Re: Golden Section and


[ Follow Ups ] [ Post Followup ] [ TubeNet BBS ] [ FAQ ]

Posted by Rick Denney on February 28, 2004 at 23:44:25:

In Reply to: Golden Section and posted by Sam Gnagey on February 28, 2004 at 22:00:25:

Sam, I think you are nuts. But I'm nuts enough to want to respond.

The Golden Mean is actually a mathematical constant, defined by

phi + 1 = phi * phi, which can be solved quadratically to prove that phi = (1 + sqrt(5))/2, which you can do on your calculator.

The notion is that if you take the Fibinacci series, where every pair of numbers in the series adds up to the next number, you get

0,1,1,2,3,5,8,13,21,34,55,89...

If you divide any number by the previous number, you get a value that, as you work your way up the list, settles down to phi.

1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.6667
8/5 = 1.6
13/8 = 1.625
21/13 = 1.615
34/21 = 1.619
55/34 = 1.6176
89/55 = 1.6182
...
1.618033989...

It's the only series of numbers that is both additive and multiplicative, which means it grows both arithmetically and geometrically. Beyond the first six or seven numbers, every number is 1.618 times the previous number (that's the multiplicative part).

It also has graphical significance, in that if you take a rectangle of sides equal to any two numbers in the Fibinacci series, and draw an arc from the long side to make a new long side, the new long side is the next number on the series. If you connect all these arcs together, you get a spiral that is very similar to the pattern of a nautilus shell.

The reason it's similar to the nautilus shell is because the snail adds to his volume geometrically each year, and as the volume increases geometrically, the diameter adds. So, additive growth patterns often follow the Fibinacci series, and it is therefore used to model additive growth patterns in nature.

The Greeks thought rectangles made using pairs of number on the series were inherently beautiful, and used them in art and architecture routinely.

With that background, how might it affect tuba design?

It might make it look pretty, but I don't think it has a lot of acoustical relevance. Using the nautilus spiral as the model for the branches of a tuba would make a mighty strange looking instrument--it would have to start small. If the first part of the tuba bugle had a radius of 1 foot, a BBb bugle would have a bell pointing the same direction as the mouthpiece (sorta like a Helicon), and would make a single 360-degree turn (a Helicon makes two turns). It would be 5' wide and 8' tall, plus half the diameter of the bell. It would look kind of like a very strange lur. Dr. Seuss would approve.

If you had a BBb bugle with 630 degrees of loop (so that the bell would point up), the base unit of the series would be 2.6 inches (possibly barely enough to wrap around one side of the your head on the first turn), and the centerline of the outer branch would form a rectangle 55" by 89" (yes, that's a golden rectangle). Furthermore, the only way for the bell to point up would be if the tuba was wider than tall, which would be mighty strange.

If you made the tuba a giant nautilus shell with a mouthpiece stuck in the middle of it, so that the outer skin of the next inner loop was the inner skin of the next outer loop, the taper would grow at the same rate as the curvature of the branch. In fact, the diameter of the branch would equal its radius. That's a mighty fast taper!

Forgetting the spiral, you could just make the tuba 1.618 times as high as it is wide, though the bell would extend out from the rectangle. I think that would amount to no more than an inside joke for people who are nuts, like you and me.

Rick "who can't believe how much time he just wasted" Denney


Follow Ups: