circle of fifths and scales

We touched on the circle of fifths before in this post. The circle of fifths is just that: notes seperated by a fifth (C - G -D - A, etc) and arranged in a circle because (in equal tempered tuning, at any rate) the fifths lead back to the starting point after all 12 tones have been accounted for (...Eb - Bb - F - C).

The circle of fifths (henceforth COF) has many uses, but one I've been playing around with lately is examining how scales look -- i.e. what shapes they take when the collections of notes are joined one to another as in the sequence of a scale. E.g. here's a whole tone scale:

And a chromatic scale:

Not surprisingly the 2 scales above take symmetric shapes when graphically displayed (the scales are symmetric in terms of their construction: comprised of 1/2 steps or whole steps). Surprising -- to me, at any rate -- is that when the major scale is displayed it also forms a symmetric shape:

As does the melodic minor.

Some asymmetric shapes: the harmonic minor:

and the neapolitan minor:

You can do these on your own, of course. Some other symmetric scales: the neapolitan major, the major pentatonic, the double harmonic. Some asymmetric ones: the hungarian minor and the harmonic major.

Lastly we can observe intervals -- actually we've already done this with the chromatic and whole-tone scales (min2 and Maj2 respectively). Here in one diagram are min3, Maj3 and P5:

The remaining intervals can all be found simply by going the other way round: C to Ab can be seen as a Maj3 down or as a min6 up. C to F is a P5 down or a P4 up, and so on...