This image represents one of the remaining problems with the Zathras timestretcher I wrote. When synthesizing a new wave, we want to do that fast, so we use an FFT to generate on average 632 sines at the same time. The problem is that whenever a wave has a frequency that does not match any Fourier bin then we need to ‘fix’ it. That is done by applying a phase modulation to it. yet because the Fourier synthesis requires circular waves, the endpoints must match (that is be a multiple of 2Pi). When the wave is modulated with a non 2pi multiple this requirement is not satisfied. The result is that we set out an frequency path (the blue line, with only 8 points), and then assume that the final synthesized wave will be equally linear. The red line shows how this is not the case.
At the moment, to solve this we overlap a lot of these windows so the error fades away in the background. Yet a metallic ring remains.
A second solution is the application of an appropriate window (E.g: Kaiser Bessel); which will push the entire error into the endpoints.