Posted by Rick Denney on February 29, 2004 at 14:38:00:
In Reply to: Golden Section and posted by Sam Gnagey on February 28, 2004 at 22:00:25:
A private correspondent took issue with my "curt dismissal" of the application of the Golden Mean to musical instruments, and of my suggestion of shared insanity with Sam.
I've come to the conclusion that people largely still be believe in magic (as opposed to the supernatural, which isn't the same thing at all). I'm generally distrustful of magic. One aspect of magic is the notion that there are guiding numbers that rule our lives. This point of view is consistent with general notions even of conspiracies that control our lives. But my observation of science (and of people) suggests that there are some relationships that repeat, but they repeat because they are the natural outgrowth of how things work, not because they are some underlying law guiding how they work. As relates to conspiracies, the notion that our lives are controlled by all-powerful and sinister human forces is belied by the general incompetence of humans to get away with grand conspiracies for very long. Are the leaders of the Tri-Lateral Commissions smarter than Andrew Fastow and the other Enron executives? Probably not. Grand conspiracies tend strongly to expose themselves, if for no other reason than they are grand. Small conspiracies that are unrelated can often add up to the same overall effect, but without the sinister guidance.
When we see a ratio that seems to appear frequently in nature, we have to ask if it is a cause or an effect. I believe that the Golden Mean is an effect and not a cause. That the nautilus shell follows the Golden Mean spiral is a result of both growing additively. The arithmetic is descriptive in both cases, just like compound interest. Likewise the other artifacts of nature that seem to follow from the Fibinacci series. That's why those who analyze additive growth use it as a model--because it describes, not because it controls.
I have read several books on the physics and design of musical instruments, including several written by Benade and an important one written by Fletcher and Rossing. In those books, I didn't see any arithmetic that would relate design to the Golden Mean. That doesn't mean that using it won't make an instrument more balanced looking. After all, the Greeks liked it because it was a balanced proportion that was pleasing to the eye, and since Greek art and architecture is the foundation of western art and architecture, our eyes are trained to appreciate it. That doesn't mean, however, that there are no cases where a 1.7:1 ratio won't be found as an artifact of some other arithmetic.
Photographers (such as myself) talk about the Golden Mean from time to time. But let's think about art. One characteristic of many good compositions is that the subject is off-center--typically about a third of the way in from the edge of the frame. "About a third" is a broad characterization, not a rule, and the notion that the recripocal of the Golden Mean is "about a third" less than 1 is probably just an interesting coincidence.
The common aspect ratios in photographic frames are 3:2 (35mm and 6x9), 5:4 (4x5, 8x10, 6x4.5), 4:3 (television), 7:6 (approximately similar to 5:4), 2:1 (6x12), 17:6, 16:9 (HDTV, which is quite close to 2:1), and square (6x6). The most popular are 3:2, 4:3, 5:4, and square, only one of which is even close to the Golden Mean. I've seen good and bad art in all these formats, as have most of us whether we know it or not.
In the end, what looks good is a matter of taste. People like rectangles, but the most pleasing aspect ratio and composition depends on the subject and the vision of the artist.
So, at the end of the day do we like the Golden Mean because the Greeks used it as the basis for western art, or did the Greeks use it because it has some special sub-conscious (or magic) importance? I vote for the former.
Rick "who enjoys math but who is not into numerology" Denney