Coscale Symmetries

Continuing from the previous post about reflexive symmetries within and between scales, here we look at another kind of symmetry which I call the "coscale relationship". As with the previous post, at this stage this material is purely theoretical.

Definition and Example

We can think of a scale in a particular key as a collection of notes, a subset of the Chromatic scale in the same key. The coscale of this scale is the scale whose notes are the notes that the original scale left out.

Take the Major scale as an obvious example. Its notes are

C   D   E   F   G   A   B

These are all the white notes on the piano; the notes that are left out are the black notes, which are

C#  D#  F#  G#  A# 

This certainly isn't a scale in the key of C, but if we look at its interval map (23223) we can see that it's a member of the Common Pentatonic group.

We'll say that two scale groups exhibit "coscale symmetry" if any scale from one group has this relationship with a scale in another group.

Note that only a handful of hexatonic scale groups can be symmetrical within themselves. For example, the Whole Tone scale group (which consists of only one scale) is its own coscale, since the interval map of the left over notes is the same as the interval map of the notes in the original scale.

More Examples

Let's look at the common scales to find their coscale groups. We already know that the Major and Common Pentatonic are coscales of each other, so what about the Harmonic Minor?

The easiest way to proceed seems to be the same way as above. A Harmonic Minor contains the notes

A   B   C   D   E   F   G#

and so its coscale group can be generated by the notes

A#  C#  D#  F#  G

The interval map of this set of pitches is 32313, an exotic scale that I call the 4d+b6maj scale. Here are the CAGED fingerings as they appear in the Encyclopoedia:

We can do the same thing with A Melodic Minor, which contains the notes

A   B   C   D   E   F#   G#

and so its coscale group can be generated by the notes

A#  C#  D#  F  G

which has the interval map 32223, which is the scale I call the 1min+4maj scale but that's more commonly known as the Minor 6 pentatonic.

So the Harmonic Minor group has a coscale relationship with an exotic pentatonic, whereas the Melodic Minor's coscale group is a fairly common blues scale. There doesn't seem to be much rhyme or reason here.

Applications

As with reflexive symmetry, it's not yet clear to me what applications, if any, this relationship between scale groups might have. Once you've noticed it I suppose the fingerings of coscales have an interesting relationship, but I doubt they could be used to assist with learning.

Coscales played together cover all 12 notes of the chromatic scale, and this is where I think there might be a musical application. Playing A Melodic Minor followed by A# Minor 6 Pentatonic, for instance, gives all 12 pitches. The analysis in the Encyclopoedia tells us we can further simplify this by thinking of it as the Melodic Minor followed by the b2 minor and #4 major triads. This may help with organising fully-chromatic material in a familiar-sounding way. At this point, however, that's just speculation. I might post some lessons later with example licks derived in this way so you can see what you think.

Finally, there's a classificatory function that all such relationships have, and it was this that led me to investigate coscales as part of the organisation of hexatonic scales in the new, expanded edition of the Encyclopoedia.