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So most synthesizers provide us with four standard wave forms:
My question is twofold:
Why is a wave with a fundamental and all even harmonics not included in this canonical set of waves, and what would one sound like?
On most synthesizers with two oscillators this is easily achieved by setting one oscillator to a sine wave (to produce the first harmonic) and setting the other oscillator to a sawtooth wave and tuning it one octave higher (to produce the even harmonics).
As mentioned in the comments, hearing just the second oscillator will simply sound like a saw wave one octave higher, but adding any of the odd harmonics will tell your brain that the saw is in fact the even harmonics of a sound one octave lower.
You can use this as a trick to play sounds that appear to be lower than what the synthesizer is capable of producing. E.g. if you tune one oscillator to the lowest setting, and then tune the second oscillator one octave and a major third higher, you get the ratio 1/2.5, which your brain then interprets as the ration 2/5, and that makes it sound like the even and multiple-of-five harmonics of a note one octave lower than the lowest setting. This is especially effective if the synth has a ring-modulator; e.g. try it on the Korg MS-20 with oscillator one set to 32' and oscillator two set to 16' and tuned up from C to E.
You can also create a sound where you have all harmonics, but can mix the balance of odd and even harmonics, by setting one oscillator to a square wave (to produce the odd harmonics) and a second oscillator to a sawtooth wave one octave higher (to produce the even harmonics). The volume balance of the two oscillators then controls the odd/even harmonic content of the sound. This is also what happens when you switch on the sub-oscillator on a synth like the Roland Juno-106.
There are even synths that use stacked square waves, each one octave higher than the previous one, to achieve a sort of additive synthesis. E.g. on the Roland SH-7, there is a section in oscillator one where you can set the level of five square waves. These five sliders then effectively control these harmonics:
32' -> 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41... 16' -> 2 6 10 14 18 22 26 30 34 38 8' -> 4 12 20 28 36 4' -> 8 24 40 2' -> 16
With what you have it's actually easy to generate what you're looking for. I think what you want is a periodic signal (wave) with a given fundamental frequency f0 and harmonics at 2xf0, 4xf0, etc. This is equivalent to generating a wave with odd and even harmonics (e.g., a sawtooth wave) at twice the desired fundamental frequency, i.e. at 2xf0, and adding a pure sinusoid at frequency f0. This will give you frequency components at f0 and at its even multiples.
One simple example of a wave with only even harmonics (apart from the fundamental) is a half-wave rectified sinusoid:
This is quite easy to see: it is equal to the sum of the fundamental sinusoid and a full-wave rectified sine, which has twice the fundamental frequency.
The square wave has the first harmonic with an amplitude of one, the third harmonic with an amplitude of 1/3, the fifth harmonic with an amplitude of 1/5 and so on... If you in a similar way take only the even harmonics, that is: the second with amplitude 1/2, the fourth with amplitude 1/4 and so on... You get a triangular wave with double frequency of the fundamental. If you, in a similar way, takes all the harmonics, you get a triangular wave with the same frequency as the fundamental.
