In many hours of doodling, I found that nested Stars of David make an interesting way of visualizing the musical octave. In particular, they’re a good way of visualizing the even divisions of the octave: as two sets of six notes (the two whole tone scales), as three sets of four notes (the three fully diminished seventh chords), and as four sets of three notes (the four augmented triads).
In mathematical discussions of music it’s common to represent the octave as a clock face (see my previous post on that), and it’s common to visualize note sets as polygons inside the clock face. But when you try to represent the whole tone scales, diminished seventh chords, and augmented triads all at once, you get two overlapping hexagons combined with three overlapping squares, combined with four overlapping equilateral triangles: a tangled mess! It’s an interesting information design problem to try to represent all of those relationships clearly, in the same diagram. I think nested Stars of David do that in a very beautiful way. The idea is to place a note at each of the outer points of the Star of David, progressing clockwise, with C at the top; then draw another star inside the main star. Each line segment on the outside of the larger star represents a half step between two notes, and each triangle edge represents a major third.
Here you can see the two whole tone scales represented in yellow (B, C#, D#, F, G, A) and light blue (C, D, E, F#, G#, A#). You can “ascend” each scale by moving clockwise from one point to the next point of the same color:
Here you can see the four augmented triads as triangles in light blue (C, E, G#), fuschia (D, F#, A#), light green (A, C#, F), and yellow (B, D#, G). Comparing this diagram with the first one, you’ll see that each whole tone scale contains two of the augmented triads:
And here you can see the three fully diminished seventh chords as rhombuses, represented in light blue (C, D#, F#, A), fuschia (C#, E, G, A#) and yellow (B, D, F, G#). Each one has a combination of notes from all four augmented triads:
Note that the coloring scheme is somewhat arbitrary–if you’re one of those people who has strong synaesthetic associations between notes and colors, these diagrams might be painful to look at (sorry!). For everyone else, hopefully they’ll be fun to check out.
From these diagrams you can deduce some interesting musical facts. For example, did you know that if you look at any augmented triad and any fully diminished seventh chord, they’ll have exactly one note in common? In fact, every one of the twelve tones can be identified as the common tone in one specific augmented/diminished pairing. The way to see this in the star diagram is to notice that each of the four main triangles (the two that make the outer star and the two that make the inner star) share a vertex with exactly one of the three rhombuses. For example, here is A aug in green and A# dim7 in fuschia. The common tone (shared vertex) is C#: