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Pentatonic Scales: Theory and Applications

Module by: Mathias Lang. E-mail the author

Summary: Pentatonic scales are used a lot in jazz and pop music, for composition as well as for improvisation. Their structure and their practical application are discussed.

Pentatonic Scales: Theory and Applications

Theory

Pentatonic scales contain five notes per octave. A popular example is the major pentatonic scale (e.g., starting on C):

C – D – E – G – A – (C)

The five notes of this pentatonic scale are taken from the seven notes of the C major scale:

CDE – F – GA – B – (C)

Note that the notes F and B cause the C major scale to contain half steps. Since these notes are left out in the pentatonic scale, the major pentatonic scale does not contain any half steps. This creates an open sound and reduces the degree of dissonance when played harmonically. This latter property makes this type of pentatonic scales especially suited for improvisation. If all notes of the C major pentatonic scale are sounded simultaneously, the result is a C6/9 chord.1

Let's have a look at the interval structure of this scale:

M2 – M2 – m3 – M2 – m3

M2 stands for ‘major second’ (i.e., two half steps) and m3 for ‘minor third’ (three half steps). As we can see, there are only two types of intervals in this scale: major seconds and minor thirds. Most pentatonic scales commonly used in Jazz and in popular music contain only these two intervals.

The circle below visualizes the interval structure of the major pentatonic scale. Moving clockwise, we go through all 12 notes of the chromatic scale (represented as points on the circle) and the black ones are the ones contained in the scale. The names of the notes are only shown for the purpose of explanation. What is important is the location of the black points in relation to each other; their relative location represents the interval structure of the scale, regardless of the key.

Figure 1: Interval structure of the major pentatonic scale
Figure 1 (graphics1.png)

If we were to play the scale starting from the second note – in our case the D – we would get another pentatonic scale with a different interval structure (see Figure 1):

M2 – m3 – M2 – m3 – M2

The scale can be started on any scale tone to obtain a new scale. Scales derived in this way are called ‘modes’ of the original scale. Of course, they can be transposed to any key. The circle shown above represents a family of 5 pentatonic scales which are modes of each other. The interval structure shown by the circle is independent of the key. A popular pentatonic scale is obtained by starting the major pentatonic scale on the fifth note (in the key of C major this would be the A): this scale is called minor pentatonic scale and it is used a lot in blues and rock improvisation.

Let's see how many different pentatonic scales containing only two types of intervals – major seconds and minor thirds – are possible. Let m be the (yet unknown) number of major second intervals and let n be the number of minor third intervals in the scale. A pentatonic scale has a total of five intervals and all of them together must span the range of one octave (i.e., 12 half tones). This can be expressed by the following two equations:

(1) 2m+3n=122m+3n=12 size 12{2m+3n=12} {}

(2) m+n=5m+n=5 size 12{m+n=5} {}

Equation (1) says that the number of half steps spanned by m minor seconds ( =2m=2m size 12{ {}=2m} {}) plus the number of half steps spanned by n minor thirds ( =3n=3n size 12{ {}=3n} {}) must equal one octave (= 12 half steps). Equation (2) says that the total number of intervals in the scale must equal 5 because we are considering pentatonic scales. The only solution to this set of equations is m=3m=3 size 12{m=3} {} and n=2n=2 size 12{n=2} {}, i.e., all pentatonic scales containing only major seconds and minor thirds must have 3 major seconds and 2 minor thirds, just like the major and minor pentatonic scales presented above.

The next question we can ask is: how many scales of this type exist? Above we have already shown five scales (the five modes of the major pentatonic scale as shown in Figure 1). But are there more than these 5 scales? Some combinatorics gives the answer. There are 53 5 3 possibilities to distribute three major second intervals over the total number of five intervals of the pentatonic scale.2 Consequently, there are 53=10 5 3 10 different scales. So there are five more scales than we have seen so far. Just as before, these five new scales can be derived as the modes of only one pentatonic scale. An example of such a pentatonic scale with an interval structure different from the one presented earlier is:

C – D – E – F# – A – (C)

Sounding all notes of this pentatonic scale simultaneously gives a D9 dominant chord. This is why the second mode of this scale (starting on D) is referred to as the dominant pentatonic scale. Note that this scale can be derived from the C major pentatonic scale by replacing the G with an F#.

The interval structure of this scale is

M2 – M2 – M2 – m3 – m3

Note that this interval structure cannot be obtained by using a mode of the major pentatonic scale. The interval structure is shown in Figure 2:

Figure 2: Interval structure of the dominant pentatonic scale and its modes
Figure 2 (graphics2.png)

No note names have been attached to the circle to focus on the interval structure and to emphasize its independence of the key. The major scale (and its modes) and this scale (and its modes) represent all possible pentatonic scales containing only major second and minor third intervals.

Of course there are many more pentatonic scales if we allow other intervals than major seconds and minor thirds. These scales are just less popular in Jazz, Blues, and Rock music. Let’s nevertheless see what we find if we also allow major third intervals in a pentatonic scale. In this case, we can come up with the following scale (and its modes):

C – E – F# – G# – A# – (C)

Its interval structure is3

M3 – M2 – M2 – M2 – M2

where ‘M3’ stands for ‘major third’. This scale is basically a whole tone scale with one note left out (in this case the D). If we also allow half steps then the number of possibilities becomes quite large. Many pentatonic scales used in Asia contain one or two half steps (e.g., Pelog, Hirajoshi, Kumoi). Table 1 presents an overview of the number of possibilities. Each row refers to a set of scales containing a certain combination of intervals. The rightmost column gives the number of possible scales for each combination of intervals. The numbers of scales are given as multiples of five, because there are always five modes of each scale. The numbers in Table 1 can be obtained by applying basic combinatorics.

Table 1: possible interval combinations for pentatonic scales
# m2 intervals # M2 intervals # m3 intervals # M3 intervals # possible scales
0 3 2 0 2 * 5
0 4 0 1 1 * 5
1 1 3 0 4 * 5
1 2 1 1 12 * 5
2 0 2 1 6 * 5
2 1 0 2 6 * 5

Table 1: possible interval combinations for pentatonic scales

Table 1 shows all possible interval combinations for pentatonic scales with intervals not greater than a major third. Note that there are no pentatonic scales of this type with more than 2 half steps (minor seconds, ‘m2’ in Table 1) because then the largest interval would need to be greater than a major third. The major and the dominant pentatonic scales (and their modes) and its modes are represented by the first row in Table 1. The Kumoi pentatonic scale used in Japan belongs to the group of scales described by the fourth row in Table 1. The Pelog (Indonesia) and Hirajoshi (Japan) scales belong to the group shown in the last row of Table 1.

Applications

Let's now have a look at how pentatonic scales can be used in practice. We will consider two scale families: the major pentatonic scale and its modes, and the dominant pentatonic scale and its modes. These are the ones that are used very frequently in Jazz and popular music.

The major pentatonic scale can be played over a major triad with the same root as the scale, e.g., a C major pentatonic scale can be played over a C major chord. It can also be played over this chord if tensions are added to the chord. The scale could be played over a C6 chord, a Cmaj7 chord, a C6/9 chord and a Cmaj9 chord etc. This use of the pentatonic scale creates very little tension and may sound a little colorless in certain musical contexts. A scale that sounds more interesting over the same chords is the G major pentatonic scale. Let’s have a look at the notes of the scale:

G – A – B – D – E

Over a C major chord it creates the tensions 6, 7, and 9. Note that it does not contain the root of the chord (C). It is basically a C major pentatonic scale with the C replaced by the B, i.e. one more tension is added at the cost of the root note. Yet another scale which is frequently used over the same chords is a D major pentatonic scale:

D – E – F# - A – B

Used over a C major chord, this scale creates the tensions 6, 7, 9 and #11. This scale neither contains the root (C) nor the fifth (G) of the chord, but it adds the tension #11, which is used quite frequently in modern jazz.

Let’s now look at minor chords: e.g., Dm, Dm7, Dm9, Dm11, Dm6. The most obvious choice is the minor pentatonic scale, i.e. the fifth mode of the major pentatonic scale. The D minor pentatonic scale consists of the following notes:

D – F – G – A – C

Apart from the basic D minor triad (D – F – A), it contains the b7 and the 11. Another, more colorful option over a D minor chord is the A minor pentatonic scale:

A – C – D – E – G

Here, the minor third of Dm (the F) is left out and the tension 9 (the E) is added instead. This implies a Dm9 sound. For a Dm6 chord, there are two more options. One is the E minor pentatonic scale:

E – G – A – B – D

Over a Dm6 chord it creates the tensions 6, 9, and 11. Another option is the G dominant pentatonic scale:

G – A – B – D – F

It just replaces the E of the E minor pentatonic scale (the 9 of Dm) with an F (the third of Dm).

Another important group of chords are dominant chords. They come in two flavors: unaltered and altered. Unaltered dominant chords can have the tensions 9, #11, and 13, and altered dominant chords can have the tensions b5, #5, b9, #9, #11, b13 (where the #11 and the b13 are enharmonically the same as the b5 and the #5, respectively). Over an unaltered G7 chord, the G dominant scale creates the tension 9 (the A):

G – A – B – D – F

Higher tensions are obtained by using the A dominant scale:

A – B – C# - E – G

It does not contain the b7 but it adds the #11 (C#) and the 13 (E). The G major pentatonic scale could also be played over a G dominant chord but since it does not emphasize the dominant sound of the chord it sounds less interesting.

Over an altered G dominant chord we can use the Db major pentatonic scale:

Db – Eb – F – Ab – Bb

It contains the b7, b9, #9, #11, and b13 of the G dominant chord and in this way it effectively emphasizes the altered sound of the chord. Another possibility is the Db dominant pentatonic scale which differs from the Db major pentatonic scale by only one note: the Bb is replaced by a B, which means that an altered note (the #9) is replaced by the third of the G dominant chord.

As a final note on dominant chords, let’s also look at dominant chords with a suspended fourth: e.g., G7sus4 or F/G (an F major triad over a G root). For such a chord it sounds better to use other scales than for a G7 chord with no suspended fourth. The reason for this is that a scale for G7sus4 should contain the suspended fourth of the chord (the C) in order to underline the chord’s character. However, for a G7 chord with no suspended fourth, the note C sounds dissonant and can better be replaced by the third of the chord (the B). There are two good sounding choices for a G7sus4 chord: the F major pentatonic scale (F – G – A – C – D), which is actually a G9sus4 chord when sounded together, and the C major pentatonic scale (C – D – E – G – A), in which the F (the b7 of G7) is replaced by the E (the 13 of G7).

The last category of chords that we will consider here are half-diminished chords, e.g. Bm7(b5). Two good choices for improvising over this chord are the G dominant pentatonic scale and the G major pentatonic scale:

G – A – B – D – F (G dominant pentatonic)

G – A – B – D – E (G major pentatonic)

The only difference is that in the G major pentatonic scale, the F (i.e. the b5 of Bm7(b5)) is replaced by E (i.e. the 11 of Bm7(b5)). If all notes of the G dominant pentatonic scale are sounded together, the resulting chord can be interpreted as a Bm7(b5, b13) chord. This means that the G dominant pentatonic scale is a bit closer to the actual chord whereas the G major pentatonic scale adds a higher tension (the 11) at the expense of a basic chord tone (the b5). Another pentatonic scale that could be used over a Bm7(b5) chord is the F major pentatonic scale:

F – G – A – C – D

It is also very similar to the G dominant pentatonic scale, but it contains a C instead of a B. The C is the b9 of the Bm7(b5) chord and is consequently an ‘avoid note’. For this reason, the F major pentatonic scale is less suited and more difficult to use over a Bm7(b5) chord than the two other scales discussed above.

Table 2 summarizes the chord types and the most useful pentatonic scale over these chords.

Table 2: pentatonic scales for different chord types
chord type pentatonic scales
Cmaj7(9,#11,13) C major
  G major
  D major
Dm7(9,11) D minor
  A minor
Dm6 E minor
  G dominant
G7(9,#11,13) G dominant
  A dominant
G7(b9,#9,#11,b13) Db major
  Db dominant
G7sus4 F major
  C major
Bm7(b5) G dominant
  G major

Let’s now apply pentatonic scales to a very common chord pattern in Jazz: the 2 – 5 – 1 progression. The name of this progression indicates that it consists of the chords built on the scale degrees 2, 5 and 1 of the major or minor (seven-tone) scale. A 2 – 5 – 1 progression in C major is:

Dm7 – G7 – Cmaj7

If we want to use pentatonic scales for improvising over this progression, the most obvious choice would be D minor pentatonic over Dm7, G dominant pentatonic over G7, and C major pentatonic over Cmaj7. If we wanted to add more color, we could also choose A minor pentatonic (Dm7), A dominant pentatonic (G7), and G major pentatonic (Cmaj7). So far we have assumed that the G7 chord is not altered. If the G7 chord is altered and if we would like to use a lydian sound over the Cmaj7 chord (i.e., we want to include the #11 of C major), we could use the following chromatically ascending sequence of scales: A minor pentatonic (Dm7), Db major (= Bb minor) pentatonic (G7 altered), and D major (= B minor) pentatonic (Cmaj7).

In minor, a typical 2 – 5 – 1 progression would be:

Bm7(b5) – E7(b9,b13) – Am9

You could improvise over this progression by using the G dominant pentatonic scale over Bm7(b5), the Bb major (= G minor) pentatonic scale over E7(b9,b13), and the E minor (= G major) pentatonic scale over Am9.

Note that I have only mentioned the most common choices here and that you might come up with other pentatonic scales that may sound even more pleasing to you over certain chords or chord progressions. I just wanted to summarize what I consider the basic options that every Jazz musician should know because they are used so frequently. To a beginner, the information given here may seem overwhelming because of the large number of possibilities. The solution to this initial problem is, as far as I can see, to relate each pentatonic scale to the chord over which it is used. This means that if you play a Bb major pentatonic scale over an altered E7 chord, you shouldn’t think of it as a Bb major pentatonic scale but you should try to see its relation to E7 and consider it a subset of the possible notes over E7. The names of the scales in Table 2 are often not related to the root of the corresponding chord (e.g., G major pentatonic over Cmaj7, F major pentatonic over G7sus4 etc.) but this is just to keep the names of the scales as simple as possible. When playing, just try to see and hear the chord on your instrument and view the scale notes according to their function with respect to that chord (i.e., the third, the b7, the #11 etc.). With a lot of practice and ear training you will finally ‘automatically’ choose an appropriate pentatonic scale which suits your purposes.

Footnotes

  1. Other common interpretations of this chord are Am7(add11) or D9sus4.
  2. nk n k is the binomial coefficient defined as nk=n!(nk)!k! n k n n k k . With n=5 n 5 and k=3 k 3 we obtain 53=5!(53)!3!=5!2!3!=5×4×3×22×3×2=10 5 3 5 5 3 3 5 2 3 5 4 3 2 2 3 2 10 .
  3. Technically speaking, the interval between A# and C is a diminished third, but for the sake of simplicity I call it ‘M2’ here, indicating that it is simply an interval of two half steps.

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