WARNING - YOU WILL NEED QUIET TIME TO SERIOUSLY READ THE FOLLOWING
Lots of folks try to wrap their heads around scales.
I'm sure it's been all over these forums (I've seen quite a few already in my short time here), and you can find it constantly everywhere there is some community or resource for music - some sort of question about trying to wrap their head around scales.
There are thousands (probably millions at this point) of write-up's attempting to help people understand this.
They'll explain it in many ways, but two things that I find rather consistent is that they will explain them in music theory formats that people are not yet familiar with, and/or they will digress into tangents of relativism prematurely.
Thirdly, all have one thing in common that I've seen: They don't really dig at the underlying pattern that governs all scales, but instead spend the time going over the upper layer of scales themselves.
This means someone has to remember crap like, "Minor is like Major, but with the 3rd, 6th, and 7th flat", etc...
While that is easy enough when you are talking about two scales in total, once you start throwing out all the names of scales at someone learning, which themselves are confusing taxonomically (meaning, the naming conventions aren't really good at describing the function of the scale; they appear to learners as rather arbitrary unless you get into music history)
, you end up with a dizzying set of archetypes to recall by memory or you have to refer to a chart regularly to get your navigation settings - which really gets in the way of the artistic flow of the music composition process.
While the upper-snub might be the idea that someone shouldn't compose until they can wield such information, that's just not going to happen and shouldn't - in my opinion - be required.
As close to Toaster Level Easy as is possible is what information dissemination should aim for; that's my motto. Don't demand users get better, demand the information gets better.
Furthermore, I prefer looking at any complex system by determining what principally governs the root of all subsequent layers of expression and then employing that, rather than learning the external layers and working backwards - like fractals.
And quite like fractals, music theory actually does have a core pattern that all of this spawns from.
We're going to refer to the chart at the bottom a lot, but I will go through different parts of it while explaining.
I will begin with a description of the semitone (so absolute beginners don't get lost), then move on to the core pattern, then show it in the standard set of scales, followed by expanded scales, and then a final thoughts wrap up.
This is not to say that you can make every conceivable scale using this method, no. That is not the point.
The point here is to show a beginning and amateur approach to wielding standard sets of scales regularly employed in Western music without having to strain memorization or slow down creative processes.
Because most people around here are using DAWs, I will show things when needed on a Piano Roll styling rather than a normal horizontal piano.
I will also outline notes as flats or sharps correctly first, but then only as sharps following because most DAWs don't have flats in the piano roll and instead just have sharp annotations by default.
In our use, we will employ the standard definition of a Semitone.
The Semitone is most easily outlined by looking at a piano roll.
In this diagram (Fig 1) we have an example of ONE (1) SEMITONE and TWO (2) SEMITONE intervals.
For our purposes here, the easiest way to think of semitone intervals is "one semitone equals one key on a piano roll" regardless if that key is white or black.
When we say to move ONE semitone, that means we start on a key (let's say C) and go ONE key up or down (we'll say up for this example; which would be C#).
So when we see 2 1, that is not a list of TWO keys (notes), but instead is a list of THREE keys (notes).
That is because if we start on C and then go TWO (2) semitones UP, then we would be on the D key.
If we then go 1 semitone from there, then we would be on D#.
As such, we understand that an interval number is not a note itself, but describes the distance BETWEEN two notes (think of it like counting the valley's between your knuckles - you have 3 valleys, so you therefore have 4 knuckles).
THE CORE PATTERN
Now that we have an agreed understanding of our term of "semitone" and what concretely we mean when saying that inside of your DAW environment, we can take a look at the core pattern that will underpin everything else.
The pattern is expressed in SEMITONE.
That's it. That's the core pattern.
There's an extra tag-along.
We'll refer to this extra pair as the APPENDAGE.
Now, THIS [2 1] will move around and that's kind of what describes the scale: where this extra pair of intervals (3 notes) is within the alignment of the core pattern of [2 2 1 2 2]
Imagine a stereoscope toy (if you don't know what that is, look it up), but imagine that it doesn't have a spinning wheel.
Instead, it has a single horizontal card. On that card are the numbers: [2 1] [2 2 1 2 2]
Now, imagine that card is a strip of paper and wrapped into the stereoscope so that the beginning and end are joined in a loop on both sides of the ribbon, so that you have [2 1] [2 2 1 2 2] [2 1] [2 2 1 2 2]
Further imagine that at all times 7 numbers will be visible when you look in and slide around the ribbon.
And lastly imagine that there is a center mark on the lens of the stereoscope toy so that you always know where "center" is.
When you pull that ribbon back and forth, you will see the pattern of those intervals move left and right and each time you move the whole set one number placeholder to the right or left, you have CHANGED SCALE.
This is what is being represented in the first part of the chart at the bottom which shows Major, Dorian, Phrygian, Lydian, Mixolydian, Minor, and Locrian scales.
The red part that is boxed in is the core pattern [2 2 1 2 2]. The remaining [2 1] in green is the appendage.
All that's happening as we move to each scale is that the whole pattern set is sliding to the left by one number placement holder each time until we have made our way from Major (which starts the migration with the appendage at the right of the core pattern) to Minor (which ends the migration with the appendage at the left of the core pattern).
Locrian is the scale where the core pattern is dead in the middle and basically is the loop point between Major and Minor in only ONE shuffle to the right or the left (if that didn't make sense, let me know).
You COULD also draw up this same diagram so that Locrian is in the middle vertically and your set is then as the following image:
From this perspective, UP from Locrian would be shuffling to the RIGHT, while DOWN from Locrian would be moving to the LEFT.
This, then, is a reasonable definition of the CORE PATTERN and its APPENDAGE.
THE STANDARD SET OF SCALES
The standard set (those outlined in the CORE PATTERN) reveal a numerical rule:
RULE 1: the SUM of the INTERVAL VALUES must equal 12.
All of the standard set accomplish this.
Further, the standard set also accomplishes a SUM of 12 vertically as well. This BALANCED relationship between all symmetries of the scales is what causes these to be related and typically thought of as your "standard scales".
When you look at each standard set and count how many times each value occurs, you end up at another starting point rule:
RULE 2: TWO 1's and five 2's must exist in each scale.
We then add a third rule that isn't visible from this chart, but does indeed exist implicitly in the nature of the intervals themselves.
RULE 3: Each letter must exist ONCE in the scale.
Now, with all three rules together, we can outline any scale that we feel like without knowing its name.
We simply must start on any given note on the piano roll key and then pick any order of five 2's and two 1's AS LONG AS we have made every note exist ONCE in our total set of 7 notes whose total interval arrangement sums to 12.
That is the fundamental bottom line.
That might sounds hard to some folks, but it's very easy.
Keep in mind that your CORE PATTERN is [2 2 1 2 2] and you're just really figuring out where you want to slap in the APPENDAGE of [2 1], so you just say I'm going start on the SECOND 2 of the CORE PATTERN.
So that means our ENTIRE pattern is [2 1 2 2] [2 1]  (<-this last "2" was moved over to the end because we pushed our core pattern to the left by one slot).
So, let's start on C.
We now need to go up TWO semitones, so that would land us at D.
Next, we move ONE semitone up so that we're at D# on the DAW.
However, according to the rule, we need to have one of each letter, so that can't be D# since we ALREADY have a D, so we know that should be Eb.
So we have C, D, Eb so far (C, D, D# on your DAW).
Next, we move another 2 from Eb and land at F.
Then we go another 2 from F and land at G.
Then we move 2 from G and land at A.
Now we have C, D, Eb, F, G, A (C, D, D#, F, G, A on your DAW).
So then we go 1 from A to an A#, but according to the rule of ONE letter, we change that to a Bb (same key on the keyboard, though, remember).
So now we have:
Regular Expression: C, D, Eb, F, G, A, Bb
DAW Expression: C, D, D#, F, G, A, A#
That's it. We now have 7 notes and that means we have a standard scale!
What about that last 2, you might be asking. We only did [2 1 2 2] [2 1] so far. Where's the last ?
That is our interval for getting back to C.
So our full description of [2 1 2 2] [2 1]  starting on C is:
Regular Expression: C, D, Eb, F, G, A, Bb, C
DAW Expression: C, D, D#, F, G, A, A#, C
TADA! You just made C Dorian using SCALE BY NUMBERS (like paint by numbers, but for music theory)!
THE EXPANDED SET OF SCALES
Now that we have a foundation beneath us we can play around and ruin the rules a bit and get more interesting scales.
With this set, the whole rule of "two 1's and five 2's" goes out the door.
HOWEVER, not entirely.
With augmented scales, there are still an attendance to that math. You just DESTROY a bit to make things happen, and we all love destruction!
The new rule:
RULE 2.Amendment 1: You can have 3's at the expense of turning a 2 and a 1 into that 3*.
* = and some other exceptions in melodic, harmonic, and diminished that we'll get into
That sounds confusing possibly, but it's really simple.
Take the Major scale:
[2 2 1 2 2] [2 1]
Now, I want to make that into a Major Pentatonic scale.
So I take the 1 and 2 and collapse them into a 3.
[2 2 3 2] [2 1]
Now I take the other 2 and 1 set and collapse them.
[2 2 3 2] 
Now, the minor version of this behaves the same way the minor does in the standard set; as a mirror of major:
 [2 2 3 2]
That's because the regular Minor is
[2 1] [2 2 1 2 2]
So that becomes
 [2 2 1 2 2]
 [2 2 3 2]
How do you know which 1 and 2 to collapse?
Within the CORE PATTERN, you collapse ALWAYS [...2 1...], so the 2 before the 1 is what collapses with the 2 and in their place is left a 3.
In the APPENDAGE you collapse ALWAYS both (even separate).
This will generate you any given pentatonic that you want.
For Melodic's, the only difference here is that we broke the CORE PATTERN and APPENDAGE definitions.
Now it's [2 1 2 2] for the CORE PATTERN and [2 2 1] for the APPENDAGE, but this follows the same rules in every other way.
You can think of Melodics as Interval Inversions of their normal setup.
You'll notice that in Melodics, Major's APPENDAGE is to the left of the CORE PATTERN, while Minor is on the right.
You'll also notice that if I do this
[2 2 1 2] [1 2 2]
[2 2 1 2] [2 2 1]
You can easily spot that the APPENDAGE is defined by its inversion of the Interval Pattern of the three intervals of the standard Major scale.
(if that doesn't make sense, let me know)
Harmonics combine both rules above and throw us into a blender.
Our CORE PATTERN of [2 2 1 2 2] is now collapsed into a three stolen from a 2 and made into [2 1 3 1 2].
Our APPENDAGE becomes [2 1] and our APPENDAGE inverts to define the difference between Major or Minor while everything else remains the same.
So a Major Harmonic has an APPENDAGE of [2 1], while a Minor Harmonic has an APPENDAGE of [1 2].
Lastly of the "But! But!" exceptions is the Diminished Octatonic.
Here we throw all rules out except for the first rule that all values must tally to 12.
In fact, that is the ONLY rule that remains consistent through all - that they all end up tallying to 12.
Here the rule is dead simple, you ONLY have a one CORE PATTERN and no APPENDAGE.
That pattern is [2 1]. You just repeat that endlessly until you have 8 total notes.
Note: on the form below that's been referenced in the above discussion, you'll notice a section called "SYMMETRICAL COUNTERPART VIEW OF SEMITONE PATTERNS".
This shows which scale is the mirror of which other scale in regards to SEMITONE INTERVALS.
You'll note that LOCRIAN is its OWN mirror and has not counterpart.
That's why it is the CENTER of the migration between MAJOR and MINOR (see above chart where LOCRIAN is shown in the center).
Also, Also Wic:
I don't pretend that this solves EVERY scale knowledge problem; again, that's really not the point.
The primary point is to make it a bit easier to start with for MOST of your common scale conceptions.
Mostly, you just need [2 2 1 2 2] and [2 1] and to move around where you are starting, and making sure whatever you move off of the left, you append to the end of the right.
That's my 2 nuyen,
I'm interested what folks will discuss here.
Long Chart For Reference Up Above: