Tetrachord scales on the guitar

Recently, while listening to a talk on standard guitar tuning, I was introduced to the concept of tetrachords as a way of navigating the guitar fretboard. This was a fascinating conceptual shift which motivated me to explore further the application of tetrachords to scales on the guitar neck. I had always wanted to expand my scale knowledge and this was a method that gels well with systems software thinking, being grounded in composability, abstraction and symmetry.

The goal of the text below is to derive 3-octave guitar scales from tetrachords. Tetrachords are 4-note scale fragments spanning a perfect fourth interval. Viewing the guitar fretboard through this lens appears to confer a number of advantages, including:

• Building scales from music theory fundamentals. Other methods for deriving scales start with learned patterns such as CAGED, 3NPS, etc. In cases such as CAGED, there is an organizing principle, namely the respective chord shapes, below we build directly on the scale definitions. All scale memorization is based around awareness of key anchor ports on the fretboard, the perfect intervals [1, 4, 5, 8].
• Composability. 7-note scales are built out of pairs of tetrachords, of which only a handful are commonly needed. Tetrachord building blocks are snapped together to form scales like Lego or Tetris pieces.
• Scalability. Knowing 5 tetrachords gives 25 derived 7-note scales. Learning scales sub-linearly, O(sqrt(N)) with the number of scales. This advantage is tempered by the limited number of scales and modes that are contextually relevant.
• Minimizing rote memorization. It's not even necessary to remember that a major scale consists of the WWHWWWH intervals. Instead, only the (shorter) major tetrachord intervals WWH needs to be remembered and the fact that a major scale consists of two concatenated major tetrachords. The tetrachord names used below also assist here, acting as a mnemonic device for their associated scales.
• Range. The entire fretboard and guitar range can be covered with reusable patterns.
• Improvisation. Tetrachords can be used as an organizing principle for improvisation. Tetrachord intervals can imply [1][2] tendencies in melodic movement.

The derived scales below follow logically from tetrachord definitions. While some efficiency considerations are made, e.g. reducing position shifts with the left hand and awkward stretches, these scale patterns are not always optimal for playability. As an early intermediate-level guitar player, my own experience so far is that the win from tetrachordal thinking on the guitar is having a consistent and efficient method for understanding music theory on the fretboard.

All diagrams below are generated via a Python script, which provides an executable specification of the tetrachords and scales.

Tetrachord definitions

Tetrachords can be defined as 4-note scale fragments covering a perfect fourth interval, e.g. the intervals 1 ♭2 ♭3 4. A helpful introduction to the history and function of tetrachords can be found here.

Example — the major and melodic minor scales

The major tetrachord consists of the whole/half steps WWH, covering the intervals 1 2 3 4. If another major tetrachord is concatenated to this sequence with a whole step between the two major tetrachords, we end up with the diatonic major scale. This can be visualized as the steps WWH-W-WWH, where the bold sequences are major tetrachords. Note the symmetry and that the tritone between the P4 and P5 intervals acts as a mid-point.

In general, scales have the form lower tetrachord-W-upper tetrachord.

On the guitar fretboard, when applied as a single-string scale starting on low G, the major scale is:

In the diagrams above and below, each type of tetrachord has a distinct color and a slightly darker note indicates the beginning of the tetrachord.

Another example is the melodic minor scale, where the minor tetrachord WHW is combined with the major tetrachord as minor tetrachord-W-major tetrachord to give the steps WHW-W-WWH:

Tetrachord building blocks

Rather than use whole/half step notation, e.g. WWH, we switch to half-steps below, e.g. 221. Halfstep notation supports larger intervals, e.g. 131.

We now introduce 4 tetrachords. The terminology is similar to that used here , except instead of "upper minor" the name "Phrygian" is used. The rationale for this is that by associating names of tetrachords with scales they typify, they are somewhat easier to learn and remember. Names are helpful in learning scales, for example a Phrygian scale is two concatenated Phrygian tetrachords, the harmonic minor scale is a harmonic tetrachord concatenated to a minor tetrachord.

Tetrachord	Halfsteps	Intervals as lower tetrachord	Intervals as upper tetrachord
Major	2 2 1	1 2 3 4	5 6 7 8
Minor	2 1 2	1 2 ♭3 4	5 6 ♭7 8
Harmonic	1 3 1	1 ♭2 3 4	5 ♭6 7 8
Phrygian	1 2 2	1 ♭2 ♭3 4	5 ♭6 ♭7 8

Each tetrachord can be split across two strings of the guitar as 4 distinct shapes, which are cataloged below for each tetrachord. Due to the similarity to Tetris pieces, they are named with tetromino-like terminology.

It may appear there is a lot to remember (16 distinct patterns) and commit to muscle memory. However, in practice it is only necessary to know:

• The I-shape, the defining intervals of the tetrachord.
• Some combination of Z-shape and J-shape, depending on scale and how optimized for playability scales need to be (discussed more below). J-shapes that span 5 frets involve larger stretches or shifts, so might be less preferable.

The R-shape is simply a truncated I-shape, where fretboard topology gives the fact that the P4 will always be directly below the P1 (or the P8 directly below the P5). Muscle memory for the I-shape should apply to the R-shape. The same should be true for the J-shape, acting as a suffix for the I-shape on the second string, but this seems anecdotally harder to capture through muscle memory.

Major tetrachord shapes

I-major	R-major	Z-major	J-major

Minor tetrachord shapes

I-minor	R-minor	Z-minor	J-minor

Harmonic tetrachord shapes

I-harmonic	R-harmonic	Z-harmonic	J-harmonic

Phrygian tetrachord shapes

I-Phrygian	R-Phrygian	Z-Phrygian	J-Phrygian

Deriving scales from tetrachords

A selection of scales built from the 4 tetrachords above are provided in the table below. There are others that can be derived from these tetrachords that are not listed, e.g. harmonic major.

The only thing in this table that requires commitment to memory is the mapping from scale name to lower and upper tetrachord name, which is simplified by the naming conventions and/or general music theory knowledge.

Scale / Mode	Lower tetrachord	Upper tetrachord	Halfsteps	Intervals
Major	Major	Major	221-2-221	1 2 3 4 5 6 7
Mixolydian	Major	Minor	221-2-212	1 2 3 4 5 6 ♭7
Dorian	Minor	Minor	212-2-212	1 2 ♭3 4 5 6 ♭7
Natural minor	Minor	Phrygian	212-2-122	1 2 ♭3 4 5 ♭6 ♭7
Harmonic minor	Minor	Harmonic	212-2-131	1 2 ♭3 4 5 ♭6 7
Melodic minor	Minor	Major	212-2-221	1 2 ♭3 4 5 6 7
Neapolitan minor	Phrygian	Harmonic	122-2-131	1 ♭2 ♭3 4 5 ♭6 7
Phrygian	Phrygian	Phrygian	122-2-122	1 ♭2 ♭3 4 5 ♭6 ♭7
Phrygian dominant	Harmonic	Phrygian	131-2-122	1 ♭2 3 4 5 ♭6 ♭7
Double harmonic	Harmonic	Harmonic	131-2-131	1 ♭2 3 4 5 ♭6 7

Single-string scales

Single-string scales can be derived by placing an I-shaped lower tetrachord next to an I-shaped upper tetrachord, with a whole step between the P4 end of the lower tetrachord and the P5 start of the upper tetrachord. For example, the harmonic minor single-string scale:

Fretboard diagrams for single-string scales are provided here.

3/2 scales

Single-string scales are a nice way to visualize a scale, but require constant shifting to play and only cover a single octave over 12 frets. In this section we derive 3-octave scales that make use of all strings and the length of the fretboard. They retain the visualization advantages of linear scales and have fewer shifts while playing within an octave.

The scales combine tetrachords across strings. An octave of each scale spans two strings, the lower string containing the lower tetrachord and the upper string containing the upper tetrachord. 3 pairs of strings cover the 3-octave range on a 6-string guitar. This 3 sets of 2 strings method of applying tetrachords to the fretboard is discussed here.

A starting point for this approach is to view the fretboard through patterns of perfect intervals. For example, starting on the low G we have P1, P4 and P5 immediately as:

The same pattern can be extended to the next two strings:

If we attempt to do the same to the top two strings on the guitar, we see a hiccup on the B-string:

The same pattern is applied across each of the 3 pairs of strings, but since standard tuning has a P4 interval between all strings except between the G-string and B-string, where the interval is M3, we have to adjust for a B-string skew. On a guitar with all-fourths tuning, this skew would not occur and the patterns would tile consistently across all 3 pairs of strings.

The perfect intervals above form a scaffolding on which 3/2 scales can be built.

All the 3-octave scales below involve application of patterns across pairs of strings with a B-string halfstep adjustment. In code, it's easier to create scale patterns assuming all-fourths tuning internally and then adjust for the standard tuning irregularity at the presentation layer, as done in the Python script

The major scale can be constructed by taking an R-major shape and placing this at the starting position:

Another R-major shape can then be placed at P5 to form a complete major scale.

This can be repeated on the next two strings:

And also on the next pair of strings, accounting for the B-string hiccup:

Note that the final R-major is flattened to an I-major shape since we have run out of strings.

The full set of 3/2 scales built on R-shape tetrachords is here.

Optimization

The 3/2 R-shape scales provide a logical relationship between tetrachords and scales that can be visualized on the fretboard easily with the split between lower/upper tetrachord clearly visible. They are somewhat awkward to play however, when compared to patterns such as CAGED, since they involve large stretches. The major scale above has as stretch of 5 frets on the first string and then spans 7 frets on the next string.

Some optimization goals / constraints:

• Support the playing of a full octave without position shift.
• Minimize stretches and hand movement in a position.
• No more than 4 notes per string.

Scales below freely also use Z and J tetrachord shapes in addition to the R-shape to achieve this property. The shape choices were arrived at via manual experiments with alternative shapes and evaluating against the above constraints. Some observations:

• Notes separated by a halfstep interval have strong affinity for remaining on the same string as each other. This is why the harmonic tetrachord always uses the Z-shape, the minor tetrachord never uses the Z-shape and the Phrygian tetrachord never has the J-shape. R-major is never used in the optimized scale patterns.
• Only major and minor tetrachords benefit from the J-shape and it only appears as the upper tetrachord.
• For the set of 10 scales, it's necessary to remember a total of 5 additional tetrachord shapes (3 * Z + 2 * J) beyond the tetrachord definition given by the I-shape.
• In general, there is no frame rule. The choice of tetrachord shape is not a pure function of position but can depend on the complementary tetrachord in the scale and its shape.
• While learning these scales requires additional rote memorization, it's still easy to see the intervalic relationships due to the scaffolding of diagonal perfect interval shapes.
• We've probably arrived at a similar place and set of considerations that led to other 3-octave scale patterns such as Segovia scales. The differences probably reflect a different choice of constraints.
• Taking the first octave of the E-pattern in CAGED, tiling it on the perfect interval scaffolding and adjusting for flats as per the scale definition, with some wrapping rules, probably would also work as a method of deriving these scales. This would not have the chunking vantage point that tetrachords bring to scales.

Major

• Lower tetrachord: Z-major
• Upper tetrachord: J-major

Mixolydian

• Lower tetrachord: Z-major
• Upper tetrachord: J-minor

Natural minor

• Lower tetrachord: R-minor
• Upper tetrachord: Z-phrygian

Harmonic minor

• Lower tetrachord: R-minor
• Upper tetrachord: Z-harmonic

Melodic minor

• Lower tetrachord: R-minor
• Upper tetrachord: Z-major

Dorian

• Lower tetrachord: R-minor
• Upper tetrachord: R-minor

Phrygian

• Lower tetrachord: R-phrygian
• Upper tetrachord: Z-phrygian

Phrygian dominant

• Lower tetrachord: Z-harmonic
• Upper tetrachord: Z-phrygian

Neapolitan minor

• Lower tetrachord: R-phrygian
• Upper tetrachord: Z-harmonic

Double harmonic

• Lower tetrachord: Z-harmonic
• Upper tetrachord: Z-harmonic