The simplest and most effective optical illusion ever is the Necker cube. Which side is in front? The answer is both and neither. Very Zen.
As I’ve been gathering musical simples, I’ve been trying to figure out the best way to categorize them. There are melodic simples, otherwise known as riffs, hooks, and licks. There are rhythmic simples, otherwise known as beats, claves, and rhythm necklaces. And then there are the simples that combine a beat with a melody. Alex came up with the term “compound simples” for this last group. You might argue that all melodic simples are compound, because they all combine pitches and rhythms. But unless the rhythm stands on its own independent of the pitches, I don’t consider it to be a musical simple.
Here’s the first set of compound simples I’ve transcribed. Click each score to view the interactive Noteflight version.
Queen, “We Will Rock You“
The simplest simple of them all. If I needed to teach someone the difference between eighth notes and quarter notes, I’d use the stomp/clap pattern.
The melody is good for introducing the concept of rests, since you have to count your way through the gap between “rock you” and the next “we will.” Continue reading
While I was doing some examination of rhythm necklaces and scale necklaces, I noticed a symmetry among the major scale modes: Lydian mode and Locrian mode are mirror images of each other, both on the chromatic circle and the circle of fifths. Here’s Lydian above and Locrian below:
Does this geometric relationship mean anything musically? Turns out that it does.
I have a thing for circular rhythm visualizations. So I was naturally pretty excited to learn that Meara O’Reilly and Sam Tarakajian were making an app inspired by the circular drum pattern analyses of Godfried Toussaint, who helped me understand mathematically why son clave is so awesome. The app is called Rhythm Necklace, and I got to beta test it for a few weeks before it came out. As you can see from the screencaps below, it is super futuristic.
The app is delightful by itself, but it really gets to be miraculous when you use it as a wireless MIDI controller for Ableton. Here’s some music I’ve made that way.
I was expecting to use this thing as a way to sequence drums. Instead, its real value turns out to be that it’s a way to perform melodies in real time. Continue reading
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.’
One of the best discoveries I made while researching my thesis is the mathematician Godfried Toussaint. While the bookshelves groan with mathematical analyses of western harmony, Toussaint is the rare scholar who uses the same tools to understand Afro-Cuban rhythms. He’s especially interested in the rhythm known to Latin musicians as 3-2 son clave, to Ghanaians as the kpanlogo bell pattern, and to rock musicians as the Bo Diddley beat. Toussaint calls it “The Rhythm that Conquered the World” in his paper of the same name. Here it is as programmed by me on a drum machine:
The image behind the SoundCloud player is my preferred circular notation for son clave. Here are eight different more conventional representations as rendered by Toussaint:
My last post discussed how we should be deriving music theory from empirical observation of what people like using ethnomusicology. Another good strategy would be to derive music theory from observation of what’s going on between our ears. Daniel Shawcross Wilkerson has attempted just that in his essay, Harmony Explained: Progress Towards A Scientific Theory of Music. The essay has an endearingly old-timey subtitle:
The Major Scale, The Standard Chord Dictionary, and The Difference of Feeling Between The Major and Minor Triads Explained from the First Principles of Physics and Computation; The Theory of Helmholtz Shown To Be Incomplete and The Theory of Terhardt and Some Others Considered
Wilkerson begins with the observation that music theory books read like medical texts from the middle ages: “they contain unjustified superstition, non-reasoning, and funny symbols glorified by Latin phrases.” We can do better.
Wilkerson proposes that we derive a theory of harmony from first principles drawn from our understanding of how the brain processes audio signals. We evolved to be able to detect sounds with natural harmonics, because those usually come from significant sources, like the throats of other animals. Musical harmony is our way of gratifying our harmonic-series detectors.
The Quora question that prompted this post asks:
Why has music been historically the most abstract art form?
We can see highly developed musical forms in renaissance polyphony and baroque counterpoint. The secular forms of this music is often non-programmatic or “absolute music.” In contrast to this, the paintings and sculpture of those times are often representational. Did music start as representational but merely move to a more abstract art form than other types of arts sooner? Does it lend it self to this sort of abstraction more easily?
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 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.
The brain is a pattern-recognition machine. We like repetition and symmetry because they engage our pattern-recognizers. But we only like patterns up to a point. Once we’ve recognized and memorized the pattern, we get bored and stop paying attention. If the pattern changes or breaks, it grabs our attention again. And if the pattern-breaking happens repetitively, recursively forming a new pattern, we find that extremely gratifying.