William M. Briggs

Statistician to the Stars!

Stock market crash, hearing aids, F-train music, and the boom-chhh Combinatoric Theory of Finite Musical Variety

The stock market is crashing and may have even bottomed out. Naturally, people are beginning to look around for buying opportunities. I have the perfect one.

Hearing aids.

Any company that sells or markets hearing aids is positioned for rapid growth over the next ten to thirty years. My scientific estimate is that a dollar invested today will bring twenty a few years from now. I give you this hot tip for free, my friends, just for being a loyal reader of this blog. You simply cannot go wrong.

My evidence for this stunning opportunity is based on my experiences commuting via the F train. My insight arrived suddenly, when yesterday I was surrounded by at least seven people wearing Thinking Suppression Devices (TSDs). I had to give up on reading James Fitzjames Stephen because each commuter’s device was louder than the others’. Everybody was wearing earphones, but even still, I was assailed by various booms and crashes which were clearly audible to anybody—to anybody not wearing a TSD themselves. These people are obviously going to lose a significant part of their hearing, if not go deaf, hence my advice about buying stock in hearing aid companies.

But I decided to make this an opportunity instead of despairing and following my usual approach of looking at my fellow passengers like they had just stepped in what they were listening to.

Now, the only distinct sounds I could make out—and I listened attentively—were various combinations of booms and chhhs (electronic cymbal crashes). This was my first important clue.

I asked the guy sitting next to me, who was fiddling with his TSD, “How many songs does that thing hold?” He said a number in the thousands, and that he had just about filled it up and was looking into getting a better model. I asked, “Got any Duke Ellington on it?” When he said “Who?”, I had my second clue, and realized that my boomchhh observation was not an accident. I therefore told him that he didn’t need a larger device, because there were only about 100 pop songs in the world anyway.

“Yes,” I sagaciously intoned, “based on my scientific measurements of your music—this is only an estimate, to be confirmed later, mind you—there are only about 100 possible songs that can be made. You don’t need a more expensive device. Think of the money I have just saved you!”

The look he gave me told me he was deeply aware of my genius. The fact that he edged toward the door as fast as he could, so that he could jump out at the next stop told me how excited he was to pass on his new knowledge to his friends and family.

It was on that train ride that I devised the Combinatorial Theory of Finite Musical Variety. This theory states that there are exactly, and only, 124 possible pop songs. Here is how it is proved.

Our first piece of evidence is that every popular song is comprised solely of combinations of booms and chhhs. There are other trivial elements, related to the outmoded and ancient theories of melody, harmony, point and counterpoint, and lyricism, but these can be, like they now always are, ignored.

The second piece of evidence is that, to build a pop song, you must have either three, four, or five booms and chhhs in combination and that at least one of these must be a special boom. Examples:

  • boom boom chhh
  • boom boom boom chhh
  • boom chhh chhh boom chhh

Ready? Let’s start counting. Begin with the three combination, with one mandatory boom. Suppose first that the boom is at the end. The other two slots must be filled with booms and chhhs. We could have two booms, two chhhs, or one of each with either leading the way. That’s 4 possibilities. But then we remember that we have the mandatory boom at the end, which could be placed in any of the three available slots (it could have come first, second, or third), and then the other booms and chhhs would fill in the other two slots. This makes 3 x 4 = 12 possibilities.

What if there were 4 slots and one mandatory boom? Suppose again that the regulated boom is at the end, and the booms and chhhs fill in the other three slots. They could all be booms or all chhhs, or various combinations. We could list them all, but that gets to be a pain, so let’s use some math. We have three slots and we could put 0, 1, 2, or 3 booms in those slots, with chhhs filling in the blanks.

There is a formula for this called choose. If we say “3 choose 0”, we say we have three slots and we choose no booms to go in them. The choose formula tells you the number of ways we can do this. It should be obvious to you that the only way to place no booms out of three is 1; namely, you choose no booms. Formally, the answer is

   n!
—————–
(n-k)! k!

where n! is read “n factorial” and means “n x n-1 x n-2 x … x 2 x 1”. In our example, n = 3 for three slots. At first, k = 0 for no booms. The formula works out to be

  3 x 2 x 1
———————–
(3 x 2 x 1) x 0!

where you have to know that “0! = 1” (there are good, technical, mathematical reasons for this, which I’ll skip). There answer equals 1. Then you work this formula out for k = 1, k = 2, and k = 3 because we want to know how many different booms go into the 3 slots for each possible number of booms.

The answer turns out to be 1 + 3 + 3 + 1 = 8 for k = 0, k = 1, k = 2, and k = 3. But again we have to remember that one mandatory boom could have gone into any of the four slots, so we have to multiply the answer by 4 to get 32.

Thus far, we have 12 + 32 = 44 possible songs, but we still have to calculate the number of possibilities if there were 5 slots and one mandatory boom. The math is exactly the same, except n = 4, and k = 0, k = 1, k = 2, k = 3, and k = 4. But it turns out that there is a simplification when you sum up all possible combinations of 2 things (booms and chhhs). The sum of “n choose 0” + “n choose 1” + … + “n choose n” = 2n, which is very easy to calculate.

If n = 4 then 24 works out to 16 different songs with the mandatory boom at the end, but there are 5 possible places for that boom, so we have 5 x 16 = 80 combinations.

In total, then, we have 12 + 32 + 80 = 124.

That’s it, friends! There are only 124 possible pop songs. I was therefore right to harangue my train-mate with this wisdom so that he would not waste his money on a device that holds more than this. It is therefore a matter of great curiosity how people could not have recognized this before now, a fact which can only tell us of the deleterious effect of pop music on the brain.

Incidentally, a corollary to the theory is: Each pop song must be played as loudly as possible, not only to annoy those around you, but to destroy your hearing as quickly as possible.

Incidentally, incidentally, I cannot vouch that the math here is 100% accurate. This is because, as I write this, the beaujolais nouveau arrived in stores this afternoon, and I am liberally sampling this year’s offerings.

25 Comments

  1. Briggs
    That is one of your funniest posts, on a par with “oh for a carreau!”
    The sound effects are hilarious when read out by speech software. I’ll have to read several times if I am to take anything in about the maths though. There was a scary word in the title as well which will need investigation.

    I have to admit to having such a device, but I have ‘proper’ neat headphones that don’t leak sound. It’s important to remember not to break into song. It’s embarrassing when people on trains can hear what you’re listening to. I don’t even like people to see the front of the I-pod. It’s like someone looking at your plate when you’re eating.
    …and the hard stares don’t work, I’ve tried this method.

  2. Qma loves Duke Ellington. What a BSer you are! (This is meant to be a compliment.)

    If there are three available slots and each could be filled with a boom or chhh,, shouldn’t there be 2x2x2=8 possibilities? Therefore, we have a total of 23+ 24 + 25 = 56. I might have missed something here. It’s not the point here anyway. 🙂

  3. Oops, my html tag code didn’t work. 2^3+ 2^4 + 2^5 = 56

  4. Briggs

    November 21, 2008 at 9:43 am

    QMA,

    To your first question: Sure, that’s why I said “The answer turns out to be 1 + 3 + 3 + 1 = 8 for k = 0, k = 1, k = 2, and k = 3.”

    Your last equation is wrong. We don’t have 2^5 possibilities in the last case. Don’t forget that one slot of the 5 must be a boom, which limits the combinations.

    You still get an A because you love the Duke.

  5. Here’s a wonderful two-fer:

    Marcus Roberts In Honor of Duke

  6. The rant for classical music against pop continues anabated, now with even more wit and sarcasm than ever! In the next instalment of Classical Rocks!, mr. William “the PhD” Briggs will formally take on Queen and Pink Floyd, and show how their music is actually more related to Global Warming than anyone could have ever imagined. Next! On Faux!

  7. Anyway Luis, about them there fairies.

  8. I got asked once why my iPhone doesn’t have any “good music” on it. At the time, I had some Bach cello concertos, a little Mingus, some Davis, and a lot of early Beethoven on there. All in near-lossless formats, of course.

    When asked why I use near-lossless codecs instead of low-quality MP3s, I responded with, “Because I actually LISTEN to my music.” That of course resulted in confusion.

    It’s bad enough that most music on iPods is absolutely terrible. But even the people with GOOD music on their iPods usually encode it in low quality crap formats. Ugh.

    By the way, Matt, we do need to account for singing in the equation. Male or female, at the very least.

  9. Ari, that’s all right and perfect and all, but I also think you do are too perfectionist. I mean, I can’t really discern the difference between my Beethoven CD disk and my MP3. If you don’t like 64kbs, then make 196 or even 256kbs. Lossless? Bokay, but then you lost any reason to be snobby smug about it, it’s called wasted space.

    It’s like encoding all your image files in BMP format, for chrissakes. Or your home videos on original AVI files.

    And yeah, I got my feelings hurt because you’ve just called me vulgar. As if.

  10. One word: Batteries!

  11. The error in the calculation is that sometimes, even in pop music, a slot is left unoccupied by either a boom or a chhh. It’s called dynamics.

  12. Luis,

    Pfft. I never called you vulgar. Just that you lack my snobby smug hearing abilities. 🙂

  13. JA:
    I agree – or perhaps “power storage devices”. Any tips? Is anyone working on a bonafide winner in this space?

  14. Briggs

    November 21, 2008 at 4:53 pm

    Masmit,

    If there’s silence, I’ve never heard it. Maybe it’s perceived silence due to successive booms mixing in the ear drum, the first seeps through the skull and bouncing back, and the second cancels the first one out.

  15. Bernie
    I bought a thing called Tune-juice in the States, They saw me coming.. It ran on a nine volt battery and didn’t appear to even slow the discharge rate of the I-pod let alone charge it. I’d bought loads of batteries to go with it as well, useless thing.
    Briggs
    0! = 1…?
    Hmm
    Well, the marker pen was out, I tried to work this out and ended up with ever larger numbers flipping from plus to minus. This could be one of those “it just is” moments.
    I had similar difficulty with “n to the power 0 = 1”.
    That’s clear now, thank goodness.
    It’s time for Cava. For I am a serious mathematician.
    Ari
    What’s PFFT?
    Luis
    I’m only playing about the fairies.

  16. I wear two digital hearing aids and can vouch for their effectiveness in compensating for hearing loss (in my case from gunfire, not rap). In addition to aiding hearing, they have a wonderful ancillary virtue: the ability to NOT HEAR. One touch of the tiny switch, and bingo — they transmute into ear plugs. I can then withdraw from all kinds of unwanted intrusions: airplane noise, politicians’ promises and pandering, street music, jack hammers,…. the possibilities are endless. My secret. Don’t tell.

  17. What about the fairies, Joy? Do they also make booms and chhhs? I can’t wait to hear them! Or is it that you replied in the other post and I didn’t see it?

  18. Luis
    Bless, no, just the banter of repetition.

    Who knows what they’d like, Maybe Punk? Thrash metal? or Tchaikovski, It’s a mystery.
    I’m going to swear,
    James Blunt, “ If I can’t hear the music” from all the lost souls is appropriate to the subject of this post, (although not my fave)
    There’s Bond and military drums going on in this piece of music.
    He is hated by many, mostly men, who squirm at the raw emotion and cheek of his catchy melody.
    (I’m guessing the guys are just jealous.)
    “you’re beautiful provoked the biggest negative reaction, always from men. What’s that about?
    Some of us love to sing, and you can’t sing to Duck Elliot or Beethoven.

  19. Briggs, I don’t say the gaps are necessarily silent – the missing boom or chhh may be replaced with a vocal ‘uhuh’, ‘yo’ or perhaps a subliminal satanic message. 🙂

    Joy, the reason James Blunt is generally disliked by men is that his only real skill is in having identified a number of phrases that have the reliable power to disintegrate knicker elastic.

    He reminds me of a guy I was in a band with, many years ago, who always told girls that before becoming a musician, he had been a pediatric nurse, and how much he missed the sick kiddies (he had actually been a lathe operator). Blunt has simply figured out how to convey that sort of thing in song.

  20. What an old cynic you are, Masmit. Your mate had the wrong line.
    I’m sure you’ll find that there is no direct correlation between good composing and knicker elastic robustness. If it is, it’s logaRhythmic.
    James can be a little maudlin but has written some gems. “High” was his best.
    I don’t have the hots for him though. A certain other James could be in with a chance.

  21. “you can’t sing to Duck Elliot or Beethoven”; yeah, it don’t mean a thing….

  22. Joy, you’re probably right, and I’m just an old cynic. Still can’t stand the guy…

    However, my old pal’s line was a big winner for him, he was well known for, well, this is a respectable blog, but you know what I mean. 🙂

  23. Dearieme,
    Not sure if I know how that goes.
    “Fat Planet” by Leftfield is the type of ‘music’ that may have leaked from those headphones. I love this track, but unless it is played with full effect of the bass it loses the richness of sound,. It was used in a famous ad in the cinema. Any normal speaker system or set of headphones doesn’t do it justice. This has to be the lowest sound I’ve ever heard. It’s a shame that it doesn’t translate to normal home speakers or headphones. I can’t remember what the film was but I still remember the ad.
    The piece sounds crass missing its more important lower notes. It’s said that sounds this low effect the autonomic nervous system. I would say all sounds have an effect more than we realise.
    It’s a long way from Beethoven. It may be evocative but it’s not sophisticated; the opposite in fact.
    The ‘geeks’ that rendered such a sound possible are the ones who deserve the credit for this, not the drummer boy, and I love it.
    Maybe I just loved the ad, who knows.

  24. Dearieme
    Sorry that’s “Phat Planet” not “fat” my mistake.

  25. Does listening to The Economist, CATO Institue, FT and BBC podcasts, for example, negate the TSD label?

    I find them very inetersting in the gym and car.

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