The circle of fifths just blew my mind
May 7th, 2012 at 6:17 pm (Mathematics, Music)
In tandem with violin practice, I’m working my way through Practical Theory Complete: A Self-Instruction Music Theory Course. It starts out REALLY basic, with simple notation and rhythms, but works all the way up to composing your own song (!). I just hit lesson 39 (of 86) and my brain exploded.
I’d heard about the “circle of fifths” before, but had never delved into what it actually meant. What it provides is a nifty arrangement of the various (major) keys, anchored by the key of C, that reveals patterns in the progression of sharps and flats that comprise each key’s signature. Check out this awesome magic:
Starting from the key of C, if you go up a fifth, you reach G. The key of G introduces one sharp, F#. Up another fifth from G, you get D, which in addition to F# also features C#. And so on. (The order of keys G-D-A-E are easy to remember for violin players, since those are the four fifth-separated strings on the instrument.) Going down from C a fifth, you get F. The key of F introduces one flat, B♭. Down another fifth is the key of B♭, which adds E♭. And so on.
This defines a linear relationship between C and the keys “above” it as well as “below”; but positioning them on a circle reveals a bit more of the magic: three of these keys are redundant (or “enharmonic”: they sound the same but are notated differently). This diagram shows that G♭ and F# are the same key; my workbook’s diagram also shows that D♭ and C# are enharmonic, as are C♭ and B. And hey, look on any keyboard and what do you see? These key pairs are, in fact, literally the same key.
Want more magic? What’s going on here is modular arithmetic! Not mod 7, but mod 13: the set of values includes { C, C#, D, D#, E, F, F#, G, G#, A, A#, B, B#, C }. For each key, the major scale is traditionally given as WWHWWWH, where W = “whole step” and H = “half step”. But let’s instead view a scale starting on x as the sequence
So the key of C contains { C, D, E, F, G, A, B, C }; C+12 = C in this modular land. Now if we go up a fifth and examine the key of G, that’s the same as adding 5 to all entries. The key of G is therefore
{ G, A, B, C, D, E, F#, G } after doing the addition mod 13.
That is, it’s as if we jumped 5 items forward, but then the extra whole-step in the second tetrachord threw off the pattern and caused the F to become an F#. If you move on to the key of D, the first four notes again are unchanged (with respect to the key of G): { D, E, F#, G }, but then we have to shift one note in the second tetrachord again, yielding { A, B, C#, D }. In this way, the sharps keep building on themselves, and the new sharps introduced in each key alternate. The sharp order is F#, C#, G#, D#, A#, E#, B# (see the pattern?). A similar process explains the progression of flats going “down” from C.
This relationship seems also to explain the conventional structure in how key signatures are written. The key of B major has five sharps, which are C#, D#, F#, G#, A# if you write them in ascending order, but F#, C#, G#, D#, A# if you write them in this circle-of-fifths-inspired order. And that seems to be just what one does (see right).
Patterns! Math! Music! And of course, at the heart of this magic is… physics. :)