Circle of fifths and roots of two

A chromatic scale in Western music divides an octave into 12 parts. There are slightly different ways of partitioning the octave into 12 parts, and the various approaches have long and subtle histories. This post will look at the root of the differences.

An octave is a ratio of 2 to 1. Suppose a string of a certain tension and length produces an A when plucked. If you make the string twice as tight, or keep the same tension and cut the string in half, the string will sound the A an octave higher. The new sound will vibrate the air twice as many times per second.

A fifth is a ratio of 3 to 2 in the same way that an octave is a ratio of 2 to 1. So if we start with an A 440 (a pitch that vibrates at 440 Hz, 440 vibrations per second) then the E a fifth above the A vibrates at 660 Hz.

We can go up by fifths and down by octaves to produce every note in the chromatic scale. For example, if we go up another fifth from the E 660 we get a B 990. Then if we go down an octave to B 495 we have the B one step above the A 440. This says that a “second,” such as the interval from A to B, is a ratio of 9 to 8. Next we could produce the F# by going up a fifth from B, etc. This progression of notes is called the circle of fifths.

Next we take a different approach. Every time we go up by a half-step in the chromatic scale, we increase the pitch by a ratio r.  When we do this 12 times we go up an octave, so r12 must be 2. This says r is the 12th root of 2. If we start with an A 440, the pitch n half steps higher must be 2n/12 times 440.

Now we have two ways of going up a fifth. The first approach says a fifth is a ratio of 3 to 2. Since a fifth is seven half-steps, the second approach says that a fifth is a ratio of 27/12 to 1. If these are equal, then we’ve proven that 27/12 equals 3/2. Unfortunately, that’s not exactly true, though it is a good approximation because 27/12 = 1.498. The ratio of 3/2 is called a “perfect” fifth to distinguish it from the ratio 1.498. The difference between perfect fifths and ordinary fifths is small, but it compounds when you use perfect fifths to construct every pitch.

The approach making every note via perfect fifths and octaves is known as Pythagorean tuning. The approach using the 12th root of 2 is known as equal temperament. Since 1.498 is not the same as 1.5, the two approaches produce different tuning systems. There are various compromises that try to preserve aspects of both systems. Each set of compromises produces a different tuning system. And in fact, the Pythagorean tuning system is a little more complicated than described above because it too involves some compromise.

Related post: Circle of fifths and number theory

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7 comments on “Circle of fifths and roots of two
  1. John,

    This is the cleanest description of the difference between Pythagorean tuning and equal temperament I’ve read, from a mathematical perspective, in any case.

    Will

  2. Daniel Black says:

    Great timing, too. A relatively recent Nova ScienceNow podcast covered musical autotuning, being put to great (ab)use in the last decade not only to ameliorate poor talent, but to hide it completely behind trendy distortion patterns. Of course, the story’s not too deep, but I have to imagine that in addition to some FFT or such, this sort of math has to come into play.

    I’m happy to be disabused of this notion, say, by a post on The Endeavor? +)

  3. Bruce says:

    Thanks to J.S. Bach for making the case for equal temperament!

  4. I have seen at least three different internet sites that explained 440 Hz was the piano’s Middle C. This is the first I have seen of 440 Hz being an A. I havent studied much in the way of music theory, and Im not accusing you of being wrong, but I recommend checking your sources on that because you are the first inconsistency I have seen in four.

  5. @cogito: http://lmgtfy.com/?q=440+hz&l=1

    @Bruce: I know very little but this area, but I was under the impression that Bach favored a well temperament system, distinct from equal temperament. I just read an interesting article on the subject – The centuries-old struggle to play in tune – http://www.slate.com/id/2250793/pagenum/all/

  6. Schneider says:

    The easiest, the shiniest and the most fantastic way to describe how different the two kinds of tuning are :)

  7. kevin says:

    the 12th root of 2 is purely mathematical and geometric and involves higher dimensions. going up in frequency on a chromatic scale also explains dimensions of space time. for example: 1, 12th rt2, (12th rt2)^2=6th rt2 so on so forth. the sixth step is (12th rt2)^6 or the square root of 2. the square root of 2 is half an octave up. the square root of 2 is how you double the area of a square. take a 1×1 square. it has an area of 1. an area of 2 = 1.41 x 1.41 . back to the chromatic scale the third note is (12th rt2)^3 or the 4th root of 2, the 4th note is the (12th rt2)^4 or the cube root of 2. to double the volume of a cube we multiply the edges by the cube root of 2. a 1 x 1 x 1 cube has a volume of 1 , a cube with edge lengths of 1,259 x 1.259 x 1.259 has a volume of 2. the fourth root of 2 does this for 4 dimensional hypercubes. the 6th root of 2 does this for 6 dimensionional cubes if they exist. and the 12th root of 2 does this for 12 dimensional cubes. if we do all this math using circles with a diameter of these numbers , the octave is double the frequency and 1 half the wavelength. two circles would fit inside one. the yin yang!! this is how furier transforms work. its also why chords work. but space works this way too. if you put 2 bowling pins on the ground the first one 10 meters away, and the second one 20 meters away the second one will appear 1/4 the size. (the inverse square law) 1/ the distance squared. if the second pin is placed 14.14 meters away the square root of the first distance then it will appear 1/2 the size. furthermore to double the area of any regular polygon multiply the diameter by the square root of 2. in electrical enginering to find the dc equivalent power for an alternating current sinesoidial wave we multiply the peaks by the RMS or root mean square (1/ the square root of 2) or 1/2 the square root of 2 or 0.707! if you had a square that was 1 x 1 and drew a circle that exactly fit around it its radius would be 0.707. the radius wrapped around the circle occupies 57.29551 degrees. the radian number. 57.29551 degrees x pi = 180 . and so 2 pi r =360 degrees. the formula for a circle.

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