√½ instead of the golden ratio

Status
This old topic is closed. If you want to reopen this topic, contact a moderator using the "Report Post" button.
As far as I've been able to discern there is an Architectural Golden Rule and a Acoustical Golden Rule. I always use the Architectural model because that is what was contained in the first speaker design book I even used.

But is is for visual appeal, not acoustic perfection, though it comes very close to acoustic perfection too.

The rule is simple -

1.4:1.0:.707

If you divide 1 by .707 you get 1.414.

So, the actual ratio is 1.414 : 1.00 : 0.707

The also correspond to trig functions, and are also the SqRt(0.5) and the SqRt(2).

So, a visually appealing cabinet would have the ratios of 14" H x 10" wide x 7" deep, or 24" x 20" x 14".

My cabinets are 26" x 18" x 13", which is slightly off from the ratios at 1.44 : 1.0: 0.722, but I don't think we need to haggle over a few fractions of an inch. The perfect dimensions would be 25.45" x 18" x 12.73"; I figure I'm close enough.

Notice that using this ratio, the depth come extremely close to being a multiple of the height.

Steve/bluewizard
 
Yeah I had noticed .414 before in other things. I had always thought of it as 22.5 tan though. Same with √½ I always think of it as 45 sine.

My monitors (wharfedale diamond 8.2) I believe use the golden ratio or are close being 364MM x 212MM x 322MM . And if you take a program like Atrise Golden Section you can kind of see how they used it in the placement of the phase plug and tweeter.

I had always used both just as a way to be less cliche than the rule of thirds. It's not used as much as the rule of thirds in photos and designs. But I also notice these numbers come up in dsp and audio processing.
 
diyAudio Chief Moderator
Joined 2002
Paid Member
I have used 0.7 when 1.618 wasn't applicable due to constrains. It works. Still less ripple on impulse graph than near symmetry. Unfortunately haven't saved any photos. Not as good as with 1.618. Near enough though.
 
Why the golden ratio?

In some cases, there is mathematical justification for choosing the golden ratio.

If we want the sequences of modes in the two directions to have few near coincidences, we want to avoid a simple rational ratio, but instead want the most irrational one we can get.

One specific way of defining "the most irrational", is to say that it is the number with continued fraction expansion that converges most slowly. Any exact rational has a finite continued fraction expansion over the integers, but for an irrational the expansion carries on. If we want the expansion to converge slowly, we should never divide by a large number. So if we consider

g = 1/(1+1/(1+1/(1 .......

which looks like the most slowly converging expansion possible, then you can show that
g = 1/2 + sqrt(5)/2

Neat, huh?
 
frugal-phile™
Joined 2001
Paid Member
I've only recently started using the golden ratio heavily. Any irrational number will usually do.

One has to be careful not to use root(2) twice in the same box. The ratio of the dimensions in the box in post 2 (0.707:1:1.414) can also be expressed as 1:1.414:2. That is the kind of ratio you want to avoid.

dave
 
Re: Why the golden ratio?

PigletsDad said:
In some cases, there is mathematical justification for choosing the golden ratio.
...

which looks like the most slowly converging expansion possible, then you can show that
g = 1/2 + sqrt(5)/2

Neat, huh?

Just a quick clarification.

Is that -

g = [0.5 + SqRt(5)] / 2

or...

g = 0.5 + [SqRt(5)/2]

Just curious.

As I said in my initial post, I got my formula out of a speaker design book, but again it said this was for aesthetic appearance. From an architectural perspective, this is a shape that is visually appealing.

Now keep in mind, this ratio seems to be from the 'bookshelf' era. Modern tower speakers seem to have abandon these ratios all together, though perhaps not the concepts the rule is based on.


Also, can we have a consensus that this -

0.618 : 1 : 1.618

is indeed the correct and true Golden Ratio.

Steve/bluewizard
 
Agreed. From a technical standpoint, any irrational fraction will do. i.e.

Golden Ration / Fibonacci ==> 1/phi ==> 1/1.618.....

natural exponent ==> 1/e ==> 1/2.71828....

Pi ==> 1/pi ==> 1/3.14159....

Any of these will ensure that room / reflection nodes are not reinforcing, but the Golden ratio tends to be the most convenient to work from.
 
planet10 said:
I've only recently started using the golden ratio heavily. Any irrational number will usually do.

One has to be careful not to use root(2) twice in the same box. The ratio of the dimensions in the box in post 2 (0.707:1:1.414) can also be expressed as 1:1.414:2. That is the kind of ratio you want to avoid.

dave

To play the Devil's Advocate, according to Mssrs. Bolt, Barenek, and Newman, their published nomograph of acceptable room ratios, ergo speaker box ratios, shows a 1.0:1.5:2.0 mean, putting the 1.0:1.414:2.0 Architectural ratio well within acceptability and rather surprisingly when I first saw it, the 1.0:1.618:2.618 Golden Mean well outside.

Since I typically design high aspect ratio cabs, this issue only applies to its width/depth where the Golden mean is one of many acceptable ratios, so typically suggest it as it's apparently the most familiar.

GM
 
So what about speaker placement? I just eyed this setup to where the height of my head is and put it off center so it wasn't a third but where I guessed the golden section of the room was.

Today I was curious as to how close I was and wanted to check other dimensions that I didn't even think about or consider - height and width.

The Golden section of the height of my room is 58.7" and I checked and my phase plug is a fraction of a centimeter away from 58".

The Golden section of the width of the room is around 73. And damn if that isn't my favorite spot to listen to the speakers in - a little in front of the sweet spot. I in no way did this on purpose with the set up. if I was shooting for that spot I would have probably put the speakers too close to the wall. That was my only aim with where I set them was to get the bass boost out of the speakers and pull them off the wall.

Now the one I did sort of guess at was the length. I was off about 8". The golden section is 102" and I am at 110".

I was just curious. I think I like where I have my speakers set, but do you think you derive benefit in the same way from speaker placement within the room as you do from the dimensions of the box and placement of drivers?

Oh and I think the dimensions of my room might be close to 45sine+1 or the golden ratio.
 
Hello,

For what it's worth, you'll find in attached file a graph I draw many years ago. This graph resulted from a study I did of the Rayleigh formula for the resonance modes of rectangular boxes (including rooms).

The colors and yellow numbers give the maximum value of the difference of frequency between 2 following resonances (we want to minimize that parameter).

Points on the red line corresponds to optimal values. You should try to design your boxes and room with such length/heigh and width/heigh ratios corresponding to a point on the red line.

Best regards from Paris, France

Jean-Michel Le Cléac'h
 

Attachments

  • rayleigh_min.gif
    rayleigh_min.gif
    89.3 KB · Views: 172
Status
This old topic is closed. If you want to reopen this topic, contact a moderator using the "Report Post" button.