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GEN14

Stores a polynomial whose coefficients derive from Chebyshevs of the second kind.

Uses Chebyshev coefficients to generate stored polynomial functions which, under waveshaping, can be used to split a sinusoid into harmonic partials having a pre-definable spectrum.

Syntax

f # time size 14 xint xamp h0 h1 h2 ...

Initialization

size -- number of points in the table. Must be a power of 2 or a power-of-2 plus 1 (see f statement). The normal value is power-of-2 plus 1.

xint -- provides the left and right values [-xint, +xint] of the x interval over which the polynomial is to be drawn. These subroutines both call GEN03 to draw their functions; the p5 value here is therefore expanded to a negative-positive p5, p6 pair before GEN03 is actually called. The normal value is 1.

xamp -- amplitude scaling factor of the sinusoid input that is expected to produce the following spectrum.

h0, h1, h2, etc. -- relative strength of partials 0 (DC), 1 (fundamental), 2 ... that will result when a sinusoid of amplitude

xamp * int(size/2)/xint

is waveshaped using this function table. These values thus describe a frequency spectrum associated with a particular factor xamp of the input signal.

📝 Note

  • GEN13 is the function generator normally employed in standard waveshaping. It stores a polynomial whose coefficients derive from the Chebyshev polynomials of the first kind, so that a driving sinusoid of strength xamp will exhibit the specified spectrum at output. Note that the evolution of this spectrum is generally not linear with varying xamp. However, it is bandlimited (the only partials to appear will be those specified at generation time); and the partials will tend to occur and to develop in ascending order (the lower partials dominating at low xamp, and the spectral richness increasing for higher values of xamp). A negative hn value implies a 180 degree phase shift of that partial; the requested full-amplitude spectrum will not be affected by this shift, although the evolution of several of its component partials may be. The pattern +,+,-,-,+,+,... for h0,h1,h2... will minimize the normalization problem for low xamp values (see above), but does not necessarily provide the smoothest pattern of evolution.
  • GEN14 stores a polynomial whose coefficients derive from Chebyshevs of the second kind.

Examples

Here is an example of the GEN14 generator. It uses the file gen14.csd.

Example of the GEN14 generator.
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<CsoundSynthesizer>
<CsOptions>
; Select audio/midi flags here according to platform
-odac  ;;;realtime audio out
;-iadc    ;;;uncomment -iadc if realtime audio input is needed too
; For Non-realtime ouput leave only the line below:
; -o gen14.wav -W ;;; for file output any platform
</CsOptions>
<CsInstruments>

sr = 44100 
ksmps = 32 
nchnls = 2 
0dbfs  = 1 
;after the example from The Csound Book, page 83

instr 1 ;compare results from GEN13 & GEN14

iwshpfn = p6    
inrmfn  = p7    
aswp    linseg  0.01, p3*.5, .49, p3*.5, 0.01           ;index sweep function
aindex  poscil  aswp, p5, 2                             ;sound to waveshape
atable  tablei  aindex, iwshpfn, 1, .5                  ;waveshape index
anrm    tablei  aswp*2, inrmfn, 1                       ;normalization 
aenv    linen   p4, .01, p3, .02                        ;amplitude envelope
asig    = (atable*anrm)*aenv                            ;normalize and impose envelope
asig    dcblock2 asig                                   ;get rid of DC
        outs    asig, asig

endin           
</CsInstruments>
<CsScore>

f 2 0 8192 10 1                 ;sine wave

f 28  0   4097 13  1 1 1 0 .8 0 .5 0 .2         ;waveshaping function: GEN13 - odd harmonics
f 280 0   2049 4   28 1                         ;normalization function for f28
f 29  0   4097 14  1 1 1 0 .8 0 .5 0 .2         ;waveshaping function: GEN14 - same harmonics
f 290 0   2049 4   29 1                         ;normalization function for f29


f 30  0   4097 13  1 1 0 1 0 .6 0 .4 0 .1       ;waveshaping function: GEN13 - even harmonics
f 301 0   2049 4   30 1                         ;normalization function for f30
f 31  0   4097 14  1 1 0 1 0 .6 0 .4 0 .1       ;waveshaping function: GEN13 - even harmonics
f 310 0   2049 4   31 1                         ;normalization function for 31
s
i1 0   3   .7   440 28  280 
i1 4   .   .7    .  29  290
i1 8   .   .7    .  30  301
i1 12  3   .7    .  31  310

e
</CsScore>
</CsoundSynthesizer>

These are the diagrams of the waveforms of the GEN14 routines, as used in the example:

f28 0 4097 13 1 1 1 0 .8 0 .5 0 .2 - waveshaping function: GEN13, odd harmonics
f28 0 4097 13 1 1 1 0 .8 0 .5 0 .2 - waveshaping function: GEN13, odd harmonics
f29 0 4097 14 1 1 1 0 .8 0 .5 0 .2 - waveshaping function: GEN14, the same odd harmonics
f29 0 4097 14 1 1 1 0 .8 0 .5 0 .2 - waveshaping function: GEN14, the same odd harmonics
f30 0 4097 13 1 1 0 1 0 .6 0 .4 0 .1 - waveshaping function: GEN13, even harmonics
f30 0 4097 13 1 1 0 1 0 .6 0 .4 0 .1 - waveshaping function: GEN13, even harmonics
f31 0 4097 14 1 1 0 1 0 .6 0 .4 0 .1 - waveshaping function: GEN14, the same even harmonics
f31 0 4097 14 1 1 0 1 0 .6 0 .4 0 .1 - waveshaping function: GEN14, the same even harmonics

See Also

GEN03, GEN13, and GEN15.

Information about the Chebyshev polynomials on Wikipedia: http://en.wikipedia.org/wiki/Chebyshev_polynomials