 ## Sums of Orbits

Another way to translate intervals into a shape representation is based on the fact that if you take two sinewaves and delay one by 1/4 of one wavelength and use one to control position horizontally and one to control position vertically, the result is a circle. If your waves are more complicated (i.e. the sum of two pitches at an interval) then the shape gets more complicated in a way that seems to express more about the sonic experience of the interval than the lissajous curves above. A single note produces a circle. Very consonant intervals like a perfect 5th and perfect 4th produce very simple shapes. More complicated intervals produce more complicated shapes. In addition, you can add multiple intervals to represent more complex chords. In the demonstration below, the number keys from 0 to 9 and the letter keys a and b represent the 12 notes of an octave. Press these keys in any combination to see the result. Press more than 2 keys for complex harmonies. (Note that your computer may not respond properly when you hold large numbers of keys down.)

• perfect 5th: 0 and 7
• perfect 4th: 0 and 5
• major 3rd: 0 and 4
• major 6th: 0 and 9
• minor 3rd: 0 and 3
• minor 6th: 0 and 8
• tritone: 0 and 6
• minor 7th: 0 and a
• whole tone: 0 and 2
• major 7th: 0 and b
• semi tone: 0 and 1
• tonic triad: 0, 4, and 7 (the most consonant chord in a musical key)

(click on the image once to have the key strokes control the image)

While this seems to hold some promise, there are some complications. First off, remember that the equal tempered scale offers approximations of harmonically related frequencies. Images of this type are static if the relation is precisely harmonic, but rotate and shift when there is some error. The error is almost imperceptible to the human ear, but it has a much larger impact visually. You can see the problem this presents if you press each whole-tone combination in turn, from 0 and 2 to 1 and 3 to 2 and 4 and so on. You will notice that most of these resemble each other except that each shows a different rotation rate. But the 4 and 6 combination looks very different, because the error is sufficient to make a large visual difference. The problem has to do with the assumed reference for the harmonic relationship. This example presents all intervals relative to the the equivalent of the key of 'C'. The '0' key represents the tonic (the first note in the scale of this key), and all intervals are represented in relationship to the tonic which means that the harmonic error between the tonic and the lower of the two notes in the interval adds to any harmonic errors in the interval, causing the visual difference. The difference is potentially deceptive because in isolation, each whole-tone interval sounds identical except for the shift in pitch.