to be true for a chord to be Major. Once we establish this, we can now parallel or compare other types of chords to it.
To do this, we use sharp (#) and flat (b) symbols for the comparisons to what is in the Major [1-7]. Any tone in between the normal numbers have a flat and a sharp name. Example: the tone that is 4 half steps away from the root is called the 3. The tone that is 3 half steps away is called a flat-3 [we lowered the normal 3 to a b3]. That same tone can also be called a sharp-2 [we raised the tone 2 half steps away, the 2, to a #2].
This is independent of tonal names - even though flats or sharps can be in the names of the tones. Examples: C Major is C E G. C minor is C Eb G. Eb is the flat 3rd and has a flat in the name. A Major is A C# E, while Am is A C E (C# has been lowered to C - is a flat 3rd - yet there is no flat in the name. It is a natural).
For more on this, check out Derivative and Parallel.