If one stands before the task of moving something back and forth or up and down (for instance, the pickup in a string model). Then the sine way is not a good solution, because the area of the direction reversal has lingered much too long. A triangular wave is, in turn, too abrupt on the reversal points.
On the search for a means between both, I hit upon the 'figure eight' or, as the mathematicians call it, the 'lemniscate'.
Apart from the fact that the horizontal form of it serves as the sign of infinity and also has other mystical meanings, friends who are skilled in manual crafts have assured me that this movement is the most organic one for many activities: wiping, sanding, polishing, etc.

If one observes the movement in the longitudinal axis, an expeditious course appears, similar to the triangular wave with a natural delay in the reversal spot, but not as long as by a sine wave.

From a tonal perspective, one observes in one channel (on an axis) the fundamental tone with uneven overtones, in the other channel the even overtones. The formulas for it originate from the Swiss mathematician Jakob Bernoulli (1655-1705). In one simplified form for a generator, it reads as follows:

x = cos(phi)/(sin(phi)²+1); y = cos(phi)*sin(phi)/(sin(phi)²+1)

(for those who want to know ...)
It is available in this form in VASP and can be observed in the xy representation (hotkey 'x' to view.)

gen.achter 400hz

The function is also present as a generator in AMP. As a standard, only the transverse movement will be outputted, in the complex mode, however, the full loop movement (longitudinal axis left, lateral axis right).

An application in the string model to control the two pickups:

k1=osc.chord (s,cyc=4,dim=128,ini=drag)
k2=gen.achter (c)
k3=rmo (s,*k1) "remove offset - one could actually put it
out=test.wav (s,*k3) "immediately into the string model
dur: 10
k1.tens: 0.4
k1.elast: 0.4
k1.damp: 5 [damp]
k1.smooth: 5 [damp]
k1.inidur: 0.001
k1.iniamp: 1
k1.inipos: 4
k1.pu.0: 90 (c=k2.0,cm=lin,ca=30)
k1.pu.1: 90 (c=k2.1,cm=lin,ca=30)
k2.freq: 0.5

There are also other, simpler variants of figure eight loops. However, they are ordinarily based on different sine waves in both channels and thus offer us no solution for the stated field of duty.

* * *

It is no great effort to also execute this function as waveshape (waveshape.achter), or to replace the circular motion in the low-frequency noise noise.band, which is so effective for controls, with a lemniscate. However, the function appears to me to be so fundamental as a movement type that a general inclusion is called for. At least for the generators that are based on phase modulation, it is not difficult to replace the circular motion, which underlies all of them, with a lemniscate. Consequently, in the current version of AMP there are the variants:


In mono and the normal multi-track modes, a triangular-like wave of the x-axis will be used; in complex (c) modus, the complex sample vectors depict, with variable speed, a lemniscate.


(c) Günther Rabl 2010

The lemniscate is considered to be a special case of the 'Cassinic Curves', which I recently found an illustration of. (The photograph of an interference pattern of a crystal, I believe).

Hans  Hauswaldt: Aragonite, ca. 1910, shiny collodion paper<BR>
                                ©  Albertina, Vienna

Hans Hauswaldt: Aragonite, ca. 1910, shiny collodion paper
© Albertina, Vienna