## Bourbaki and How To Write Numbers, A Trifle

So my attempt at keeping the Reading the Comics posts to Sunday has crashed and burned again. This time for a good reason. As you might have read between the lines on my humor blog, I spent the past week on holiday and just didn’t have time to write stuff. I barely had time to read my comics. I’ll get around to it this week.

In the meanwhile then I’d like to point people to the MathsByAGirl blog. The blog recently had an essay on Nicolas Bourbaki, who’s among the most famous mathematicians of the 20th century. Bourbaki is also someone with a tremendous and controversial legacy, one that I expect to touch on as I catch up on last week’s comics. If you don’t know the secret of Bourbaki then do go over and learn it. If you do, well, go over and read anyway. The author’s wondering whether to write more about Bourbaki’s mathematics and while I’m all in favor of that more people should say.

And as I promised a trifle, let me point to something from my own humor blog. How To Write Out Numbers is an older trifle based on everyone’s love for copy-editing standards. I had forgotten I wrote it before digging it up for a week of self-glorifying posts last week. I hope folks around here like it too.

Oh, one more thing: it’s the anniversary of the publishing of an admirable but incorrect proof of the four-color map theorem. It would take another century to get right. As I said Thursday, the five-color map theorem is easy. it’s that last color that’s hard.

Vacations are grand but there is always that comfortable day or two once you’re back home.

## Listening To Vermilion Sands

My Beloved is reading J G Ballard’s Vermillion Sands; early in one of the book’s stories is a character wondering if an odd sound comes from one of the musical … let’s call it instruments, one with a 24-octave range. We both thought, wow, that’s a lot of range. Is it a range any instrument could have?

As we weren’t near our computers this turned into a mental arithmetic problem. It’s solvable in principle because, if you know the frequency of one note, then you know the frequency of its counterpart one octave higher (it’s double that), and one octave lower (it’s half that). It’s not solvable, at this point, because we don’t have any information about what the range is supposed to be. So here’s roughly how we worked it out.

The note A above middle C is 440 Hertz, or at least you can use that for tuning ever since the International Standards Organization set that as a tuning standard in 1953. (As with any basically arbitrary standard this particular choice is debatable, although, goodness but this page advocating a 432 Hertz standard for A doesn’t do itself any favors by noting that “440 Hz is the unnatural standard tuning frequency, removed from the symmetry of sacred vibrations and overtones that has declared war on the subconscious mind of Western Man” and, yes, Nikola Tesla and Joseph Goebbels turn up in the article because you might otherwise imagine taking it seriously.) Anyway, it doesn’t matter; 440 is just convenient as it’s a number definitely in hearing range.

So I’m adding the assumption that 440 Hz is probably in the instrument’s range. And I’ll work on the assumption that it’s right in the middle of the range, that is, that we should be able to go down twelve octaves and up twelve octaves, and see if that assumption leads me to any problems. And now I’ve got the problem defined well enough to answer: is 440 divided by two to the twelfth power in human hearing range? Is 440 times two to the twelfth power in range?

I’m not dividing 440 by two a dozen times; I might manage that with pencil and paper but not in my head. But I also don’t need to. Two raised to the tenth power is pretty close to 1,000, as anyone who’s noticed that the common logarithm of two is 0.3 could work out. Remembering a couple approximations like that are key to doing any kind of real mental arithmetic; it’s all about turning the problem you’re interested in into one you can do without writing it down.

Another key to this sort of mental arithmetic is noticing that two to the 12th power is equal to two to the second power (that is, four) times two to the tenth power (approximately 1,000). In algebra class this was fed to you as something like “ax + y = (ax)(a y)”, and it’s the trick that makes logarithms a concept that works.

Getting back to the question, 440 divided by two twelve times over is going to be about 440 divided by 4,000, which is going to be close enough to one-tenth Hertz. There’s no point working it out to any more exact answer, since this is definitely below the range of human hearing; I think the lower bound is usually around ten to thirty Hertz.

Well, no matter; maybe the range of the instrument starts higher up and keeps on going. To see if there’s any room, what’s the frequency of a note twelve octaves above the 440-Hertz A?

That’s going to be 440 Hertz times 4,000, which to make it simpler I’ll say is something more than 400 times 4000. The four times four is easy, and there’s five zeroes in there, so, that suggests an upper range on the high side of 1,600,000 Hertz. Again, I’m not positive the upper limit of human hearing but I’m confident it’s not more than about 30,000 Hertz, and I leave space below for people who know what it is exactly to say. There’s just no fitting 24 octaves into the human hearing range.

So! Was Ballard just putting stuff into his science fiction story without checking whether the numbers make that plausible, if you can imagine a science fiction author doing such a thing?

It’s conceivable. It’s also possible Ballard was trying to establish the character was a pretentious audiophile snob who imagines himself capable of hearing things that no, in fact, can’t be discerned. However, based on the setting … the instruments producing music in this story (and other stories in the book), set in the far future, include singing plants and musical arachnids and other things that indicate not just technology but biology has changed rather considerably. If it’s possible to engineer a lobster that can sing over a 24 octave range, it’s presumably possible to engineer a person who can listen to it.