posted 10 March 2002 16:34            

tchebyshevoscillator.kym

The advantage of a Tchebyshev oscillator is that you can directly specify the amplitudes of the harmonics that you desire (up to 18 in this Sound). The script at the head of this Sound block chain generates polynomial coefficients corresponding to these amplitudes, normalizes the whole to a max amplitude of 1, and then feeds these coefficients to a PolynomialWaveshaper.

You have to excite a Tchebyshev wavetable with a full amplitude sinewave oscillator. After wavetable conversion of the sinewave, you can attenuate the result to your desired amplitude.

This is a lot like additive synthesis, but the addition is performed up front in the script, and you have no control over the individual harmonic phases -- only amplitudes. But instead of using a whole bank of oscillators, you can get by with only one -- the exciter sinewave at your chosen frequency. The polynomial wavetable makes that one oscillator look like a whole bank of oscillators. So this sound is very cheap on DSP resources.

The Sound attached here asks you to place your amplitudes in an array. Instead of raw numbers you could use HotValues from VCP sliders.

Enjoy!

- DM

[Oh yeah... just because you can specify so many harmonics (ultimately as many as you like, just make the array larger and add the new coefficients to the wavetable), you must beware of aliasing at higher frequencies. The highest permissible harmonic at any given frequency is no higher than the half-samplerate divided by the frequency. So at 4 KHz with 48 KHz sampling, you should use no more than 6 harmonics if you want to avoid aliasing.]

[There was a mistake in the original submission, now corrected. I am working on trying to create an anti-aliased version, but so far Smalltalk a la Capybara is thwarting my efforts...]

posted 11 March 2002 01:17            

If you put an oscilloscope on the output of this Sound, you will NOT see a square wave, despite its spectrum being identical to a square wave. It sounds like a square wave, but due to the odd phase relationships given to the harmonics by this technique, what you actually see is a kind of peaked waveform above and below the zero amplitude line. Except for this unusual phasing between harmonics, it is otherwise identical to a square wave with the example amplitudes provided.

But as so often happens with Kyma, one thing leads to another, and before you know it, I have discovered even more gold in that phonebook. Even after more than a year of living with Kyma, I am still discovering new things about it.

I happened onto the subject of granular frequency scaling, and played with their trombone samples. The results are quite astonishing. But then I took a sample from a French Horn that I recorded last year with a friend playing scales, and applied this same technique. The results are quite amazing and chilling. I have never heard a sampler do this well!

If anyone needs a French Horn, let me know and I'll clear it with my friend Ray. He plays horn for the Cleveland Auxiliary Symphony -- a kind of amateur extension of the Cleveland Orchestra.