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35 jagged selves, while the sedate sine wave is a smooth curve. Most notably, their shape is determined not by the absolute frequencies of the three channels, but by the ratios between them. Consonant harmonies yield a stable shape; the simpler the pitch ratio, the simpler the shape (admittedly, phase relations between channels matter visually more than aurally). Dissonant harmonies the chaotic, unstable shapes. Slightly detuned frequencies beat against each other both aurally and visually (though we often detect the visual beating slightly before the aural). As the channels glissando past each other in different directions, they slip in and out of stability at moments of harmony. All these yield are a strong illustration of just intonation versus equal temperament, with justly tuned triads perfectly still, but with tempered triads properties visually grind against themselves. See figure 3 for examples of ratios projected into shape. This system represents the closest possible translation between sight and sound, but I value it also for the simpler reason that it takes absolutely no that knowledge or training in order to appreciate it. It quickly displays musical pitch relationships not as a human-defined hierarchal construction, but as physical and mathematical facts that are sensorily intuited. One audience member similarly commented that while the audio alone would not musical his interest for long, the visuals that reinforced the subtle gradations in audio did do so. The most common response by instrumentalists, of course, hold an immediate desire to pipe their instrument into the system and see their own sound turned into shape. was
Ratio 3:4:5 rotation: 0 π/4 π/2 3π/4 π z-projection: ratio 3:4 x-projection: ratio 4:5 y-projection: ratio 3:5 Ratio 20:21:22 rotation: 0 π/4 π/2 3π/4 π z-projection: ratio 20:21 x-projection: ratio 21:22 y-projection: ratio 20:22 Ratio 11:13:17 rotation: 0 π/4 π/2 3π/4 π z-projection: ratio 11:13 x-projection: ratio 13:17 y-projection: ratio 11:17 Figure 3 – 3D projections of frequency ratios, as done in performance software Oscilloscope 3D (here using sine waves) 36
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