HARMONIC MATHEMATICS

Basic theory & application to audio signals

This theoretical technique, inspired by a principle discovered by John Whitney, US film maker (see his book ‘Digital Harmony’ McGraw Hill (1980?)), is a form of mathematics devised for the production of musical, visual and audio/visual art forms. It employs the rapidity and complexity of computer processing to express hitherto too-painstaking elaborations of detail, and unattainable speeds and precisions of music composition or performance. I have applied forms of the math. to create computer graphics videos, musical scores and recordings, and in some cases audio-visual videos with mathematical correspondence of the two media. The form of this exposition will be via a series of works of mine, simultaneously exemplifying my techniques as they evolved. At root, the math is a way of controlling a one (or two or more) dimensional array of parameters over time.

A Hundred point “radius”

Imagine 100 points (or more, or less) forming the radial line of an invisible circle. Imagine each point, prepared to set off on a circular motion at a distance from the circle’s centre which is equal to its initial distance from it. If moving at the same angular velocity, the points would move in rotation like a stick connected to the circle centre. But- now start off again with each of the 100 points moving with an angular velocity proportional to an index number from 1 to 100 assigned to each point, numbered from one end to the other (it matters not from which end – each option has interesting and beautiful outcomes). This is best timed to last between 10 and 30 minutes (I feel). At first, a winding spiral dominates, but winds itself to ‘break-up’ point at which stage crystallisations and dissolutions of order begin to occur.(These are ‘the good bits’).

At the end of the cycle, point 1 (‘speed 1 also) will have done one complete circle orbit, point 2 – 2 circles, point 3 – 3 circles etc. up to 100 orbits for point 100. So the cycle at this point would begin again from a line identical to the position of the start. At the ‘halfway’ stage, all, the odd numbered points will have covered n/2 (n divided by 2) laps (where n is the point’s index number), which will be ‘an integer plus one half’ orbits, forming a line halfway around the circle from the start position. The even numbered index points will all be at their start position. In fact, any fraction up to a denominator of about a fifth or a sixth of the number of points plotted will yield a figure like this halfway shape- but having a number of projecting radial arms equal to the fraction’s denominator. This results in a quite breathtaking display of simple forms emerging from and dissolving into the teeming of moving points.

This is John Whitney’s “Differential Dynamics” as he calls it. Very magnificent it is too. He has used and examined it in a variety of ways in his films, although he cuts up and edits sequences derived in this way into a very human-developed artistry. I enjoyed, and myself prefer, setting up a complex form on the computer, and then simply running and enjoying it in real-time.

Other elementary ways in which the principle can be demonstrated:

A line of points may be drawn along the leftmost (say), column of a screen. 256 points (say), are each assigned a row of of the screen to move along to the right .Upon reaching the far end of the screen, the points may either reappear at the opposite edge to move again from the left, or , as an alternative system, each one could ‘BOUNCE’ , i.e. travel backwards, and then eventually forwards again. Making the ppoints move at speeds proportional to their respective index number, as described earlier gives rise to comparable effects to the circle system, only with vertical instead of radial alignments at the crystallisation points.

The Jump Across To Sound.

I then wondered how to make sounds like this. With the ‘BOUNCE’ technique described above, I used a digital synthesizer to convert a 256-point system of parameters into a dynamically changing waveform timbre, playing a low B as a drone. Starting with a pure sine wave , (treated as a list of 256 numbers as wave table data), each point of it is made to travel up to its maximum value and then bounce down to minimum and the back up again etc. The sinewave form had index numbers assigned to each point (for speed assignments) from 0 (!) at the left (say) end, to 255 at the right, I found that a sonic journey (through 400,000 timbres) was produced, easily fascinating enough – even on one pitch, which journeyed through complex timbral ‘states’ formed from many combinations and amplitudes, gradually changing, of 128 harmonics. An ‘orchestral’, 7-octave sheet of tone-anchored from one, low fundamental.

Sound number two

A modification of the assignment system of speeds to points was then developed. Rather than always having the linear array of points moving at speeds proportional to their “index number” , it became desirable to assign the speeds in different ways. Such as having the middle points in the sequence travelling fastest, and those at each end slowest. Or vice versa. This gave rise to the concept of a MODIFIER shape. In the elementary system described before, a graphical shape of posistion in sequence plotted against speed was a ramp form (either ascending or descending). If fastest (or slowest) speeds are assigned to the middle, this can be called a VEE (because of its shape) or TRIANGLE modifier. In another variant, fastest (or slowest) speeds are assigned to the 1/4 and 3/4 positions along the sequence of movable points.

The most successful sound/visual combination (on the video piece called ‘Triangles’) was a triangle wave starting point (this now referred to as the SEED form, with ‘two triangles’ as a modifier. (Two ‘mountains’ with maximum speed at 1/4 and 3/4 points. The two halves of the image were set to move in different directions at the start. Many interesting forms arise, which suggest Inca designs. The timbral and graphic sequences are each characteristic of a particular quality, right through the whole cycle.

SEQUENCE technique- more general than BOUNCE

I wanted then to make the motions do better than simply go up and down and back to where they started. I conceived of a sequence of values (i.e. positions) through which each point moves, but, again, at speeds related to assigned integer values, which are in turn in whole number ratio (and hence harmonic) to each other. Hitherto, each point moved up, bounced, moved down, bounced, and moved back to starting point. A harmonically constructed complex timbre form, (eg a timbre with harmonics 1,2,3,4,6,12 with amplitudes gradually decreasing) is used instead to control the movement sequence of each point. This 6-harmonic timbre gives rise to a sequence of intricate, smooth bobbing motions, each point gradually rolling in and out of alignment with others. The points’ motion in this more complex way I call SEQUENCE technique. The crystallisation points are spectacular in different ways, the way in which they are approached are also, being not simply an ‘overtaking’ process, but also having unrolling and twirling motions, implying a 3-dimensional interpretation. Such a one, “Helix” is on one video recording (which, along with ‘Triangles’ is one of the seven audio ‘ShapeTapes’ based on harmonic maths applied to the timbre of a drone, which have been researched in application to healing and relaxation by Isobel McGilvray, and have been selling by international mail order since 1987). ‘Helix’ uses a timbre, comprised of those harmonics 1,2,3,4,6 and12, as a sequence. The points start off as a horizontal line (silence), and each point moves through the sequence at a rate proprtional to its distance from the left hand end. (i.e. a ramp modifier). The timbral journey resulting consists of the 6 harmonics “themselves going on a journey” up and down the 7-octave spectrum of that synthesizer, at a speed proportional to the no. of the harmonic.(ie at speeds 1,2,3,4,6,12).

Lawrence Ball 24/5/1999