I’m not arguing here that everyone loves Mozart, or that I’m about to explain what all humans enjoy all the time. But I can say with confidence that this little bit of Mozart goes a long way toward explaining what most humans enjoy most of the time. The four bars I’m talking about are these, from “Eine Kleine Nachtmusik.”
What these four bars of music demonstrate is that humans like:
- Breaks in the repetition
- Repetition of the breaks in the repetition
- Breaks in the repetition of the breaks in the repetition
- Recursive layers of patterns of breaks and repetitions
In order to prove this to you, I’m going to talk you through these eighteen notes one at a time.
The same Disquiet Junto project that spawned this wildly recursive remix also involved a few more people remixing my remix. Here’s a family tree of the three first generation source tracks, the seven second generation remixes of those tracks, and the three third generation remixes of the second generation remixes.
You can hear the three third-generation metaremixes below.
Here’s a strange and interesting thing that happened to me. The assignment for Disquiet Junto project 233 was to remix three tracks. The assignment for Junto project 234 was to metaremix one of the remixes from project 233. One of the people whose remix I metaremixed was listening to my track and accidentally had it playing in two different browser tabs simultaneously. He liked how it sounded, so he did a metametaremix with two copies of my metaremix offset by a few beats. It came out amazing!
Musical repetition has become a repeating theme of this blog. Seems appropriate, right? This post looks at a wonderful book by Elizabeth Hellmuth Margulis, called On Repeat: How Music Plays The Mind. It investigates the reasons why we love repetition in music. You can also read long excerpts at Aeon Magazine.
Here’s the nub of Margulis’ argument:
The simple act of repetition can serve as a quasi-magical agent of musicalisation. Instead of asking: ‘What is music?’ we might have an easier time asking: ‘What do we hear as music?’ And a remarkably large part of the answer appears to be: ‘I know it when I hear it again.’
Gödel, Escher, Bach by Douglas Hofstadter describes and defines the concept of recursion, and discusses its applications in computer science, consciousness, art, music, biology and various other fields.
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.
I always enjoy when hip-hop artists sample themselves. It makes the music recursive, and for me, “recursive” is synonymous with “good.” You can hear self-sampling in “Nas Is Like” by Nas, “The Score” by the Fugees and many songs by Eric B and Rakim. The most recent self-sampling track to cross my radar is “Unbelievable” by Biggie Smalls, from his album Ready To Die. Here’s the instrumental.
Music is richly mathematical, and an understanding of one subject can be a great help in understanding the other.
Geometry and angles
My masters thesis is devoted in part to a method for teaching math concepts using a drum machine organized on a radial grid. Placing rhythms on a circle gives a good multisensory window into ratios and angles.
The brain turns out to be adept at decomposing sinusoids into their component frequencies. We can’t necessarily consciously compare the partials of a sound, but we certainly do it unconsciously — that’s how we’re able to distinguish different timbres, and is probably the basis for our sense of consonance and dissonance. If two pitches share a lot of overtones, we tend to hear them as consonant, at least here in the western world. There’s a strong case to be made that overlapping overtone series is the basis of all of western music theory.
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.
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. You also need logarithms to understand decibels and loudness perception.
Music is really just a way of applying symmetry to events in time. See this delightful paper by Vi Hart about symmetry and transformations in the musical plane.
I have a theory that what people find most interesting in music is self-reference, recursion and fractal-like scale-invariance. Rhythms based on powers of two are a great way to get this kind of recursion because they can be compounded or subdivided so easily. A bar of four can be treated as two bars of two, or half of a bar of eight. You can further subdivide your bars into quarter, eighth and sixteenth notes. You can group your bars of four or eight into four or eight or sixteen-bar phrases. Here’s a visual representation of this kind of powers-of-two recursion:
In 1987 I remember having my ears grabbed by this track on the radio called “Pump Up The Volume” by MARRS.
Now that mashups are so common, this track doesn’t sound particularly remarkable. But in seventh grade it was startling to hear a house music track full of random samples. “Pump Up The Volume” was part of the same UK dance music movement that spawned the KLF’s “Doctorin’ The Tardis” and “Rush” by Big Audio Dynamite. I wasn’t enough of a hip-hop head in 1987 to recognize where the phrase in the title comes from, but now I do, it’s from “I Know You Got Soul” by Eric B and Rakim. Listen at 0:43:
If I had to pick a single track to explain to an alien or time traveler what hip-hop is and why it’s so awesome, I think I’d pick “Nas Is Like.”
Nas has a great flow full of powerful imagery, but what truly sets this track apart for me is DJ Premier’s production. It’s a complex web of samples and scratches that tie together so seamlessly as to be much greater than the sum of their parts. A lot of the samples are from other songs by Nas himself. Here’s a diagram of all the samples, click to see it bigger: