Am 22.09.2016 um 12:18 schrieb André Michelle:

How do I detect discontinuities? It is easy to see when printed visually but I do not see how I can approach this with code. Do I need the ‘complete’ function at once and check or can I do it in runtime for each sample. I think so since you suggest that I can jump around within the function without alias? Because that would sound like a solution I wanted to have from the very beginning.

You "detect" them they way you construct them.
For instance you have a phase ramp, say from -.5 to .5, you know that the discontinuity happens when your phase + frequency_step is > 0.5, and it happens in that fraction of a sample when the phase would be 0.5, so it happens at (phase + frequency_step - 0.5)/frequency_step fraction

I do not quite get this: C(1). Does it mean I have C(n) values of the function where C(1) is the second value?

It's about differentiability and smoothness
"The function f is said to be of differentiability class Ck if the derivatives f′, f′′, ..., f(k) exist and are continuous"
But you can ignore this for now...

What frequency does the integrated sync function has?

It has the same bandlimit as your waveform should have

What is a 'fraction of a sample'?
The jump in your sawtooth waveform happens within a fraction of a sample time, as explained above When you read your wavetables you also read them at fractions of a sample (and interpolate to get the value at this fraction of a sample), all your signal exists also between samples.

I am missing to many aspects of your suggestion. Any hints where to learn about this would be appreciated.

I also have a question: what is the benefit of having a synthesizer in a webbrowser?

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