transforming the circle of fifths

Yesterday we looked at the circle of fifths (COF) and regarded how some scales and intervals looked when displayed on that circle. But as interesting as the COF is, it isn't the only way to display all 12 pitch classes: another intuitive way to do so would be to arrange the pitches chromatically. Here's how a circle of minor 2nds (COM2) looks:


It might not be obvious at first glance, but there's an interesting relationship between the COM2 and the COF. To start to see it let's look at how the whole tone scale -- or major 2nds -- looks when displayed on the COM2:

And let's recall how the same collection of tones looks on the COF:

They're the same shape! The two circles differ only by the tones not a part of this collection, and they differ in a very systematic way. The easiest way to see the pattern is like this:

that is, the tones not a part of the major 2nd collection simply swap places with their tritone.

Another way -- a more arithmetical way -- to understand this transformation is to use a formula:

1. Convert all the tones to numbers as in 12-tone music (C=0, Db=1, D=2 and so on).
2. Multiply all tones by 7 mod 12 (i.e. multiply by 7 and if the number is 12 or bigger subtract 12: if it is still bigger subtract 12 again, and so on until the number in question is less than 12).

Some examples:

C [0 M7 mod 12 = 0 x 7 = 0 =] C
Db [1 M7 mod 12 = 1 x 7 = 7 =] G
D [2 M7 mod 12 = 2 x 7 = 14 -12 = 2 =] D
A [8 M7 mod 12 = 8 x 7 = 56 (mod12) = 8 =] Eb

This multiplicative operation is a way of extending the basic operations of 12 tone music (retrograde, inversion, inversion retrograde), but here we have a graphic display.

Lastly we can see how some scales look when displayed on the COM2. Here's a major scale:
which is still a symmetric shape, though far different from the one it takes in the COF.

And here's how perfect 5ths look:

This shape is more visible evidence of a strong correspondence between the two circles under question...if this shape is mapped onto a COF one then has a chromatic scale: