Description

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

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 a simple example of the GEN14 routine. It uses the file gen14.csd. It creates a function which, under waveshaping, will split a sinusoid into 3 odd-harmonic partials of relative strength 5:3:1. Here is its diagram:

Diagram of the waveform generated by GEN14.

<CsoundSynthesizer>
<CsOptions>
; Select audio/midi flags here according to platform
; Audio out   Audio in
-odac           -iadc    ;;;RT audio I/O
; For Non-realtime ouput leave only the line below:
; -o gen14.wav -W ;;; for file output any platform
</CsOptions>
<CsInstruments>

; Initialize the global variables.
sr = 44100
kr = 4410
ksmps = 10
nchnls = 1

; Instrument #1.
instr 1
  ; Create an index over the length of our entire note.
  kcps init 1/p3
  kndx phasor kcps

  ; Read Table #1 with our index.
  ifn = 1
  ixmode = 1
  kval table kndx, ifn, ixmode

  ; Generate a sine waveform, use our Table #1 value to
  ; vary its frequency by 100 Hz from its base frequency.
  ibasefreq = 440
  kfreq = kval * 100
  a1 oscil 20000, ibasefreq + kfreq, 2
  out a1
endin


</CsInstruments>
<CsScore>

; Table #1: a polynomial function (using GEN14).
f 1 0 1025 14 1 1 0 5 0 3 0 1
; Table #2, a sine wave.
f 2 0 16384 10 1

; Play Instrument #1 for 2 seconds.
i 1 0 2
e


</CsScore>
</CsoundSynthesizer>

See Also

GEN03, GEN13, and GEN15.

Credits

Example written by Kevin Conder