Question: My question relates to the chord pattern formulas.

For this Example we are commonly given these two formula patterns as one of the steps to construct chords in a key. I have been self teaching/learning and can't seem to find any information on the web to explain how these patterns are derived?

For the Major scale = {Maj, Min, Min, Maj, Maj, Min, Dim} is used to build the chords in a key

For the Minor scale = {Min, Dim, Maj, Min, Min, Maj, Maj} is used to build the chords in a key

Anyone who can shed light on this part I would really appreciate.
Thanks

Answer: 
The patterns derive from the three note triads built on each tone of the gamut of letters `A B C D E F G`. You call these the *diatonic* triads...

[![enter image description here](https://i.stack.imgur.com/J17pu.png)](https://i.stack.imgur.com/J17pu.png)

...notice that the pattern is really *one pattern of seven chords* where major and minor are just segments within that seven chord pattern.

The seven chord pattern can theoretically ascend or descend infinitely which is represented visually more easily as a circle...

[![enter image description here](https://i.stack.imgur.com/bcaOI.png)](https://i.stack.imgur.com/bcaOI.png)
I think it's important to understand the structure of the major scale and the answer given by Son of Fire makes this explicit. You are not really given a formula for the chords in a key in my opinion, you are given the definition of a key, and a formula for building a chord. From these two kernels the rest follow. And since major and natural minor are related it's no surprise that the set of chords is the same, just in a different sequence because natural minor scales and major scales are related, relative minor starts on the 6th degree of the corresponding major.

This follows for all the modes, as pointed out in other answers. If you ask what are the chords that naturally occur in Phrygian just start the major scale sequence on the 3rd degree and keep going.

It might be more interesting to look at melodic minor, or some other exotic, non-Western scale. The (jazz) melodic minor is W-H-W-W-W-H, e.g. A mm would be {A, B, C, D, E, F#, G#, A}. The triads that naturally occur here are: A-, B-, C+, D, E, F#dim, G#dim, A-.

Try applying the formula to other examples and you will become proficient at it and understand the relationship.