I grew up in the eighties. Cassette recorders were just starting to be ordinary household gear. My sister and I made a bunch of random tapes as kids, not knowing what we were doing or why, just that it was fun. We also taped songs we liked off the radio. We waited until the song we wanted came on, and then held up the tape recorder to the radio speaker. Go ahead and laugh, millenials, but this was such a widespread practice among my generation that there’s a whole Facebook group devoted to it.

Recording to a single-track tape from a single mic was the only way to record music until 1955. In the single-track era, music was recorded more or less the same way it was performed for an audience. There was a single mic in the middle of the room, and everybody played into it simultaneously. The only “mixing” was done by placing quieter instruments closer to the mic and louder ones further away. Recording as an art form unto itself came into being with the invention of multitrack tape, which made it possible to record different sounds non-simultaneously.

Multitrack is an enormously big deal for recorded music. It enables you to capture ideal performances more easily, since you record each voice or instrument in isolation from the others. An error on one track can be fixed while leaving the others intact. Multitrack also opened the door for mixing, since you can manipulate the volume and tone of each sound independently of the others. This might not seem like such a big deal, but that’s because we’re all so used to spectacularly high-tech sculpting of sound. When I listen to old jazz records, the bass is a vague muffled presence buried in the murk of the low end. It took until the sixties for recording engineers to really figure out how to make the bass jump out of the speakers; now we take for granted that it’ll be as crisp and defined as any other sound.

Even with all the flexibility it offers, tape recording is still relatively unforgiving. I recorded a few songs on tape with my first band in college. Correcting mistakes was tedious and took considerable skill and timing on the engineer’s part.

Since 1997 or so, everything I’ve recorded has been on the computer. There are some pros and cons. The major con is sound quality. Tape is analog. The waveforms it captures are infinitely smooth and continuous. By converting the continuous electrical signal from the microphones or instruments into digital files, you necessarily sacrifice some signal quality.

So that’s the bad news. For me, and for most recording musicians at this point, the good news enormously outweighs the bad news. Digital recording is cheap and constantly getting cheaper. Good quality audio tape is expensive; hard drive space costs next to nothing. A computer costs a heck of a lot less than a decent tape recording console and you can use it for other purposes. But cost is only the tip of the iceberg. The really big deal with the computer is that it visualizes music, turning it into screen objects that you can drag, drop and otherwise manipulate the same way you’d manipulate words in a word processing document. For a visual thinker like me, this is a transformative and revelatory change. It’s radically easier to do complex edits on the computer screen than keeping track of a bunch of pieces of identical-looking tape.

The other big deal about digital audio is perfect copying fidelity and endless editing. Every time you copy a tape, the sound quality degrades a little. Also, as tape ages, it chemically degrades. Digital audio files are highly robust. They’re just long lists of numbers, so you can copy them flawlessly and endlessly across any data storage medium. You can edit digital audio non-destructively, so you can try out ideas to your heart’s content without ever harming or losing your original tracks. Digital audio is also nice and portable. You can lay down basic tracks in your basement, overdub more sounds in someone else’s bedroom and then mix and master in a million dollar studio. And while there’s no undo with tape overdubs, you can effectively undo anything you do on the computer.

Music is intellectually a lot easier than it looks. The big challenge for me, and for most would-be musicians I encounter, is anxiety. We have a crippling fear of being judged, and when we’re doing a recording, the panel of potential judges is enormous. Digital recording has done a lot to reduce my anxiety in front of the microphone. Knowing that nothing is carved in marble takes a lot of the pressure off. I’m much likelier to lay down a perfect take or a cool new idea if I’m feeling relaxed, and recording in my apartment on a computer is as relaxing as it gets.

I’ve been recording an acoustic singer-songwriter’s album for the past year. Aside from the vocals and guitar, everything on the album is fake: the bass, the drums, the percussion and keyboards. The vocals and guitar are processed using Auto-tune, digital EQ and reverb and compression, and various other tricks. The “performances” are stitched together from many different takes, with sections repeated and individual notes corrected for timing and volume and decay. None of these techniques are unusual in the age of computer recording. Some people feel that the computer is harming musicianship by making it so easy to sculpt a flawless performance. My feeling is that the computer just shifts the locus of creative work from the original performance to the editing process.

After doing enough of my own projects using the full digital toolkit, I started questioning the wisdom of recording instrumental performances at all, when it’s so much easier to use sampling and synthesis. The turning point came while working with a soul/R&B band called Love Child. The singer and I were writing and arranging songs using samples, drum machines and all the other hip-hop tools. We gave these tracks to the band to teach them the parts. I made charts too, but the tracks were better for conveying the vibe and nuance we were after. We had a bunch of ace musicians in the band, but they never sounded as good as our sample-based tracks. We’d meticulously sequence a bassline, and then the bassist wouldn’t play it exactly. He’d do variations and little improvs, the usual embellishments that musicians add almost unconsciously. The problem wasn’t his ideas, they were all good. The problem was that by straying away from the extremely sparse parts we were writing, he was deflating the tension, turning our hip-hop feel into a generic-sounding funk.

So it went with all the musicians. Also, it was a logistical nightmare getting everyone together, and it cost a fortune. Eventually we asked ourselves, why are we doing this? The songs sound better on the laptop, why don’t we just commit ourselves to life in electronic world? So we started doing gigs with just the laptop and singers, and it sounded terrific. I feel bad for contributing to the rapid drying up of gigs all musicians are facing in the computer era. But meanwhile, we were going for a sound, and the human beings weren’t giving it to us.

Samples and loops give you a lot of freedom. They also carry their own constraints. When you use, say, two bars of a Miles Davis tune in a particular scale with particular chords to a particular beat played on particular instruments, that forces you to fit the rest of your musical elements to fit. This constraint is a stupendously valuable songwriting tool. Repeating the loop identically is easy and varying it is hard. So by default, sample-based music uses a lot of repetition, and you have to justify each variation because it takes so much more effort than another copy and paste. You’d think this would be true with live musicians too, but it’s not. Getting a band to play a loop without variation is just about impossible. I’ve tried many times, everyone gets bored or feels the need to express themselves. We in the western musical tradition undervalue repetition, and having the computer encourage it has improved my writing and arranging enormously.

Sampling is such a useful framework for structuring musical ideas, now I take a sampling approach to live recordings of instruments whenever I can. If I’m doing a rock track with Barbara Singer, we’ll record a take of her flailing freely away at the guitar over a beat, and then find the best bar or two and loop them. If we need a variation or another section, we’ll use the second-best bar or two, and maybe the third. The less material we use, the better it sounds.

In the future I would wish for a more porous barrier between the recording artist and the listener. It’s been a bottomless source of pleasure for me to remix and mash up other people’s recordings. With all due respect to my fellow musicians, I know what I like better than they do. For the vast majority of recordings I have, I’d rather hear the key musical ideas repeated identically in groups of four or eight over hip-hop beats. If recording artists don’t want to oblige me by structuring stuff that way, I can just edit their music to suit myself. It would be a lot easier to do this if I had access to the individual tracks. A few, very few, artists release tracks with the stems separated out. I wish for the day when it’s standard practice.

Update: for hilarious insight into the process of making a top ten hit in 1988, don’t miss the KLF’s Manual.

Let’s say you’d never heard of the Beatles, and I played you “Lucy In The Sky With Diamonds,” “Within You Without You,” and “When I’m Sixty-Four.” You wouldn’t have any reason to think they were written and recorded by the same people. They weren’t. The three songs are effectively solo John Lennon, George Harrison and Paul McCartney tunes, respectively. It’s wonderful to imagine that a single group of humans working together could have produced such wildly disparate sounds, and it was a royal bummer for me to find out that during long stretches of the Sgt Pepper’s sessions, the Beatles weren’t even talking to each other.

I think the late Beatles are still so relevant because they remind people my age and younger of our divorced parents. Their albums are extremely well-made art produced by a group of people in a failed and dysfunctional relationship. Yet the product bears a collective name, creating the illusion of a unified creative team. For legal reasons, the songwriting credits are mostly Lennon/McCartney, even after the two stopped writing and recording in the same room. It’s like how my mom retains my dad’s last name decades after their divorce and remarriage to other people. The mental process of trying to resolve the jagged stylistic contradictions in Sgt Pepper is familiar to me, it’s like squaring the conflicting values and loyalties of my parents and stepparents. Late Beatles albums are more like mixtapes than albums by a band.

I was always was more of a Beatles guy than a Stones guy. Like me, the Beatles didn’t remotely hate their parents. Not the way rock stars usually do; not the way the Stones did. The Beatles revered their parents. They wrote songs for and about them. It’s mostly McCartney doing these songs, but my favorite John Lennon song ever is about his mother Julia. The Beatles were kid-friendly, too. Could you imagine the Stones writing “Yellow Submarine” or “Octopus’ Garden”?

Most rock musicians turn their angst into hedonistic defiance or anger. The Beatles turned most of their angst into wistfulness. Even when their music pushed boundaries, it mostly did so in a relatively polite, restrained way. Maybe the band kept so much composure in their later years because instead of playing in rowdy bars, they were performing for George Martin and the BBC engineers in their coats and ties. These straightlaced British civil servants were the only listeners present for most of the band’s live music-making after 1965, along with Yoko Ono. The Beatles’ poker face is uptight by rock standards, but it makes perfect sense for professionals in a high-tech work setting.

The advances in recording technology that gave the late Beatles albums so much of their imaginative sweep also contributed to their feeling of alienation. In the early years, the band recorded by getting together in a room and playing live to single-track tape. By the end, Paul McCartney could use multitracking to play every instrument on “Back In The USSR” and “Birthday”, as if he was Prince recording “When Doves Cry.” The tape collage stuff like “Revolution 9″ and the end of “Strawberry Fields Forever” is more like Aphex Twin than Chuck Berry. And the instrumentation moved steadily into synth and sampler territory. The flutes at the beginning of “Strawberry Fields” aren’t real, they’re tape samples in a Mellotron. Here’s a video about this early sampling keyboard – thanks, Nick Seaver.

The famous medley that ends Abbey Road is a sixteen-minute DJ mix of leftovers from the White Album and Let It Be. It was carefully edited into a seamless suite by McCartney and George Martin. The medley can’t exist outside of the recording medium. The Beatles never played it live, and to my knowledge no one else has either. How would you even approach it? I learned the first chunk on the guitar and it was a whole music education unto itself, but my rendition is not going to make you forget the original.

Given how electronic their sound was, it’s a shame that the Beatles have never allowed anyone to sample them. If they had been born twenty years later, they might well have tried their hand at loops and breakbeats. Their early songs are collages of Chuck Berry and Carl Perkins and Buddy Holly. The later, more ambitious songs feel more “original” only because the source material for the collaged is more diverse. Wikipedia says:

According to Lennon, “Because” was inspired by Ludwig van Beethoven‘s “Moonlight Sonata“. “Yoko was playing Beethoven’s ‘Moonlight Sonata’ on the piano … I said, ‘Can you play those chords backwards?’, and wrote ‘Because’ around them.

Another Beatles classical remix is “Blackbird.” It includes a fragment of Bach‘s Bourrée in E minor. It’s the ascending G major part, a loop that runs through the song. These guys are a narural fit for sample culture.

Not like people are waiting for permission to sample the Beatles. The white half of Gnarls Barkley, Danger Mouse, made his first big splash by combining Jay-Z‘s Black Album with the White Album into his breathtakingly copyright-infringing Grey Album. While no one is officially allowed to sample the Fab Four, some people have been allowed to use pieces of cover versions.

Common’s song “Forever Begins”, produced by Kanye West, samples a cover of “She’s Leaving Home” by Syreeta. The line “Father snores as his wife gets into…” loops under the verses. The sample cuts off “her dressing gown.” It’s a strange thing to rap over, but it works. (The track also uses another perfect sample, Steve Gadd’s snare drum intro to “Fifty Ways to Leave Your Lover” by Paul Simon.)

Here’s a screen shot of Beatles Rock Band – click through to see the fascinating vocal notation more clearly. It’s a combination of MIDI and standard music notation.

So what do you say, Beatles copyright holders? How about loosening up the restrictions a little? People are remixing the tunes anyway. Why not get in front of the situation and put the stems on iTunes or Amazon? Nothing can ever replace those albums, but why should the story end there? “Forever Begins” doesn’t take anything away from “She’s Leaving Home” any more than “Because” takes away from the Moonlight Sonata. We the fans have been remixing the songs in our heads for years anyway. Why not let us do it with computers too?

Here’s a remix/cover/mashup of the Beatles’ “Tomorrow Never Knows” combined with “Galang” by M.I.A. and “Slide” by Missy Elliot. Vocals by Babsy Singer, production and game controller synth by me.

Tomorrow Never Knows by ethanhein

Vocalist Dieter Meier recorded the words “oh oh, chicka chicka” and “oh yeah” at a relatively normal pitch into the sampler, and keyboardist Boris Blank played them back lower and slowed down. There are also some cool sampled Tarzan yells and Lord Of The Rings synthesized men’s chorus. This track could have been recorded last week.

We think of sampling as this high-tech modern practice, but analog sampling keyboards go back to the early fifties. The first one was the Chamberlin. It played short tape recordings of a few different instruments when you pressed the keys. The Chamberlin has a much more famous descendant (some might say ripoff), the Mellotron. Here’s a little branding 101: don’t name your invention after yourself, unless you have a cool name like Robert Moog. Pick something retrofuture and groovy. The Mellotron sounds like something from Sleeper that you use between the Orb and the Orgasmatron. The intro of the Beatles’  “Strawberry Fields Forever” is Paul McCartney playing sampled flutes on a Mellotron.

Analog tape isn’t a great sample medium. The mechanisms are delicate and bulky. The tape decays over time. The little motors have to be running at exactly the right speed for the notes to play back in tune. Sampling keyboards didn’t really take off until the invention of inexpensive digital audio. Now that computers can play back audio recordings and perform all kinds of strange mathematical operations on them in real time, anything with a processor and a sound card can act as a sampler. Even high-end cell phones can perform the same functions as Ferris Bueller’s E-mu.

Some sampled instruments work better than others. A MIDI interface can only capture certain aspects of your performance: which note you played, how loud you played it, how long you held it. You can add some other performance data with the sustain pedal, with the pitch or mod wheels, and maybe with a few other parameters. That’s not nearly enough data dimensionality to convey all the infinitesimal nuances of the way a violin bow or guitar pick grips and releases a string. Stringed instruments sound extremely fake when played on a sampling keyboard. The fakeness has its own charms, but that’s a whole different instrument unto itself. Piano works well as a MIDI instrument since it practically was one to begin with. Any keyboard instrument translates well to MIDI. Massed orchestral instruments work better than solo ones. Horn samples can work okay if you don’t mind monotonous phrasing. Again, sometimes the robotic sound has its own quality. I mostly prefer more purely electronic sounds like abstract synths and samples of other songs.

One of my most entertaining experiments with sampling was called the Sikoratron. It’s a Reason patch I made using samples of Catherine Sikora, who I played with in a jazz group. To build my sample library, I recorded every member of the horn section doing solo improvisations. Catherine recorded these long, angular Coltrane-esque sax lines. By mapping different phrases to different regions of the keyboard, I could play my own far-out Catherine solos. The results were unpredictable, since the tonality of the phrases didn’t necessarily match the key that triggered them. The Sikoratron gave the best results when my non-keyboard playing friends explored it intuitively with their index fingers.

The full surrealism of MIDI is only just revealing itself. You can map sampled sounds to just about any physical action. Jerry Garcia used a MIDI guitar to play synthesized flute and such with the Grateful Dead. MIDI guitar such a cool idea in theory, since the guitar is already an amazing analog synth controller. Unfortunately, you can’t just slap a MIDI pickup unto an electric guitar, because it won’t track as accurately as you would want. You need to get an expensive special guitar made of a futuristic carbon composite. Fine if you’re Jerry Garcia, lame if you’re a normal person.

When I play electronic music live, I do my sample triggering with a video game controller. It’s more limited than a full MIDI keyboard, but for my stuff that’s a virtue. I see the future of MIDI belonging to game controllers. Check out this Synthtopia post on the world’s strangest MIDI controllers. Behold the Drumpants:

Other videos show people controlling synths and samples using pennies, a laser, a robotic exoskeleton, a sheet of paper, a driver’s license, hamsters and other odd things. Music looks like it’s going to continue to be fun in the future.

Your voice can produce a smooth continuum of pitches. To sing, you eliminate most of those possibilities, vibrating your mouth and throat only at certain frequencies, the pitches of the melody. Auto-tune helps by shifting the voice’s frequency to the closest desired piano-key pitch.

Towards the bottom left is a knob labeled Retune Speed. Even the best singers waver around their intended pitch for a few milliseconds before converging on it. If you correct away that wavering, the result sounds artificial. So Auto-tune can be set to delay its effects. Slower retune speeds allow more human-sounding shakiness to pass through the filter. If you set the retune speed to zero, there’s no wavering allowed, and you get the robo-vocal sound beloved by T-Pain, Kanye West and Lil Wayne. It’s more widely known as the Cher Effect, because a lot of people first encountered it in her song “Believe.”

Hear the Cher Effect at work in my mashup of the Beatles, M.I.A. and Missy Elliot with vocals by Barbara Singer. The exotic melisma comes from Auto-tuning Barbara to the mixolydian mode.

Tomorrow Never Knows by ethanhein

There’s a persistent and false story that Cher used a vocoder for “Believe.” The producers lied in interviews, not wanting to give away their trade secret. Auto-tune isn’t exactly a software vocoder, but it’s based on the same math.

Music-theoretically, the interesting part of Auto-tune is the center of the window, listing the twelve pitches on a piano. By default, Auto-tune is set to the chromatic scale, all the piano keys, starting on C:

To Auto-tune yourself in C major, you need to remove C sharp, D sharp, F sharp, G sharp and A sharp. (There are no flats for some reason.)

This is a lot like the way you set up a xylophone or marimba for a beginner. By taking the bars for the undesired notes off, you make it impossible to play anything wrong.

A bit of fun for music nerds: the notes you omit from the C major scale, the black keys on the piano, form the F sharp major pentatonic and E flat minor pentatonic scales.

To make the C natural minor scale,  you omit C sharp, E, F sharp, A and B.

To get the other minor scales, you’d just need to toggle the sixth and seventh notes differently. For C dorian you’d leave A in and remove G#.

Here’s how you’d set up the C blues scale:

“A Love Supreme” by John Coltrane starts with this fanfare on the notes B, E and F sharp.

In the key of C, the fanfare’s three pitches are C, F and G.

This is one of my favorite Auto-tune settings. It sounds amazing on speech.

It’s even more fun to strip the pitch set down even further, taking out the F and even the G for maximum posthuman warbling. Being limited to a smaller group of pitches forces you to concentrate on rhythmic patterns. Check out how cool it sounds when we Auto-tune Barbara to just the root note for the second verse, or just the fifth on the outtro:

Revival Revival – Those Shoes Never Scared by ethanhein

Auto-tune’s binary representation of the combinational possibilities of music theory is similar to the way I learned how to conceive my chords and scales in jazz training. You can derive any scale or chord by starting with the chromatic scale and omitting the wrong notes.

Sometimes you want to be constrained to a traditional scale, but more often the blend of pitches you want is more idiosyncratic. In major keys, you very often want to use the minor third and sometimes minor sixth. For a blues feel in any key, you can include the flat fifth. Omitting the fourth and seventh from a major or minor scale makes a dissonance-free pentatonic.

Here’s how you can build your own scales, chords and pitch groups in the key of C, in or out of Auto-tune.

C: The root or tonic. Probably leave it on.

C#: The flat second. Leave on for Middle Eastern music, turn off for Western.

D: Second. Usually leave on, except for Middle Eastern music.

D#: Minor third. Leave on for tragedy and blues.

E: Major third. Leave on for happy, turn off for sad.

F: Fourth. Probably leave on unless you’re making major pentatonic.

F#: Flat fifth, sharp fourth. Leave on for blues and exotica.

G: Fifth. Probably leave on, though try turning it off for fun.

G#: Minor sixth. Leave on for tragic feel.

A: Major sixth. Leave on for bright/happy feel.

A#: Minor seventh. Leave on for blues, minor, rock, or mixolydian.

B: Major seventh. Leave on for major and harmonic minor, brightness, and suspense.

See more scales in Auto-tune representation.

Symmetry and patterns register on the ears and eyes differently, but there’s substantial and intriguing overlap. If you illuminate every other note on the list, you get the whole tone scale. If you alternate skipping a note, not skipping the next one, skipping the one after that and so on, you get the diminished scale. The symmetries of those scales announce themselves to the ear immediately, though you may not be able to figure out what specifically the symmetries are.

I’m pretty psyched about the convergence of the mind’s eye and the mind’s ear. All digital music-making tools have a synaesthetic element. For visual thinkers like me, the computer’s music visualization tools have opened up some big new swaths of sonic terrain.

Any interesting related music visualization systems out there? Hit me in the comments.

Turning sound into electricity

At the physical level, a sound is a rhythmic vibration of air molecules. Your ears can detect subtle changes in the air pressure, and can reconstruct good guesses about what might be agitating the air to produce those changes. When the air pressure fluctuates in a steady sine-wave pattern, you hear a musical pitch. The faster the fluctuation, the higher the pitch. This video by Vi Hart explains:

Microphones work a lot like your ears. They contain pieces of metal that vibrate in response to the vibrations of the air, generating a fluctuating electromagnetic disturbance.

Analog recording media store the fluctuating electric current as fluctuations in something else. In a vinyl record, the fluctuations are encoded in the squiggly physical shape of the sides of the groove. Magnetic tape stores the fluctuations in the alignment of tiny magnetic particles embedded in the plastic.

Turning electricity into numbers

The computer takes in fluctuating electric current and turns it into numbers. Long, long, lists of numbers. It does this with a specialized device called an analog-to-digital converter.

The analog-to-digital converter in the computer’s sound card has a clock, like the one synchronizing the activities of the computer generally. At each clock pulse, the converter takes a reading of the current on the input wire and finds the closest numerical value out of a finite set of choices. The more choices the converter has, the more accurately it can represent the input signal. A two-bit converter would only be able to tell if the input wire was on or off. A four-bit converter could sixteen different possible readings. An eight-bit converter could store 256 different readings. The more bits you have, the more accurately your digital readings can represent the original input signal. The standard for CD-quality audio is 16 bits, which means that each reading can take one of 65,536 different values. Higher bit depths are also possible — very fancy equipment can record 24-bit, 48-bit or even 96-bit audio.

Bit depth is only one factor in determining the quality of your digital audio. The other is sampling rate, the frequency with which the converter takes its readings.  If the converter takes its readings more often, it gets more accurate results. Standard CD-quality audio is recorded at 44,100 samples per second. This sounds like an incredible speed, but CPU clocks routinely operate thousands of times faster than that.

The image below shows a four-bit analog-to-digital converter in action.

The red line shows the amplitude of the input wire’s voltage over time. The sixteen horizontal grey lines are the different voltage levels the converter can detect. It takes four bits of data to specify the sixteen different possible values. The tick marks on the horizontal axis are clock pulses. Each reading gets stored as a binary number. The list of numbers makes up the digital audio file.

To play digital sounds, the computer sends its list of numbers to a digital-to-analog converter. To produce sound on speakers or headphones, the converter sends pulses at the appropriate voltages out on the wires, which your ear averages out into a pretty good reconstruction of the original sine wave.

Digital audio editing

Once you have your current stored as numbers, you can do a lot of cool stuff. Any sound in any digital medium is basically a spreadsheet with two extremely long columns, one for each stereo channel. In 16-bit audio, the numbers in the columns range from zero to 65,535 (2^16 – 1.) One second of stereo CD-quality audio is two lists of 44,100 numbers each. If the values of the numbers range smoothly along a sine wave that cycles four hundred forty times per second, you hear a computery beep playing concert A. If the numbers fluctuate along the pattern you get from superimposing the sine wave with another one that cycles six hundred sixty times per second, you hear two computery beeps a perfect fifth apart. Add in another sine wave doing eight hundred eighty cycles per second and you get the I-V-I power chord beloved by rock and roll.

All of the audio editing and processing that happens in Pro Tools and programs like it boils down to systematic mathematical operations on your lists of numbers. Auto-tune looks for sine wave patterns and alters them so they snap to the closest piano-key frequency. At the transistor level, Auto-tune is no different from Microsoft Excel, except that it acts a lot faster on bigger lists of numbers. Copying and pasting repeated sounds is the same procedure for the computer as copying and pasting a list of numbers or a string of text.

You need a fast computer with a capacious storage capacity to do serious audio work, but we’re lucky enough to live in an era when even a garden-variety laptop can handle stupendously complex audio tasks.

The great miracle of music for me is not any particular technique or piece or performer, but just the fact that it exists at all. A single linear wave can encode all the rich complexity of all the sounds we hear. This wave is as easily translated into numbers as dollars can be translated into pizzas. The complete works of Bach or Coltrane can be encoded as simple two-dimensional waveforms. All that music is two-dimensional curves, voltage vs time, or air pressure or guitar body flexion vs time.

Our brains are stupendously adept at detecting patterns of patterns of patterns in the linear waveform of air pressure, deconstructing and comparing the component sounds that went into it. If there are multiple frequencies present simultaneously in the pattern of vibrations, we can distinguish them and, with a little training, detect the ratios between them. I feel like we’ve barely begun to scratch the surface of the artistic possibilities of mathematical operations on numerical audio data.