GEN12

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

Syntax

f # time size 12 xint

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

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.

Examples

Here is an example of the GEN12 generator. It uses the file gen12.csd.

<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 gen12.wav -W ;;; for file output any platform
</CsOptions>
<CsInstruments>
;example from the Csound Book, page 87
sr = 44100
ksmps = 32
nchnls = 2
0dbfs  = 1

instr 1

idur    =       p3
iamp    =       p4
icarfrq =       p5
imodfrq =       p6          
aenv    expseg  .01, idur*.1, iamp, idur*.8, iamp*.75, idur*.1, .01
i1      =       p7*imodfrq                      ;p7=modulation index start
i2      =       p8*imodfrq                      ;p8=modulation index end
adev    line    i1, idur, i2                    ;modulation frequency
aindex  line    p7, idur, p8                    ;modulation index

ar      linseg  1, .1, p9, p3-.2, p10, .1, 1    ; r value envelope: p9-p10 =exp. partial strength parameter start and end
amp1    =       (aindex*(ar+(1/ar)))/2
afmod   oscili  amp1, imodfrq, 1                ;FM modulator (sine)
atab    =       (aindex*(ar-(1/ar)))/2          ;index to table
alook   tablei  atab, 37                        ;table lookup to GEN12
aamod   oscili  atab, adev, 2                   ;am modulator (cosine)
aamod   =       (exp(alook+aamod))*aenv
acar    oscili  aamod, afmod+icarfrq, 1         ;AFM (carrier)
asig    balance acar, aenv
        outs    asig, asig

endin       

</CsInstruments>
<CsScore>
f 1     0   8192    10  1               
f 2     0   8192    9   1 1 90          
f37 0   1024    -12 40          ;Bessel function-defined from 0 to 40

i 1  0  2  .2  800  800    1    6   .1   2
i 1  +  .  .   1900 147    8    1    4   .2
i 1  .  .  .   1100 380    2    9   .5   2
i 1  .  10 .   100  100    11   3   .2   5
s
i 1  0  1 .1   200 100     1    6   .1   2
i 1  +  .  <   <    <      <    <   <    <
i 1  +  .  .   <    <      <    <   <    <
i 1  +  .  .   <    <      <    <   <    <
i 1  +  .  .   <    <      <    <   <    <
i 1  +  .  .   <    <      <    <   <    <
i 1  +  .  .   <    <      <    <   <    <
i 1  +  10 .2  800  800    9    1   .9   6
s
i 1  0  11 .25 50   51     1    6   .1   2
i 1  1  9  .05 700  401    1    6   .1   2
i 1  2  8  .   900  147    8    1    4   .2
i 1  3  7  .   1100 381    2    9   .5   2
i 1  4  6  .   200  102    11   3   .2   5
i 1  5  6  .   800  803    9    1   .9   6
e
</CsScore>
</CsoundSynthesizer>

This is the diagram of the waveform of the GEN12 routines, as used in the example:

f 37 0 1024 -12 40 - Bessel function-defined from 0 to 40

Credits

Example is, with minor modifications, taken from The Csound Book (page 87).