I was wondering if there is some way to generate 2nd and 3th order (even?) harmonics with a micromodular... I was reading some info on emulations of analogue tube distortion and tape-saturation and it seems an important part of these effects are generating these harmonics and mixing them with the original audio... Anyone?

Frequency smear can be quite important too (in tape saturation), you might be able to get that by lowpass filtering a signal poinlessly, eg. 20 Khz lowpass, high Q, but the nords filters might be too good. Theres probably a much better way though, and I'd like to hear it 'cos I'm trying to do this now, in a plugin.

The NM could certainly do with some more elaborate and subtle distortion and waveshaping modules, like e.g. a harmonic waveshaper Kyma-style.

There's a good chapter on harmonic waveshaping in Hal Chamberlin's book (Musical Applications of Microprocessors, second edition page 473-480).

Sadly my library only has the 1st eg., but I found a good explanation online (about 1/2 way down the page).

Nonlinear Synthesis: Frequency Modulation & Waveshaping

A - peak amplitude

fc - carrier frequency in hertz

fm - modulating frequency in hertz

delta_f - frequency deviation in hertz

I - index of modulation = delta_f/fm

The spectrum of a frequency modulated waveform consists of many sinusoidal components

Carrier frequency at fc with amplitude J0(I)

Sidebands at fc + kfm (k = ..., -3, -2, -1, 1, 2, 3) with amplitude Jk(I).

Bessel functions of the first kind of order n are solutions to Bessel's differential equation: x2y'' + xy' + (x2 - n2)y = 0

Solution: Jn(x) = SUMk=0..inf[((-1)k(x/2)n+2k)/(k!(n+k)!)

Also: Jn+1(x) = (2n/x)Jn(x) - Jn-1(x)

J-k(I) = -Jk(I) for odd k

J-k(I) = Jk(I) for even k

Aliasing can occur since fc - kfm can be negative and fc + kfm can be greater than R/2.

Amp at 440 Hz = J0(2) * 10000; Amp at 880 Hz = J1(2) * 10000; Amp at 0 Hz = J-1(2) * 10000; etc.

; p4 = amplitude

; p5 = carrier frequency

; p6 = modulating frequency

; p7 = index of modulation

f1 0 2048 10 1

i1 0 2 20000 440 440 3

e

H = fm/fc

H = 1/2 (frequency component appears between 0 Hz and carrier frequency: fc-fm)

H = 2 (odd harmonics appear only)

H = 1/sqrt(2) (inharmonic sounds like bells)

Horn (H=1, I = 5)

Bassoon (H=1/5, I = 1.5)

Clarinet (H=2/3, I = 4 to 6)

Gong (H = sqrt(2), I = 10)

Waveshaping is a technique of distortion which creates complex spectra from simple tones by explicitly altering the shape of the original waveform Eavh waveshaping instrument is characterized by its transfer function

Let the transfer function of a waveshaping instrument be F(x), defined for x between -1 and 1. The new waveform is given by solving F(x) given x is an output from a unit-amplitude sinusoidal waveform. The process of waveshaping is equivalent to composition of functions.

If the input of a waveshaper is a sinusoid of unit amplitude, then the amplitudes of the harmonics of the output waveform can be determined based on Pascal's triangle. (SEE HANDOUT) Given a transfer function F(x) = cnxn + cn-1xn-1 + ... + c1x + c0 where x is the output of a unit-amplitude sinusoid, the output waveform from F(x) has harmonics hj(j=0,1,...,n) whose amplitudes are given by the formulas:

h0 = SUMk=0..n(ckH[k,0]/2k)

hj = SUMk=0..n(ckH[k,0]/2k-1) for j=1,...,n where H[k,j] is the table entry in Pascal's triangle (HANDOUT) at row k, column j (counting from 0)

F(x) = 4x5 + 6x4 - (11/3)x3 + (5/4)x - (7/4)

h0 = 1/2 (DC or average power), h1 = 1, h2 = 3, h3 = 1/3, h4 = 3/4, h5 = ¼

In order to determine the actual transfer function needed to generate a desired output waveform, a technique called spectral matching is used. Chebyshev polynomials are used to reconstruct the original transfer function.

The Chebyshev polynomial of the first kind of order k is defined by Tk(x) = cos(k arccos x)

Also, Tn+1(x) = 2xTn(x) - Tn-1(x)

Thus, T0(x) = 1, T1(x) = x, T2(x) = 2x2 - 1, etc. (SEE HANDOUT)

If hj is the amplitude of the jth harmonic of an audio waveform (with n harmonics) output by a waveshaper, then the transfer function required to generate this waveform is F(x) = h0T0(x) + h1T1(x) + ... + hnTn(x).

h0 = 1/2, h1 = 1, h2 = 3, h3 = 1/3, h4 = 3/4, h5 = 1/4

F(x) = 4x5 + 6x4 - (11/3)x3 + (5/4)x - (7/4)

If a transfer function is an even function, then the output will consist of only even harmonics. An odd transfer function yields an output with odd harmonics only. The more extreme the change in slope is for a transfer function, the greater the number and intensity of harmonics that will be in the output waveform. The highest harmonic of the output waveform must be nf0, for an input unit-amplitude sinusoid, where f0 is the frequency of the sinusoid and n is the highest degree of F(x), the transfer function.

ar table andx, ifn

Extracts sample at index andx from function table number ifn.

GEN09 - sum of sinusoids, specifying each sinusoid's partial (harmonic) number, amplitude (strength) and phase (in degrees)

f # time size 9 pna stra phsa pnb strb phsb ...

GEN13 - generate transfer function F(x) defined from -1 to 1 in wavetable given amplitude of desired audio spectrum

f # time size 13 1 1 h0 h1 h2 h3 ... hn

GEN03 - generate transfer function F(x) defined from -1 to 1 in wavetable given coefficients of transfer function equation

f # time size 3 -1 1 c0 c1 c2 ... cn

; p4 = amplitude of note

; p5 = fundamental frequency

f1 0 256 9 1 1 90 ; cosine wave (sine offset by 90 degrees)

f2 0 512 13 1 1 0.5 1 3 0.3333 0.75 0.25 ; transfer function using harmonics

OR

f2 0 512 3 -1 1 -1.75 1.25 0 -3.6667 6 4 ; transfer function using coefficients

i1 0 5 20000 440

I'm afriad I could only just follow the stuff on the webpage, I found an easier technique though:

The hp is to remove the DC offset and the lp filter is for smearing (aparently). Should be doable on the nord, but I'm at work so I can't try it. Should work with exp/log waveshaper, but atan might be better, and really you want:

Like this? Keys or external, but no knobs on the external here... Got playing with the fm n stuff...