For reasons that needn’t concern us here I’ve been reflecting on the Circle of Fifths recently. For those who are not familiar with this concept it’s a way to describe the relationship between the 12 tones of the chromatic scale in music. Amongst other things it can be used to work out the flats and sharps in a key signature or, vice versa, given the number of flats or sharps in a piece, the key signature itself.
The dim part of this comes from thinking about how to determine the number of sharps in a given major key. The sequence of keys corresponding to zero sharps, one sharp, two sharps etc. is pretty easy to remember. You start with C, which has zero sharps, then the next key is a fifth away, which is F and has one sharp, then back to D (one tone up from C) which has two sharps, then up to G (one tone up from F) which has three sharps and so on. The full sequence for most use cases is therefore C-G-D-A-E-B (it does continue but you’re into keys with six and more sharps at that point, which is unusual to say the least). The thing I have only just come to realise is that, rather than having to remember the sequence of sharps (which is F, FC, FCG, FCGA, FCGAD for the keys of G major, D major, A major, E major and B major) you can work them out from the fact that they are one tone down from the key signature. So, F, which is the one sharp that appears in the key of G major, is one tone down from G. C, which is the one you add to the key signature when moving from G major to D major, is one tone down from D. And so on. Simple but that has only just occurred to me.