GEN12 — Generates the log of a modified Bessel function of the second kind.


This generates the log of a modified Bessel function of the second kind, order 0, suitable for use in amplitude-modulated FM.


f # time size 12 xint


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 -- specifies the x interval [0 to +xint] over which the function is defined.

[Note] Note

  • This subroutine draws the natural log of a modified Bessel function of the second kind, order 0 (commonly written as I subscript 0), over the x-interval requested. The call should have rescaling inhibited.

  • The function is useful as an amplitude scaling factor in cycle-synchronous amplitude-modulated FM. (See Palamin & Palamin, J. Audio Eng. Soc., 36/9, Sept. 1988, pp.671-684.) The algorithm is interesting because it permits the normally symmetric FM spectrum to be made asymmetric around a frequency other than the carrier, and is thereby useful for formant positioning. By using a table lookup index of I(r - 1/r), where I is the FM modulation index and r is an exponential parameter affecting partial strengths, the Palamin algorithm becomes relatively efficient, requiring only oscil's, table lookups, and a single exp call.


Here is a simple example of the GEN12 routine. It uses the file gen12.csd. It generates the function ln(I0(x)) from 0 to 20. Here is its diagram:

Diagram of the waveform generated by GEN12.

Diagram of the waveform generated by GEN12.

Example 562. A simple example of the GEN12 routine.

See the sections Real-time Audio and Command Line Flags for more information on using command line flags.

; 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 gen12.wav -W ;;; for file output any platform

; 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
  kamp tablei kndx, ifn, ixmode

  ; Create a sine wave, use the Table #1 values to control
  ; the amplitude. This creates a sound with a long attack.
  a1 oscil kamp*30000, 440, 2
  out a1


; Table #1: a modified Bessel function (using GEN12).
f 1 0 2049 12 20
; Table #2, a sine wave.
f 2 0 16384 10 1

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



Example written by Kevin Conder