2 Motivation to question and background

Motivation to this question, short version: I feel this is a good new way to look at functions while retaining the time-domain toolset of real analysis.

More motivation for this question: I am wondering whether this sort of analysis tool could become a generalization of the time domain. We have a lot of tools for analyzing functions in the frequency domain, and we have a lot of tools in the time domain - however the results they yield are somewhat disjoint. The way this 'spinning graph' effect depends directly on the frequency content of the function at hand. It could make it possible to do time domain based analysis of some form of frequency content, or rather 'slope content': what I am refering to here is the fact that in the time domain it's easier to think in terms of slope rather than the frequency. The two are tied together (a function with a sharp rising edge will probably have a lot of high-frequency content) but not directly (a simple 1 Hz sinusoid with enough amplitude can have higher slope on an interval than a 200 Hz triangle wave). This tool that arises from simply frequency-shifting the function gives a new way to look at time-domain signals, and so I think it could give rise to interesting questions - I am extremely surprised that this is not mentioned anywhere in literature: it feels like a very direct generalization of the time domain display of a signal and can be applied to pretty much every real-valued function out there giving us new information about it. On the one hand it is very useful for periodic functions, on the other hand the way the graph seems to 'coil up' when the frequency shift is increased can make it useful for the analysis of functions which are not periodic. I strongly feel that the analysis of 'where the graph is on that spinning cylinder' can provide new information about functions.

Background: I had come across this effect when studying the musical properties of stretched-harmonic waveform synthesis, motivated by the fact that real instruments tend to have slightly inharmonic rather than harmonic frequency content, whereas electronic synthesizers tend to have purely harmonic timbre. This got me asking some questions not really related to music!

1

What is this effect in Fourier/additive synthesis called?

Hi, I have re-synthesized a cyclic function additively, and I added a fixed offset to the frequency of each partial. So if the function was $\sum a_{n} sin(2 \pi x * n)$ and its frequencies were $n*f_{B}$ where n is the number of the partial and f_B is the base frequency of the function, then now the frequencies of the partials are $n*f_{B}+f_{o}$. The function becomes an almost-periodic function. The waveform seems to be 'rotating' around the time axis.

Is this effect described somewhere? If so, where, and what is its name?

Here are animations of this happening: rotation.rar, 2MB

By the end of the first animation the function is a bit rippled, those ripples are only created by errors in the model. It should in fact look smooth like in the beginning.

In the 2nd movie, at the beginning I raise and then lower the offset frequency of the partials (f_o), and then set it to 0. On 0:32 I reset the resynthesis model (this is irrelevant to the effect, but it explains why the waveform suddenly changes). Later I set f_o to +1 Hz and 0 Hz, alternating the setting.

In the 3rd and 4th animations I alternate f_o between +1 Hz and 0 Hz.

It is as though the function graph is drawn on a 3-dimensional glass cylinder that rotates around the time axis. To avoid confusion, I define that if f_o is positive, then the direction of the rotation is positive. Note that the points of the graph don't seem to be equidistant from the axis of rotation - some seem to be orbiting at smaller and some at bigger radii. This is not really exposed in the movies, but if you start going up very far with f_o, the graph becomes 'twisted', as if the 'cylinder' were becoming twisted around its axis of rotation.

Thanks!