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

All of our behavior results from a bunch of molecules bouncing around according to the laws of quantum mechanics. Seen that way, we don’t have any more free will than pebbles being tumbled down a river. We think we have free will because we can’t predict the future, and because our immediate experience is full of so much ambiguity.

Free will is an illusion, but it’s a powerful, persistent and useful illusion. The inherently complex and chaotic nature of our brains prevents us from being able to predict our own actions. We’re even worse at predicting events caused by the emergent complexities of the interactions between large groups of people other people.

Way down at the quantum level, we may well be living in a deterministic universe. But because we have no way of perceiving all the intricate quantum interactions doing the determining, for all practical purposes, we might as well pretend to have free will.

Original post on Quora

I’ve always wondered where the Egyptian melody came from. It turns out to be hundreds of years of old, and goes by many different names. You can find an excellent capsule history of it in William Benzon’s book Beethoven’s Anvil. The context is a discussion of a Louis Armstrong recording from 1928 called “Tight Like This.” Listen at 2:04 as Louis quotes the “Egyptian” melody and varies it a few times.

Benzon knows the Egyptian melody from childhood. He quotes different sets of lyrics, such as:

All the girls in France do the hokey pokey dance,
and the way they shake is enough to kill a snake

Another variation:

On the planet Mars all the women smoke cigars.
Every puff they take is enough to kill a snake.
When the snake is dead they put flowers on its head.
When the flowers die they say 1969!

The tune has been known in America as the “hookie-kookie dance” or the “hoochie-coochie dance.” It came to fame when it accompanied a belly dancer at the 1893 Chicago World’s Columbian Exposition, and afterwards it became something of a hit. The melody was copyrighted under various names early in the 20th century, including “Dance Of The Midway,” “Coochi-Coochi Polka” and “The Streets Of Cairo.” (Thank you, Eunji Choi, for pointing me to this last tune’s Wikipedia page.)

The Egyptian melody appears in the widely-studied Arban’s Complete Conservatory Method For Trumpet from 1864, under the title “Arabian Song.” Arban almost certainly didn’t write it; it’s one of many “representative ethnic songs” in the book learned from the folk tradition. The tune is related to an Arabic or Algerian melody called “Kradoutja” that had been circulating around France since the 1600s. Who knows if the tune in Arban’s book is an actual middle eastern folk song, or a European mutation of “Kradoutja,” or what.

The Egyptian melody also gets quoted a lot in performances of “Sheik of Araby,” for example as performed here by the Beatles for their unsuccessful Decca audition in 1962.

Steve Martin uses the melody at the beginning of “King Tut.” (no embedding.)

They Might Be Giants use the tune in “Istanbul” for the line “Even old New York was once New Amsterdam.”

A more recent quotation — “Who’s That? Brooown!” by Das Racist:

This story is a perfect illustration of how musical memes evolve the way organisms do. It has a similar evolutionary history to the “Asian riff,” another stereotypically ethnic musical meme.

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

My musician friends use the term “jazz face” to describe the sometimes ridiculous expressions they have when they’re most deeply immersed in the music. Think also of Michael Jordan sticking his tongue out in the heat of play. And consider the fact that some people need to pace in order to think, or gesticulate, or perform repetitive manual tasks like knitting or splitting wood.

Original post on Quora

Carl Sagan sets out the idea in The Dragons Of Eden.

Human intelligence is fundamentally indebted to the millions of years our ancestors spent aloft in the trees. And after we returned to the savannahs and abandoned the trees, did we long for those great graceful leaps and ecstatic moments of weightlessness in the shafts of sunlight of the forest roof? Is the startle reflex of human infants today to prevent falling from the treetops? Are our nighttime dreams of flying and our daytime passion for flight, as exemplified in the lives of Leonardo da Vinci or Konstantin Tsiolkovskii, nostalgic reminiscences of those days gone by in the branches of the high forest? Could the pervasive dreams and common fears of “monsters,” which children develop shortly after they are able to talk, be evolutionary vestiges of quite adaptive baboon-like responses to dragons and owls?

Here Sagan doesn’t mean dragons in the fantasy sense; he means giant lizards and other hominid predators.

The seeming fact that mammals and birds both dream while their common ancestor, the reptiles, do not is surely noteworthy. Major evolution beyond the reptiles has been accompanied by and perhaps requires dreams. The electrically distinctive sleep of birds is episodic and brief. If they dream, they dream for only about a second at a time. But birds are, in an evolutionary sense, much closer to reptiles than mammals are. If we knew only about mammals, the argument would be more shaky; but when both major taxonomic groups that have evolved from the reptiles find themselves compelled to dream, we must take the coincidence seriously. Why should an animal that has evolved from a reptile have to dream while other animals do not? Could it be because the reptilian brain is still present and functioning?

In his book Beethoven’s Anvil, William Benzon goes deeper.

Jaak Panksepp, a specialist in the neuroscience of emotion, observes that while dream sleep is almost universal among mammals, it is lacking in fish and reptiles and only sporadic in birds. Suggesting that dream sleep is unlikely to have evolved from nothing, Panksepp speculations that the relevant core brain mechanisms “originally controlled a prmitive form of waking arousal. With the evolution higher brain areas, a newer and more efficient waking mechanism may have been needed.” Among these higher-brain areas, of course, is the neocortex, the chief repository of learning.

Panksepp goes on to speculate that dream mechanisms “originally mediated the selective arousal of emotionality. Prior to the emergence of complex cognitive strategies, animals may have generated most of their behavior from primary-process pscho-behavioral routines that we now recognize as the the primitive emotional systems… These simple-minded behavioral solutions were eventually supersseded by more sophisticated cognitive approaches that required not only more neocortex but also new arousal mechanisms to sustain efficient waking functions within those emergine brain areas.”

Dreaming is thus a kind of neural palimpsest: the evolutionary vestige of the system that once regulated the waking state but has since been overwritten. Panksepp concludes by suggesting that dreaming now serves to integrate the emotional impulses of old brain systems with the cognitive capacities of the new brain systems.

I would like to recast Panksepp’s speculation ins a more colorful way: when we dream, the ancient animals within go out romping in the neocortex. The core brain systems of our reptilian heritage treat the neocortex as an environment and set out to explore it. Imagine the neocortex as some lush jungle setting or peraps a grassy savanna, in which a small animal may bask in the sun, pursue prey, and enjoy a warm meal. To put it another way, dreaming is your inner lizard running free in the dance hall of the mind.

Benzon conjectures that music, at its best, is a kind of waking dream, a speculation I find highly plausible

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

Our smaller-brained ancestors survived fine on the African savannah for millions of years. The growth in our cranial size has been explosively sudden in evolutionary terms. Susan Blackmore has a theory that the bigger brains aren’t necessarily for our sole benefit. She thinks we’re co-evolving with memes, information viruses that are symbiotic and/or parasitic with our brains the way our gut flora and fauna are symbiotic and/or parasitic with our digestive tracts. She posts a sexual selection theory: having more memes in your head signals greater reproductive fitness, and bigger brains can hold more memes, setting off the same kind of self-perpetuating cycle that resulted in birds of paradise having absurdly long tails.

The big brain has been an advantage against other megafauna during very recent evolutionary history, but it might not be that big an advantage in the long term. Consider this: brainless microbes have been around for four billion years, surviving asteroid impacts so big that the oceans boiled away. There have been jellyfish with simple neural nets for six hundred million years, and cockroaches with tiny brains incapable of learning for four hundred million years. There have been humans for only two hundred thousand years, and we’ve shown evidence of abstract thought for only the past forty thousand. In this geological eyeblink, we’ve exploded our population and our global range, so good for us, but we’ve also caused a global extinction event that might eventually wipe ourselves out too. If we do, the microbes and cockroaches will barely even notice that we were here.

Original 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.