and the

Golden Ratio

M.Ed. Mathematics

Author of

A student recently asked if I had any insights into why the "Amen Break" is so popular in some modern music. "What's that?" I had to admit that I hadn't heard of it. But when we listened to it, a drum break beat, I recognized the rhythm right away. I'd heard it in television commercials and the music of James Brown, and elsewhere. You probably have too. I became intrigued when I saw an image of the audio waves themselves because I immediately recognized the Golden Ratio in the structure of its timing. And I was surprised to find an even deeper relationship to the structure of the human body.

The "Amen Break" is 5.20 seconds long in four bars. It was created and performed by Gregory Cylvester Coleman in the 1960's funk-and-soul band The Winstons in their song "Amen, Brother". (They've never received royalties from the hundreds of bands which have sampled and used their creation.)

A history of the Amen Break can be found here, and the term googled here and here.

A video explaining the Amen Break can be found here.

What I first noticed in the wave picture are the distinct peaks representing the beat. I considered the biggest peaks near each end as the limits of a whole.

Having looked at the geometry of the Golden Ratio a great deal, and its expressions in worldwide art, I have a decent sense of its place along a line. The Amen Break had that feel. For a quick check I used homemade Golden Ratio calipers to examine the peaks. Indeed, peaks pop up at Golden Ratio intervals, as do smaller peaks within them, reminsicent of the fractal structures in nature.

For more exact visual analysis I examined the wave image in my computer, in which I have a palatte of geometric forms and proportions for quickly identifying an object's ratios. Sure enough, Golden Ratio relationships were indicated among the different peaks. Am I seeing things? You decide. But the appearance of the Golden Ratio may help explain its popularity.

To appreciate this relationship between the Golden Ratio and sound, it's worthwhile to consider some of the ideal, eternal, unchanging principles of Golden relationships which can only be approximated in nature, and byartists, architects and musicians. I'm not going to re-teach here everything there is to know about this wonderful ratio since some great websites already tell you everything you want to know, such as this and this. Most pertinently, information about the appearance of the Golden Ratio in worldwide music, such as in the work of the classical composers Mozart, Beethoven, Bartok, Debussy and Satie can be found here.

For our interests, the main principle to appreciate about the Golden Ratio is this: A whole line may be divided in such a way that the length of the whole relates to its large part in the same way that the large part relates to the small part. In other words, the same relationship appears on different scales, comprehending a mathematically balanced whole.

This visual linear relationship has only one solution, when
the large and small parts are in the numerical relationship of
1.6180339... to 1 (symbolized by the Greek letter *Phi*).
Simultaneously, the whole relates to the large part this same
way. Our wrist divides our forearm and hand at the point of the
Golden Ratio, and our finger bones form an expanding Golden Ratio
sequence. It extends through the entire body (as I've illustrated
in *A Beginner's Guide To Constructing The Universe* and
the *Constructing
The Universe Activity Books*). What the Golden Ratio does
is provide balance among the parts and whole, allowing a harmonious
elegance to emerge among the changing relationships among the
parts, maintaining a balanced whole, whether in the motions of
a dancer or the growth of a tree.

Another relevant point to help us understand the Amen Break
is how this Golden Ratio balance expands to maintain its balance.
It's a remarkable mathematical truth that when unity is added
to the Golden Ratio, the result is the Golden Ratio *squared*!
This occurs among no other ratios. (Try it on a calculator to
see that 1.6180339 times itself is the same as adding one to it!)

This process can continue to produce an expanding sequence
of powers of the Golden Ratio, each power always equaling the
sum of its two previous powers. (All this is expanded upon in
*Constructing The Universe Activity Book Volume
3*). So on the computer I simply laid down lengths divided
at the Golden Ratio (yellow and blue lengths are in Golden Ratio),
and Golden Proportion expansions, to see how many peaks appear
simultaneously at Golden divisions. Here's what I saw:

Further dividing the whole length (between extreme beat peaks) and subdividing those parts in Golden Ratio relationships results in these images. They don't account for all minor peaks, although in their approximation of the ideal many are very close. But the timing betwen the major peaks show clear correspondence. Investigators of the Golden Ratio in art have remarked that even without "knowing" anything about this ratio, it is what creative people intuitively discover and aim for and express in many media anyway. Its harmonious wholeness and balance satisfies something deep within us.

When I saw the divisions and subdivisions of the whole indicate
Golden Ratios in an expanding sequence, I immediately thought
of the expanding Golden Ratio seen in the dimensions of the ideal
human body as discovered and used in art and architecture by the
ancient Greeks. Here are pictures taken from my books (*Golden
Ratio Man* drawn by Libby
Reid) showing a few of the Golden Ratio relationships in the
harmonious whole called our body.

Because the ideal body's parts are in Golden Ratio relationships
(everyone is stretched or compressed) each two levels (beginning
at the brow downward) add to equal the length of the next level.
I've turned the Amen Beat's waves vertically and put them next
to the levels of the human body. It seems to me that the popularity
of the Amen Break must have something to do with this appearance
of the Golden Ratio in its timing structure. And furthermore,
its expanding (and subdividing) Golden Ratios resemble the Golden
divisions of the human body *but in audible form*. I suppose
it might appear in music most conducive to dancing. And also singing
and chanting, since a great deal of sacred architecture around
the world had been consciously designed in the Golden Ratio to
make such use of it's self-reflecting internal harmonious relationships.

Human bodies and created sounds, like flowers, crystals and galaxies, can never exactly equal any ideal mathematical template. But the major wave peaks of the Amen Break, and many of its smaller ones, seem reasonably close to being an expression of the fractal nature of the wonderful Golden Ratio. Maybe it's just an apparent coincidence or maybe it's like thousands other approximations of the Golden Ratio that are not consciously planned but do appear in human designs based on the refined sensibilities of some artists, architects and designers. I wonder what it would sound like if it was more precisely proportioned to the ideal, but I also know that slight differences are what make it human and alive.

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Michael S. Schneider

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