posted 24 February 2003 02:20            

anharmonia.kym

The original sound produces a synthetic spectrum based on 16 partials of a sawtooth waveform. The CloudBank Resynthesis then creates Grains in each of these 16 spectral tracks, each grain having an amplitude envelope of a Gaussian multiplied by !Cycles of an internal sawtooth waveform filtered to provide only the fundamental and its octave. The net effect is to produce a grain spectrum that averages to a pair of Gaussian spectral ranges centered on the fundamental and its octave with half amplitude. One pair of these is placed on top of each of the synthetic spectral lines with amplitudes provided by the synthetic spectrum.

As is, the CloudBank Resynthesis Sound produces very emotive string-like sounds, especially in the lower registers.

Okay... now I have always been fascinated by anharmonic spectra as produced by metal plates, circular membranes, and clamped circular plates. (metal bars too!). I first got interested years ago after reading about these interesting spectra in an article by Wendy Carlos back in the days when she was experimenting with additive synthesis and the Indonesian Gamelan sounds.

I took the GrainCloud-Resynthesis sound as a starting point, added a low pass filter to the output to rid myself of excessive high harmonics, and then implemented a number of synthetic spectra.

Here you will find 1/N amplitude sequences for frequencies corresponding to circular membranes, circular plates, the zeros of the Bessel Functions J0 and J1 (just for the heck of it!), and one last Sound which implements the spectrum of a rectangular plate.

That rectangular plate allows you to control the width (!A) and height (!B) of the plate (ranges 0-10), as well as the amplitudes for each of the first 9 normal modes, labeled !A10, !A01, !A20, !A11, !A02, !A30, !A21, !A12, and !A03. The first index labels the wavenumber on the !A dimension, while the second index labels the wavenumber on the !B dimension.

A physical model of plates and membranes would produce amplitude sequences of these normal modes that satisfy particular boundary conditions -- such as where you are striking the membrane or plate. Here, I cheaped out and simply provide an ad-hoc amplitude sequence.

Playing around with the rectangular membrane example allows you to generate many interesting anharmonic spectra, as well as some nearly harmonic anharmonic spectra.

In two dimensional systems such as these, the number of harmonics between any two frequencies increases as the square of the frequency. We cheap out by using only 9 of the lowest normal modes, but there are clearly many more remaining than you find in a one dimensional system like a string or a bar. Feel free to implement additional harmonics (for increased realism?).

All of these Sounds resynthesize the anharmonic spectra with the Gaussian enveloped filtered sawtooth, so despite the anharmonic spectra, a certain amount of harmony is reintroduced by the choice of the resynthesis waveform. Try other waveforms if you dare.

I don't get to play with my Kyma as often as I would like, and after more than 2 years with it, I'm still just scratching the surface.

Cheers,

- DM

[Woops! The lack of precedence in the syntax of Smalltalk gave rise to a mistake in the computation of the rectangular membrane frequencies. This is now fixed in the above attachment. Extra parentheses are needed to overcome the lack of syntactic precedence... ]