Circle Of Fifths

The circle of fifths is an artificial harmonic construct of western music based on a rather peculiar mathematical feature. It's an interesting pattern itself, and an interesting example of a historical compromise between verity and practicality.....

The basic intervals of music are derived from the overtone series. A bowed string produces a sawtooth wave which can be described as the sum of a series of sine waves. If the frequency of the sawtooth wave is x, the resultant sine waves are in a series at frequencies of x,2x,3x,4x......the ratios between these correspond to certain musical intervals:

 1:1  =  Unison
 2:1  =  Octave
 3:2  =  Perfect Fifth
 4:3  =  Perfect Fourth
 5:4  =  Major Third
 6:5  =  Minor Third

Those of you with music training will see that a triad can be described as:

6:5:4

So the derivation of a triad is based on a very simple physical principle. At a certain point in western musical history, musicians began to favor modulation, the progress from one key to another. The most common interval for doing so was via the fifth, since that's the strongest interval after the octave. (Western music considers notes an octave apart to be synonymous, so pitch values can always be divided by powers of two. It is standard practice to express these ratios as a value between 1:1 and 2:1. You will also see them expressed as fractions.) Pythagoras had already experimented with this, discovering that modulating five times by the interval 3:2 yielded a note very close to the third - 81:32. The interval between them, 81:80, is called the Pythagorean comma.

It wasn't long before western musical science discovered that if you modulated up by a fifth 12 times, you reached the interval 531441:4096, which reduces to 129.75, and further by octaves to 1.01. This is mighty close to a unison. So it was decided to squeeze all of the fifths by a certain amount (see SqueezingTheFifth?, which is beyond my math skills.) The upshot of all this is that the fifth on a piano is slightly askew, and the third is additionally 81:80 sharp. It is this tension of the third, and a similar, more dramatic tension in the seventh (7:6 vs. a skewed, misused 9:8) that gave birth to the blues.

What's particularly fascinating, is that the study of this piece of musical history is limited to a few specialists in early music. The twelve-tone equal tempered scale is taken as gospel by the overwhelming majority of western musicians.

However, this is changing rapidly, as computer-based music simplifies the job of retuning. For more on principles of tuning, visit the Just intonation Network, at http://www.dnai.com/~jinetwk/index.html.

-- EricMoon

Eric's notes prompted me to run some numbers in excel so that I might see the circle for myself. Modulating by 3:2 I took to mean increasing frequency by a factor of 3/2 or 1.5. So I got the sequence ...

440 660 990 1485 2227.5 3341.25 5011.875 7517.813 11276.72 16915.08 25372.62 38058.93 57088.39

Now, with a tempered scale, 12 halfsteps will bring you up an octave, so I figured each halfstep increases by 2^(1/12). Modulating by a fifth was very close to moving up seven halfsteps, which would be 2^(7/12) or 1.498307. Repeating the above calculation I get ...

440 659.2551 987.7666 1479.978 2217.461 3322.438 4978.032 7458.62 11175.3 16744.04 25087.71 37589.09 56320

Now I wondered what notes these frequencies might be? (Never mind that some are quite high - I could have started lower.) If I named the notes with numbers, 0 to 11, and reused these names each octave, then I can convert frequency to tempered note number with the formula MOD(LOG(freq, 2^(1/12)), 12). For the tempered cycle I get the repeating sequence ...

0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5 and back to 0

The note number formula will yield fractional notes when a frequency doesn't fall exactly on a tempered note. For the original sequence I got notes ...

0.00, 7.02, 2.04, 9.06, 4.08, 11.10, 6.12, 1.14, 8.16, 3.18, 10.20, 5.22 and back to 0.23

You can see an accumulation of a couple of hundredths of a halfstep with each modulation. Continuing around for another cycle one gets more accumulation ...

0.23, 7.25, 2.27, 9.29, 4.31, 11.33, 6.35, 1.37, 8.39, 3.41, 10.43, 5.45 and back to 0.47

That is the problem that tempering the scale solved. But at what price? Other solutions include staying within a subset of the notes, having lots more notes in an octave, or having a more agile definition of a note than I provide. Eric's links lead to mention of micro-tuning midi devices. Perhaps keyboard instruments are now smart enough to find useful alternatives to the tempered compromise. -- WardCunningham

Um, how sure are you about the statement that "it is the well-tempered scale which is nearly universal today"? Maybe I have tin ears (in fact, I'm sure I do), but all the pianos I've played or heard seem to me to be equally tempered except in so far as they're out of tune. -- GarethMcCaughan

Unless the piano is so perfectly in tune that you can count the beats of the various intervals, well-tempered tuning (at least in some of its variants) is extraordinarily difficult to distinguish from equal-tempering.

My piano is not well tuned, I must confess. It's temperament varies with the temperature. -- DavidBrantley

One of my pet peeves is the claim (oft repeated by people who ought to know better) that JS Bach advocated equal temperament. See http://www.bachfaq.org/welltemp.html for a brief sketch of the issue.

CategoryMusic