[sdiy] Better waveforms of our nature

[Donald Tillman]

> On Oct 16, 2016, at 4:56 PM, Ove Ridé <nitro2k01 at gmail.com> wrote:
> 
> I wondered why the even harmonic waves looked so "weird". Naively, the
> even waveform should be 2f+4f+6f+8f+10f... Ie, just the same sawtooth
> shifted one octave up. Then I realized you had included the
> fundamental as well, which is technically not an even harmonic.

While the fundamental is technically a harmonic, it's in a unique position; it defines the frequency of the waveform, and all the regular harmonics are based upon it.

Interpreting the phrase "even harmonics" as eliminating the fundamental makes a different waveform, at a different frequency, that doesn't contribute to a waveform palette, and doesn't have any especially interesting musical applications.

So leaving the fundamental in is a big win.  And you can always subtract a sine wave from any of the waveforms (odd, even, or all harmonics) if you want to diminish or eliminate the fundamental.  (If the phases are correct.) 

And this is for music, so we're used to it.  It's like a CMaj7 chord without the C root becomes an E minor; both the pitch and the character of the chord are different.

> The waveform may be useful in its own right, but in practice it's nothing
> but a sawtooth wave with a sine suboscillator. Till, how did you
> implement the even harmonic waves in the oscillators that you built?

"In practice" I'm using the waveforms for music.  I do in fact build the evenangle wave from a sine and a double frequency sawtooth, with the phases correct.  But I use a Quadrature Trapezoid VCO core, so I do things differently.  

> I was also wondering about the seemingly arbitrary choice of using
> cosine instead sine for all the mellow waveforms.

It's not arbitrary at all.  We've already got the triangle wave.

While you can certainly use any phase relationship (sines, cosines, any polarity, 30 degrees offset, golden angle offset, etc.) it's good to be somewhat consistent, to choose a waveform shape that we can relate to and talk about, and one that can be easily implemented in hardware.

I highly recommend the Grapher app that comes with Macintoshes to try out lots of phase and harmonic combinations.

  -- Don

--
Don Tillman
Palo Alto, California
don at till.com
http://www.till.com