The "Golden" Ratio?
I try to consider every aspect of loudspeaker design when I make one. Before I make anything, the ratio of dimensions has to be determined. Now, I understand some (most?) people don't care about the "golden ratio," but the most baffling fact is that I've found several ratios out there that claim to be "golden." On the Index of Resources section on their website, Parts Express says that it is 2.6/1/1.6 (h/w/d). But according to an old Radio Shack book, it's 1.6/1/0.6 (h/w/d). They do look different, don't they?
I generally want the depth of my enclosures to be considerably larger than the width, so I trust PE more. (After all, who's to believe the Shack?) However, PE's ratio seems to apply only to dual-woofer or MTM designs (the Joe D'Appolito configuration). I mean, for a two-way with a tiny midbass, following that ratio means you can't even fit the darn midbass on the front!!! Surely no tower or floorstander follows that convention. And with the advent of plasma, some HT speakers are now designed to complement them. (The ratio doesn't matter as long as it's not deep!)
So, for the time being, I'm using average dimension ratios from "normal" commercial systems as a starting point. (Count out the B brand!) I'm really looking for a good ratio suitable for a small two-driver two-way.
Comments?
I try to consider every aspect of loudspeaker design when I make one. Before I make anything, the ratio of dimensions has to be determined. Now, I understand some (most?) people don't care about the "golden ratio," but the most baffling fact is that I've found several ratios out there that claim to be "golden." On the Index of Resources section on their website, Parts Express says that it is 2.6/1/1.6 (h/w/d). But according to an old Radio Shack book, it's 1.6/1/0.6 (h/w/d). They do look different, don't they?
I generally want the depth of my enclosures to be considerably larger than the width, so I trust PE more. (After all, who's to believe the Shack?) However, PE's ratio seems to apply only to dual-woofer or MTM designs (the Joe D'Appolito configuration). I mean, for a two-way with a tiny midbass, following that ratio means you can't even fit the darn midbass on the front!!! Surely no tower or floorstander follows that convention. And with the advent of plasma, some HT speakers are now designed to complement them. (The ratio doesn't matter as long as it's not deep!)
So, for the time being, I'm using average dimension ratios from "normal" commercial systems as a starting point. (Count out the B brand!) I'm really looking for a good ratio suitable for a small two-driver two-way.
Comments?
This is about the golden ratio only, I know next to nothing about
speaker design.
The golden ratio is just a numeric constant, but an irrational one
so it can't be written down exactly. Hence, it is usually
approximated by a rational number, that is as a ratio between
two numbers. 2/3 is a crude approximation, 3/5 a sligthly better
one, 5/8 yet a better on. In fact, you can get arbitrarily close to
the golden ratio by taking the ratio of subsequent pairs of
numbers in the Fibanacci number series which starts as
1,1,2,3,5,8,13,21,...
and where each number is the sum of the two preceeding ones
in the series. You see from this series where the approximations
I gave above come from. The golden ratio has a strange tendency
to pop up in various situations as diverse as art, optimizing
certain data structures for computer programs and, obviously,
loudspeaker design. BTW, Fibonacci invented his number series
in an attempt to find a mathematical model for how quickly
rabbits multiply.
Back to the speakers then. I don't really know, but I suppose
what you want is a box dimension AxBxC s.t. B/A is the golden
ratio and C/B is the golden ratio. So in the case of 1.6/1/0.6
you referred to we have 1/1.6 = 5/8 which is a "standard"
approximation of the golden ratio, and we have 0.6/1= 3/5,
which is also a standard approsimation, but a cruder one. If
you really want golden ratios, you should probably adjust
at least one of these ratios so they have the same "degree"
of approximation, ie. both being 5/8 for instance. While the
increasingly refined approximations given by the Fibonacci
series give increasinly lower error, they do have a tendency to
alternate between positive and negative error. Hence, if you
use different ratios for the approximations of the box dimensions
you increase the error unnecessarily. Then what
precision you need, or if golden ratios are even good for speaker
design, that you have to ask somebody else.
speaker design.
The golden ratio is just a numeric constant, but an irrational one
so it can't be written down exactly. Hence, it is usually
approximated by a rational number, that is as a ratio between
two numbers. 2/3 is a crude approximation, 3/5 a sligthly better
one, 5/8 yet a better on. In fact, you can get arbitrarily close to
the golden ratio by taking the ratio of subsequent pairs of
numbers in the Fibanacci number series which starts as
1,1,2,3,5,8,13,21,...
and where each number is the sum of the two preceeding ones
in the series. You see from this series where the approximations
I gave above come from. The golden ratio has a strange tendency
to pop up in various situations as diverse as art, optimizing
certain data structures for computer programs and, obviously,
loudspeaker design. BTW, Fibonacci invented his number series
in an attempt to find a mathematical model for how quickly
rabbits multiply.
Back to the speakers then. I don't really know, but I suppose
what you want is a box dimension AxBxC s.t. B/A is the golden
ratio and C/B is the golden ratio. So in the case of 1.6/1/0.6
you referred to we have 1/1.6 = 5/8 which is a "standard"
approximation of the golden ratio, and we have 0.6/1= 3/5,
which is also a standard approsimation, but a cruder one. If
you really want golden ratios, you should probably adjust
at least one of these ratios so they have the same "degree"
of approximation, ie. both being 5/8 for instance. While the
increasingly refined approximations given by the Fibonacci
series give increasinly lower error, they do have a tendency to
alternate between positive and negative error. Hence, if you
use different ratios for the approximations of the box dimensions
you increase the error unnecessarily. Then what
precision you need, or if golden ratios are even good for speaker
design, that you have to ask somebody else.
Christer,
If I can pick your brain a bit more, the reason for the use of golden ratio's in speaker dimensions is that it would avoid standing waves appearing of the same wavelength or multiples of it in more than one dimension. Meaning that there should not be a standing wave in the width of say L, while at the same time there would be a standing wave of 2L in one of the other dimensions, L of course being determined by the fact that it would be an integral of the dimension.
Do you agree that this golden ratio in speaker dimensions is a good one to avoid these multiple-wavelength standing waves? Could there be a better one?
Jan Didden
If I can pick your brain a bit more, the reason for the use of golden ratio's in speaker dimensions is that it would avoid standing waves appearing of the same wavelength or multiples of it in more than one dimension. Meaning that there should not be a standing wave in the width of say L, while at the same time there would be a standing wave of 2L in one of the other dimensions, L of course being determined by the fact that it would be an integral of the dimension.
Do you agree that this golden ratio in speaker dimensions is a good one to avoid these multiple-wavelength standing waves? Could there be a better one?
Jan Didden
Fibonacci
Christer,
I like your clarification of the golden rule with the link to Fibonacci. This is based on the simple mathematical equation b:a = (b+a):b. Some fiddling around with this equation shows that b:a = 2/(sqrt(5)-1) which is 1.6180339887498948482045868343656.
This requires an highly accurate sawing-machine and skills.
Steven
Christer,
I like your clarification of the golden rule with the link to Fibonacci. This is based on the simple mathematical equation b:a = (b+a):b. Some fiddling around with this equation shows that b:a = 2/(sqrt(5)-1) which is 1.6180339887498948482045868343656.
This requires an highly accurate sawing-machine and skills.
Steven
Re: Fibonacci
Yes, I thought that was the reason. Or rather, I suppose we
want to find a relationship s.t. for each pair a,b of the three sides
of the box, we never have the case that both a and b are both a
multiple of the wavelength for any frequency in the audio band.
Or maybe multiple of half the wavelength is more appropriate?
It's been very long since I took the course on wave propagation.
I don't know if the golden ratio is guaranteed to achieve this
and if so if it is the only number achieveing it. If it does not
guarantee this condition, then maybe it is still the best choice
or maybe it isn't really known if that is the case. I am not a
mathematician so I have no clever answer here. However, I
wouldn't be the least surprised if it has been proven that the
golden ratio is the optimal value. That number as well as the
Fibonacci series has an almost magical tendency to appear even
where you least expect it.
Then, let's not forget that this is only the theory and since
element positions must be taken into account,
elements are not mathematical points etc. the best practical
values may very well be off from the theoretical best ones.
You ought to get yourself a better sawing-machine, then you may
use this approximation instead
http://www.cs.arizona.edu/icon/oddsends/phi.htm
Edit: As i said, the golden ratio has a tendency to show up
where you least expect it
http://www.golden-ratio.co.uk/
janneman said:
Yes, I thought that was the reason. Or rather, I suppose we
want to find a relationship s.t. for each pair a,b of the three sides
of the box, we never have the case that both a and b are both a
multiple of the wavelength for any frequency in the audio band.
Or maybe multiple of half the wavelength is more appropriate?
It's been very long since I took the course on wave propagation.
I don't know if the golden ratio is guaranteed to achieve this
and if so if it is the only number achieveing it. If it does not
guarantee this condition, then maybe it is still the best choice
or maybe it isn't really known if that is the case. I am not a
mathematician so I have no clever answer here. However, I
wouldn't be the least surprised if it has been proven that the
golden ratio is the optimal value. That number as well as the
Fibonacci series has an almost magical tendency to appear even
where you least expect it.
Then, let's not forget that this is only the theory and since
element positions must be taken into account,
elements are not mathematical points etc. the best practical
values may very well be off from the theoretical best ones.
Steven said:Some fiddling around with this equation shows that b:a = 2/(sqrt(5)-1) which is 1.6180339887498948482045868343656.
This requires an highly accurate sawing-machine and skills.
You ought to get yourself a better sawing-machine, then you may
use this approximation instead
http://www.cs.arizona.edu/icon/oddsends/phi.htm
Edit: As i said, the golden ratio has a tendency to show up
where you least expect it
http://www.golden-ratio.co.uk/
>If you dont like this shape then you play around by halving one ratio :
giving h=1.6, w=1, d=1.3, suitable for a two way.
====
Right, while the golden ratio collapses at the 8th harmonic, this one makes it to the 10th.
====
>According to the radio shack book i have you can use either
.6 : 1 : 1.6
or
.79 : 1 : 1.25
====
There's actually numerous acceptable acoustic ratios, some better than the golden ratio, but in most cases, any ratio that works out to the 4th or 5th harmonic is sufficient.
The asthetically pleasing 1:1.4:2 often seen in architecture is good to the 11th harmonic, and 1:1.26:1.41, 1:1.17:1.47, and 1:1.45:2.10 are (were?) popular speaker cab ratios.
If you can find a copy of Bolt, Beranek, and Newman's acceptable room ratio graph you can choose many more, or just plug whatever numbers you want to use in a room mode calculator to see how well they work.
====
>The ratios are much more applicable to room modes in acoustics.
In speakers you can damp the internal modes into insignificance.
====
I agree it's moot WRT the typical small two way monitors currently in vogue, but this often requires too much stuffing IMO if 'fullrange' drivers are used and/or the cab is fairly large.
GM
giving h=1.6, w=1, d=1.3, suitable for a two way.
====
Right, while the golden ratio collapses at the 8th harmonic, this one makes it to the 10th.
====
>According to the radio shack book i have you can use either
.6 : 1 : 1.6
or
.79 : 1 : 1.25
====
There's actually numerous acceptable acoustic ratios, some better than the golden ratio, but in most cases, any ratio that works out to the 4th or 5th harmonic is sufficient.
The asthetically pleasing 1:1.4:2 often seen in architecture is good to the 11th harmonic, and 1:1.26:1.41, 1:1.17:1.47, and 1:1.45:2.10 are (were?) popular speaker cab ratios.
If you can find a copy of Bolt, Beranek, and Newman's acceptable room ratio graph you can choose many more, or just plug whatever numbers you want to use in a room mode calculator to see how well they work.
====
>The ratios are much more applicable to room modes in acoustics.
In speakers you can damp the internal modes into insignificance.
====
I agree it's moot WRT the typical small two way monitors currently in vogue, but this often requires too much stuffing IMO if 'fullrange' drivers are used and/or the cab is fairly large.
GM
Small speakers ignore "golden" ratio?
I've realized that most small speakers ignore the golden ratio altogether (either for aesthetics or because of the fact that using such a ratio means the driver(s) cannot fit on the front panel). In fact, I've seen small speakers that are square or are nearly square in two dimensions (usually width and depth).
My question is: does an enclosure with a dimension 1.5 times larger than another just as bad?
I've realized that most small speakers ignore the golden ratio altogether (either for aesthetics or because of the fact that using such a ratio means the driver(s) cannot fit on the front panel). In fact, I've seen small speakers that are square or are nearly square in two dimensions (usually width and depth).
My question is: does an enclosure with a dimension 1.5 times larger than another just as bad?
Just as bad as what? Any dimension which is a the same as another will have the same standing wave fundamental and harmonics. A dimension which is easily divisble by another dimension will have some clashing standing wave harmormics. So 1.5 times is not as bad as having two the same, but is bad nevertheless.
See how the 2nd and 3d, 4th and 6th harmonics clash for your 1.5 times example. But it's not nearly as bad as having two dimensions equal
The smaller the dimension, the higher the standing wave frequencies. At 7cm for example, the standing wave fundamental is 2.4kHz. I think (though not sure) that for a reason related to the high frequency, small enclsosures do not have the same concern of standing waves (less enegry perhaps = more easily suppressed by stuffing/frequency handled by tweeter?)
"Thanks for the interesting info here, everyone. Maybe I'll have to look into devising my own ratio for each type of speaker (i.e. mini monitor, dual-woofer, etc.). About that Radio Shack book, sardonx, I probably have the older edition..."
I have Advanced Speaker Systems from radio shack.. and yeah i guess this might be newer than the one you have because i had one before this too... an older version (probably the one you have) but it got torn apart from laying around the floor too long.. lol. This book teaches lots of stuff.. nothing too in-depth but a useful book to have.
I have Advanced Speaker Systems from radio shack.. and yeah i guess this might be newer than the one you have because i had one before this too... an older version (probably the one you have) but it got torn apart from laying around the floor too long.. lol. This book teaches lots of stuff.. nothing too in-depth but a useful book to have.
- Status
- This old topic is closed. If you want to reopen this topic, contact a moderator using the "Report Post" button.