Numbers
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Part 1 - Deriving Major Scales from Chromatic Scales with The Pattern
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Numbers and music... seems a bit incongruous on the surface; music is so beautiful and free and
wonderful and artistic, and numbers are so... oh, they're just a pain.
But if I said "10 fingers, 10 toes", that's a use of numbers that's not so bad. Counting... that's
pretty easy. Let's pretty much just count, then, when we use numbers for music.
And since counting by 1's is sometimes boring, let's do a little addition and subtraction, too, just
so we can skip around from here to there a little quicker.
That's all we need. Then we can start to put numbers on "where we are" in the music, and we get two
things from it: a (numeric) name for "where we are," and if we push just a little beyond, we get a
sense of that position relative to other places we might be.
Here's a chromatic scale, with ordinary musical note names; these represent the notes you'd get by
playing each key on a keyboard (including the black ones), or each fret on a guitar or bass, or each
incremental note on most other instruments:
C C# D D# E F F# G G# A A# B C C# D D# E F F# G G# A A# B C
That's two octaves worth of notes, and after we go up twelve notes, we find another one of the same
name that's an octave higher. We'd be playing a "chromatic scale".
Normally, especially in western music other than jazz, we pick a subset of the notes that seem to
fit together pretty well when played in a sequence. It turns out the piano is made with a strong
assumption, here, with the white keys making up the "C major scale", which is one of these
nice-sounding subsets. (And it turns out due to somewhat more complex math that we've promised not
to talk about, that there are REASONS these notes seem to work well and sound well together. It's
not purely subjective.)
So here's what we're left with to make a C major scale, and sorta blank out the notes we don't want
to use right now, and we'll just keep one octave of it, from C to shining C:
C _ D _ E F _ G _ A _ B C
Notice the pattern of this scale: we take the first chromatic note, C, and then we skip a note, C#.
Then we take the D, skip the D#. Then we take the E and F, and skip the F#, take the G, skip the
G#, take A, skip A#, take B, and the octave C.
That's a rather wordy description of this pattern, accurate though it may be. There's another way
to describe this pattern: start at the first note, C, on the chromatic scale, and then move 2, 2, 1,
2, 2, 2, 1, taking all those notes:
C C# D D# E F F# G G# A A# B C C# D D# E F F# G G# A A# B C
r 2 2 1 2 2 2 1
The little "r" means "root".
Here's a very important point -- make sure this is melted into your brain: it's not the notes
themselves that make the major scale work right, it's the PATTERN. It's the spacing of the notes,
the distances between them and from one note to another.
Here's the very important implication of that point: we can start anywhere we want, and as long as
we use the same pattern, we'll get a major scale. Here's an E major scale:
C C# D D# E F F# G G# A A# B C C# D D# E F F# G G# A A# B C
r 2 2 1 2 2 2 1
So we have
E _ F# _ G# A _ B _ C# _ D# E
If you check, you'll see that the pattern -- the spacing -- is still 2, 2, 1, 2, 2, 2, 1.
You can start anywhere you want. In some cases, we'll use flats marked with (b) rather than sharps
marked with (#), but that's another lesson for another day. (Some "Safe" scales that formally use
only sharps are G, D, A, E, B, and F#. C is neutral. The others use flats.)
C C# D D# E F F# G G# A A# B C C# D D# E F F# G G# A A# B C
r 2 2 1 2 2 2 1
which gives us the G major scale:
G A B C D E F# G
So... we can make any major scale we want, by using the pattern to pick a subset of notes from the
chromatic scale.
Exercises:
Remember, the frets on your guitar give you a chromatic scale; let's pick a subset
of those notes and make major scales:
Pick any string on your guitar. Play that open string, and then move up 2 frets and
play that, and 2 more, then 1, 2, 2, 2, 1 and play all of those notes. Sounds good, eh?
Do another string.
Do it again, but instead of starting with the open string, pick some fret somewhere on
it (not *too* high up), and start there as the root. Then move and play 2, 2, 1, 2, 2, 2, 1.
Just for fun, do two strings at the same time!
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Part 2 - Numbering the Major Scale to Make a Generic Major Scale
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Now, let's move beyond how we made the scale to begin with, and just take the scale itself. The
relationships between the notes are always the same, no matter what root note we start on. There
are eight notes (including the octave), and we could, if we wanted to, just label them 1 through 8.
So in a RELATIVE world, all of these scales are (relatively!) the same, but with different names:
C D E F G A B C
E F# G# A B C# D# E
G A B C D E F# G
1 2 3 4 5 6 7 8
Now, as it happens, when we change keys, the chords keep the same internal relationships, too.
So does the melody. So does everything else.
For example:
The C major chord, as the root chord for the key of C, is made of the notes C, E, and G -- the
first, third, and fifth notes of the major scale. (Check it on the chart above.)
The E major chord is made of the notes E, G#, and B -- the first, third, and fifth notes of the
major scale. (Check it on the chart.)
The G major chord is made of the notes G, B, and D -- the first, third, and fifth notes of the
major scale. (Check it.)
The "generic" major chord is made of the notes 1, 3, and 5 -- the first, third, and fifth of the
major scale.
In other words, we can think in terms of note names, but if we start moving the key around, we end
up with a jumble of note names, but if we think in terms of the numbered notes, we can think
generically, always using the basic set of numbers. Later, when we're having inspiration, we can
decide which key we want to play it, and we can change the key at a moment's notice.
Similarly, for the melody, we could write it out using note names, but we could also write it out
using numbers, and then later on, decide which key we want to use, pick another key and do that one
from the same patters, and so on.
So: using numbers as note names gives a generic scale that can then be assigned to any key when
the need arises.
Exercises:
Play a scale again the way you did in the previous exercise, but name the notes out
loud with numbers.
Play it again, starting at "8" and working downwards!
Find a "G" note somewhere on your guitar fretboard. Play a major scale upwards from
there, on the same string, and name the notes as you go.
Play the G scale again, from the octave down the first note.
Play the E major scale and name the notes as you go.
Play the E major scale backwards, and name the notes as you go.
Do that all again, because you're not quite comfortable with it, yet! You're done
when it feels easy.
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Part 3 - Making Chords
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We've already touched on creating the root major chord for a scale by taking the first, third,
and fifth notes of the scale. These notes can be rearranged in different octaves, and they can be
in any order, and they can be duplicated... picking different "inversions" of the notes -- octaves
and ordering and so on -- make the chord "feel" different, but they don't change what chord it is,
musically speaking.
Let's pursue this chord-making idea, because it's absolutely, vitally important if you want to
make music with freedom.
The important question: how do we make ALL the chords for a given key?
As it happens, we just use another pattern. (Patterns on top of patterns.) This is a remarkably
easy one, considering what wonders come out of it musically. Simple, powerful:
Start somewhere, and take three notes, every other note on the scale.
That's it. That's how you make a chord.
So here's a C major scale, and we'll extract seven chords from it before we run into ourselves
again at the octave and start over with the same stuff. (We'll need to extend into the next
octave to get a few more notes for the upper chords.)
Here's the scale:
C D E F G A B C D E F G
And here are the notes we want to pick to make the chords, one per line:
C E G C The root chord Major
D F A Dm The second chord minor
E G B Em The third chord minor
F A C F The fourth chord Major
G B D G The fifth chord Major
A C E Am The sixth chord minor
B D F Bdim The seventh chord diminished
C E G C And we're home again at the root.
Notice how the first note of the "triple" becomes the basic name of the chord, and then to determine
it's full name, we just have to figure out whether it's major or minor. Major chords are unmarked, and
minor chords are marked with a small "m".
Whether it's major or minor depends on the spacing between the first two notes in the chord: four
frets (where the chromatic pattern from a previous lesson would be 2, 2) gives a major chord, and
three frets (where the chromatic pattern would be 2, 1 or 1, 2) gives a minor chord. The way the
scale works out, when the spacing between the first two notes is 4 frets, the spacing between the
second two notes is 3 frets, and vice-versa. So it's KINDA like there's a major/minor or
minor/major being put together to make the whole chord -- it nearly always turns out to be seven
frets from the first note of the chord to last note of the chord. The diminished turns out to be
three and three, so it's a minor/minor, if you will. (The weirdo of the bunch.)
Of course, we're working with patterns, so we can do this for every key. Here's G, without the
pretty diagonal:
G B D G The root chord Major
A C E Am The second chord minor
B D F# Bm The third chord. minor
C E G C The fourth chord. Major
D F# A D The fifth chord. Major
E G B Em the sixth chord. minor
F# A C F#dim The seventh chord dimished
G B D G And we're home again at the root.
Now, notice something else very important, here: If you look DOWN the first column, you see the
scale, with each note of the scale becoming chord's basic name.
But look at the SECOND column: it's still all following the r, 2, 2, 1, 2, 2, 2, 1 pattern, but it
starts in the middle on B -- the third note, and finish again at the octave B.
Similarly, the third column is still the G major scale, but it starts and ends on D.
What that means is this: If I play the notes of the G major scale starting on G, and at the same
time, you play the notes of the G major scale, starting and ending on B, and then our best friend
plays the notes of the G major scale starting and ending on D, we will all together play the seven
basic chords that can be extracted from the G major scale.
If we were singing them, we'd probably call them "harmony". Since we're plyaing them, we probably
call them "chords".
But hey! Since my guitar has more than three strings, I can, in fact, play the G major scale starting
on G, and at the same time starting on B, and again on D, all at the same time, myself!
And since all of this works in any key, we can use generic numbers instead of note names, and choose
the key at our leisure, with the situation calls for it.
1 3 5 1 The root chord Major
2 4 6 2m The second chord minor
3 5 7 3m The third chord minor
4 6 1 4 The fourth chord Major
5 7 2 5 The fifth chord Major
6 1 3 6m The minor chord minor
7 2 4 7#dim The seventh chord diminished
1 3 5 1 And we're back at the root again, an octave higher.
Now the "counting" pattern for each note going down the columns is even more obvious.
(We might use the terms "root, third, fifth" to talk about the three notes in the chord, no matter
which chord we're looking at. Technically, this is only strictly correct for the root chord,
but if we think "chord-centrically", it's a relaxation of the rules that can simplify things.)
Exercises:
Make some chords!
List the chords in the key of E major on paper, naming the three notes that make
up each chord. Give it's number name, and whether it's major or minor. (So, for
example, the F# is 2 and it's minor.)
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Conclusion
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Think generically, and choose a key. Put your capo anywhere you want, if you use one. The
SPACING between the notes in the scale, the chords, the melody, the harmonies, always stays
the same. Numbers as names let us look at them generically, and think a little easier about
the distances between them.
Exercises:
Do the key of A major, all yourself! One step at a time. First, get oriented:
Make sure you know the names of the strings. You need to be able to find your
way around.
The strings are, from low to high, fattest to thinnest, E A D G B E. Some people
say "Eat apples daily, grow big ears." There are many variations of the mnemonic;
google if you want the good and bad ones.
Make sure you know the chromatic scale. You need to be able to find your way around.
You don't necessarily need to know this *fast* yet, but you should know it
*correctly*. The basic rule is that for the note names A through G, each
one is followed by a sharp except for B to C, and E to F.
Map the chromatic scales onto the strings.
Put the two together. Starting with the open low E string, for example, play
and name each note, going up one fret at a time. "E" for the open string.
Play the first fret and say "F". Play the second fret and say "F sharp."
And so on, to the 12th fret. Then do the A string, and the D string, and
so on.
Now, map out the goal:
Starting with the chromatic scale, and the pattern, pick out the notes that make up
the A major scale. Write them down.
Play the notes of the A major scale on the A string of your guitar.
For each note on the A major, make a chord by taking the appropriate three notes.
Write them down with the proper chord name.
And make sure everything is working as expected:
Play each chord. As you play the chord, look at each string, and figure out
what note you're playing on that string. That note *should* be part of the
chord; it should be one of the three that you wrote down.
If it's not, there are two likely explanations:
Maybe that string should not be played as part of the chord form. Check a book or
other reference.
Or... maybe you're not really playing the actual chord, but some variation of it.
In other words, some people modify the chord forms to make them easier to play, or
to make them sound a little more interesting. They're not really playing the chord
that we've named -- they're playing a variation of that chord that should be given a
different name. For example, if you found a "B" note in an A major chord form, it's
really not an A major, it's an "A2," because the B is the second note of the scale,
and we name the chord to indicate this variation.
It is good to know the pure, basic chord forms, and the go from there. In other
words, it would be good to know the chord forms with only the three specific notes
that should make up that basic chord. The inversion doesn't matter -- the ordering,
the octaves the notes are in, the repeated notes -- as long as it's "those three."