Scalar Tetrachords

There are a host of skills required for effective improvising on the guitar. Ultimately, improvising music is not entirely a reductive process, and yet this site is premised on the notion that it is helpful to isolate and develop reductive skills. In this instance, I will focus on scales, and in particular, two important abilities for guitarists to develop: 1) the ability to play long scale lines, extending if need be from the top of the instrument to the bottom, and vice-versa; 2) the ability to arbitrarily transition from one scale to another, regardless of position within the scale, and with good voice-leading. From a pedagogical perspective, I have heard both of these skills emphasized in the teachings of Barry Harris, and they are also a key part of my studies with Mark Sherman.

As I began my own practice focused on these two areas, I found myself wondering why both of these skills are so difficult to develop on the guitar in particular, as opposed to, for example on the piano?

I think that part of the reason lies in the fact that we guitarists tend to have an over-emphasis on practicing scales strictly in position. While it is true that scale practice in a single position leads to good muscle memory and fast execution, it places an obvious limit on range, with a typical position providing at best just over two octaves. Position playing also tends to "chunk" a scale into the entire position, making it difficult to change to any arbitrary scale at any point within the scale.

In my own practice, I have come to understand that working on scales in four-note groups, or tetrachords, helps to solve both problems. In this article, I will focus on the tetrachords themselves, how they are constructed, and how they are combined to create scales. I will leave the details of how to effectively practice them for further articles and lessons. In other words, this article is more focused on the theory rather than the practice of this area.

Constructing scalar tetrachords: intervallic constraints

What makes a tetrachord "scale-like" or scalar? The two key properties of scales are that they are constructed of small intervals, and they continue in one direction. Let's dive in and start constructing tetrachords by listing some constraints on which small intervals may be used. Let's define the small intervals as being intervals of 1, 2, or 3 semitones. But to begin with, let's look at tetrachords formed by intervals of 1 or 2 semitones.

Constraint 1: An interval of 1 is allowed, but may not be repeated and must be followed by a different interval; put another way, the interval subset i(1-1) is not allowed.

Constraint 2: An interval of 2 is allowed, and may also be repeated.

With only these two constraints in play, we can generate a group of five tetrachords which combine to form many of the most important scales in jazz: the diatonic modes, melodic minor modes, diminished scale and wholetone scale. For each tetrachord, I will show 1) a proposed name for the tetrachord, 2) the ordered interval set used to construct the tetrachord, 3) an ordered pitch set of the tetrachord using integer notation, 4) a comment on the symmetrical properties of the tetrachord.

Now let's revisit our first constraint about intervals of 1. If we now allow immediate repetitions of interval 1, we add three more tetrachords. This brings our total to eight tetrachords, which is the total number of tetrachords one can construct out of the intervals 1 and 2.

Constraint 1a: An interval of 1 is allowed, and also may now be repeated.

With these three added tetrachords, we can now also construct the chromatic scale, as well as the family of bebop scales. Together with the scales we added previously, this now covers almost all of the most widely-used scales in jazz. However, we still have not been able to construct the pentatonic scales, or the harmonic minor and harmonic major scales. In order to add those scales, we will need additional tetrachords that include the interval of 3 semitones, so let's address that now, with a new constraint.

Constraint 3: An interval of 3 is allowed, but may only be used once in the tetrachord.

Constructing scales with the tetrachords

Until now, we have been thinking of tetrachords as entities in themselves, so now let's look at how the various tetrachords combine to form scales and modes.

Diatonic modes

Figure 1 diatonic modes

Melodic minor modes

Figure 2 Melodic minor modes

The diminished scale

The diminished scale has only two modes. Some people in jazz pedagogy like to call the two modes the "half-whole" and the "whole-half," and some go even further and call these two separate scales to be used on different chords, e.g., use the "half-whole" on a dominant 7th chord, the the "whole-half" on a diminished 7th chord. As you may have guessed, I do not care for that way of thinking, because it obscures the relationship of the dominant and diminished 7th chords, and also makes the diminished scale seem far less interesting than it is--which is a topic that I will explore in depth elsewhere on this site.

Figure 3 diminished modes

Because the diminished scale is an octatonic scale, two tetrachords align perfectly with one octave of the scale. That means that the next octave will repeat exactly the previous two tetrachords. But, as we see, each mode consists of two repetitions of the same tetrachord, so the next octave will repeat the same tetrachord, and so on until you reach the top of your instrument. So, for example, three octaves of mode 1 would proceed as diminished - diminished - diminished - diminished - diminished - diminished. And likewise, three octaves of mode 2 would proceed as minor - minor - minor - minor - minor - minor. This is so conceptually simple, and physically simple, that it makes long diminished scale runs actually quite easy to play, and yet it is not usually taught this way.

The wholetone scale

Since the wholetone scale is a hexatonic scale, two tetrachords will "overshoot" the octave.

Figure 4 wholetone modes

Harmonic minor modes

Figure 5 harmonic minor modes

Harmonic major modes

Figure 6 harmonic major modes

Pentatonic scale modes

The chromatic scale

Because the chromatic scale is a dodecaphonic scale, it takes three tetrachords rather than two to exhaust all the notes in the scale. And, like the wholetone scale, it has a single scalar generator, the interval 1, which means it has only one mode.

The bebop scale