Hey Akira—you're quite welcome! Glad you're having fun exploring these techniques. (FWIW, when I use this technique, I almost always allow a multiplication factor of 0—being able to use the shaping process as a means of imparting articulation/dynamic control can be very interesting/useful—I can understand some reasons Buchla might have decided against this possibility, but no need to follow his logic exactly. It's more interesting to go a bit further.).Thanks ryangaston
Also interesting reading about the implementation of control over 24 Polynomial coefficients in the 400 and 32 in the 700. Assuming the 259e has a higher order of coefficients involved in the red bank.
Are any resources around for choosing useful
combinations of Polynomial coefficients? I've read that alternating the signs of even and/or odd harmonics avoids some of the phase cancellation. It seems having a single odd number Polynomial is equivalent to basic wavefolding, and neighbouring this with some even-ordered coefficient give some offset similarly to symmetry.
It's certainly interesting how you can get continuous evolution over the range of the index rather than a single folding pattern!
Anyway—to be clear, there's never a limit on the order of polynomials that you could average on any of these instruments, in that it is always possible to define the transfer function on a point-by-point basis. It's just that on the later 400 software and on the 700, there was an alternative data entry method that allowed you to specify the balance of Chebyshev polynomials rather than NEEDING to define everything on a point-by-point basis (it's much faster if you're just looking for a specific combination of relatively low-order overtones!).
So on the 400, for instance, you COULD always plot the 47th Chebyshev polynomial on paper, and then map it out in the waveshaper by entering the correct values for each point in the table. The biggest practical limitation in that respect is about the number of indices/points available in the table itself—effectively limiting the viable resolution for more complex shapes (higher order polynomials). It's possible that the restriction of 24 polynomials for the alternative input method on the 400 and 32 on the 700 was decided because of this, but I'm not sure right off the bat, so don't quote me on that—but my point is that, at some point, e.g. 64 indices isn't going to give you enough resolution to effectively plot out specific shapes without losing some detail.
As for recommended resources...your ears are the best resource, because it's really all subjective! As for some practical thoughts, alternating between positive and negative coefficients CAN help reduce unwanted amplitude buildup as you sum different coefficients. But you can also just sum together your desired coefficients, and then assess the maximum absolute value of all positions in the array in order to extrapolate an amplitude scaling factor. E.G., if you realize the maximum absolute value resulting from your particular combination of polynomials is 3.57, you can divide all values by 3.57 in order to normalize the total range—or just divide the audio output of the process by the same factor. But that's just one tactic!
My personal preferred approach is to forego the direct use of Chebyshev polynomials altogether, and instead focus on different means of generating arbitrary transfer functions. Some of the default tables that Buchla used, for instance, DO conform to precise combinations of low-order Chebyshev polynomials or individual polynomials, but many of them DON'T: they're just arbitrary (but interesting-souding) shapes. It's my understanding that many of the transfer functions in the 259e were constructed more so in this way: not thinking exclusively about the polynomials, but also thinking about the timbral implications of different types of continuity/discontinuity in the shaping table.
Anyway, feel free to share your findings, and by all means, anyone else feel free to chime in! I love this technique and I'm happy that people are picking up on it Image may be NSFW.
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Statistics: Posted by ryangaston — Wed Jan 22, 2025 8:49 pm