Harmony

This page contains the most surprising fact in the medium: the visual arrangement and the chord are the same thing. Not analogous. The same.

Copies are a frequency ratio

Draw one shape at three positions around the screen.

The beam now traces the shape three times per lap of the position pattern. So the shape's waveform repeats three times for every one repeat of the position waveform. That is a 3:1 frequency ratio — and a 3:1 ratio is an interval you can hear.

Duplicating a shape is playing a chord. Raise the copy count in Shape Morph with the audio up and you can hear the interval arrive as the copies appear. OSC-1 names the interval next to the count, so you can watch the arithmetic and hear the result at the same time.

This generalises. Fractal arrangements stack ratios inside ratios, giving stacked intervals. Three-dimensional and higher-dimensional duplication patterns do the same thing with more axes. And it works in reverse: if you want a particular interval, you know how many copies to draw.

Harmony is literally resonance. On a scope you can see the resonance.

Why the image wobbles: just intonation vs equal temperament

Here is the practical consequence, and the reason experienced practitioners care about tuning far more than most musicians do.

A figure sits still only when the frequency ratios are exact whole numbers. If X repeats exactly twice for every once of Y, the pattern closes perfectly and the image is frozen. If the ratio is 2.001:1, the pattern does not quite close, so each lap starts a fraction later than the last — and the figure slowly rotates or crawls.

Standard 12-tone equal temperament, which almost all Western music uses, deliberately does not use whole-number ratios. It divides the octave into twelve equal steps so every key sounds the same, and pays for that by approximating every interval except the octave:

A 0.1% error on a fifth gives a figure that looks nearly stable, with a slow drift you might not notice. A 0.8% error on a major third gives a visibly unstable image. And in a chord of three or more notes the errors compound rather than cancel.

Just intonation uses the exact ratios instead. The intervals are what the visual arrangement already implies, so the figures close and the picture sits still.

The trade is real and worth stating: just intonation sounds pure in the key it is tuned to and progressively less good as you move away from it. This is exactly the compromise equal temperament was invented to escape. For oscilloscope work the trade usually goes the other way, because a wobbling image is a more obvious defect than a slightly restricted key palette.

Switching it on

The synth has a TUNING panel with two controls:

Set ROOT to the key you are working in — that is the whole trick, and getting it wrong is the main way just intonation sounds worse rather than better.

The clearest demonstration is a major third. Hold C and E together in equal temperament and watch the figure turn slowly; switch to JUST with ROOT on C and it stops. Nothing else changed. The E moved by about a seventh of a semitone, which is barely audible as pitch and completely obvious as motion.

Switching tuning does not disturb notes that are already sounding — the change applies from the next note you play, so you can flip it mid-performance without a click or a stuck note.

Practical consequences

1. If a figure drifts slowly, suspect tuning before you suspect anything else. A drift with a period of several seconds is a tiny frequency error, and tiny frequency errors are what equal temperament is made of.
2. Simple ratios are your stable notes. Octaves, fifths and fourths hold still most easily. Thirds are where instability becomes visible.
3. Chords of three or more are the hard case. Errors stack. A triad that sounds fine can look distinctly unsettled.
4. Use the visual as a tuning meter. This is a genuine perk: a scope shows you interval purity directly. When the figure stops moving, you are exactly in tune. No other instrument gives you that.

The other direction: reading a figure

Because ratios are visible, you can identify what you are hearing by counting.

On a Lissajous figure, count the lobes along each axis: lobes across versus lobes up gives you the frequency ratio directly. Three across and two up is a 3:2 — a fifth. That is a working skill, not a curiosity; it is how people verify tuning by eye.

Detune slightly on purpose and the figure begins to rotate, at a speed proportional to the error. Small deliberate detuning is a legitimate compositional device — a slow, controlled tumble instead of a static image.

Related

• Where oscilloscope music's chords come from — the illustrated version
• Multiple objects — the copies that produce these ratios
• Synth — the built-in instrument