Just Use a Saw

I discovered something profoundly usesless today: When you combine a square wave at amplitude 2x, pitch y with a sawtooth at amplitude x, pitch 2y, the result is a sawtooth at amplitude 2x, pitch y, when the two waves are in phase with each other.

square(2a,p) + saw(a,2p) = saw(2a,p)

I can prove it using the harmonic series, but first I should warn that I am not a good mathematician, and there may be errors here, and the above proposition may be false. Please let me know if you see any errors; children will starve if any flaws go unnoticed.

A square wave (http://en.wikipedia.org/wiki/Square_wave) contains every odd-integer harmonic n at amplitude a/n where a is the amplitude of the fundamental (the first harmonic). (Please pardon the infinitely recursive definition... the fundamental is at volume a.)

A sawtooth wave (http://en.wikipedia.org/wiki/Sawtooth_wave) contains every integer harmonic n at amplitude a/n where a is the amplitude of the fundamental.

Now, to sum the amplitudes of the harmonics of the 2 waves, we can express one wave's harmonics in terms of the other wave. Since the square wave in this proposition is lower in frequency and double the amplitude of the sawtooth, I will call its frequency the fundamental (first harmonic) for this example. (After all, it _is_ the fundamental if both are played together anyway, but that's irrelevant to this proof.)

Since the frequency of the sawtooth wave is double that of the square wave, its nth harmonic is the (2n)th harmonic of the square wave. Therefore, it will provide all the even-numbered harmonics. Since its amplitude is half that of the square wave, its fundamental (the 2nd harmonic of the square wave) will sound at 1/2 the amplitude of the square wave, its 2nd harmonic (the 4th harmonic of the quare wave) will sound at 1/2 _that_ amplitude, its 3rd harmonic (the 6th harmonic of the square wave) will sound at 1/3 its amplitude (1/6 that of the square wave), etc.

So, the square wave provides all odd-numbered harmonics n at amplitude 1/n, and the sawtooth provides all even-numbered harmonics _of our resulting wave_ at 1/n. Therefore, by combining a square wave and a sawtooth at twice the frequency and half the amplitude, we have all integer harmonics at volume 1/n, which is a sawtooth at the pitch and amplitude of the contributing square wave.

I guess there is one application to this: On a synthesizer, using a sawtooth oscillator combined with a square oscillator one octave lower at the same volume is useless. One octave lower because that's half the frequency, and the same volume because that's twice the ampltidue and frequency times amplitdue equals volume. So the moral of the story: Just use a saw!