Der übermassige Drieklang

As I've mentioned before, many symmetrical musical structures are rich in canonic possibilities. For the next few days, I'll explore one of these structures -- the whole tone scale.

This scale, to state the obvious, is made up of all whole steps. Its extremely symmetrical nature makes it easy to have exact imitation and exact inversions -- on any scale degree -- without leaving the scale.

However, this nature giveth and it taketh away. What we gain in imitative facility, we lose in intervallic variety. There are plenty of whole steps, major thirds, tritones, minor sixths and minor sevenths. In fact each of these intervals exists on each scale degree but there are absolutely no half steps, minor thirds perfect fourths, perfect fifths, major sixths or major sevenths. Not just a paucity of these intervals -- none, period.

It is easy to create harmonies of augmented triads but very difficult to imply much else. There are no major or minor triads, at all and progression by fifths is impossible.



So, to today's canon...a puzzle canon.

(click on image to enlarge)



This is a strange little canon patterned after Bach's Trias Harmonica but using the whole-tone scale instead of the major scale. Instead of the major triad, it generates the augmented triad. Since Bach, a German, named his canon in Latin, I feel compelled to name mine in German: Der übermassige Drieklang.

This is offered in the same spirit as Bach's little canon -- a simple demonstration of contrapuntal/harmonic facts about triads and traditional voice leading.

Here is the solution:
(click on image to enlarge)


Generating a gently fluctuating augmented triad in a whole tone environment is no great feat. Tomorrow, and for the next few days we'll see what else I can squeeze out of this scale.