You need to know how major scales are constructed. (With the chromatic scale and the WWhWWWh pattern.) If you don't, i suggest you read this first.
The notes in scales are sometimes numbered in music theory. This would look as follows with the C-major scale:
C D E F G A B
1 2 3 4 5 6 7
Some lessons about scales give you scalepatterns with fingerings and these numbers. Although these numbers may come in handy if you want to explain how a major chord is constructed for example, it is in my opinion NOT handy when learning scales! Because these numbers tell us nothing about where to find that note on the fingerboard.
That's why we going to number the scales a bit more handier. It's usefull for understanding chords, practicing scale-patterns (and inventing your own scale fingering patterns).
To explain everything we're going to construct a C-Major scale.
First, write down the chromatic scale.
Now number the chromatic scale, calling the root note C number 1.
Construct a C-major scale using pattern of whole and half steps:
1 semitone = 1 fret = the distance between 1 note of the chromatic scale = a half step (+h)
2 semitones = 2 frets = the distance between 2 notes of the chromatic scale = a whole step (+W)
Instead of using the +W +W +h +W +W +W +h pattern you could say +2 +2 +1 +2 +2 +2 +1, and with the root note C as number 1, you would also get:
C D E F G A B C
1 3 5 6 8 10 12 1
What you need to remember to figure out scale patterns (instead of memorizing them without knowing what you're actually doing), is that a major scale consists of the numbers 1 3 5 6 8 10 12 1, where number 1 is the root note.
Now let's play the C-scale only on the A-string, starting on the with the C on the 3rd fret. A|-3-....Call this note in your head number 1, from this point you should have no trouble finding the other notes of the major scale, numbers 3 5 6 8 10 12.
The system relies on counting instead of memorizing notes and patterns.You can just call a note say number 8 and as long as you play only notes 1 3 5 6 8 10 12 you'll be playing a major scale, although you don't know which.
Counting on the same string is easy, because one fret is one number is one semitone. But how do I count if I move to another string? I think you already know the answer, but for those who don't i'm going to explain it:
If you ever tuned a guitar manual, you probably used this method: First you tune the low E string, then you fret it on the 5th fret and tune the A-string until they match. Then you fret the 5th fret on the A string and tune the D string until they match, etc.
Write down the notes of the strings (EADGBE) with the corresponding numbers of the chromatic scale you wrote down earlier. Now look at the differences between those numbers: they match the frets you use when tuning your guitar!
E A D G B E
5 10 3 8 12 5
+5= +5= +5= +4= +5=
So if you stay on the 3rd fret, and you move from the A-string to the D-string, you add 5. This gives you 1+5=6, and 6 or an F. (Count 1-3-5-6: 4th note = C-D-E-F: 4th note).
Now i'm going to show you a couple of tricks where you can apply this counting:
01. Octaves: Ever heard of the trick that if you go 2 strings and 2 frets up, you'll have the same note only an octave higher? (except if you cross the B string, then it's 2 strings and 3 frets). This is easy to understand if you use the numbering system.
An octave is 12 semitones, so you must add 12 to whatever note you're at. 12 = 5+5+2 = two strings + two frets. But if you cross the B string, which is +4 instead of +5, you'll get 12 = +5 +4 +3, that's the two strings and three frets!
02. Creating a scale fingering: Let's make G-major scale fingering around the 5th fret! First, locate a number 1 note (a G that is!) somewhere around the 5th fret. Found it? (5th fret, D string) That's note number 1. Fret is with finger number 1 also. We'll go up from here first:
1 -> 3, finger number 3 at the 7th fret!
3 -> 5, finger number..uh... we're out of fingers! Luckily, +2 = +5-3, up one string, 3 to the right!
That gives us fret number 4. (Change fingering! finger 1 on fret 4, 2 on 5, etc) 5 -> 6, That's easy, one to the right. Second finger, G string, 5th fret. continue, you get the idea. When you are on the high E-string you can go all the way down. Eventually you will have played something like this.
You learn from this to do things like "+2 = +5-3" or "-2 = -5+3" faster.
Also, if you want to use other scales you can use different numbers. For example, the minor scale has the pattern +W+h+W+W+h+W+W which gives the numbers.
1 3 4 6 8 9 11 or
10 12 1 3 5 8 12, if you say number 10 is the root (Cmajor = Aminor, In C-major is A number 10)
The Major pentonic is:
1 3 5 8 10 (leaves out the 6 and the 12 from the major)
The Minor pentonic is:
1 4 6 8 11 (leaves out the 3 and the 9 from the major)
10 1 3 5 8 again, if you say number 10 is the root (that Cmajor pentonic = Aminor pentonic)
03. Chords. Major chords use the 1st, 5th and 8th notes (where 1 is the root of course). In general music theory it said they are the the 1st, 3rd and 5th note, but as I said in the beginning of this lesson: this is an other way of counting. (Naming the 7 notes of a scale 1-7, without paying attention to the distance in semitones between them). If you look at the C-major scale you see that the 1st, 3rd and 5th notes are C, E, G, and these are numbers 1,5,8 on the chromatic scale numbering.
Minor chords uses the 1st, 4th and 8th notes. Other chords types I don't know, but you can figure out their numbers by looking up a chord, write down the notes it has, then write down a chromatic scale with numbers and see what numbers match the notes.
Power chords have the pattern 133xx, which matches the notes 1,8,1. They are almost a major chord.
You can invent your own chords, analyse known chords, turn major into minor (locate note number 5 and lower them into a number 4!), etc. I also found out my own sort of powerchords: 431xxx, x431xx this is just as the powerchords movable, the notes are 1,5,8: a major chord. if you use 421xxx you have 1,4,8: a minor chord.
Of course you can't cross the B-string, just like powerchords. (Although you could think of things like xx432x)
We'll, that's all i could think of now. I found it helped me understand patterns and the fretboard better, but I don't know if it really helps learning scales. I enjoyed finding out my own powerchord thingies, and maybe there more to understand or discover with this numbering. Let me know what you think about it.