Golden Showers or Golden Ratio: What Sounds Better?

The golden ratio is considered the most irrational number because it is basically the answer to the question of how do you construct a number that is as far away as possible from every rational number imaginable.

This explains it really well. Both the graphical explanation using sunflower patterns and the derivation with formulas is beyond what I could do here.


The golden ratio also ends up being the solution to the question of how do you add numbers to a set while keeping the new number as far away as possible from every multiple of every number previously added.
Being close to a multiple of what was previously added is what forms spokes in the sunflower explanation in the video.

My understanding is that any pair of openings in a mesh is going to form an interference pattern that is defined by the distance between the openings, right? So if many openings have similar distances from each other, their similar interference patterns would add up, whereas if every possible pairwise distance was different enough, you get many minute effects that don't add up to anything. And if the distances are multiples of each other they would still harmonize.

While golden ratio patterns don't explicitly optimize pairwise distance, they seem to be so good at being irregular and not lining up that they should still effectively prevent pairwise distances being identical or multiples of each other.

 

Let me provide a simpler example.

Let's say you have two wavelengths λ1 and λ2, with corresponding fundamental frequencies f=v/λ and nth harmonic frequencies n*f=n*v/λ. (v is the speed of sound)

If λ1 and λ2 are identical, then their fundamental and all harmonics are equal.

If λ2 is half of λ1, so
λ2=1/2 * λ1, and f1=v/λ1 and f2=v/λ2, then
f2 = v/λ2 = 2*v/λ1 = 2*f1,
meaning that the fundamental frequency of λ2 is equal to the second harmonic of λ1, its second harmonic equal to the fourth harmonic of λ1 and so on.

If λ2 is two thirds of λ1, then the second harmonic of λ2 is equal to the third harmonic of λ1 because:
λ2=2/3 * λ1
f2=v/λ2=3*v/2*λ1=3/2 * f1
2 * f2 = 2 * (3/2 * f1) = 3 * f1

And it goes like this for every fraction of integers. If there is fraction n/m so that
λ2=n/m * λ1
then the nth harmonic of λ2 is going to line up with the mth harmonic of λ1.

If you wanted to choose your two wavelengths so that not only do none of the harmonic frequencies of λ2 line up with the harmonics of λ1, but you want all of your harmonics to stay as far away as possible from the other wavelengths harmonics, then the optimal solution is the irrational number that stays as far away as possible from every possible fraction of integers (every rational number), and that is the golden ratio.

 

Yeah, that's why I ultimately ended up selling my Z1R for a VC. It was hard to justify its performance for the money -- that treble peak could cause a good chunk of music to be oddly sibilant.

Speaking of mental masturbation, I'd enjoy reading about some measured observations on this topic. I'm sure I could scour Google Scholar to find something to the similar effect, though. Maybe just not applied directly with headphones (we're a pretty niche, OCD bunch).

 

Only if those Golden Showers are coming from a Golden Shlong. Maybe more SBAF branding is in order.

 

Oh I know: Golden Shlongs for number 1 winners, Silver Dicks for those who come in second place, Brass Balls for the 3rd position, and Golden Showers for honorary mentions.

 

i think it's amazing that @waquo knows all that stuff, but when it comes to come to old-fart speaker design and marketing-man pretty grills, I'm absolutely with @purr1n. Especially as I got thrown out of high-school maths and don't understand any of it.

Maybe I could go for golden mean pretty girls? Hmmm... Might be a bit Picasso?