GEN12
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## GEN12

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

## Description

This 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 opcode. It uses the file gen12.csd.

Example 989. Example of the GEN12 opcode.

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

    <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).

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