Tha attached paper has an interesting idea at the end: a dualquat harmonic oscillator which can describe
You may recall that Shepard's original paper on pitch circularity discusses helical representations for
pitch perception. http://psycnet.apa.org/journals/rev/89/4/305/
This strongly suggests that dualquats could also be used to operationalize motion in musical contexts.
The pitch one is obvious because the mapping is clear and we have lots of well known ways to think of translation and rotation
in this context (e.g. arpeggiated cadences, vamping) but I think there may be some interesting cases where we can use this
for fancier rhythmic things too. For example, in your work on polytempo you can define certain requirements such as events lining up in time as configurations of points on different rigid bodies in motion (one for each meter) that have to be colinear. My intution is that
interpolation with dualquat's will give you better results than the current scheme in tempocurver. It may at the very least give you hints as to how to represent what you are composing in a 3d GUI. I often feel that phase unwrapping would be better understood if people could see it in 3D. With care the information we need can be obtained by casting shadows as I did with Amar for the 3D SDIF editor.
There is of course the question of how to integrate time into the model. You could start by parameterizing the screw parameters as a function of n.deltaT in discrete time signal processing fashion and then drive the system with my relaxation functions (shifted and time scaled sawtooths).
I hope this is good enough. I am slightly afraid that we might want to go all out and do tensors because I have seen time introduced in another paper by forming linear sums of two tensors to represent a reference frame from which the dualquats emerge. I am hoping that dualquats do most of the heavy lifting and that we will be happy with using o.'s existing vector functions with lambda() to define the operations to endow vectors with the appropriate constraints and operations to become tensors.
The code will be less obvious because we don't have operator overloading but most of the papers on tensors end up looking like APL which is not very clear either.
Having developed critiques of periodicity in one essay and being a fan of the point- and line-free geometries of Whitehead, Spencer Brown and others, I am rather embarrassed to be suggesting this strong push into dual-quats, the power tools for leveraging the Greek idolatry of the circle and line. Maybe the way to deal with ghosts is to dance with them?