Tetrachords

[Jean-Michel G]

Musicians rarely care much about tetrachords (except violin players!). Yes, this is a somewhat nerdy subject, but as we will see it has very practical applications.

The word “tetrachord” comes from the Ancient Greek and literally means “four strings”; it presumably refers to an old lyre-like instrument with – you guessed it: four strings…
For that reason I would rather spell it “tetraCord” (like we do in French), since it has nothing to do with “chords” and a lot more to do with “cords”. But I double- and triple-checked and all references are positive that it should be spelled with “ch”. Oh well!

In music theory, a tetrachord is usually defined as a set of four consecutive notes spanning an interval of a fourth (diminished, perfect or augmented).

A tetrachord is most easily designated by its internal intervallic structure. For example, the tetrachord [C D E F] would be designated by [2 2 1] because there are two semitones between C and D, two semitones between D and E, and only 1 semitone between E and F. Similarly, [D E F G] would be [2 1 2].
A tetrachord can also be represented by the interval of each note with respect to the first note, e.g. [1 2 b3 4].

1. Tetrachord structures
Let’s take the C major scale running from tonic to octave: C D E F G A B C; we can divide that scale into two tetrachords: [C D E F][G A B C].
As you can see, the structure of the two tetrachords is the same: the intervals between the notes is [2 2 1], and there is an interval of a whole tone between the two tetrachords.
So we can represent this scale with [2 2 1] 2 [2 2 1].

A tetrachord whose structure is [2 2 1] is called “Ionian” or “Major” or “Dominant”.
So the major scale consists of two Major tetrachords separated by a W.

Clearly, [2 2 1] is not the only possible structural organisation of a tetrachord. In fact, limiting the discussion to the equally tempered scale, we have the following possibilities:

– tetrachords spanning a perfect fourth (five semitones)
Major/Dominant: [2 2 1], e.g. [C D E F]
Minor/Dorian: [2 1 2], e.g. [C D Eb F]
Harmonic/Gypsy: [1 3 1], e.g. [C Db E F]
Phrygian/Locrian: [1 2 2], e.g. [C Db Eb F]
[3 1 1], e.g. [C D# E F]
[1 1 3], e.g. [C Db Ebb F]

– tetrachords spanning an augmented fourth (six semitones)
[3 2 1], e.g. [C D# E# F#]
[3 1 2], e.g [C D# E F#]
Lydian/Whole tone: [2 2 2], e.g [C D E F#]
[1 3 2], e.g. [C Db E F#]
[2 1 3], e.g. [C D Eb F#]
[1 2 3], e.g. [C Db Eb F#]

– tetrachords spanning a diminished fourth (four semitones)
Diminished: [1 2 1], e.g. [C Db Eb Fbb]

2. Scale construction
If we take the upper tetrachord of the C major scale, i.e. [G A B C] and add another Major tetrachord behind it, we get [G A B C][D E F# G] which is the G major scale.
Doing the same again yields [D E F# G][A B C# D]. And then [A B C# D][E F# G# A]. Etc.
As you can see, we get the sequence of sharps in the circle of fifths: C, G, D, A, …. This is not really a surprise, since each new tonic is a fifth higher than the previous tonic.

We can also take the first tetrachord [C D E F] and add a Major tetrachord in front of it; we then get [F G A Bb] [C D E F]. Repeating this will produce the circle of fifths in the flat sequence: C, F, Bb, Eb, …

Here is a list of the most common scales and their tetrachord constituents:
C Major = C Major tetrachord + G Major tetrachord
C Dorian = C Minor tetrachord + G Minor tetrachord
C Phrygian = C Phrygian tetrachord + G Phrygian tetrachord
C Lydian = C Lydian tetrachord + G Major tetrachord
C Mixolydian = C Major tetrachord + G Minor tetrachord
C Aeolian = C Minor tetrachord + G Phrygian tetrachord
C Locrian = C Locrian tetrachord + Gb Lydian tetrachord
C minor = C Minor tetrachord + G Phrygian tetrachord
C melodic minor = C Minor tetrachord + G Minor tetrachord
C harmonic minor = C Minor tetrachord + G Gypsy tetrachord

You can of course combine tetrachords any way you want in order to build your own scales; some of those scales do actually exist as such in Western music, as we have seen. But other combinations will sound very exotic, like [1 2 2] 1 [1 2 3], e.g. [C Db Eb F] [Gb Abb Bbb C]. Many of the most exotic scales approximate Eastern modes (Indian, Persian, …), but they are only approximations because those modes don’t use the equal temperament so they cannot be played exactly in our musical system.

3. Tetrachords for improvisation
Suppose we have te following chord progressions:
|A |F#m7 |G7 |E7 |A |

A quick harmonic analysis gives: |I |vi7 |bVII7 |V7 |I |
This is A major, except for the G7 chord which is clearly not diatonic to A major.
The interesting question is of course: how do we improvise over that G7 chord?
The chord-scale relationship theory tells us that the Mixolydian mode is one of the easiest way to improvise over a dominant 7th chord.

So, if we treat that G7 as the signature chord of the G Mixolydian scale, we get [G A B C] [D E F G].
Restricting our note choice to one of the two tetrachords, we can use the G major tetrachord or the D minor tetrachord; the first choice uses the root, the 9th, the 3rd and the 11th – and that 11th may not sound very well. The second choice uses the 5th, 6th, b7th and the root – and that should sound great. So we’ll use that in order to avoid the 11th.
The notes we use to improvise over that G7 chord are [D E F G].

We can apply the same logic to all the other chords:
A: we can pick the A major tetrachord (with the 11th) or the E major tetrachord (a better choice): [E F# G# A]
F#m: the F# minor tetrachord will work just fine [F# G# A B]
E7: the B minor tetrachord is a good choice: [B C# D E]

So, for each chord we are left with a choice of four notes (and only four notes!) to improvise with; this is similar to Brian’s concept of “four note boxes”, but much more general.

[sunjamr]

Yep, they’re totally not “chords”, just “cords”. Looks to me like this is another way to keep track of the note intervals when doing improv. Incidentally, in the last example (improvisation), that G7 chord hurts my ears. For some reason, a G6 sounds nicer.

Sunjamr Steve

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