Tuning By Eye
Consonant and dissonant fireflies
Whenever I try to see what music looks like, it always feels like it’s just trying to run away from me.
I want to capture the sounds in a jar so I can stop and really look. If I could just get them to stand still, I could compare them to one to another, like living creatures. But time just refuses to leave them alone.
Cycles of the simplest possible unit of tone, the sine wave… they just fly by. And even when I can run fast enough to catch up to them, their shape is never what I expect; if they really move in “cycles,” why do they always pass by me from left to right? Shouldn’t the beginning connect to the end?
I think back to high school trigonometry, and how they told me that these waveforms are actually just representations of circular movement, plotted against time.
What if I could collapse the time, and just see the cycle?
It might be like catching a firefly in a jar:
It leaves a trail of light that looks like a line at first, but a slight rotation reveals that it’s actually a circle: a cycle of vibration, moving so fast that I see and hear a constant tone — an afterglow with the illusion of total stasis.
Combined with another simple tone (in, for example, the proportion of 2:1 above), the new composite “interval” is only a tiny bit more complex. A single loop forms, and the full length of the cycle is doubled. But what I see is still the static glow of a solid shape. This form is really useful for examining sound and music up-close.
I can even rotate the jar and examine this firefly from all sides. But it quickly escapes.
Maximum Clarity
When intervals are in whole-number ratio proportions, they will sync up in cycles. All the cycles above are fast enough to escape the “refresh rate” of our eyes and ears, and they appear frozen. We perceive clarity and stasis.
The simpler the ratio between tones of an interval, the more likely it is to sound and look completely stable, and yet we still see varying degrees of complexity within, like frost patterns on ice.
In this example, can you hear the different complexities as you see them?1 This is the crux of consonance: the simpler the ratio, the more consonant the interval (and less complex the visual pattern).
But even with ratios in whole-number proportions (those “justly tuned intervals”), if the cycles become long enough, slow enough, they escape below our sensation of stable tone. Instead, we hear hear (and see) friction — beating.
Toward a more perfect unison
This interval is small enough that it has the quality of a single tone — a very nervous unison. It beats. At first those beats are very fast and barely perceptible.
But by slowly decreasing the pitch of the upper tone, the beating becomes progressively slower and more obvious — until it completely aligns with the lower tone, and a “perfect” stasis is reached. If you ever hear a musician talking about “tuning out the beats,” this is what that looks like.
The Scope
The jar we’re using to hold these fireflies of tone is an oscilloscope,2 set to “XY” mode. Typically this device would plot cycles of voltage against time, creating the familiar left-to-right “waveforms” like sine, triangle, pulse, or sawtooth.
But in XY (or “Lissajous”) mode, the dimension of time is removed, and one signal is plotted as a function of the other. The resulting visuals are known as Lissajous curves.
In engineering, these are most often used to identify problems — usually with the aim of resolving them. But in music, I find that oscilloscopes (in general, but especially Lissajous curves), are great tools for learning about different types of consonances and dissonances. They can be used to give the ears a rest and make comparisons between otherwise subtle distinctions of tuning, by eye.
Rational and irrational fireflies
In contrast to the clarity of a justly-tuned interval, a “tempered” interval is a beautiful ideal of complexity: its cycles will never sync up, and thus always contain some amount of dissonance — it will always beat.
While personally I find living in a musical world of non-stop friction less and less satisfying (and am thus increasingly attracted to just intonation, especially on the piano), tempered intervals have their own kind of idealism and beauty. After all, they have been the entire palate of colors in western music for several hundred years now.
Comparing the twelve irrational intervals of an equal tempered scale to a justly-tuned counterpart can be tricky to hear when played with pure sine tones.
But you can easily hear the difference with your eyes:
Resolution
The line between consonance and dissonance, argued over for hundreds of years, is subjective, and personal. It changes with the culture, the musical context, and even with the evolution of physiology. But no matter how you quantify it, dissonance is movement — it is various types of friction, mathematically analogous to complexity.
If a frictionless life were possible, would you choose it?
Some might. With no tension, there is no resolution… and the absence of both might be exactly what some people are looking for. Others might gravitate toward the exact opposite: an all-consuming friction that results in its own kind of stasis: a permanent state of dissonance that also never resolves.
To see these extremes manifested in the real world, I can’t help but think of the current state of the technology industry, trying to sell the world a frictionless life, and the irony of using a constant state of chaos and disruption as its method.
I wouldn’t even try to guess what the right balance between consonance and dissonance is (in life, or in music!). But the chance to examine it is essential — and if I can find a way to freeze time for a moment, I can take a closer look.
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