I highly recommend you do not read the article, "The Amen Break and the Golden Ratio", by Michael S. Schneider, because it is 100% unadulterated clap-trap, a pure waste of your time, and the point of this article is too discourage people from continuing to waste other people's time by linking to it.
Indeed, the article is hosted on a site shilling a book which, although it has largely positive reviews on Amazon, has some telling 1-star reviews calling it e.g. "Muddled Mathematical Mysticism". If the Amen Break article is reflective of the quality of the material in the book, the people rating it highly have been hoodwinked.
The Amen Break is 5.2 seconds of solo drum performance which has reportedly been widely sampled in popular music. Musically, it consists of four measures (bars) in 4/4 time. The four measures are:
If you click through to the audio linked from Schneider's article, you will not hear the full Amen Break. You will instead hear what appears to be measure #2 looped four times.
This is, admittedly, the common use of the Amen Break. Indeed, the real significance of the Amen Break was probably nothing magical about it, just that it was a decent, funky beat which could be found with no accompanying music, allowing it to be sampled and reused as a pure beat in a way that the drums in most recordings could not be.
Also, you will find many people claiming things sample the Amen Break when they do not (as far as I can tell); they simply have similar beats. The thing is, there is only a little that is unique to measure #2; imitating it is indistinguishable from imitating thousands of previously recorded drum beats.
So if the Golden Ratio were going to explain the popularity of the Amen Break, it should probably be describing properties of the part that everyone samples, and everyone has heard.
However, that is not the part that Schneider studies.
Here is a waveform of the full Amen Break and surrounding material; the audio file was downloaded from wikipedia:
Here is just the 16 beats of the drum solo:
You can see the bass guitar in the first beat, which looks different from all the rest of the solo; unlike the drum sounds which all decay rapidly, the bass guitar sustains, causing height throughout the beat. Because the bass guitar is a a tonal instrument it has a periodic waveform, and the pitch is low enough that the period is actually visible in the image as a pattern of regular spikes. (The more-sustained-than-normal drum hit near the end is a non-tonal cymbal with a long decay time.)
Here are the four beats of the second measure:
You can see regular peaks on the eighth notes (each one is half a beat); powers of two are the most common number we find in music, and the Amen Break is no exception.
This is based around the classic rock drumming pattern: snare on 3 & 7, kick on 1 & 5, with some kicks altered and rearranged for interest but the snare beats inviolate--moving the snare beats usually makes it a drum fill, not a drum beat (although there are exceptions).
There are additional peaks approximately halfway between the others: softer snare drum hits between 4&5 and 5&6, and a kick drum between 6&7. The extra kick drum hit is actually shifted a bit to the right. It is not clear if this is an intentional "swing" on the part of the drummer, or just failure to hit the kick drum fast enough--the in-between snare drums are not offset that way, so I'm uncertain it's intentional. And anyway, although it's the closest thing I've seen to a golden ratio in this measure, it's not the golden ratio; with pixel spacings of 88 pixels and 58 pixels, it's about 1.52, not 1.62. (1.5 means the extra kick falls almost exactly 2/3rds through the eighth note, which is a straight triplet swing, not a subtle swing at all.)
Update: To clarify, every single one of the beats above falls on a "simple" rational time within the measure (measured relative to the entire measure). All the eighth notes occur at fractions with 8 in the denominator. The two extra snare hits are at 7/16 and 9/16. The extra kick may have been meant for 11/16, but it actually falls at about 17/24. And just to point out how stupid this whole exercise is, the golden ratio is around 5/8 of the way through; 8/5ths is within 1% of the golden ratio; finding 5/8ths in a rhythm is not particularly hard, but measuring accurately enough to find that extra 1% probably is. End update.
So that's the Amen Break that everyone's familiar with.
Schneider finds the Golden Ratio in this waveform:
Looking at that, it's very hard to figure out musically what's going on in it, but it sure doesn't seem to match up with the audio he links to (which appears to be a loop of measure #2, above).
Hunting through the full Amen Break waveform, I located a match to his:
Here I've left off the initial snippet from his wave, to make the match clearer (and I've left on a tiny bit of the next beat from the song so you can see a little context; that's the end of the break, where the band comes in).
What part of the song is this? It starts on the 3rd eighth note of the third measure, and his end point is the 8th eight note of the fourth measure (not the end of the fourth measure). It's the syncopated part of the drum break, starting from and ending in a weird place.
This is nonsensical musically. It is nonsensical from a listener's perspective.
The period in which he's finding the golden ratio is just a totally non-musical-choice of subsection of the Amen Break (and a part that is hardly ever sampled or imitated). It's not even an even number of beats long; it is 6 and a half quarter notes in length, i.e. 6 1/2 beats long.
If you listened to that in a loop it would sound bizarre. (It's actually a sort of "odd-time signature" thing I personally like a lot, but it's the opposite of popular. The nice powers-of-two ruler-marking pattern of measure 2 is the quintessence of popular.)
Here's what he claims to have done:
I considered the biggest peaks near each end as the limits of a whole.
He doesn't explain why he chose such a weird subsection to start with before finding the biggest peaks near each end.
Except, wait a minute, let me go ahead and include the bit I left out at the beginning of my waveform:
Let's look at Schneider's again:
Schneider claims he's chosen to analyze from the highest peak to the highest peak, but this is clearly false. In his waveform, he's truncated the front off the previous beat so it looks shorter than the next one. Also, he finds the golden ratio by ending on the last peak of his waveform, which we can see is nestled between two higher peaks.
So he isn't even following his claimed methodology on the inexplicably weird subsection of the Amen Break he's chosen to look at (without explanation or even acknowledgement). This is enough for me to conclude without hedging:
There's no truth here, just bulshit. The data has been cherry-picked to generate a seeming result that occurs simply by chance.
He's either a kook or a charlatan.
Dont waste people's time with it.
The kick on 6 is the loudest drum hit in the entire break, and the snare has been moved from 7 to 8, and the kick in the next measure at 1 is missing--it makes the whole thing feel like a normal drum beat with kick-on-5, snare-on-7, kick-on-1 pattern has been delayed by a single eighth note, which is a very weird effect.