Intervals and Scales

Figure 1

Not very musical, is it?

Guitars, pianos, and most other Western instruments are designed around the evenly tempered scale.  The place where musical theory meets physics is the octave.  An octave simply represents a halving or doubling of frequency; that is, if you play an A 440 (that is, an A note whose frequency is 440 cycles per second) on a piano, the next A note you encounter, moving from left to right, will have a frequency of 880 cycles per second.  Move left instead, and the next A note will have a frequency of 220 cycles per second.  The more cycles per second, the higher the pitch of the note.  That’s all the mathematics we’ll need for our understanding of musical theory.

Our basic Western scale divides each octave into twelve equal segments called steps.  Hence, all of the scales we’ll be considering are, in reality, simply different variations on this set of twelve notes. 

We can envision the evenly tempered scale as a series of twelve cells, each of which represents the smallest division we can use in creating our music, as shown in Table 1.

Table 1

More pleasing results can be produced by omitting some of these notes, and playing only those that remain.

The first musical theory most of us were exposed to as children was the familiar do re mi fa so la tee do.  Let’s see, that’s seven notes out of a total of twelve, but which ones?  The answer depends on what key you want your do re me scale to be in.  In C, the corresponding notes are: C, D, E, F,G,A,B.  Which ones were omitted?  Obviously, C#, Eb, F#,Ab, and Bb, right?  (These omitted notes may just as accurately be named: Db, D#, Gb, G#, and A#. These situations in which one tone has two legitimate names is referred to as “enharmonic spellings.”  The rules as to which spelling is preferred are somewhat subtle, and beyond the scope of this book.  See Table 2.)

Enharmonic Equivalency

Table 2

Let’s look at this on the fretboard and see which notes were included and which were left out:

Figure 2

If the distance between any two circles on the same string as a half step, then we can see that there are two half steps, or one full step between C and D, another full step between D and E, only a half step between E and F, a full step between F and G, a full step between G and A, a full step between A and B, and finally, a half step between B and the C an octave above were we began.

Musicians centuries ago reached the same conclusion, and as a result, they came up with a scheme for playing some of the notes and not others.  This compartmentalization is shown in Table 3:

Table 3

Rather than using a big fretboard diagram every time we want to illustrate a particular pattern of half and full steps, suppose we write it like so:

C=D=E-F=G=A=B-C

Where “=” indicates a full step, and “-“ denotes a half step.  (Looking forward to when we may have occasion to refer to an augmented interval of three half steps, we’ll use the following symbol: “º”.)

This is what we refer to as the diatonic major scale, or simply the Major Scale.  (This is not 100% accurate since, as we shall see, any scale which has a major third degree can technically be referred to as ‘major’.)  Another way of referring to this particular sequence of steps and half-steps is the Ionian Mode.

If you’ve ever sang the “Do-Re-Mi” song, then you know the sound of the Ionian mode, but just to make sure, try playing it on your guitar, using the scale pattern shown in Figure 3.

Figure 3

Start and end on a ‘C’ note, at least until you start to get the sound of the scale you’re your ears.  You’ll have to stretch a bit to get that ‘B’ note. 

The rest of the C-Major scale notes in this position are shown in Figure 4.

Figure 4

Track 1 on the audio CD contains me improvising over a steady C Major chord.  Track 2 contains just the background C Major jam, so you can play over it yourself.  On the data portion of the CD, Ch1-1.CMP is a looping file for the Jammer™ application that you can also use for a ‘backup band’.

If this seems familiar, it’s because it looks a lot like the pentatonic minor scale in the key of ‘A’, which was probably the first lead ‘box’ you ever learned (Figure 5).

Figure 5

To understand how this ‘coincidence’ occurs, let’s play the same ‘C’ scale shown in Figure 4, but this time, play it over an Am6 chord.  (This will be tracks 3 and 4 on the audio CD, and Ch1-2.CMP on the computer partition). 

Amazing, huh?  You’re playing the same notes as before, but now, played against an Am chord instead of a ‘C’, they convey on an entirely different feeling.  We’ll be exploring modes in a latter chapter, but for now, we’ll just observe that the key of Am is the relative minor to the key of C Major.  More precisely, you’re playing the Aeolian mode, also known as the Natural Minor.  Suppose we rearrange our C scale, keeping the intervals in the same relative positions, but start on A instead of C:

A=B-C=D=E-F=G=A

As you can see, we still have a full step between the first and the second notes, but now our first half-step falls between note two and note three.  The distance from 3 to 4 is now a full step, rather than a half step.  In other words, it appears as if our third note was moved a half step to the left.  Thus, our 3^rd is flatted.  In fact, the presence of the flatted third is what makes this a minor scale.  Moving on, the distance from 4 to 5 is still a full step, but the distance from 5 to 6 is a half step.  By inspection, we can see that our 6^th has been flatted as well.  Finally, we can see that our seventh has also been flatted.  Consequently, we can expect it to have a minor kind of flavor. 

Suppose we compare this to an A major scale, that is, a scale starting and ending on the A note, which preserves the original step/half-step pattern of the C major scale.  This A Major scale contains the following notes (Table 4):

Table 4

Ever notice how music written in the key of A has 3 sharps at the left end of every staff?  Well, this is where they come from.

 On the fretboard, this looks like Figure 6:

Figure 6

As you see by comparing Figure 6 with Figure 4, the fingering of the A Major scale is quite different from the A minor scale, at least when played from the same fifth-fret root position.  If you want to use a more comfortable fingering for the A Major scale, try the pattern shown in Figure 7.

Figure 7

This is one of the peculiarities of the guitar—certain positions are more convenient for some scales than for others.

Looking at this in terms of intervals may make things clearer:

Figure 8

Figure 8 Shows our schizophrenic scale when played in a C Major context.

Figure 9

In Figure 9, on the other hand, we see the intervals as they appear from the perspective of Am.  In other words, the same notes take on different roles depending upon the harmonic context. 

When practicing a new scale, I generally know the notes of the scale, so I start out by taking Grid A (Appendix A: Fretboard Grids), and filling in the notes in a single convenient area of the fretboard using a highlighting pen.  I then play it against an appropriate chord or progression.  Once I have internalized the sound, I transfer the pattern to Grid B, filling in the interval numbers by hand.  This allows me to gain an understanding of what I’m doing as I practice.  Since the intervals will be the same in every key, once I have the pattern down, and have memorized all of the intervals, it’s a simple matter to transpose to any key, as well as to other areas on the fretboard.  This approach will become clearer as work our way through the examples in this book.

Harmony is an immense subject, and since this is basically not a theoretical treatise, we’ll only develop as much harmonic theory as is essential for understanding the material at hand.

Basically, you harmonize a scale by stacking thirds.  What this means is that you start on the first note, and take a ‘pick one, skip one’ approach until you have created a chord, which is by definition a combination of three or more notes.  For example, the first intrinsic chord in the key of C is created by taking C, skipping D, Taking E, skipping F, and taking G.  This gives you a C Major chord, consisting of the notes C, E, and G.  Numerically, you have a 1-note, a 3-note, and a 5-note.  This is generally referred as the root, the 3^rd, and the 5^th.  If you were to continue, you’d skip the A (6^th) and take the B (7^th).  This would yield a C Maj7 chord. You can continue doing this, skipping the octave, and taking the 2^nd of the next iteration of the scale.  This is usually referred to as a 9^th, with the next one being an 4^th/11^th, and then the 6^th/13^th.  That’s as far as chords are generally specified.

We can repeat this process, only starting on the second note of the C scale: D.  In this case what you end up with is:

D, F, A, C

What kind of chord do you suppose this is?  Is it a D Maj7?  The answer is no, and here’s why: In order to understand what these notes mean in the context of the chord’s root note (D), we must compare it to a D scale.   The notes of the D major scale are:

D=E=F#-G=A=B=C#-D

In our mystery chord, we have D, which is clearly the root, but now, instead of an F#, we have an F.  Remember when we said before that a flatted 3^rd indicates a minor scale?  Well, the same holds true in chords.  The ‘A’ is right out of the D scale, giving us a 5^th, however, the ‘C’, in terms of the D Major scale, is a flatted seventh.  This combination of 1,b3,4,b7 is known as a minor seventh chord.

I’ll leave it as an exercise for you to go through the C Major scale, stacking thirds starting on each degree, and comparing the resulting intervals to the major scale beginning on that same degree.  The results are as follows:

The I chord in any key is always a Major 7^th  (spelled: 1,3,5,7).

The II chord in any key is always a minor 7^th  (spelled: 1, b3, 5, b7).

The III chord in any key is always a minor 7^th .

The IV chord in any key is always a Major 7^th  

The V chord in any key is always Dominant 7^th (spelled: 1,3,5,b7)..

The VI chord in any key is always a minor 7^th.

The VII chord in any key is always a half-diminished (spelled 1, b3, b5, b7).

There are, of course, other forms of chords than those listed above, such as:

            C+ (augmented),

            Cdim7 (diminished 7^th),

            C7#5b9,

            And innumerable other alterations.

For detailed examples of how these chords are derived, and what kinds of scales can be used over them, check out my book: Exotic Scales: New Horizons for Jazz Improvisation (ISBN: 1931055602).  By the time we finish the current volume, you should have a pretty good idea how to figure these out on your own.

The main concept you should take away from this chapter is that so long as you play scale tones against chords built solely from those scale tones, you will never have to worry about hitting ‘sour’ notes. 

Tracks 5 and 6 on the audio CD contain a progression built solely of chord tones in the key of C, first with an example solo which uses only the C major scale, and then with just background tracks so you can practice yourself..   Ch1-3.CMP on the computer partition contains a Jammer™ background track you can also use for practice.  The progression, by the way, is: CMaj7-Em7-Am7-Dm7-FMaj7-Bm7b5-G7-CMaj7.  How did I come up with this particular sequence?  Did I just pull them out of the air?  Well, in fact, there are some rules, known as Rules of Cadence, which deal with combining chords in ways that are most musical and contribute to moving the tune forward.  These rules are quite simple, and will serve you well in your songwriting career:

I leave it as an exercise to confirm that my sample progression indeed follows these rules.

If you’d like to try this scale in its ‘A’ Natural Minor context, tracks 7 and 8 contain the following A-minor progression:

Am7-Bm7(b5)-CMaj7-Am7-Dm7-Am7-G7-Am7-FMaj7(#11)-Am7-Em7-Am7

One thing you’ll probably notice as you experiment is that while none of the notes you play sound bad, some sound stronger than others.  We’ll examine why this is, and one way to get around it, in the next chapter. 

The reason that minor pentatonic we all rely upon so heavily works so well is that you can just plow ahead and play it over every chord of most common blues progressions.  But have you ever wondered precisely why this is so?  Let’s have a look at our precious fifth position A blues box (Figure 10) for a minute:

Figure 10

We can see by inspection that it closely resembles our A Aeolian or Natural minor scale (Figure 3), with a few notes missing.  In terms of the A Major scale (A=B=C#-D=E=F#=G#-A), we can see that we have a root, a minor 3^rd , a 4^th (also known as an 11^th), a 5^th, and a minor 7^th  (five notes, hence, pentatonic.)  The flatted third, as discussed in the previous chapter, means this is a 5-note or pentatonic minor scale.  (In fact, it is only one of many other pentatonic minor scales, but it is the one with which most people are familiar.)

Going back to our intervallic view, Figure 11 shows us precisely which intervals comprise this handy scale:

Figure 11

In order to understand why this particular scale always sounds so good in a blues setting, let’s take a look at the quintessential blues form: the 12-bar blues.  This consists, of a I chord, a IV chord, and a V chord, over a 12-bar repeating progression.  There are innumerable variations on the 12-bar blues, but a typical one looks like this (in the key of A):

Table 5

Tracks 9 and 10 on the audio CD contain this progression, with and without a lead track, respectively.  Ch2-1.CMP on the computer partition contains a Jammer™ background track of this same progression.

A useful variation on the pentatonic minor is the blues scale, which adds a #4 to the scale, as shown in Figure 13.  That altered tone, known as the tritone, gives the blues its characteristically boisterous sound.

Figure 12

Let’s have a look at the notes that comprise each chord in the progression, and compare them to the notes we’re hearing when we play our pentatonic minor scale.

The I chord is A7.  As we saw earlier, the notes of the A Major scale are:

A=B=C#-D=E=F#=G#-A

Since a dominant seventh (e.g., A7) chord consists of the root, major third, perfect fifth, and minor seventh degrees, our A7 chord is spelled: A,C#,E,G.  When we play an A pentatonic minor, we’re playing  A, C, D, E, and G (plus the Eb in the case of the blues scale).  Hence, when we’re playing over the A7 chord we’re playing the root, the 4^th/11^th, the fifth, and the minor 7^th right from the chord.  These are all strong tones.  In addition, we’re playing a C (natural) against the chord’s C#, which really makes the solo’s minor tonality stand out.  (Remember, if you played only notes directly from the A7 chord [i.e., an A7 arpeggio], you’d be creating a major/dominant sound, rather then the tangy blues tones we’re looking for.)

The IV chord is D7.  The D major scale is spelled:

D=E=F#-G=A=B=C#-D

The D7 chord consists of D, F#, A, C.

When compared to our D7 chord, the A of our pentatonic minor is the perfect fifth, the C gives us our minor 7^th, the D gives us the chord’s root, the E is the 2^nd/9^th, and the G gives us the 4^th/11^th.  In this case we have all chord tones.  The Eb of the blues scale is a b9, in the context of D7.  That’s a common tone in jazz improvisation, and gives a hip, sophisticated sound when played over this chord.

Finally, the V chord is an E7.   The E major scale consists of the following notes:

E=F#=G#-A=B=C#=D#-E

The E7 chord is spelled: E, G#, B, D.  In terms of this E7 tonality, our A pentatonic minor yields a 4^th/11^th (A),  a minor 6^th  (C), the minor 7^th (D), the root (E), and a minor 3^rd (G).  Here again we have some nice pungent tones present: The minor 3^rd emphasizes the minor tonality against the G# of the E choird.  The minor 6^th, which is not present in the E7 chord, implies a natural minor (Aeolian) scale.  The 11^th is a nice strong extension, and the minor 7^th strengthens the dominant nature of the E7 chord.  Finally, that Eb from the blues scale is a Major 7^th, which makes for a pungent clash with the D (minor seventh) of the chord.

To sum this all up graphically,

Table 6

Intervals shown in parentheses are extensions provided by the solo notes, which are not in the unextended chords.  The italicized degrees are those provided by the blues scale (the tritone), and how they fit into the background chords.

Somewhere along your musical career, someone may have shown you that you can slide the minor pentatonic down by three frets and achieve a totally different sound.  What you’d be playing in this circumstance is the major pentatonic scale.  For example, if you were playing the minor pentatonic at the fifth position, as shown in Figure 11, sliding down three frets would put this same pattern at the second fret, as shown in Figure 13.

Figure 13

Lets analyze this as we did before, in order to understand why this simple approach sounds good over a blues progression.

First of all, we can see that our major pentatonic scale includes the notes: A, B, C#, E, and F#.  In terms of the A Major scale, these represent the root, the 2^nd/9^th, the Major 3^rd, the 5^th, and the major 6^th.    Hence, against our A7 chord, the root and the 5^th tend to solidify the tonal center, while the 9^th and the Major 6^th add some interesting extensions.  That Major seventh sounds real tangy against the minor 7^th of the dominant chord, which gives you just the right sense of tension. 

When played against the D7 chord (D, F#, A, C), our major pentatonic yields a perfect 5^th, a major 6^th, a minor 7^th, a 2^nd/9^th, and a major third; all chord tones or common extensions.  Bear in mind that when you play an extension over a harmonic background, you are actually creating extended harmony, in much the same way as experienced jazz players often omit the root from their chords, secure in the knowledge that the bass is playing the root.  The overall harmonic content is the combination of all these different elements.

Over the E7 (the V) chord of our progression, our major pentatonic scales delivers A (the 4^th/11^th), B (the 5^th), C# (a major 6^th), E (the root), and F# (the 2^nd/9^th).  Here again, we have all chord tones or common harmonic extensions.  Hence, both the minor pentatonic and the major pentatonic will always sound good against each chord of a progression consisting of the I, IV, and V chords.

Track 11 on the audio CD demonstrates the major pentatonic played over the 12-Bar Blues progression.  (Since Track 10 provides the 12-bar background without a lead, there’s no point repeating it.)  As before, here’s a graphical layout

Table 7

This approach should be studied and fully internalized, since it can be used to analyze how any scale will work against any harmonic setting.