How do I add odd or even harmonics to a floating point signal?
Do I have to use tanh or sin?
What I'm trying to do is achieve some very simple distortion effects, but I'm having a hard time finding exact references. What I'd like is something similar to what the Culture Vulture does by adding odd and even harmonics in its pentode and triode settings. The float value is a single sample in a sample flow.
What your distortion box does is apply a non-linear transfer function to the signal: output = function(input) or y = f(x). You're just applying the same function to every individual input sample to get the corresponding output sample.
When your input signal is a sine wave, a specific type of distortion is produced called harmonic distortion. All of the new tones created by the distortion are perfect harmonics of the input signal:
So yes, if you want to add odd harmonics, put your signal through an odd-symmetric transfer function like y = tanh(x) or y = x^3.
If you want to add only even harmonics, put your signal through a transfer function that's even symmetric plus an identity function, to keep the original fundamental. Something like y = x + x^4 or y = x + abs(x). The x + keeps the fundamental that would otherwise be destroyed, while the x^4 is even-symmetric and produces only even harmonics (including DC, which you probably want to remove afterwards with a high-pass filter).
Even symmetry:
Transfer function with even symmetry:
Original signal in gray, with distorted signal in blue and spectrum of distorted signal showing only even harmonics and no fundamental:
Odd symmetry:
Transfer function with odd symmetry:
Original signal in gray, with distorted signal in blue and spectrum of distorted signal showing only odd harmonics, including fundamental:
Even symmetry + fundamental:
Transfer function with even symmetry plus identity function:
Original signal in gray, with distorted signal in blue and spectrum of distorted signal showing even harmonics plus fundamental:
This is what people are talking about when they say that a distortion box "adds odd harmonics", but it's not really accurate. The problem is that harmonic distortion only exists for sine wave input. Most people play instruments, not sine waves, so their input signal has multiple sine wave components. In that case, you get intermodulation distortion, not harmonic distortion, and these rules about odd and even harmonics no longer apply. For instance, applying a full-wave rectifier (even symmetry) to the following signals:
So the output spectrum depends strongly on the input signal, not the distortion device, and whenever someone says "our amplifier/effect produces more-musical even-order harmonics", you should take it with a grain of salt.
(There is some truth to the claim that sounds with even harmonics are "more musical" than sounds with only odd harmonics, but these spectra aren't actually being produced here, as explained above, and this claim is only valid in the context of Western scales anyway. Odd-harmonic sounds (square waves, clarinets, etc.) are more consonant on a Bohlen–Pierce musical scale based around the 3:1 ratio instead of the 2:1 octave.)
Another thing to remember is that digital non-linear processes can cause aliasing, which can be badly audible. See Is there such a thing as band-limited non-linear distortion?
What you trying to achieve is called distortion. This techniques used when you want to add some harmonics to given signal. You have 2 basic methods to do this: waveshaping and ring modulation.I'll try to explain first one.
Waveshaping allows you to make distortion via use of specially selected function. One of useful methods is Chebyshev polynomials. They have a very important property when filing through them harmonic signal with unit amplitude (for example, a sine wave), we obtain the same signal, only a few times higher. Frequency multiplier will depend on the order of the polynomial. All polynomials looks like this:
$$ \ y = f(x) =d_0 + d_1x + d_2x^2 + d_3x^3 +… + d_Nx^N; $$
In our case, each element generates a harmonica, and then they all add up. View of each member is determined by the following recurrence relation:
$$ T_{k+1}(x) = 2xT_k(x) – T_{k–1}(x); $$
In it, each member is determined based on the previous one, it all begins with a zero, in our case it is equal to one, and the first, which is equal to x( but you can change it, ofcourse) $$ T_0(x) = 1; $$
$$ T_1(x) = x; $$
Knowing them, you can determine the third and forth: $$ T_2(x) = 2x*x – 1 = 2x^2 – 1; $$
$$ T_3(x) = 2x(2x^2 – 1) – x = 4x^3 – 3x; $$
As you might guess, the second term - the first harmonic, and the third - the second and so on.
Another feature of the Chebyshev polynomials, when through them gives a signal whose amplitude is less than the unit, the output is less saturated sound with harmonics. This allows to create overdrive effect.
After all, your signal is an array of floating points, you can choose any part of your array and apply to them Chebyshev polynomials, which will create additional harmonics. And using $sin$ functions will be good enough for this.
