Update: there’s a lively discussion of this post happening on Synthtopia’s Facebook page.
Musical repetition has become a repeating theme of this blog. Seems appropriate, right? This post looks at a wonderful article by Elizabeth Hellmuth Margulis, investigating the reasons why we love repetition in music in Aeon Magazine.
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 are eight different representations of it as rendered by Toussaint:
And here it is in my preferred circular notation:
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.
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.
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.
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:
The blues is a good entry path for beginner guitarists. If you learn the standard fifteen chords and the blues scale, you’ll be well on your way. However, there’s one crucial piece of additional music vocabulary you need to do the blues justice, and that’s diminished chords.
To make a diminished chord, you start on any note, go up a minor third, then another, then another. Here are the notes in C diminished — the scale tones are in red.
Here are some good guitar fingerings for diminished chords.
Diminished chords are highly symmetrical, which gives them a peculiar property. The circle above shows C diminished, but the same notes also make Eb, F# and A diminished. The only difference between these four chords is their respective bass notes. This symmetry means that there are only four diminished chords total. The diagrams below show what I mean. On the left is the circle of fifths; on the right is the circle of half-steps. Each square is a diminished chord.
I follow science news the way normal dudes follow sports. If you’re geekily inclined like me, you may have heard that the particle physics people are getting closer to producing the Higgs boson. You may have wondered what that is exactly, and why you should care. The science press has nicknamed the Higgs “the God particle,” which is poetic but doesn’t move me any closer to understanding. Here’s my best effort to wrap my head around the idea — maybe you’ll find it helpful, or at least entertaining. If you’re a real scientist and want to clarify or correct anything I’m saying here, please jump in on the comments.
I’ve had a lot of music teachers, formal and informal. The best one has been the computer. It mindlessly plays anything I tell it to, over and over. Hearing an idea played back on a continuous loop tells me quickly if it’s good or not. If the idea is bad, I immediately get annoyed, and if it’s good, I’ll cheerfully listen to it loop for hours.
There’s something in the cumulative experience of a loop that makes it greater than the sum of the individual listens. Good loops create a meditative, trance-like state, like Buddhist mantras you can dance to. As far as I’m concerned, if it’s the right groove, there’s no such thing as too much repetition. Take “Hey Jude” by the Beatles.
At the end, they repeat “Naah, na na nanana naah, nanana naah, hey Jude” over and over for four minutes. I could listen to it for forty minutes. Why don’t I get bored? Continue reading