How Octaves work

[jmusser]

I have been reading RGs "Distortion 101", and some things make sense, and some don't. If I understand up octaves correctly, you get an up octave when you rectify the wave form above the reference signal. Half, Full, or Bridge rectification, would enhance the 1st octave up effect, but would not make a second and third octave up. Correct? It looks like you would have to reference the second octave up off of the first octave up and so on, to give you the next octave up (however this is done). I would guess that down octaves would work the same, only the signal is rectified under the reference, instead of on top. Correct? As I'm reading along, RG uses the reference of  440 cps, and then if I understand correctly, 220 cps would be one octave down, 110cps would be 2 octaves down, etc. So, I guess this would be the opposite for up octave. In the middle of this, you have crap that gets introduced into the mix called "intermodulation distortion" which gives you mathematical harmonics that have nothing to do with the original signal tone wise. These I have heard above the 10th or 12th fret on up octaves. I do not hear trash with the down octaves, but this may be due to the lower frequencies. I don't know, because the gymnastics of down octaves are not really talked about much. Am I way off base here, or am I in the ball park on what I think I know?

[R.G.]

If I understand up octaves correctly, you get an up octave when you rectify the wave form above the reference signal.

I was simplifying pretty heavily. An octave is perceived when there are two frequencies with a ratio of 2:1. That is, 220Hz is one octave *down* from middle A of 440Hz and it one octave *up* from 110Hz, low A, the A string on your guitar.

Your ear has an affinity for octaves. If there is even a hint of 2:1 frequencies, the ear goes !GaDing! Octave!

Half, Full, or Bridge rectification, would enhance the 1st octave up effect, but would not make a second and third octave up. Correct?

It gets complicated. Full wave rectification gives you an octave up, plus a slew of other stuff. There is indeed 4x the base frequency, 6x, and 8x in there. There is also a bunch of 3x, 5x, 7x, etc. to infinity. The way this really works is tied up in the sound spectrum.

In the beginning, there was the sine wave. This is a single frequency, no other included. It is mathematically pure, and the ear hears it as a single, fairly plain and uninteresting tone. Some flutes come close to sines. Any other repetitive waveform can be created by mixing different amounts of the basic frequency, 2x the basic frequency, 3x, 4x... and so on to infinity.

In math terms, Wform = a*f + b*2f + c*3f + d*4f + e*5f +...
where a, b, c, d, e... are just the amounts in the mix. A sine has a=1, and b,c,d... = 0.

It is possible to make up sounds from non-integer-related (that is, not 1x, 2x, 3x, etc.) frequencies. Your ear hears these as "clangs". Complex clangs sound like beating on a piece of metal.

The set of frequencies and magnitudes (loudnesses) of each frequency make up a spectrum for that sound.

If you take a pure sine and feed it into any nonlinear circuit, it distorts the sine. Any deviation from a perfect sine is heard in your ear as the basic sine and also the deviation, as described by the Fourier series. Your ear does a spectrum analysis and tells the brain "Hey! We got some basic sine at 110Hz, a tiny bit of 220, quite a bit of 330, some 440, ..." although its fuzzy about exactly how much of each is there. So your ear unwraps the neatly tied bundle of frequencies and tells you a crude decomposition.

Now we get to nonlinear processes. Each nonlinear process has a different output spectrum. What I just said was that if you feed a sine into a nonlinear process, it will put out a characteristic set of 1x, 2x, 3x, 4x, 5x... components. Each nonlinear process does a slightly different mix out.

Full wave rectification has the property that it does a=0 (no 1*frequency) and all of b,c,d,... are nonzero; there are components at different levels at all higher multiples of the input frequency. But the second one is the biggest, and if we listen to it, our brain says "Hey! All 2x. Ok, Ok, a little smear of other stuff."

Half wave rectification has a similar property of b is big and c,d,e,... are nonzero, but for halfwave, a is not zero. So the ear hears the frequency you fed in *and* all the rest, including a prominent second harmonic, which your brain hears as an octave.

Every nonlinear process will be different. But there is a great divide. Any nonlinear process that does exactly the same thing to positive and negative polarities of the input wave have all the even harmonics as zero. That is b, d, f, ... are all 0. Any process that does different things to the positive and negative parts of a wave produces b, d, f, ... nonzero, and we hear these as octave related.

One more thing about octaves. Ears are not mathematiclly pure devices. If something is pretty close to 2x, it says - "Hey, close enough. Brain-boss, we got an octave, and it's two different notes". If something is exactly, exactly,exactly 2x then it tends to recognize that as part of the same note, not two different sources.

It looks like you would have to reference the second octave up off of the first octave up and so on, to give you the next octave up (however this is done). I would guess that down octaves would work the same, only the signal is rectified under the reference, instead of on top. Correct?

The reference of an octave is tricky. The brain does the recognizing based on but not mathematically certainly on what is there, and sticks in what *might* be there.
For instance, if you have a tone composed of 220, 440, 660, 880, the ear/brain hears that as 220 plus some harmonics. If you have 220,330,440,550,660,770,880... the ears hears that as 110, inserting the base note that it infers from all the harmonics of 110 being there. This is how cellos make bass notes. The cello body does not couple the bass note of the string to the air, but it does a bang up job on the harmonics. The ear re-inserts the bass note that it infers. Tricky, eh?

As I'm reading along, RG uses the reference of 440 cps, and then if I understand correctly, 220 cps would be one octave down, 110cps would be 2 octaves down, etc.

That is correct.

So, I guess this would be the opposite for up octave.

That is correct. For 440, the first octave up is 880, then 1760 and so on.

In the middle of this, you have crap that gets introduced into the mix called "intermodulation distortion" which gives you mathematical harmonics that have nothing to do with the original signal tone wise.

Correct. Non-harmonically related tones, heard as harshness or clanging.

These I have heard above the 10th or 12th fret on up octaves. I do not hear trash with the down octaves, but this may be due to the lower frequencies. I don't know, because the gymnastics of down octaves are not really talked about much. Am I way off base here, or am I in the ball park on what I think I know?

You're doing good.

[jmusser]

Wow! I was not expecting that explanation at all. Sound, I thought was cut and dried as to what you we're hearing. I would never have thought in my wildest dreams that a lot of what you're hearing is implied and tonal trickery. I have never studied sound perse' , because I never really thought there was anything to study. I feel really ignorant right now. I get what you're saying, but the problem is......I get what your saying :shock: It's like you just told me that my body is composed mostly of yarn and paper clips! it isn't is it?