Analysing the musical 'circle of fifths'

I'll start with a detour, but bear with me.

It's fun to read children's stories to children. David is just starting to read, so he's starting to learn the concept of rhyming. This means I sometimes explain the rhymes in the story. It then comes back to me how much of the enjoyable flow of these stories is simply rhythm and rhyme, which in turn are complex terms for 'has the right number of syllables in each line' and 'ends with the same sound'.

Of course, this is all the same stuff I did at GCSE, but I never really cared for poetry. When it's a fun story for small children, I haven't bothered to look under the hood at how it's all put together. When I do so, it kind of takes the joy out of it!

Similarly, I have the score for Beethoven's 9th kicking around somewhere. It's amazing how something so magical can, on the page, boil down to what is an awful lot of repetetive, boilerplate-like structure. Working within that framework, there's still enough freedom to create pieces that are fantastic or dross, but having the framework revealed is very odd.

I find the structure of classical music particularly odd, since some things are driven by convention, and some by more fundamental concerns, and it's sometimes a bit unclear which is which. For example, particular intervals sound good because they're very close to simple frequency ratios. At the other end of the spectrum, there are standard forms, like the sonata, which are arbitrary, but provide a framework to build in, and listen in. Inbetween, you have things like the number of notes in an octave, which is semi-arbitrary - the precise number of notes is just something we've learnt to get used to, but the choices are constrained by choosing a number of notes so that our scale can include those simple ratio intervals.

So, in classical music (especially harmony) there's quite a lot of things to learn that 'just are', and it's difficult to tell if they sound good because we're used to it, or because they're fundamental. One thing you learn is the 'circle of fifths' - C major has no sharps or flats, and if you repeatedly go up a fifth you add a black note each time, passing through G major, D major, A major etc. When you run out of black sharp notes the other notes get sharpened, until you work through all the scales, and finally your adding of sharps turns out to be the removal of flats, and you end up at C major again.

Is this something that only works for a conventional major scale, or would it work with whatever interval structure we chose?

Let's start with why we cover all the possible starting notes by repeatedly moving up a fifth. There are 12 tones - an octave has eight 'white notes', including the repeated note, so that's seven notes, plus the five 'black notes'. A fifth is actually an increment of seven semi-tones. Seven and twelve are co-prime, so repeatedly adding a fifth (as addition of seven modulo twelve) will cover all the values. So, depending on the interval chosen and the number of notes we fit in our 'octave', this coverage may or may not happen.

Next, why does moving up a fifth add a sharp? Well, the intervals for a major scale are 'T T S T T T S', where 'T' is a tone (two semitones) and 'S' is a semi-tone. If we rotate the sequence along four elements, to start on the fifth, we get 'T T S T T S T'. The change between playing C major notes starting on G and playing G major is in the final two steps - whether we do a tone followed by a semitone, or vice versa, and that's basically whether the last note is sharpened or not.

What other schemes would work like this? Well (assuming even tempering, so that all 'semi-tone' equivalents are the same size), if your 'scale' consists of a set of 'tones' and 'semitones', the requirement is that there is a rotation of the scale that differs from the original scale by swapping around a neighbouring pair of 'S' and 'T'. For a scheme like the major scale, with a scale of 5 tones and 2 semitones, I think the only scheme where this works is a rotation of the major intervals (which is why the normal circle of fifths doesn't work for minor keys). For scales constructed differently, there are other possibilities.

Finally, why, when we start on C major, do we sharpen the notes whose sharps are black notes first? This is simply because our cycle of scales moves up a fifth each time, so that we're doing an 'almost half octave' step each time. The black notes are in two groups (of size two and three), and we alternate sharpening a note from each group sequentially.

Overall, what does this tell us? We could construct a very similar system with, e.g. decades and a 'circle of sixths', with a group of three black notes then four black notes. Alternatively, by tweaking the intervals used, you could get very different systems where, for example, different base notes share exactly the same notes in their scales. This probably makes modulation somewhat interesting.

At the end, I'm once again left wondering what is really fundamental to music, and what's arbitrary (but fascinating) structure.

Posted 2012-12-13.