What is the Major Semiquintal Axis?

The major semiquintal axis is a pitch collection which functions as a kind of "outer structure" within which the major scale (and many other scales) resides. It is generated by stacking neutral thirds (7 quarter tones, 350 cents) instead of by stacking perfect fifths (14 quarter tones, 700 cents). It allows for chord progressions of double the "resolution" of the major scale.

The reason I'm calling this an "axis" is to attempt to differentiate it from any expectations that a musician might have of a scale. To a musician, a scale may be something that they can meaningfully play melodies from directly, whereas I tend to treat the MSA differently. There are other reasons, too. More on this later. For now, let's dive into the structure of the MSA.

Formula

The formula for building the MSA stepwise in quarter tones is as follows:

1 - 3 - 3 - 1 - 2 - 1 - 3 - 1 - 2 - 1 - 3 - 1 - 2

In C

The MSA in C is as follows:

C - C+ - D - Ed - E - F - F+ - G - G+ - Ad - A - Bd - B

Note that in the above notation, "+" indicates "half-sharp" and "d" indicates "half-flat".

An interesting consequence of building the MSA with a generator that is half the size of the one we use for the major scale is that we are provided with intermediary modes.

C  - Ionian
C+ - Semiphrygian 
D  - Dorian 
Ed - Semimixolydian 
E  - Phrygian 
F  - Lydian 
F+ - Semiaeolian  
G  - Mixolydian 
G+ - Semilocrian 
Ad - Semilydian 
A  - Aeolian 
Bd - Semidorian 
B  - Locrian

It may become more immmediately obvious how these relate when we put them in neutral thirds order:

F  - Lydian 
Ad - Semilydian 
C  - Ionian
Ed - Semimixolydian 
G  - Mixolydian 
Bd - Semidorian 
D  - Dorian 
F+ - Semiaeolian  
A  - Aeolian 
C+ - Semiphrygian 
E  - Phrygian 
G+ - Semilocrian 
B  - Locrian

This can be further enriched by adding representations of modal brightness:

F  - Lydian             o++++++    (Brightest Sounding)
Ad - Semilydian         o+++++
C  - Ionian             o++++
Ed - Semimixolydian     o+++
G  - Mixolydian         o++
Bd - Semidorian         o+
D  - Dorian             o
F+ - Semiaeolian       -o
A  - Aeolian          --o
C+ - Semiphrygian    ---o
E  - Phrygian       ----o 
G+ - Semilocrian   -----o
B  - Locrian      ------o          (Darkest Sounding)

It's worth noting here that the intermediate modes are named from the perspective of the major scale, in that they are named as transitional modes from a mode closer to the major scale to one further from it. This is mainly a nod to the traditional primacy of the major scale, and also because "Semiionian" is not particularly pleasant to write or say.

Chordal Structure

As a natural result of the modal structures of the MSA, each degree is root to specific chords; some familar, others novel:

F  - Major 7 Sharp 11       (Δ7#11) 
Ad - Major 7 Half-Sharp 11  (Δ7+11) 
C  - Major 7                (Δ7) 
Ed - Half-Flat 7            (d7)
G  - Dominant               (7) 
Bd - Neutral 3 Minor 7      (N3-7) 
D  - Minor 6                (-6) 
F+ - Minor Half-Flat 6      (-d6) 
A  - Minor 7                (-7) 
C+ - Minor 7 Half-Flat 9    (-7d9)
E  - Minor 7 Flat 9         (-7b9) 
G+ - Minor 7 Half-Flat 5    (-7d5) 
B  - Minor Seven Flat Five  (-7b5)

Summary

In this short article, we have covered the fundamentals of the MSA - its construction formula, letter names in C, modal and chordal structure. In future installments, we will dive further into the various qualities of the MSA, and how they can be applied musically to achieve unique sonic expressions.

-- J.