Draw fearlessly.
Use a pen or Sharpie. No erasers, no correcting fluid. Fill the page completely as fast as you can. Use loose scribbles and gestures. Don’t sweat details. Use The Force — let go your feelings, young Skywalker. Get it right the first time or start over. Try to push each drawing to completion, but if you’re really not happy with where it’s going, toss it in the recycling and move on. Also try drawing without looking at the page. Get ready to be pleasantly surprised by the result.

Draw light, not objects.
Squint your eyes at the scene until all you can see are big blobs of light and dark. Draw those. Try to ignore boundaries of objects; let those emerge out the natural boundaries of light and shadows.

Draw repetitively.
Get a stack of 3×5 index cards. Set up a simple still-life, a bowl of fruit or whatever, and draw it on every single index card. Do it from different angles, distances, etc. Use simple lines and don’t spend more than two minutes on each one.

Draw fast.
Give yourself thirty seconds to do an entire scene. Use big newsprint and a Sharpie. Ignore details. Use wide, loopy gestures. This is an especially important exercise when you get started drawing people, where it’s easy to obsess about faces or hands or feet without ever getting to the overall pose. One out of ten 30-second drawings you do will be amazingly good.

Draw everything.
Bowls of fruit and people are always nice, but try drawing whatever’s around: rocks, tangled computer cables, a brick wall, a rumpled bedsheet, wrappers and packaging, flaking paint, a movie poster, other works of art, a comic book page, an eggbeater, the underside of a table. Imitate other artists you admire. Copy your favorites as exactly as possible.

There’s no such thing as “talent.”
Everybody who draws well got there by practicing, practicing, practicing, formally or informally. The only thing that separates you from the masters is the Gladwellian ten thousand hours. Get to it, and have fun.

Some drawings I like

• The illustrations in Roger Penrose’s book The Road To Reality are delightfully minimalist. I have no idea what the math means, but I enjoy thinking about the drawings.
• Understanding Comics by Scott McCloud is a deep dive into the nature of art through the seemingly innocuous comics medium.
• See tons of interesting art, drawn and otherwise, in my Flickr favorites.

Original post on Quora

Recursion is crucial to writing computer programs in a compact, elegant way, but it also opens the door to infinite loops and irreconcilable logical contradictions.

Self-reference makes loops possible, which is great for programming. But sometimes the computer gets stuck in those loops. XKCD gives a playful illustration of how this can happen, using ducks.

We experience these infinite loops as computer crashes. The computer isn’t “stuck” when it crashes; it’s just running the same few instructions over and over.

The computer can’t break its own loops by “stepping outside of itself;” it needs an external agent to intervene, like you hitting the reset button.

The operations of our minds are also heavily recursive and self-referential. But unlike computers, we aren’t prone to getting stuck in loops, and we seem to be unfazed by logical paradoxes. Some of us even find them beautiful. Nature is full of self-similar, “paradoxical” structures like fractals.

Biological systems are especially self-similar and fractal-like.
Our brains are full of recursive loops. The brain’s representation of itself to itself is probably the basis of our consciousness.

The profoundest truths take on the quality of strange loops, GEB’s useful shorthand for recursive paradoxes. Here’s a diagram I made of the “heterarchy” of human knowledge.

Bach isn’t the only musician to use recursion and self-reference. Hip-hop and other sample-based music use it too, in the form of artists sampling their own songs within their own songs. Here are some blog posts digging into this idea.

• Biggie Biggie Smalls Is The Illest
• Eric B and Rakim
• Nas Is Like
• In A Silent Way is a remix of itself
• Self-reference in computer programming and hip-hop
• Take it to the bridge

Hofstadter also tackles the concept of emergence, the way that an intelligent mind can arise from the interaction of unintelligent component. He compares the mind to an anthill — the collective ant colony has intelligence, even though the individual ants are dumb.

Finally, the book is the best introduction to Zen Buddhist thinking that I’ve come across. Hofstadter observes that westerners are used to thinking in terms of neat Manichean categories — profound truths are unambiguously true or false. Zen prepares the mind to deal with Gödelian paradoxes, strange loops, fractals and the like.

I don’t think I’ve ever succeeded in reading GEB from cover to cover. It’s not really that kind of book. I prefer to just open to a random page and struggle with whatever concept I find there. I recommend a similar approach.

Original post on Quora

Groove is harder to pin down in text and diagrams than chords and scales, so it doesn’t get as much written about it. That gives some folks the mistaken idea that rhythm isn’t as important as melody and harmony. The reverse is true. You can have a long, rich and satisfying guitar-playing life using nothing but the standard fifteen chords, as long as you can groove. If you can’t groove, you can learn all the chords and scales you want, but you won’t sound good.

Here’s an exercise that worked great for me when I was learning, and that I make all my students do. I call it the One Note Groove. It’s pretty simple, you just put on a repetitive beat and play one note over it. Since you don’t have to think about which notes to play, you’re free to devote your entire attention to your timekeeping, your attack, your whole sound — in other words, your groove.

Get some beats lined up

The ideal scenario for practicing groove is to have an excellent drummer or bassist handy. This isn’t too practical for most beginners, who tend to attract other beginners as jam partners. I’m all in favor of hacking it out with other newbies, but you want to develop good timekeeping habits too. Technology is the best solution for most of us.

I dislike metronomes intensely. They’re too cold and artificial, and they can be a disincentive to practicing. The only good thing about metronomes is that they prepare you to play to a click in studio situations. But then, I don’t like using click tracks while recording either. Both for practicing and recording, I prefer drum or percussion loops. They do the same job of keeping your time steady, but they also impart a feeling of groove and energy, and they result in something that sounds a lot more like music.

So where do you get good loops?

I programmed these, feel free to download and use them:

Swung/Shuffle 8th notes 90 bpm by ethanhein

Straight 8th notes 90 bpm by ethanhein

Programs like Garageband and Fruityloops come loaded with plenty of good loops. Using Ableton Live or similar software, it’s easy to make loops out of any song in your record collection. With a drum machine or any of the aforementioned programs, you can program your own loops. Learning how to program drums is a great way to get inside of the beat — it did wonders for my timekeeping.

I also recommend getting a bunch of hip-hop instrumentals. The internet is full of them. I’m particularly fond of the instrumentals from Ready To Die by Notorious B.I.G., which you can download here.

I don’t recommend doing the one-note groove over complete songs. You don’t want chord or key changes, lyrics, sections or any other musical information to distract you from the groove. The less “interesting” your loops or instrumental tracks are, the better.

Which single note do you play?

If you’re using a loop with pitched instruments in it, you’re going to have to figure out what key it’s in so your one note can be the root (or fifth, or some other note that fits well.) And how, you may ask, do you find out what the key is? Trial and error works fine. There are only twelve possible notes, and you can just systematically work your way up the low E string until you find one that sounds good. In fact, this exercise is tremendously valuable ear training in its own right.

If you’re playing to drums or percussion, you can play whatever note you want. After you get deep enough into loop playing, you might notice that even drum loops usually have a characteristic pitch to them, especially in the kick and snare drum. Try to find a note that blends well with the percussion tuning.

Getting the most from your one-note groove

Put your loop or instrumental on infinite repeat, and start playing your note. Find a simple pattern, nothing challenging. Focus on getting it in the pocket. Make it really sound smooth. Don’t push, just relax into it. See how long you can keep it steady.

Once you’ve really nailed your pattern down, try adding or removing a note, and see how long you can keep that going. Shift the pattern a beat earlier or a beat later. Try finding a pattern with an odd length and let it cycle in and out of sync with the beat.

Try to play only the downbeats. Then try playing any beat but the downbeat. Play long sustained notes and pay attention to the way they decay. Play short, staccato notes and see how percussive you can sound. Play louder and softer.

Focus on the drums. Try playing the same pattern as the kick drum, as the snare, as the hi-hats. You can learn a lot about rhythm by focusing on specific drum instruments.

If you’re struggling to keep up, slow the tempo down until it’s comfortable. In general, you should play as slow as you can stand. Focus on sounding better, not playing faster.

Make sure you aren’t rushing ahead of the beat — beginners always rush like crazy as soon as they start getting a little mastery. Try leaving a lot of space between your notes and hear how your silences interact with the groove. If your loop swings, make sure to match its feel.

The one-note groove is a highly meditative exercise. For the first few minutes, you might struggle a little to settle into your groove. Then you probably get bored. But if you keep going through the boredom, you can suddenly break free into previously unexplored regions of musical inspiration. The longer you stay focused on your groove, the more pleasurable it gets. Listen to Fela Kuti or John Coltrane stretch out on single-chord grooves for fifteen, twenty, thirty minutes, and try to emulate their relaxed intensity.

Example: the Funky Drummer Bonus Beat Reprise

One of the best tracks you can use to do the one-chord groove exercise over is the this remix of “The Funky Drummer Parts One and Two” by James Brown.

Play on the open A string. Use your left hand and the heel of your right hand to mute the string to control your notes’ duration. Get ready to feel the funk!

Repetition is a great teacher, not just of music, but of everything. Let the loop ride and see where it takes you.

Miles and Eno seem radical, but in a way, they’re just boiling the usual compositional process down to its raw essentials. Really, all composition and songwriting consist of improvising within constraints and then sequencing the best ideas into shape. Usually this improvisation happens in short spurts, inside the composer’s head or alone at an instrument. Using a recording device instead of a sheet of paper can make the process more bodily and immediate, and can help get at playful ideas that might not squeak past the mind’s internal judges and editors during the relatively slow process of writing stuff on paper. Michael Jackson wrote his best stuff by improvising into a tape recorder. There’s something about improvising a performance while being recorded that focuses the mind wonderfully.

Since 2004 I’ve been writing and recording with Barbara Singer in different configurations. The first version was her idea, a band called Blopop. She had some techno versions of pop songs programmed into her MC-909 groovebox, and the idea was that she’d sing and DJ, and I’d improvise guitar on top.

Both Barbara and I come from jazz training, and both of us felt boxed in playing standards. Free jazz wasn’t that interesting to us either; it felt too chaotic and self-indulgent, too disconnected from the musical world we live in. Babsy had the bright idea to use electronic beats and loops as the basis for improvising. Her original concept was to use pop songs as the basis for improv. We did a little performing that way, but then quickly moved into completely open-ended blowing over beats.

Brian Eno has all kinds of different systems for imposing order on his in-studio improvising. For us the system was to use the presets in Barbara’s groovebox. The generic techno grooves programmed into the box establish  a key and a vibe, so you just set the tempo and you’re off to the races. In a perfect world we would have programmed everything ourselves from scratch, but there was something wonderfully effortless and expedient about just dialing through the presets at random.

Babsy is an improv comedian, a veteran of various improv groups and a student of the Upright Citizens Brigade Theatre. We talked a lot about the improv comedy bible Truth In Comedy and how applicable it is to music too. If you’re confident, responsive to the other performers, and genuinely focused on the present moment, you really can’t do anything wrong.

Constrained improvisation is a perfect meditation exercise. I learned firsthand what the Buddhists always say, that it takes a lot of practice and discipline to be maximally effortless and intuitive. I’ve enjoyed few activities more than freeform musical improv over techno beats. Completely free improv can be a pleasure too, but it can also be a pain, since it usually devolves into formless noodling. The beats give enough structure to make the process fun. Here are some of our attempts to put the Truth In Comedy principle into action.

See

mp3 download, ipod format download

Improvisation recorded during the first time Barbara and I were ever in a room together, in the summer of 2004. Babsy is in the excellent habit of recording pretty much every note she plays or sings. I was a little taken aback when she wanted to record our first session, but went along. This isn’t edited, or even mixed. I pick a starting note at random, which turns out to be the flat seventh of the synth loop’s key. That establishes the main riff I have to work off of. This element of harmonic randomness ended up being a big part of the band’s pleasure for me, having to puzzle out a good-sounding relationship between the note I picked to start on with whatever came out of the groove box.

Warmup

mp3 download, ipod format download

Another unedited improv, recorded a month later than the one above. As the title suggests, this was just to get limbered up at the beginning of a session. It fades out once I lose the thread.

Everything We Do Is Right

mp3 download, ipod format download

Maybe our best attempt at a longer-form improv.

Window remix

mp3 download, ipod format download

Edited from over half an hour down to eight or so minutes. The original contains all these ideas, but they’re separated by some stretches of aimless wandering, and with looser repetition. I like it better this way.

Listening now, this stuff doesn’t nearly as tight or focused as our more pop and remix-oriented material we eventually moved into. But I admire the spirit of adventure behind it. My guitar playing certainly improved enormously under the pressure of all that recorded improvising. We never remotely found an audience for this music. It was too weird and avant-garde for the dance music people, not weird enough for the avant-garde, too unfocused and unpredictable for pop fans, too electronic for jazz fans. Still, I think it was a cool idea, one that I don’t think we came close to exploring completely. I’m still interested in pursuing this format further. Anybody out there game for some Eno-flavored freeform techno? Drop me a line.

I post this because I’ve been reading Coming Of Age In The Milky Way by Tim Ferris, as good a summary of the state of cosmology between two covers as a person could ask for. Thinking about the horrifying enormousness and ancientness of the universe might have depressed Woody Allen, but it has a paradoxically calming effect on me. Reading books like Ferris’ is my favorite form of meditation.

Every single illustration you’ve ever seen of the Big Bang is wrong. They all show an explosion from a central point, expanding outwards into space. The Cartoon History Of The Universe shows the pre-Bang universe as a little blob floating against a white background. When you turn the page, the blob bursts forth in a gooey-looking explosion across a two-page spread. It works great as a graphic device, but it gives the wrong idea. At the moment of the Bang, there wasn’t a central point in the universe where everything was gathered up. The explosion happened everywhere in space. More accurately, the explosion created space.

Starting at the moment of the Bang, everything in the universe got further away from everything else. From anywhere in space it would have looked like everything was being blasted away from you. The universe still looks that way; anywhere you look, all the galaxies are racing away from us. The further away they are, the faster they’re receding.

It’s hard for me to wrap my head around the idea of space being smaller. Space never had a boundary, so far as anyone can tell, so how can it have a size? It helps me to think of an eighties video game like Pac-Man or Asteroids where the screen wraps around. When you go off the right side of the screen, you reappear on the left side. It’s as if the screen is on the surface of a cylinder, so really you’re just going around in a circle.

It’s harder to imagine three-dimensional space wrapping around this way, but not impossible. There was a moment right after the Bang when the universe had a radius of ten feet. If you had been there (and weren’t instantly vaporized), you’d see the back of your own head ten feet in front of you.

Tim Ferris’ book has a nice passage that helps visualize the Bang. He asks you to imagine a staircase going backwards in time. Each step is ten times shorter than the previous one. Step one is a billion years after the beginning of time. Step two is a hundred million years after the beginning of time. Step three is ten million years, step four is a million, step five is a hundred thousand, and so on. At step one, a billion years after the Bang, everything in space is younger, hotter and closer together. At step two, a hundred million years in, the universe is mostly dark. There are a few stars, but mostly it’s just a lot of hydrogen and helium gas swirling around. At step three, ten million years in, all of space is at room temperature. At step four, there are so many electrically charged particles rubbing against each other that the entire universe is flooded with blinding white light. Very biblical.

At step six, ten thousand years in, the entire universe is the same temperature as the surface of the sun, and totally dark, since there isn’t enough space between particles for the photons to get anywhere before whacking into something and being reabsorbed. At step eleven, the entire universe is hotter than the center of the sun. Between steps seventeen and eighteen, one second after the beginning of time, the entire universe is “denser than rock and as hot as the explosion of a hydrogen bomb.” Cool!

There are so many big mysteries. Why did the Big Bang happen at all? One school of thought says, well, if it hadn’t, then we wouldn’t be here to think about it. This makes logical sense but isn’t very satisfying. Another idea is that there’s a tiny but nonzero chance of a Big Bang happening anywhere and anytime as part of the random quantum fluctuations of the universe. By this thinking, Big Bangs could be happening all the time throughout the multiverse. Scientists don’t like this idea because there’s no way to test it; all the parallel universes would be mutually inaccessible and invisible to each other.

A more interesting question, because we might find an answer someday, is why the universe has stars, galaxies and other interesting structure to it. At the moment of the Bang, space was a single super hot energy field. As space expanded and cooled down, the energy converted itself into pairs of matter and antimatter particles: quarks and antiquarks, electrons and positrons and so on. Based on what we see in the world now, it would be logical to assume that the Bang would have produced equal numbers of particles and antiparticles. And yet, that isn’t what happened. The balance was close to equal, but not exactly. For every billion antiparticles, there were around a billion and one particles. As the pairs zapped each other out of existence, the leftovers congealed into galaxies, stars, planets and us.

A ratio of a billion to a billion and one is pretty improbable-seeming, and very lucky. If the matter and antimatter in the early universe been perfectly balanced, every particle-antiparticle pair would have mutually annihilated back into energy, resulting in a universe with no objects in it, just a photon every trillion cubic miles. If the ratio had been more unequal, the amount of matter (or antimatter) left over would have overwhelmed the expansion of space, collapsing instantly into a few humungous black holes. The very slight asymmetry between matter and antimatter production is the reason we’re here. So why did this extraordinarily lucky thing happen? Nobody knows. It might have something to do with muons and the weak nuclear force, but that doesn’t explain anything, it just locates the weirdness in a particular branch of particle physics.

Lee Smolin has an attractive hypothesis. He starts with the assumption that inside every black hole is a new Big Bang. As viewed from the inside, Smolin imagines the black hole’s violent collapse as looking like a violent explosion. This is a controversial idea among the physicists, but it appeals to my intuition. Smolin suggests that every new universe formed in a black hole has slightly different physical parameters, different basic physical constants and ratios. Most of these universes will have rules of physics totally unlike ours, and will be inconceivably different. Some universes will be kind of similar to ours. A very few will be nearly identical. The ones with delicate imbalances between matter and antimatter will have stars, planets and galaxies.

Here’s where Smolin’s idea gets really fun. Universes with more stars and galaxies are likelier to have more black holes, and will thus spawn more baby universes. So the multiverse could be selecting for improbable universes like ours, the way the conditions on the earth select for improbable lifeforms.

Another idea I find appealing is Paul Steinhardt and Neil Turok’s theory that our universe has an invisible twin out there in higher-dimensional space. We violently collide with this twin every few trillion years, an event which looks from our perspective like a Big Bang. I like to imagine the twin as being the Star Trek evil universe.

I have a lot of scientists in my family, professional and amateur. I was exposed to the Big Bang theory long before the biblical creation story. The idea that the universe just came into being for no real reason suits my intuition better than the idea that there’s a grand purpose to the whole thing, a purpose that specifically involves humans. The idea of the Big Bang as a meaningless, accidental event makes perfect sense to me.

My intuition has a harder time imagining the universe as it is now. Douglas Adams was right:

Space… is big. Really big. You just won’t believe how vastly hugely mindbogglingly big it is. I mean you may think it’s a long way down the road to the chemist, but that’s just peanuts to space.

My favorite illustration of Douglas’ point is the Hubble Ultra Deep Field, a photo taken by the Hubble Space Telescope:

It looks like a regular old photo of space, but it’s not. The Hubble was deliberately pointed at a boringly “empty” region of the sky. It imaged this region carefully over many exposures to reveal whatever objects there were that might be too faint to have been noticed. In the resulting picture, nearly every singly dot is a galaxy containing billions of stars. This is where the meditation exercise gets away from me. It’s still fun though.

I leave you with some wisdom from the Cartoon History Of The Universe:

Our creation stories start with the assumption that we’re the most important thing in the world, the reason for everything else’s being. The story that science tells relegates us to the periphery. Life has mostly been smaller and simpler than us. I think it’s important to recognize that we might very easily wipe ourselves out, and the microbes will barely have noticed we were even here.

Seeing numbers with nine zeroes in them doesn’t help in imagining the time scales involved here, so I’ll also use the convention that the entire history of the Earth is taking place in one day. In this analogy, one “hour” on the clock is about 192 million years.

midnight (4,600,000,000 years ago)

Earth forms from space dust left over from the new sun.

12:21 am (4,533,000,000 years ago)

Earth collides with the planet Theia, causing rings of debris to form. The rings last for millions of years until they coalesce to form the Moon. (Check out this fun visualization of what the modern Earth would look like with rings.)

2:36 am (4,100,000,000 years ago)

After a lot of volcanic excitement, the surface of the Earth cools enough for the crust to solidify. The atmosphere and oceans form.

3:08 am (4,000,000,000 years ago)

The earliest known life appears. How? In the fifties they thought it was a lightning strike in the primordial soup. Newer theories center around RNA molecules or phospholipids forming little bubbles.

DNA emerges and quickly out-replicates whatever other self-catalyzing chemical reactions are happening. During this time, the atmosphere has no free oxygen.

3:42 am (3,900,000,000 years ago)

Late Heavy Bombardment: the rain of giant rocks from space on the Earth reaches its peak. The oceans boil away completely, more than once, but life persists underground.

5:42 am (3,500,000,000 years ago)

Lifetime of the last known common ancestor of every living thing today. Did it look like this?

8:18 am (3,000,000,000 years ago)

Photosynthesizing bacteria evolve. They release the first free oxygen into the atmosphere, which is a poison for a lot of other bacteria. The moon’s orbit is still very close to the Earth, causing tides a thousand feet high. The planet is continually wracked by hurricane-force winds.

1:01 pm (2,100,000,000 years ago)

More complex cells appear, with various organelles and other internal structures.

These cells probably start out as symbiotic communities of simpler organisms, co-evolving to become inseparable, like modern mitochondria and chloroplasts.

3:44 pm (1,200,000,000 years ago)

Sexual reproduction begins, so child cells no longer need to be exact clones of their parents. The pace of evolution picks up.

4:48 pm (1,000,000,000 years ago)

Simple multicellular organisms appear in the oceans: first colonies of algae, then seaweed.

8:06 pm (600,000,000 years ago)

Cell colonies evolve into sponges, the earliest multicellular animals. Next come jellyfish, with muscles, digestive systems with mouths, and the beginnings of simple neural nets (no brains yet.)

The ozone layer forms, filtering some of the sun’s radiation and allowing the microbes to creep out onto land without getting cooked.

9:12 pm (540,000,000 years ago)

Worm-like creatures get more specialized and complex, with circulatory systems and everything.

9:24 pm (505,000,000 years ago)

The first vertebrates emerge, jawless fish resembling lampreys.

9:30 pm (475,000,000 years ago)

Fish evolve jaws from their frontmost gill arches. They also develop stylish armor plating on their heads and thoraxes. The first plants and fungi move onto land.

9:42 pm (450,000,000 years ago)

The arthropods appear. Since their exoskeletons support and hold water in, they’re the first animals to move onto land, starting with millipedes and centipedes, then spiders and scorpions.

9:54 pm (400,000,000 years ago)

The first insects and sharks appear.

10:06 pm (360,000,000 years ago)

Some lobe-finned fish develop legs, maybe to maneuver through shallow plant-choked swamps. Some of these creatures make short trips onto land.

Plants evolve seeds that protect their embryos, enabling them to spread more quickly on land.

10:24 pm (300,000,000 years ago)

The supercontinent Pangaea forms, eventually providing the name for a great Miles Davis album.

Evolution of the amniotic egg gives rise to reptiles, who can reproduce on land. Insects resembling humungous dragonflies are the first animals to fly. The vast forests of clubmosses, horsetails, and tree ferns will eventually fossilize into coal and oil.

10:42 pm (250,000,000 years ago)

The Permian-Triassic extinction event wipes out about ninety percent of all animal species. Yikes.

10:54 pm (220,000,000 years ago)

Early reptiles diversify into crocodilians, dinosaurs, and pterosaurs. From another reptile branch, the synapsids emerge and evolve into the first tiny ancestors to mammals. (You can see the remnants of our lizard forebears in the scaly tails of rats.)

Conifers become the dominant land plants. Plant-eating animals get bigger to house the long digestive tracts it takes to digest the not-very-nutritious pine trees.

11:00 pm (200,000,000 years ago)

Dinosaurs survive a mass extinction and grow enormous. Modern amphibians emerge, including frogs and salamanders.

11:04 pm (180,000,000 years ago)

Pangaea begins to break up into big land masses. The biggest is Gondwana, made up of Antarctica, Australia, South America, Africa, and India. Antarctica is covered with lush forests. North America and Eurasia are still joined. The first true mammals appear.

11:13 pm (150,000,000 years ago)

Giant dinosaurs dominate the land, the ones we all learned about in second grade. The earliest birds evolve from therapod dinosaurs.

11:18 pm (135,000,000 years ago)

Microraptor gui, a two foot long dinosaur in Northeast China, has bird-like feathered wings on four limbs.

11:19 pm (130,000,000 years ago)

Plants evolve flowers, setting off a major burst of animal co-evolution, like bees.

11:21 pm (125,000,000 years ago)

Ancestors of placental mammals that look like mice live in small shrubs in China.

11:37 pm (75,000,000 years ago)

Lifetime of the last common ancestor of humans and mice. Birds with teeth roam the Northern Hemisphere.

11:40 pm (65,000,000 years ago)

The Cretaceous-Tertiary extinction event wipes out about half of all animal species, including all non-avian dinosaurs. It’s likely caused by a giant asteroid landing off the Yucatan Peninsula.

Without the dinosaurs, mammals can and do get bigger and more diverse. A group of small, tree-dwelling, insect-eating mammals branches out into the ancestors of primates, treeshrews, and bats.

11:44 pm (50,000,000 years ago)

Ancestors of whales walk on land like modern sea lions and swim like modern otters. Ancestors of manatees walk like hippos and also swim like otters. The ancestors of dogs, cats, bears and raccoons are meat-eating, weasel-like tree climbers.

11:47 pm (40,000,000 years ago)

Primates diverge into lemurs, tarsiers, monkeys and apes.

The earliest elephant is the size of a large pig.

11:49 pm (37,000,000 years ago)

Grasses evolve. The first dogs appear in North America.

11:53 pm (22,000,000 years ago)

India collides with Asia, pushing up the Himalayas. The first bears are the size of foxes and hunt in the tree tops.

11:54 pm (20,000,000 years ago)

The African plate collides with Asia. Gigantic animals roam South America, including eighteen-foot-long giant sloths and flying birds with twenty-foot wingspans.

11:55 pm (16,000,000 years ago)

Whales begin to use echolocation. There are sharks the size of buses.

11:56 pm (13,000,000 years ago)

Human ancestors speciate from orangutan ancestors.

11:57 pm (10,000,000 years ago)

Monkeys proliferate, and the apes go into decline. Human ancestors speciate from gorilla ancestors.

11:58 pm (5,000,000 years ago)

Volcanoes erupt and create Panama. Mammals from North America cross the isthmus and cause extinctions of South American mammals. Human ancestors speciate from chimpanzee ancestors. The largest known primates reach twelve feet tall in Asia.

11:59 pm (3,700,000 years ago)

Australopithecus afarensis leaves footprints on volcanic ash in Kenya.

11:59.04 pm (3,000,000 years ago)

Early hominins live on the savannas of Africa, where they’re hunted by giant cats.

11:59.38 pm (2,000,000 years ago)

Meat-eating Homo species coexist with Paranthropus, hominins who eat plants and termites. Homo habilis uses stone choppers in Tanzania. Broca’s area, the speech region of the modern human brain, begins to emerge.

11:59.40 pm (1,800,000 years ago)

Homo erectus emerges in Africa and migrates to other continents, mostly South Asia.

11:59.46 pm (1,750,000 years ago)

Armadillos the size of cars live in southern Peru.

11:59.76 pm (700,000 years ago)

Common genetic ancestor of humans and Neanderthals.

11:59.82 pm (500,000 years ago)

Homo erectus in China use charcoal to control fire, though they may not yet know how to create it.

11:59.94 pm (195,000 years ago)

The earliest known Homo sapiens live in Ethiopia.

11:59.95 pm (160,000 years ago)

Homo sapiens in Ethiopia practice mortuary rituals and butcher hippos.

11:59.953 pm (150,000 years ago)

“Mitochondrial Eve” lives in Africa, the last female ancestor common to all mitochondrial lineages in humans alive today.

11:59.959 pm (130,000 years ago)

Neanderthals in Europe and the Middle East begin to bury their dead and care for the sick.

11:59.969 pm (100,000 years ago)

Humans live in South Africa and Palestine, probably alongside Neanderthals.

11:59.974 pm (82,500 years ago)

Humans in Zaire fish using spear blades made from sharpened animal bones.

11:59.977 pm (74,000 years ago)

An enormous volcanoic eruption in Indonesia causes the human population to crash as six years without a summer are followed by a thousand-year-long ice age. Volcanic ash up to fifteen feet deep covers India and Pakistan.

11:59.988 pm (40,000 years ago)

Jared Diamond’s “great leap forward.” Humans paint and hunt mammoths in France. Most large mammal species disappear, probably because of the expanding human population.

11:59.990 pm (32,000 years ago)

First known sculpture in Germany. First known flute, made from a bird bone, in France.

11:59.991 pm (30,000 years ago)

Humans enter North America from Siberia, as well as the Solomon Islands and Japan. Bows and arrows are used in the Sahara. Fired ceramic animal models are made in Eastern Europe.

11:59.992 pm (27,000 years ago)

Neanderthals die out, leaving Homo sapiens and Homo floresiensis as the only hominid species. In Eastern Europe, humans invent textiles and press weaving patterns into pieces of clay before firing them.

11:59.994 pm (20,000 years ago)

Oil lamps made from animal fats on shells are used in caves in France. Bone needles are used to sew animal hides in China. Wooly mammoth bones are used to build houses in Russia.

11:59.995 pm (15,000 years ago)

The most recent Ice Age ends. Megafauna extinction begins in the Americas, again, probably because of humans.

11:59.997 pm (10,000 years ago)

The human population reaches five million, presumably causing the extinction of Homo floresiensis and the woolly mammoth. Humans domesticate gray wolves into dogs, and in the Middle East, begin to develop agriculture. Hunter-gatherers in Japan create the earliest known pottery. Humans now occupy every continent except Antarctica.

11:59.9975 pm (8,000 years ago)

Humans hybridize bread wheat from emmer wheat and goat-grass.

11:59.9980 pm (6,500 years ago)

Humans domesticate rice.

11:59.9991 pm (3,000 years ago)

Humans in Eurasia start using iron tools.

11:59.9997 pm (1,000 years ago)

The human population reaches one hundred fifty million.

11:59.99995 pm (150 years ago)

The human population reaches one billion.

11:59.99999 pm (40 years ago)

Humans walk on the moon. The human-caused extinction event accelerates.

midnight (present)

The human population approaches seven billion.

And then what? Your guess is as good as mine.

See a version of this post on Quora.

Groove and minimalism

“Once In A Lifetime” is made of layers of percussion and keyboards and guitars swirling around the central bassline, a four-bar cell that repeats almost identically under the entire song. Rock and pop are all about simplicity and repetition, and this bassline pushes both qualities as far as they can go. Byrne and Eno have a well-known love for African pop and funk, and it comes through clearest in “Once In A Lifetime.” Byrne and Eno know that if you have a really good groove happening, people will never get bored no matter how repetitive it is.

A normal American pop song is around three and a half minutes long. Three and a half minutes would barely be enough of “Once In A Lifetime.” In the video up there it’s five and a half minutes, and on The Name Of This Band Is Talking Heads, it’s six minutes. It could be forty-six, as far as I’m concerned; it’s one of those grooves, like Herbie Hancock’s “Chameleon” or James Brown’s “Funky Drummer” that you never get to the bottom of.

The harmony in “Once In A Lifetime” isn’t as minimal as the bassline, but it comes close. The entire song is just different voicings of D7 or D7sus4. As with the bassline, I’m not bored of D7 by the end, any more than I’m bored of the sruti box drone in a Ravi Shankar record.

“Lifetime” has verses, choruses and a bridge, like a normal pop song, but the sections are all harmonically identical. They differ in arrangement, as combinations of instruments enter and exit, muted and unmuted on the mixing desk. This kind of song structure was radically weird by US pop standards in 1980. Hip-hop embraced it enthusiastically, and now it’s becoming the mainstream pop standard as well.

Songwriting using improvised loops

“Lifetime” was written by loop-based improvisation in the studio over a click track, followed by many hours of mixing and tape editing. The band performed a long, simple, repetitive groove, and you can think of the finished product as the jam’s highlight reel.

Using improvisation as the basis for songwriting is nothing new. Duke Ellington and Benny Goodman routinely borrowed (or stole) improvised licks from their sidemen and incorporated them into their tunes. Some of Miles Davis’ best albums are built entirely from edited improvisations by his band. But piecing together songs out of improvisation at the level of single phrases was a pretty fresh concept in 1980.

Building songs out of live improvisation means that your song is written by people focused in the moment, with their usual self-consciousness temporarily dissolved. This is the kind of brain state in which people have their best ideas. Who knows which bandmember thought up the “Once In A Lifetime” bassline, but I’ll guarantee you that they never would have arrived at it sitting alone in a room with a pencil and paper.

Improvised lyrics

Musicians who focus mostly on lyrics, like Joni Mitchell and Leonard Cohen, tend not to engage the listener much from the neck down, much less the waist. David Byrne’s lyrics on “Lifetime” were mostly improvised over the completed instrumental track. This is a strange way to work for most of the rock musicians I know. For hip-hop, it’s a common practice, and the pop mainstream is mostly following suit.

Improvised lyrics like David Byrne’s don’t come from the verbal consciousness. They come from deeper in the intuitive mind. Talking Heads lyrics, goofy and asymmetric though they are, always have nice strong body logic. They feel good when you sing them or speak them, or speak-sing them the way David Byrne does.

You may find yourself living in a shotgun shack
You may find yourself in another part of the world
You may find yourself behind the wheel of a large automobile
You may find yourself in a beautiful house, with a beautiful wife
You may ask yourself: Well… how did I get here?

Letting the days go by, let the water hold me down
Letting the days go by, water flowing underground
Into the blue again, after the money’s gone
Once in a lifetime, water flowing underground

You may ask yourself
How do I work this?
You may ask yourself
Where is that large automobile?
You may tell yourself
This is not my beautiful house!
You may tell yourself
This is not my beautiful wife!

Then the chorus again, then:

Same as it ever was
Same as it ever was
Same as it ever was
Same as it ever was
Same as it ever was
Same as it ever was
Same as it ever was
Same as it ever was

Water dissolving and water removing
There is water at the bottom of the ocean
Under the water, carry the water
Remove the water from the bottom of the ocean

That last line is pretty much obliterated by tape loops of itself. Then the chorus again, and then the last verse:

You may ask yourself
What is that beautiful house?
You may ask yourself
Where does that highway go to?
You may ask yourself
Am I right? Am I wrong?
You may say to yourself
My god!…WHAT HAVE I DONE?

Chorus, chorus, bridge over the chorus sung half speed, out.

What does it mean? It could be gibberish, or a deep and profound statement about the existential crisis facing modern humans, or a zen koan, or a bunch of inside jokes between David Byrne and himself. I think what makes the song so cool is that it works equally well on any of those levels.

David Byrne figured out how a nerdy white guy can have soul

David Byrne’s ingenious approach to soul is to be his anxious and uptight self. By not even attempting to be cool, he becomes the coolest nerdy white guy he can be. The same strategy works great for Jason Schwartzman, Jon Stewart and Napoleon Dynamite.

Art should be fun

“Lifetime” is a piece of abstract, conceptual modern art, but it’s also totally accessible. It doesn’t require any special knowledge to enjoy it. Why can’t all modern art be fun? Why should highbrow culture make me bored? I experience enough boredom. I wish for more highbrow musicians to follow Talking Heads’ example and write fun songs you can dance to.

Brian Eno’s role

Non-musicians have a hard time imagining what the producer’s role is in music like this. Anybody who watches TV can picture a guitarist or a drummer, but you might never see a producer at work unless you’ve been in a studio. The producer’s job in electronic music is like the editor’s in a movie or TV show. Sometimes the music producer is directing the movie too, sometimes not.

Brian Eno has made an earnest effort over the years to explain his job to people. Nick Seaver tipped me off to this lecture he gave called “The Studio As Compositional Tool” from (they think) 1979. It’s worth quoting at length.

The first thing about recording is that it makes repeatable what was otherwise transient and ephemeral. Music, until about 1900, was an event that was perceived in a particular situation, and that disappeared when it was finished. There was no way of actually hearing that piece again, identically, and there was no way of knowing whether your perception was telling you it was different or whether it was different the second time you heard it. The piece disappeared when it was finished, so it was something that only existed in time.

The effect of recording is that it takes music out of the time dimension and puts it in the space dimension. As soon as you do that, you’re in a position of being able to listen again and again to a performance, to become familiar with details you most certainly had missed the first time through, and to become very fond of details that weren’t intended by the composer or the musicians.

The effect of this on the composer is that he can think in terms of supplying material that would actually be too subtle for a first listening. Around about the 1920s – or maybe that’s too early, perhaps around the ’30s – composers started thinking that their work was recordable, and they started making use of the special liberty of being recorded.

I think the first place this had a real effect was in jazz. Jazz is an improvised form, primarily, and the interesting thing about improvisations is that they become more interesting as you listen to them more times.

Speak it, Brian Eno!

I think recording created the jazz idiom, in a sense; jazz was, from 1925 onwards, a recorded medium, and from’35 onwards I guess – I’m not a jazz expert by any means – it was a medium that most people received via records. So they were listening to things that were once only improvisations for many hundreds of times, and they were hearing these details as being compositionally significant.

Jazz listeners were already been hearing these details as significant in the moment too. There is something novel to recording about about scrutinizing and memorizing improvised solos.

So, to tape recording: till about the late ’40s, recording was simply regarded as a device for transmitting a performance to an unknown audience, and the whole accent of recording technique was on making what was called a “more faithful” transmission of that experience. It began very simply, because the only control over the relative levels of sounds that went onto the machine was how far they were from the microphone – like device. The accent was on the performance, and the recording was a more or less perfect transmitter of that, through the cylinder and wax disc recording stages, until tape became the medium by which people were recording things.

The move to tape was very important, because as soon as something’s on tape, it becomes a substance which is malleable and mutable and cuttable and reversible in ways that discs aren’t. It’s hard to do anything very interesting with a disc – all you can do is play it at a different speed, probably; you can’t actually cut a groove out and make a little loop of it.

You can if you’re an ace turntablist. Tape does make it a lot easier though.

The effect of tape was that it really put music in a spatial dimension, making it possible to squeeze the music, or expand it.

Initially tape recording was a single track, all the information contained and already mixed together on that one track. Then in the mid-’50s experiments were starting with stereo, which was not significantly different. The only difference was that you had two microphones pointing to your ensemble, and you had some impression of a real acoustic sound came to you from two different sources as you listened. Then came three-track recording; it allowed the option of adding another voice or putting a string section on, or something like that. Now this is a significant step, I think; it’s the first time it was acknowledged that the performance isn’t the finished item, and that the work can be added to in the control room, or in the studio itself. For the first time composers – almost always pop composers, as very few classical composers were thinking in this form – were thinking, “Well, this is the music. What can I do with it? I’ve got this extra facility of one track.” Tricky things start getting added. Then it went to four-track after that, and the usual layout for recording a band on four-track at that time.

You should remember that everything, including the Beatles’ Sgt. Pepper’s Lonely Hearts Club Band, was done on four-track until 1968. Normally engineers would do something like this: the drums on one track, the voices spread on two tracks with the guitars and the piano, say, on one of those tracks, and then the strings and additional effects on the fourth track. This was because they were thinking in terms of mono output; eventually, it would be mixed down to one signal again, to be played on radio or whatever. When stereo came in big, it gave them a problem. When they converted to stereo, things were put in either the middle, or dramatically to one side, or you’d hear some very idiosyncratic panning.

Anyway, after four-track it moved to eight track – this was in ’68, I guess – then very quickly escalated: eight-track till ’70, 16-track from’70 to’ 74, 24-track to now when you can easily work on 48-track, for instance, and there are such things as 64-track machines. The interesting thing is that after 16-track, I would say, the differences are differences of degree, not differences of kind. Because after you get to 16-track, you have far more tracks than you need to record a conventional rock band. Even if you spread the drums across six tracks, have the bass on two, have the vocals, have the guitars, you’ve still got six tracks left. People started to think, “What shall we do with those six tracks?”

From that impulse two things happened: you got an additive approach to recording, the idea that composition is the process of adding more, which was very common in early ’70s rock (this gave rise to the well known and gladly departed orchestral rock tradition, and it also gave rise to heavy metal music – that sound can’t be got on simpler equipment); it also gave rise to the particular area that I’m involved in: in-studio composition, where you no longer come to the studio with a conception of the finished piece. Instead, you come with actually rather a bare skeleton of the piece, or perhaps with nothing at all. I often start working with no starting point. Once you become familiar with studio facilities, or even if you’re not, actually, you can begin to compose in relation to those facilities. You can begin to think in terms of putting something on, putting something else on, trying this on top of it, and so on, then taking some of the original things off, or taking a mixture of things off, and seeing what you’re left with – actually constructing a piece in the studio.

In a compositional sense this takes the making of music away from any traditional way that composers worked, as far as I’m concerned, and one becomes empirical in a way that the classical composer never was. You’re working directly with sound, and there’s no transmission loss between you and the sound – you handle it. It puts the composer in the identical position of the painter – he’s working directly with a material, working directly onto a substance, and he always retains the options to chop and change, to paint a bit out, add a piece, etc.

This like how Quincy Jones once compared a synthesizer to sculpting a pure electronic waveform.

Each channel on the mixer is a long strip. Generally at the bottom is a level control, for how loud you want that channel to play back. Next up, normally, there’s a pan control, for where you want the sound object in the stereo/quad image. Next up is an echo control, and echo is really a separate issue, which has to do with something very unique to recording: briefly, it enables you to locate something in an artifical acoustic space. There’s also equalization – a device by which you can create a timbral change in an instrument, which in rock music is especially important, because many different rock records, in my opinion, are predicated not on a structure, or a melodic line, or a rhythm, but on a sound; this is why studios and producers keep putting their names on records, because they have a lot to do with that aspect of the work. Apart from equalization, there are other facilities which are widely used, such as limiting, compression – which has the effect of altering the envelope of a note or an instrument, so you can do something I’ve been interested in, creating hybrid instruments.

Compression is quite interesting over a whole track; if you’re using severe compression and limiting at the same time, when you push one instrument up, the track is governed so that the overall level will never change. Pushing one instrument up effectively pushes the others down, so all you do is alter the ratio between the instruments where you make a move. I started to use this as a deliberate, compositional, sound-type device; it’s generally been ignored or regarded as a misuse of the equipment before, but I’ll let you judge for yourself. On Helen Thormdale from the No New York album (Antilles), I put an echo on the guitar part’s click, and used that to trigger the compression on the whole track, so it sounds like helicopter blades.

Naturally, all of these things are variable throughout the entire course of the music. These are the kinds of things that you, as a listener, don’t generally notice; some of them operate almost subliminally – they are the ambiance of a track, not the obvious aspects of the track. Those are very much the things that traditional production is concerned with. And they allow you to rearrange the priorities of the music in a large number of ways.

We’ve spoken of the transition from the ’50s concept of music to the contemporary concept of mixing. If you listen to records from the ’50s, you’ll find that all the melodic information is mixed very loud – your first impression of the piece is of melody – and the rhythmic information is mixed rather quietly. The bass is indistinct, and the bass is only playing the root note of the chord in most cases, adding some resonance. As time goes on you’ll find this spectrum, which was very wide, with vocals way up there and the bass drum way down there, beginning to compress, until at the beginning of funk it is very narrow, indeed. Things are all about equally loud.

Then, from the time of Sly and the Family Stone’s Fresh album, there’s a flip over, where the rhythm instruments, particularly the bass drum and bass, suddenly become the important instruments in the mix. A timbral change also takes place. The bass becomes a very defined instrument; by the use of amplitude control filters, the bass actually begins to take on a very vocal attack. The bass drum gains a more physical sound, and also has a click to it; generally you’ll find that bass drums are equalized very heavily, something like 1000-1500 cycles, to give a real sharp click. It becomes the loudest instrument in disco – watch the vu meter while a disco track is playing, and you’ll see the needle peak each time the bass drum hits.

Once you’re on tape, there are so many variations you can make that you don’t really.need to spend all that money hiring musicians; you can do a great deal with one piece of work. So when you buy a reggae record, there’s a 90 percent chance the drummer is Sly Dunbar. You get the impression that Sly Dunbar is chained to a studio seat somewhere in Jamaica, but in fact what happens is that his drum tracks are so interesting, they get used again and again.

This takes us to reggae, which is a very interesting music in that it’s the first that didn’t base itself around the standard approach of making work by addition. Earlier I said the contemporary studio composer is like a painter who puts things on, puts things together, tries things out, and erases them. The condition of the reggae composer is like that of the sculptor, I think. Five or six musicians play; they’re well isolated from one another. Then the thing they played, which you can regard as a kind of cube of music, is hacked away at – things are taken out, for long periods.

A guitar will appear for two strums, then never appear again; the bass will suddenly drop out, and an interesting space is created. Reggae composers have created a sense of dimension in the music, by very clever, unconventional use of echo, by leaving out instruments, and by the very open rhythmic structure of the music. Then, too, someone like Lee Perry, a producer who’s always been very intelligent as far as using the constraints of the situation goes, might find there’s hiss building up on tracks he’s used over and over. A Western engineer might get frightened by this, and use all sorts of noise reduction and filtration. Perry says, “Okay, that’s part of the sound, so we’ll just add something else to it and use it’ ” This adds an ambiance of weirdness behind what was straightforward reggae.

This has nothing to do with anything, but Brian Eno’s most widely heard work is probably the Windows 95 startup sound.

Sampling “Once In A Lifetime”

Hip-hop and electronica musicians have been drawn to the “Lifetime” groove, more for its ambiance than its beat. The highest-profile example is “It’s Alright” by Memphis Bleek and Jay-Z.

Here’s my mashup of  “Once In A Lifetime” with several hip-hop and dance tracks that sample it.

Once In A Lifetime megamix by ethanhein

The brain is a pattern recognition machine. We like repetition and symmetry. But we only like it up to a point. Once we’ve recognized and memorized the pattern, we get bored and stop paying attention. If the pattern changes or breaks, it grabs our attention again. If the pattern-breaking happens repetitively, itself forming a new pattern, we find it super gratifying.

Count to four, over and over again, at a rate of one beat every three quarters of a second or so. On beat one, play a low-pitched percussion sound. This could be a kick drum, a foot stomp, a grunt or ‘bff’ sound, or the bottom of a paint bucket, whatever. On beat three, play a higher-pitched sound: a snare drum, a clap or snap, a plosive syllable like gh or kh, the rim of a paint bucket. This pattern, alternating high and low, will get monotous quickly. To make it funky, you need to introduce some predictable unpredictability. The best way to do that is to change the placement of every alternating bass drum hit. Anticipate it by a beat, or half a beat. Delay it a beat or half a beat. Or do both: play a kick both on beat four and a little before the next beat two. Or just skip every other kick. Maybe also add an extra snare hit in an unexpected place: the very last beat of every other phrase, for instance.

For another layer of texture, add an even higher-pitched sound on every beat, or every half beat. It could be a hi-hat, a shaker, a tambourine, a plosive like ‘ts’, or the metal handle on the paint bucket. Maybe even try doubling up the tempo for a beat at the beginning or end of the phrase. And that should be it for complexity. Cluttering up a beat too much diminishes its power. The silences between the drum hits are as important as the drum hits themselves. Leaving out every other kick drum hit is an effective strategy because the ear fills in the silence by itself, and that imaginative participation is a big pleasure.

There’s a natural tendency to want to immediately start putting in variations, embellishments and fills. We get nervous that the beat isn’t going to be “interesting” enough. It’s true that patterns get monotonous after a while, but too much variation releases the tension that a good pattern can build. If the pattern gets on your nerves quickly, find a better pattern. Rather than let the air out of the balloon too early, you want to pick your symmetry-breaking spots carefully: in the last measure of a sixteen-measure phrase, or even better, a thirty-two bar phrase. Building up lots of tension gives the listener greater satisfaction when it’s finally released. And when it comes time to throw in a variation, using unexpected silences is more attention-grabbing than adding extra sounds. Eliminate the kicks, or the snares, or everything but the hi-hats, or drop the beat out entirely but keep the pulse going, and watch the crowd react with pleasure when the full beat kicks back in.

My musician friends complain that drum machines don’t sound as good as human drummers because they’re so predictable and robotic. I myself have not found this to be true, unless the human in question is a deep Jedi master like Clyde Stubblefield or Questlove. The best drummers are the ones who play the most tightly and predictably. Questlove in particular got to be as good as he is in part by listening to a lot of drum machine music, and striving to sound like one. My friend Doug, one of the best drummers I know, used to leave his metronome on for hours at a time while he did stuff around the house. The beauty of drum machines and the software that emulates them is that they discourage you from needless complexity. They make it effortless to program a two-measure pattern, and annoying to vary or embellish it.

American popular music in all of its varieties mostly agrees about the power of four:

Every beat is four sixteenth notes long.
Every measure is four beats long.
Every phrase is four or eight measures long.
Every section is four or eight phrases long.

Sometimes you see sections that are three phrases long, as in twelve-bar blues. Very rarely a more eccentric phrase length shows up. But most of the music that most of us like here in the western world is based on powers of two. Instead of breaking symmetry totally, you can create extra tension by gently bending the form. You can lay behind the beat or lean out in front of it. You can stretch out every other beat a little, a practice known to musicians as swing. Quality drum programming software lets you bend and stretch time in musical-sounding ways without fracturing the pattern and breaking the spell.

You can’t ask the computer. Computers fail when you ask them questions with no unambiguous answer. Dividing by zero is just such a question. Folklore suggests that asking the computer to divide by zero makes it spectacularly explode or something. In reality, it returns an error message or the reply Not A Number, or it gives a wrong answer, or the program terminates, or sometimes the machine falls into an infinite loop.

The internet’s favorite divide-by-zero error is the one that temporarily crippled the USS Yorktown, a Ticonderoga-class cruiser that was the test bed for the Navy’s Smart Ship program. When a crew member typed zero into a database field, the computer tried to divide by it, crashing the system badly enough to cripple the ship’s navigation systems for several hours.

Humans are smarter than computers in some ways, and we’re capable of coming up with creative answers to seemingly unanswerable questions. So what do you get when you divide something by zero? My answer draws heavily on the entertaining wikipedia article. For the sake of simplicity, let’s say we’re dividing one by zero. The math people have a crafty method for dealing with problems you can’t approach directly. You can edge closer and closer to the problem and see if you converge on an answer. So instead of dividing one by zero, you could try dividing it by smaller and smaller numbers that approach zero. One divided by one tenth is ten. One divided by one one-hundredth is a hundred. One divided by one one-thousandth is a thousand. Since one divided by one one-gazillionth is one gazillion, logic suggests that one divided by zero is going to be infinity.

It makes sense, but there’s a problem. We’ve been approaching zero from above, but we could just as easily approach it from below. When you divide one by negative one tenth, you get negative ten. One divided by negative one one-hundredth is negative one hundred. One divided by negative one gazillionth is negative one gazillion. So you could just as easily say that one divided by zero is negative infinity. Both infinity and negative infinity are equally valid answers. Here it is as a graph.

Some people interpret this graph to say that infinity and negative infinity are the same number. It’s not as crazy as it sounds. Let’s say that instead of being on the computer screen, the graph was drawn on a globe. Imagine the number line wrapped around the equator. Say the spot where the Prime Meridian crosses the equator is zero. If you’re in a rowboat bobbing in that spot in the Atlantic Ocean, enjoying the warm breeze, you can think of the positive numbers as going off along the equator to the east, and the negative numbers going off to the west. Infinity is the farthest possible point away from you on the equator to the east, and negative infinity is the farthest point away from you to the west. On the Earth, positive and negative infinity are the same place, the International Date Line in the Pacific. For this image to be totally accurate, the Earth would have to be infinitely large, but the math guys bracket that. By this thinking, one divided by zero does have a single, unambiguous answer: this mysterious number called unsigned infinity.

When you type “divide by zero” into Google images, you get a lot of stuff like this:

Our European-descended philosophical assumptions are at work here. Western thinkers prefer clear, unambiguous, yes-no dichotomies. Paradoxical and multiply-determined truths make us anxious. Some of the internet cartoons show dividing by zero ripping holes in the space-time continuum, forming black holes, or making your head explode. That much hyperbole has to conceal some pretty intense anxiety. I know these pictures are jokes, but I agree with Freud, on some level there are no jokes.