The top row shows the fundamental frequency, the one you hear as the pitch — say it’s a violin string playing A 440. The second row shows the first harmonic, the string vibrating in halves, producing A 880. The harmonic is quieter than the fundamental, so you aren’t necessarily conscious of it, but you can isolate it by lightly touching the string at its halfway point while playing. The other rows show other harmonics, vibrations of the string in integer ratios, each producing a pitch that’s an integer multiple of the fundamental frequency. The second harmonic is E 1320; the third is A 1760; the fourth is C# 2200.

In an ideal string, the harmonics would continue to get infinitely higher, beyond the range of your hearing. As the harmonics get higher, they also get quieter and subtler. Still, they all have an impact on the overall sound of the instrument. All musical instruments have overtones: winds, the human throat, speaker cones, even well-tuned drumheads.

Read more about how harmonics form the basis of western music theory.

Original post on Quora

Why has music been historically the most abstract art form?

I had an art professor in college who argued that all “representational” art is abstract, and all “abstract” art is representational. Any art has to refer back to sensory impressions of the world, internal or external, because that’s the only raw material we have to work with. Meanwhile, you’re unlikely to ever mistake a work of representational art for the object it represents. You don’t mistake photographs (or photorealistic paintings) for their subjects, and even the most “realistic” special effects in movies require willing suspension of disbelief.

Music seems more abstract than other art forms because it represents emotional states, symmetry and repetition, and other intangibles. But just because you can’t see or touch these things, doesn’t make them any less real. In preliterate societies, music was probably one of the best methods for storing and conveying complex stories and information.

Also, I dispute the idea that visual art started representational and then “progressed” toward greater abstraction. Architecture, textiles, tile work, face and body decorations and jewelry all use pattern, color and texture for their own sake, without any representational content.

Any one of the above images could pass as a music visualization or notation. I see a strong parallel between this kind of decorative art and the mathematical patterns in music — there’s the interest in interlocking patterns and symmetry for their own sake. Symmetry is a fact of the world, and both abstract art and music represent that fact clearly.

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

Music notation

Western music notation is a venerable method of visualizing music. It’s a very neat and compact system, unambiguous and digital, and not too difficult to learn. Programs like Sibelius can effortlessly translate notation to and from MIDI data, too.

But western notation has some limitations, especially for contemporary music. It doesn’t handle microtones well. It has limited ability to convey performative nuance — after a hundred years of jazz, there’s no good way to notate swing other than to just write the word “swing” at the top of the score. The key signature system works fine for major keys, but is less helpful for minor keys and modal music and is pretty much worthless for the blues.

Here’s a suggestion for how notation could improve in the future. It’s a visualization by Jon Snydal of John Coltrane’s solo in Miles Davis’ “All Blues”  (I edited it a little to be easier on the eyes.)

Snydal’s visualization is more analog than digital — it shows the exact nuances of Coltrane’s performance, with subtle shadings of pitch, timing and dynamics.

MIDI sequencers suggest further improvements over standard notation. Here’s a simplified electronic music sequencer called iNudge. Play, it’s fun:

Here’s Thelonious Monk’s tune “Four In One” as shown in standard MIDI “piano roll” view. The rectangles show not only which notes are being played and when, but exactly how long they’re held. Darker red means louder, paler pink means quieter. You can also read volume off the bars along the bottom.

MIDI is a versatile and user-friendly system. It can capture your keyboard performances, you can import scores, and you can even just draw notes onto the screen directly (my preferred method.)

The Music Animation Machine has a wonderful series of videos matching MIDI piano rolls of various classical pieces with recordings of them. Here’s Bach’s infamous Toccata and Fugue in D minor.

As software gets more sophisticated in its ability to extract pitch data from actual audio recordings, you can start manipulating them with the same ease as MIDI. Here’s a screencap of the pitch-correction program Melodyne, a close cousin of Auto-tune.

The lines show the actual sung pitches, and the orange blobs show the notes the program thinks the singer meant to hit. The blobs’ thickness shows volume. You can drag and drop the blobs and redraw the lines at will to alter the melody to your heart’s content. Melodyne even transcribes the performance to standard notation and MIDI for you.

High and low

We’ve made up our collective mind that faster frequencies should be spatially represented as being “higher,” and that slower ones should be spatially “lower.” It seems so reasonable, but really it’s totally arbitrary, and doesn’t even line up with physical experience. On the piano, the high notes are on the right and the low ones on the left. On the guitar, the “low” E string is physically located above the “high” one. The fingerings for higher and lower notes on wind instruments don’t correspond to a simple higher-lower axis either.

Absolute pitch is a straight line ladder, but pitch class is circular. The truest representation of pitch space is a helix.

Other ways to conceptualize pitch space

High and low aren’t the only metaphors we use for faster and slower vibrations. Like I said, pitch class is circular.

But the circle is really just replacing up/down with clockwise/counterclockwise. There are other ways to conceptualize pitch. We intuitively experience changing pitches as moving closer and further, or inwards and outwards. We also think of higher pitches as brighter and lower pitches as darker. Players of stringed instruments sometimes tune their upper strings a little bit too high on purpose, producing an effect known as brilliance.

Time

It’s a universal convention that notation shows time moving from left to right. But that’s not the only possible axis to use. How about forwards and backwards instead? That’s the paradigm in rhythm games like Dance Dance Revolution and Guitar Hero. The purest realization of this concept is in a game called FreQuency.

The game even allows you to construct your own remixes.

I like this tunnel metaphor and would like to see it extended into a full-blown production environment.

Waves

Pitches are sine-wave vibrations, and you can visualize them as such.

Sine waves wouldn’t make for very a helpful music notation, but they do help you understand what’s going on scientifically when you physically hear something. They’re even better animated:

See all of Wikipedia’s animated drum heads.

Waveforms

Audio editors show music as amplitude waveforms, blobs that get wider where the sound is louder. Here’s the Funky Drummer break in Recycle. The blue blobs show drum hits. These amplitude blobs don’t tell you much about the musical content except for timing and volume. But Recycle was meant for drum loops, where timing and volume are the only information you really need.

Here’s a graphic I made showing how you hear the Funky Drummer as it’s looping:

In a post on Design Observer, Rob Walker discusses the waveform as the new icon for music, replacing the stylized eighth notes or records that have done the job in the past. The SoundCloud player uses an attractive waveform graphic that helps the listener track where they are in the song by following the volume peaks. There’s even a SoundCloud group called Pretty Waveforms.

Tomorrow Never Knows by ethanhein

The waveform has the potential to move from purely functional settings to more decorative ones. Here’s a waveform-based labeling concept by Joshua Distler, showing the tracks on Post by Björk.

Music theory and networks

I’ve always thought it would be cool to use networks to conceptualize music theory, and have made a few attempts at doing so. Here’s a comparison between the circle of half-steps and the circle of fifths, which are involutes of each other:

Here’s a map of the chord progressions in “Giant Steps” by John Coltrane.

And here’s a flowchart showing how you can figure out what scale or mode you’re hearing.

It would be way cooler to have more abstract three-dimensional interactive visualizations showing how chords, scales and melodies function. Leonhard Euler showed how you can represent tonal harmony as a lattice with the topology of a torus, as shown in this animation. Red lines show major thirds, green lines show minor thirds, and blue lines show fifths:

I have ambitions of my own in this area, but so far, I lack the programming skills to realize them. Others are taking some exciting strides, though. Dmitri Tymoczko made waves for getting the first music-related article published in Science about his topological visualization methods for tonal harmony. I can’t quite wrap my head around his ideas, but they’re intriguing.

Here’s an illustration by Aniruddh Patel from his paper, “Language, Music, Syntax And The Brain.” Again, I’m not totally clear what it all means, but I plan to investigate further.

Other theorists have attempted to use color to show harmonic function. Scriabin invented a “keyboard of lights” for that purpose, though it didn’t really catch on.

Visualizing musical form and structure

I like to use simple color-coding to keep track of which section is which while working on a song. Yellow is for intros and outtros, blue is for verses, green is for choruses and orange is for instrumentals and breakdowns.

Edward Tufte shows some more sophisticated song structure visualizations on his forum:

The Shape of Song project by Martin Wattenburg shows repetition within a piece of music. Here’s his visualization of “Like A Prayer” by Madonna.

Here’s Wattenburg’s visualization of Beethoven’s “Für Elise.”

Speculation

Here’s an entertaining video showing how you can create a happening drum machine sequence using counting in binary by Niklas Roy.

Wouldn’t this graph coloring system make a cool music notation or interface?

Visual Complexity has many more ideas like this one.

I feel like we’ve barely scratched the surface of useful and attractive schemes. Are there other cool visualization methods I should know about? Hit the comments.

Updates

John Clover hipped me to this post, which overlaps heavily: Amazing Music Visualizations and Teaching

I just had the chance to play with some of Björk‘s Biophilia song/apps. Some of them are groundbreaking interactive visualizations; some are just entertaining and groovy; some are baffling but deserve points for creativity. All the way around, it’s a remarkable experiment, one that I think is going to be influential.

See this post on Quora

The picture shows a stylized nucleus with red protons and blue neutrons, surrounded by three grey electrons. It’s an attractive and iconic image. It makes a nice logo. Unfortunately, it’s also totally wrong. There’s an extent to which subatomic particles are like little marbles, but it’s a limited extent. Electrons do move around the nucleus, but they don’t do it in elliptical paths as if they’re little moons orbiting a planet. The true nature of electrons in atoms is way weirder and cooler.

Pictures are a terrible way to understand the nature of quantum particles. Music theory is much better.

Quantum particles are waves

The problem with textbook images like the one above is that they mislead you into thinking of particles as “things.” Particles aren’t things. They pop in and out of being in a rapid, flickery way that’s more like the way we think of energy. What we call “particles” are really just knots or bundles of energy fields.

Protons and electrons pull on each other the way refrigerators and magnets do. If electrons really were like little moons orbiting a planet, it seems like they could orbit at any distance, and could easily fall into the nucleus to collide with the protons. And yet, this never happens. Electrons always organize themselves into very specific spatial arrangements around the nucleus. This fact was totally mysterious until scientists started conceiving of electrons as probability waves in an energy field.

You can get a good idea of how particles really behave by looking at television static, which consists of huge numbers of electrons being fired at the screen at random. Now try to imagine “static” surrounding the nucleus of an atom, and you’ll get a much better picture of what’s going on than you get from imagining moons orbiting a planet.

When electrons are in orbit around an atom or molecule, their pattern of static isn’t random the way it is in TV static. When electrons orbit atoms, their energy fields are organized into patterns of overlapping ripples. You can explore these patterns with Paul Falstad’s interactive visualizations of the subatomic world — scroll down to the Quantum Mechanics sections for his simulated hydrogen atom. The colorful blobs show the probability of electrons being found in a particular place.

So what does this have to do with music theory? The electron field’s vibrations around an atom behave like harmonic oscillators. Electrons have harmonics, just like guitar strings do. Electron harmonics are three-dimensional instead of the one-dimensional harmonics of strings, but the underlying math is the same. These harmonics determine the arrangement and interactions of the electron wave, the same way that harmonics of a string form the basis of chords and scales. The electron field’s harmonics are called orbitals.

The physical world is made of electron harmonics

This screenshot of Falstad’s quantum harmonic oscillator applet shows the first harmonic of the electron field around an H2 molecule, two hydrogen atoms, each with one proton and one electron. This is the electron equivalent of the twelfth fret harmonic on a guitar string.

The blue blob represents the position of one electron, and the red blob is the other. At higher energy levels, the orbitals take on more complex shapes. There’s a direct analogy here to the more complex musical intervals that come from the higher harmonics in a guitar string.

You can think of the orbitals as a structure of cubbyholes, each of which can be occupied by one electron. The cubbyholes come in pairs, and electrons “prefer” to live in filled pairs of cubbyholes. All of the structure of objects and chemistry in the world arises from the way that atoms’ outer orbitals interact. If an atom’s outermost cubbyholes are unfilled, electrons from other atoms with unfilled orbitals can fill them, locking the atoms together into molecules. All solid and liquid materials are held together by this sharing of electrons between orbitals.

Here’s the molecular structure of ice, as rendered by Masakazu Matsumoto. The red balls are oxygen atoms. The blue ones are hydrogen atoms. The yellow rods represent the bonds caused by electrons shared between the oxygen and hydrogen atoms’ outermost orbitals.

The “sixness” of ice’s structure emerges from the way that hydrogen and oxygen orbitals combine to make open slots in groups of six. You can see the “sixness” repeated up at the macroscopic scale in the shape of snowflakes and frost.

If you raise the ice’s temperature to the melting point, what you’re really doing is shooting photons at the ice, knocking the electrons out of their orbitals so they can skip more freely from atom to atom. The atoms still stick together, but not as tightly, and not in so rigid an arrangement:

If you zap even more photons into the water, you can sever the bonds between the molecules completely, freeing them to bounce around independently in the state we perceive as steam. If you zap even more photons at the steam, you can rip the molecules apart and tear the electrons from the nuclei to form plasma. Even more energy will rip the nuclei into protons and neutrons, and ridiculously more energy will rip the protons and neutrons into their constituent up and down quarks. The quarks, protons, neutrons, nuclei, atoms and molecules are all vibrating energy fields with waveforms and harmonics of their own.

Whenever I’m bored, I like to try to imagine everything around me, all the matter and energy, as resonating energy fields, cohering the way pitches cohere into chords. Who says science isn’t fun?

Teaching science with music

Albert Einstein told interviewers that he often “thought in terms of musical architectures.” Einstein was an enthusiastic amateur violinist, and an early architect of quantum mechanics. These two facts are probably related.

Did Einstein make an explicit connection between musical harmonics and quantum harmonics? Maybe we’ll never know, but the connection exists, and future scientists can benefit from it. The concept of electron orbitals is really hard. When I was in high school, my (excellent) chemistry teacher told us not to even bother trying to visualize the true nature of electrons. She was right to not try to condescend to us or mislead us, but she gave up too easily. True, she didn’t have cool interactive computer visualizations, but the school did have a great music department. If I ever get a chance to teach chemistry, first I’m going to make sure the kids get some hands-on experience with harmonics. I’ll have them experience the way that it takes more energy to produce higher harmonics, and the way those higher harmonics produce more complex musical sounds. Then we’ll go back to chemistry class and I’ll bet the kids will have an easier time.

And here’s the full song — contains much explicit language.

The hook samples the line “Biggie Smalls is the illest” from “The What” on the same album. It’s twenty-three seconds in.

Sampling is a severely underappreciated songwriting tool. Even if you have moral or legal issues with sampling from others, sampling from yourself is still a good idea. Biggie’s line about himself being the illest is just part of a verse in “The What.” The producer on “Unbelievable,” the great DJ Premier, was smart enough to recognize that Biggie’s line could stand on its own as a hook. DJ Premier also produced “Nas Is Like,” and built its chorus through similar means.

The word “Unbelievable” itself comes from R Kelly, sped up a little and raised in pitch to sound female. Listen at 0:58.

Sampled vocals aside, the chopped-up keyboard part is the most musically sophisticated aspect of the track. Its original source is “Remind Me” by Patrice Rushen — I’m pretty sure it comes from the end of the solo section around 4:10.

Premier chopped up this little keyboard phrase and resequenced it beyond recognition. The result is a hip angularity that a normal keyboard player would probably not have arrived at organically.

The beat in “Unbelievable” is an old standby, “Impeach The President.”

The string ambiance in the background comes from the very odd Quincy Jones song “Kitty With The Bent Frame.” Listen at 1:08.

Quincy’s record is a favorite for hip-hop producers looking for an uneasy mood.

Here’s a diagram showing the sample genealogy of “Unbelievable.”

The meaning of self-sampling

Like I said above, self-sampling is so interesting to me because it’s recursive, self-referential. Most of the music we like is full of self-reference, and generally, the more self-referential it is, the more structured and meaningful it feels. Even simple-seeming nursery rhymes can be recursive and self-similar. Here’s a visualization by Lee Byron showing self-similarity in the nursery rhyme “Hickory Dickory Dock.”

Self-similarity makes for compelling visual art, too. One reason we find nature attractive is its rich fractal self-similarity. Here’s a leaf I photographed in my neighborhood; notice how the same veiny structure repeats itself at different size scales:

Even very simple recursive mathematical equations can produce stunningly complex, biological-looking forms, like the classic fractal known as the Mandelbrot set.

Recursion isn’t just attractive. It’s fundamental to computer science — self-reference is a key programming technique. Recursion may be essential to the very nature of consciousness itself. Some neuroscientists think that your entire sense of self emerges out of recursive self-referential loops as your brain represents different parts of itself to itself.

No wonder recursive music is so fascinating. Keep on sampling yourselves, musicians; let’s see what other recursive truths we can uncover.

Wave mechanics

The concept of orbitals in quantum mechanics made zero sense to me until I finally found out that they’re just harmonics of the electron field’s vibrations. I wasn’t at all surprised to learn that Einstein conceptualized wave mechanics in musical terms as well.

Logarithms

Octave equivalency is really just your brain’s ability to detect frequencies related by powers of two. The relationship between absolute pitches and pitch classes is an excellent doorway into logarithms generally.

Symmetry

See this delightful paper by Vi Hart about symmetry and transformations in the musical plane.

Combinatorics and graph theory

Generating diatonic chords from a scale is basically just an exercise in combinatorics. Seventeenth-century European bellringing introduced one of the earliest nontrivial results in graph theory, change or method ringing.

Discrete mathematics

The pitch continuum is, well, continuous, but tuning systems and scales are discrete. The voice, fretless stringed instruments and trombones produce continuous pitches. Keyboards, fretted string instruments and saxophones produce discrete pitches. Great intuitive preparation for the concepts of discrete vs continuous generally.

Modular arithmetic

Once you’ve dealt with the circle of fifths, and with scales and modes, extending the idea to generalized modular systems is no problem.

For example: What is the mathematical relationship between the circle of fifths and the circle of half-steps?

Recursion

Maybe you’d consider this more a computer science topic than a math topic. Regardless, the best music is very recursive. See Gödel, Escher, Bach by Douglas Hofstadter.

Some speculation

My experiences in both music and math have convinced me that music is a severely underutilized resource for math teaching. There are many ways to learn besides manipulating symbols on a page or computer screen. In his book Anathem, Neal Stephenson imagines monks solving proofs and running cellular automata by chanting melodies that evolve by systematic rules.

When I was trying to learn how wrap my head around binary numbers, I eventually just wrote a song that counts in binary from one to sixty-four and back down. It works great, and also turns out to be a highly relaxing and meditative exercise. Maybe if more kids felt relaxed and meditative in math class, they’d learn the material a lot better.

Original question on Quora

Here’s “Amen, Brother.” The famous drum break comes at 1:27.

“Amen, Brother” is an uptempo adaptation of “Amen” by Jester Hairston, written for the movie Lilies of the Field, and made famous by The Impressions (with Curtis Mayfield, before he went solo.)

“Amen, Brother” didn’t get much attention until crate-digging hip-hop producers started sampling the drum break in the 1980s. Breakbeat Lenny included it in the first volume of Ultimate Breaks and Beats. Since then, the break has become ubiquitous not just in hip-hop, but in every style of dance music. It almost single-handedly spawned entire genres of electronica, particularly especially drum ‘n’ bass and its various offshoots. The Amen shows up in rock and pop songs ranging from Oasis to Nine Inch Nails. It’s in TV theme songs and commercials. Casual music listeners have probably heard it in dozens if not hundreds of recordings. Here’s a family tree showing the most noteworthy usages.

As is so often the case in sample history, GC Coleman never got a dime from any of these uses beyond his union scale for the original recording session. He died in 2006, so there’s not much we can do for him now, but I think he at least deserves some recognition.

Inside the break

Here’s the Amen break, looped four times:

Amen break by ethanhein

Wikipedia has handy drum notation and drum machine tablature for the break.

Here’s how the break looks in the sampling program Recycle:

Each blue spiky blob is a drum hit. The vertical lines are slices I added using Recycle. Once you’ve sliced up the loop, you can play the slices back in any order and any combination using a MIDI keyboard or drum pads. You can generate an infinite variety of new loops this way.

Two documentaries on the Amen break

The usual reference for the Amen break is this twenty-minute video by Nate Harrison.

Here’s an hour-long podcast on the break by Crissy Criss:

Noteworthy Amen break samples

I’m not attempting anything resembling completeness here; these are just tracks I like or find interesting. Starting in the eighties with the old skool, here’s “King Of The Beats” by Mantronix.

In “I Desire,” Salt N Pepa mixes the Amen with drums from Aerosmith’s “Walk This Way” and the synth line from “Daisy Lady” by 7th Wonder.

Maybe the best-known hip-hop usage of the Amen is NWA’s “Straight Outta Compton” (very, very NSFW.)

The break appears in pitched-down form at the very beginning of “Informer” by Snow. What the heck is he saying in the chorus, anyway?

As I said above, the Amen is most closely associated with drum ‘n’ bass, for example “The Angels Fell” by Dillinja.

Luke Vibert did an album under the pseudonym Amen Andrews where just about every song uses a variation on the Amen break.

An artsier take on the Amen: “Girl/Boy” by Aphex Twin.

Even artsier: Amon Tobin’s “Nightlife.”

David Bowie uses the Amen for his foray into the world of drum ‘n’ bass, “Little Wonder.” It’s not one of his strongest tunes but he gets huge points for trying.

The most current hip-hop song I could find that uses the Amen is Lupe Fiasco’s “Streets On Fire.”

Many, many more examples can be found on the Amen Break Database and Whosampled.com.

The Amen break on TV

As the Nate Harrison documentary points out, the Amen pops up in quite a few TV commercials. It’s made its way into some theme songs, too, most notably the one from Futurama:

The Amen also shows up in the Powerpuff Girls theme song.

Do you still think that sampling is stealing? I don’t mean monetarily, I mean artistically. Do you think that there’s something unoriginal in all these uses of the Amen break? Do you think that the way Aphex Twin or Lupe Fiasco recontextualizes the break is somehow a lesser creative act than getting out a drum kit and playing something? You can probably guess where I stand.

Why is the Amen break so magical?

Producers talk about how funky and passionate the Amen is, how compressed and dirty the drum sounds are, how much hip syncopation it uses in its second half. But what if there’s a mathematical explanation for the break’s popularity? Michael Schneider has a theory that the Amen Break sounds so good because it’s structured around the golden ratio.

Is there anything to this theory? You be the judge.

Try it yourself

Here’s my mashup of many of the above tracks, with heavy processing of the Amen break in Recycle and Reason.

Amen Brother megamix by ethanhein

If you want to play too, the internet is full of resequenced and reshuffled variations on the Amen break available for your downloading pleasure. Here are a hundred fifty Amen loops. Here’s another forty and yet another twenty. Stick them in your favorite audio editor and have fun!

Let me tell you a story. In ninth grade math, we took a break from all the trigonometry to do a little section on probability. It wasn’t anything exotic, just the likelihood of pulling certain cards out of a deck, stuff like that. I had been a straight-A math student my whole life until that point, and I couldn’t wrap my head around probability at all. I could memorize the equations well enough, but I was used to intuitively understanding the rationale behind the equations, and with probability I just could not do it. When you flip a coin and it winds up tails, where does the heads outcome “go?” How does the coin “know” it’s supposed to converge on a fifty-fifty ratio of heads and tails as you flip it more and more times? I almost flunked the test on that unit, I was so baffled.

I forgot the whole thing until more recently when I started learning about quantum mechanics. That’s where the probability questions get to be way more philosophically disturbing than in coin-flipping and card-choosing.

The situation is summed up best by the famous double-slit experiment. In this experiment, you shine light through a screen with a pair of slits in it onto photographic film. You get a stripy interference pattern as the waves of light coming from each slit overlap each other, the way overlapping ripples in a pond do. In this case, what’s “waving” is the probability density of a given photon passing through a given slit and landing at a given spot on the screen. Just the thought of probability having a physical density gives me intense vertigo, but it gets worse.

If you shoot a single photon at a time through the slits onto the film, the result looks like this:

Each individual photon somehow “knows” that there are two slits, and that the probability waves emanating from each slit interfere with each other. So even when the photons come one at a time and don’t interact with each other at all, they still obey the same probability distribution as if you fired them all at once. What’s even weirder is that you can get this same result with any quantum particle, and even entire molecules like buckyballs.

You can interpret this situation to mean that the probability waves are some sort of physical entity, a “pilot” wave that tells each photon what to do, which somehow extends forwards and backwards in time. I find this idea repugnant. You could also interpret the experiment to mean that space and time are nonlocal in some way that lets the single photon go through both slits at once. I also find this idea repugnant.

The only explanation of the double-slit experiment that makes any sense to me is the many-worlds interpretation. Because the left-slit universe and the right-slit universes are so similar, they overlap and mutually interfere, and that’s what produces the stripy pattern. It took me some time to get used to many-worlds, but once I got comfortable with it, I’ve become much more relaxed when I think about probabilities. Now I just see the different possible outcomes as all happening in some universe, and the “probability density” is just the density of the different universes. If I flip a coin, there’s an equal number of universes where it lands heads or tails, plus a very small number of universes where all the particles in the coin spontaneously jump into the Andromeda galaxy.

I’m not a mathematician or a scientist. I’m just a humanities guy who likes math and science. So if you’re a professional in one of these fields and you can correct me or clarify what I’m trying to say here, please do, I want to understand this stuff more clearly.

Original post on Quora

Below is a diagram showing a computer that can add one plus one to get two. It’s made out of an XOR gate and an AND gate, each of which is a relatively simple bunch of transistors wired together in particular ways. Transistors are just on-off switches that can be flipped electrically, so they have no moving parts (except electrons.) The beauty part is that the output wire of one transistor can be used to flip another transistor on and off.

The “numbers” in the diagram are just voltages: “zero” is zero volts, and “one” is (I think) 2.5 volts. The numbers in a computer are encoded in binary (base two) because it’s the most convenient way to physically realize them — voltage that’s pretty close to zero can be read as zero, and a voltage that’s pretty close to 2.5 can be read as one.
This simple adder has two input wires and two output wires. The input wires each represent a single binary digit, and the output wires together represent two binary digits. If one input wire has a voltage and the other doesn’t, the equivalent of adding one plus zero, the adder returns no voltage in the “twos” digit and a voltage on the “ones” digit. If there’s a voltage on both input wires, the equivalent of adding one plus one, the adder returns a voltage on the “twos” digit and no voltage on the “ones” digit, the binary number 10, or as we know it in decimal notation, 2.

Here’s a diagram of a four-bit adder, capable of adding numbers as big as sixteen. The diagram shows how adding seven (0111) plus twelve (1100) to get nineteen (10011) would work.
Wire together enough adders and other basic logic devices, set up at the right initial voltages, and you’ve got yourself a computer.

Original post on Quora