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.
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 fully inhabit blues tonality, 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.