11 July 2008

Good's "singing logarithms"

I've previously mentioned Sanjoy Mahajan's Street Fighting Mathematics. (Yes, that's right, almost the entire sentence is links, deal with it.)

One thing I didn't mention is approximating logarithms using musical intervals, from that course. We all know 210 and 103 are roughly equal; this is the approximation that leads people to use the metric prefixes kilo-, mega-, giga-, tera- for 210, 220, 230, and 240 in computing contexts. Take 120th roots; you get 21/12 ≈ 101/40.

Now, 21/12 is the ratio corresponding to a semitone in twelve-tone equal temperament. So, for example, we know that 27/12 is approximately 3/2, because seven semitones make a perfect fifth. So log10 3/2 ≈ 7/40 = 0.175; the correct value is 0.17609... Some more complicated examples are in Mahajan's handout.

You might think "yeah, but when do I ever need to know the logarithm of something?" And that may be true; they're no longer particularly useful as an aid for calculation, except when you don't have a computer around. But I often find myself doing approximate calculations while walking, and I can't pull out a calculator or a computer! (To be honest I don't use this trick, but that's only because I have an arsenal of others.)

Is this pointless? For the most part, yes. But amusingly so.

The method is supposedly due to I. J. Good, who is annoyingly difficult to Google.

Oh, and a few facts I find myself using quite often -- (2π)1/2 ≈ 2.5, e3 ≈ 20.


Panos Ipeirotis said...

The last sentence begs for this link:

misha said...
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misha said...

Reminds me of yet another puzzle dished out by Sanjoy. Why is pi*pi so close to g, the free fall acceleration? Just because of the original definition of the meter).

Isabel Lugo said...


I asked the same question. My post "why g = π^2" is, I believe, one of the most-read posts here. It was also pretty strongly misunderstood -- a lot of people linking to it assumed I was some sort of crackpot, because they didn't bother to read it. (Most of those comments are not on my post, but on reddit, slashdot?, etc.)

misha said...

Nah, you are just a run-of-the-mill numerologist ;-)

misha said...

The pedulum meter standard had been updated because its unreliability, not only due to the amplitude of the oscillation, but also because g varies considerably with the latitude, as a result of the centrigugal force produced by the earth rotation, and also because the earth bulges out at the equator due to the same centrifutal force, which makes the dependence of g on the latitude even stronger. I'm surprised that nobody pointed it out at your old post on the topic.

misha said...
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misha said...

To tie up some lose ends of this mystery, we also have to understand where seconds came from. Why 24 hours, why 60 minutes in an hour and 60 seconds in a minute. I did some investigation a while ago, and found out that the ancient Egyptians used to count by touching the tips and the first 2 joints on their fingers by the thumb of the same hand. That gives us a dozen. Now, we got 5 fingers on each hand, therefore 60, and we got the day and the night. That about explains it.

Joe said...

The "why g = π^2" post is, if I remember correctly, the first post I read here. I probably came from a slashdot article on it. (Yes, some small subset of slashdotters do RTFA.) And I've been reading ever since. So, it's not all bad.