What Is Music? Part Three: Cycles of Fifths, Pentatonic, and More

This series of posts has become a bit longer than I intended, I’m afraid. It turns out that a question like “what is music?” is a bit more complicated than it sounds. Here’s part one and part two. There’ll be more.

So in the last post, we saw that if you take a note and double its frequency, the two notes go together so well that we can give them the same name.

What this means, in effect, is that we can think of musical notes like a clock — they “wrap around”. A C note has a frequency of 262Hz (I’m using the nearest integer, to make the sums easier, — it’s actually 261.626Hz), and if we double that to 524Hz *that* is a C, but if we halve it to 131Hz, *that* is also a C.

If you look at our graphic of a piano keyboard which we pinched off Wikipedia…

piano keyboard

You’ll see it has a repeating pattern — every eight white notes the pattern repeats. That’s the looping round we talked about.

And this means that from now on I’m going to talk about every note as if they’re all in one octave.

In the last post, we saw that if you take a starting note (the tonic) and then find a note that is one-and-a-half times its frequency, that note (which we called the fifth, or dominant) goes well with the first note.

What we can now say is that if you take that note as our starting point, you can find the pitch that’s one-and-a-half times *that*, too. If our original starting note is C (in blue), the dominant of C is G (in red), and the dominant of G is D (in green) — G is the fifth white note counting from C, and D is the fifth white note counting from G:

keyboard with C highlighted in blue, the G up from that C highlighted in red, and the D up from that G highlighted in green

But because of this wrap-around effect, we can talk about them in the same octave:

keyboard with C highlighted in blue, D in green, and G in red, all in the same octave

Now, if you play those notes together, they work well together, and we can add in two more notes — A (the fifth of D) and E (the fifth of A). I’ll highlight E in orange and A in yellow on our diagram:
keyboard with C highlighted in blue, D in green, E in orange, G in red, and A in yellow, all in the same octave

What we have here is called a pentatonic scale — pentatonic for “five tones”, because we’re using five different notes here.

All these notes go well with each other, but if you play around with them, you’ll notice that they seem to “want to” go to C. That’s because C is the tonic note from which we started our piling on of the fifths, so when you hear two or three of these notes, you start (subconsciously) to think of them in terms of their relation to C.

And this is where we get the idea of a key from. In the Western pop/classical styles of music we’re discussing (remember that when I generalise about music, the further we get from 1966 Anglo-American pop music, the less true the generalisation becomes, though it still has some validity), a tune tends to start on the tonic note — to set your expectations — and then to end on the tonic as well, so it feels like a satisfying conclusion. And in between, you will only get notes that are part of a scale that starts on that note.

(These rules are all broken, all the time, but you’re meant to notice when they’re broken, and say “ooh that musician is exciting and daring, and probably sexy too”).

The pentatonic scale (there are actually several different scales called “pentatonic”, but this is the most common), which we see here, is one of the simpler forms of scale worthy of the name, but it’s a very common scale indeed. You hear it in Indian classical music, in Irish folk song, in popular song, in hymns… when you hear that horrible racist piece of faux-“Oriental” music you get when Hollywood wants to show something’s Asian, that uses a pentatonic scale. But so does Amazing Grace, or Dirty Old Town.

This pentatonic scale appears so widely that it’s best to think of it as the foundation of all our music. However, there are leaps in the scale (between E and G, and between A and C), and people tend often to prefer to have tunes that go between notes that are close together, so we add in the notes F and B, and now we have the full seven notes of the Western octave. In the key of C:

C — I/tonic — “Doh”
D — II/supertonic — “Re”
E — III/mediant — “Mi”
F — IV/subdominant — “Fa”
G — V/dominant — “So”
A — VI/submediant — “La”
B — VII/leading tone — “Ti”

These are the white notes (and you’ll notice that the black notes, handily, make a pentatonic scale starting on F#)

(If you play much music you will notice that F also goes well with C, because C is actually the fifth of F. B goes less well with C, because it’s the fifth of E, so it’s “far away” in our cycle of fifths. Often the leading tone is not played very much in melodies, just used as a brief note between others, because of this, while the subdominant is used more often. Quite often in popular music the leading tone will be replaced with a flattened seventh — so in this case it’d be B♭ rather than B. F is the fifth of B♭, and so B♭ feels about as “close” to C as B does, and fits about as well.)

So at this point, we now know:

A scale is a collection of related notes. It starts on a tonic, a note to which all the others are related. When we say, for example, that something is in the key of C, that means it is in a scale whose tonic is C. A typical melody will both start and end on its tonic, and only contain notes in one scale.

And that’s a good place to end this. More soon.

[Or at least, it’s a *fairly* good place to end it. This bit is a little bit of extra credit. If you’ve bothered to do the sums, you’ll notice that there’s no way to arrange the notes so that the ratios between them all work perfectly as described — it’s just not mathematically possible, and if you get it right for one octave on the keyboard, all the others will sound out of tune.

The solution to this, arrived at in the 17th century, was simply to make every note a little out of tune with everything else, but not so much that anyone would notice. Bach’s The Well-Tempered Clavier is effectively a propaganda piece, to show that you can get much better music if you do that than if you do what had been done previously, which is to have some notes perfectly right and others horribly wrong.

Some musicologists have suggested that the bent or “blue” notes in the blues and jazz come from people trying to play the pentatonic scales of African folk songs on even-tempered Western instruments. Personally, I think trying to find the roots of black American music in Africa smacks a little of racism, and early 20th century black Americans probably had not much more cultural connection to Africa than I do, but I thought I should note that in this extended footnote.]

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9 Responses to What Is Music? Part Three: Cycles of Fifths, Pentatonic, and More

  1. S Fred L says:

    I am enjoying this series: thank you.

  2. Mike Taylor says:

    This is excellent. After 30 years as a practising musician, it had never occurred to me that the pentatonic scale is just a stack of fifths wrapped around the octave.

    (There is a separate discussion to be had about whether a sequence of three notes can be “racist”, but let’s leave that for another day.)

    • Andrew Hickey says:

      The sequence of notes itself is, of course, not racist, but its use pretty much always is.

    • Thomas says:

      One of the best little mini-music theory lessons I ever received was via a clip of one of Leonard Bernstein’s lectures at Harvard (which are I believe all available on YouTube at the moment) where he points out that the entire chromatic scale itself comes from arranging all of the circle of fifths by frequency–so it’s just taking C–G–D–A–E–B–F#–C#–G#–D#–A#–F–C and arranging them from high to low, and that’s how we get the order of the notes on the keyboard. Blew my mind when I first watched the video.

  3. John says:

    Fascinating Andrew. I really didn’t realise it was so scientific. I can’t wait to get to the bit about why people like Brian Wilson’s chord changes touch you so deeply, compared to other popular composers

    • Andrew Hickey says:

      That might be a bit beyond this series — and indeed beyond my powers of analysis — I’m afraid, though some of what I touch on will be *part* of that. But if I could explain it exactly, I’d be writing songs as good as Brian’s myself ;)

  4. Seth Walter says:

    I get what youre saying with that last part, but it comes across as insensitive because its not like early 20th centiry black americans had a choice about it. I would also think that their music had basis in slave music which was directly decended from African music, but I could be wrong about that.

    • Andrew Hickey says:

      I’m certainly not saying early twentieth century black Americans had much of a choice about what cultures they were and weren’t exposed to.
      My understanding, though, is that most of the music made by slaves had little connection to African music. In fact the biggest influence African music seems to have had on American music is on that “whitest” of forms, country music — through the banjo (an adaptation of an African instrument) and the claw-hammer technique of playing it, which is derived from West African music.
      But one of the great tragedies of the slave trade is how thoroughly the slave owners destroyed the original culture of the people they kidnapped. My understanding (and it may be wrong, it’s been many years since I looked into this) is that the idea that African musical forms influenced the blues, even indirectly, has been pretty much debunked. The only influence that people point to that might remain is the call-and-response form — and even that is something that’s seen in many cultures’ music.

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