Frequency ModulationFrequency ModulationFrequency ModulationFrequency Modulation• related to musical vibratorelated to musical vibrato• VibratoVibrato - Periodic variation of frequency - Periodic variation of frequency• A simple signal:A simple signal:

sample(time) = sin(2 freq(time)time)sample(time) = sin(2 freq(time)time)• with vibratowith vibrato

freq(time) = freqfreq(time) = freq11 + vib + vibwidwidsin(2 vibsin(2 vibrateratetime)time)

• vibvibwidwid = vibrato width = vibrato width• amount of vibratoamount of vibrato

• vibvibraterate = vibrato rate = vibrato rate• frequency of vibratofrequency of vibrato

VibratoVibratoVibratoVibrato• Typical vibrato values:Typical vibrato values:

• vibvibraterate = 5 Hertz = 5 Hertz• normal range:normal range:

1-6 Hertz, with slight acceleration during tone1-6 Hertz, with slight acceleration during tone

• vibvibwidwid

• minimum: 0 (none)minimum: 0 (none)• usual maximum for instruments:usual maximum for instruments:

.01*freq.01*freq11 (1%) (1%)• usual maximum for voices:usual maximum for voices:

.05*freq.05*freq11 (5%) (5%)• maximum for special effects:maximum for special effects:

SR/2 - highest frequency in signal SR/2 - highest frequency in signal

[iii:1][iii:1] Tenor Voice Vibrato Tenor Voice Vibrato[iii:1][iii:1] Tenor Voice Vibrato Tenor Voice Vibrato• Tenor voice has 5 Hertz vibrato rate and Tenor voice has 5 Hertz vibrato rate and

vibrato width of 4.5% vibrato width of 4.5%

Vibrato in CsoundVibrato in CsoundVibrato in CsoundVibrato in Csound• [iii:2][iii:2] synthesized tenor without vibrato synthesized tenor without vibrato• [iii:3] [iii:3] synthesized tenor with vibratosynthesized tenor with vibrato

• [iii:1][iii:1] the real tenor the real tenor

Vibrato in CsoundVibrato in CsoundVibrato in CsoundVibrato in Csound

• orchestra:orchestra:avibavib oscili ivibwidth, ivibrate, 1oscili ivibwidth, ivibrate, 1 ; modulator; modulator

asigasig oscili iamp, ifreqoscili iamp, ifreq+avib+avib, 2, 2 ; carrier; carrier

outout asig asig ; output; output

• score:score:f1 0 16385 -10 1f1 0 16385 -10 1 ; for modulator; for modulator

f2 0 16385 -10 1 .9 .8 whatever …f2 0 16385 -10 1 .9 .8 whatever … ; for carrier; for carrier

[iii:4][iii:4] Vibrato Rate Vibrato Rate[iii:4][iii:4] Vibrato Rate Vibrato Rate• vibvibraterate = 0 -----> 10 Hertz (over 10 seconds) = 0 -----> 10 Hertz (over 10 seconds)

• vibvibwidwid = .01 * freq = .01 * freq11 (1%) (1%)

slow vibrato fast vibrato

Vibrato RateVibrato RateVibrato RateVibrato Rate• orchestra:orchestra:;fmbasic.orc - use with fmbasic.sco

instr 1 ; increase vibrato rateidur = p3 ; (10)iamp = p4ifreq = p5ivibwidth = .01*ifreqiwave1 = 1

kvibratelinseg 0, idur, 10aenv linseg 0, .1, 1, idur-.3, 1, .2, 0avib oscili ivibwidth, kvibrate, iwave1asig oscili iamp, ifreq+avib, iwave1

out asig * aenvendin

Vibrato RateVibrato RateVibrato RateVibrato Rate

• score:score:

; fmbasic.sco – use with fmbasic.orc; fmbasic.sco – use with fmbasic.orcf1 0 16385 -10 1f1 0 16385 -10 1 ; sine; sine

; start dur amp freq; start dur amp freq

i1 0 10 10000 261.6 i1 0 10 10000 261.6 ; Middle C; Middle C

[iii:5][iii:5] Vibrato Amount Vibrato Amount[iii:5][iii:5] Vibrato Amount Vibrato Amount• vibvibraterate = 5 Hertz = 5 Hertz

• vibvibwidwid = 0 -----> .05 * freq = 0 -----> .05 * freq11 (0-5% over 0:10) (0-5% over 0:10)

small vibrato large vibrato

Vibrato AmountVibrato AmountVibrato AmountVibrato Amount• orchestra:orchestra:;fmbasic.orc - use with fmbasic.scoinstr 2 ; increase vibrato widthidur = p3 ; (10)iamp = p4ifreq = p5ivibrate= 5iwave1 = 1

kvibwidth linseg 0, idur, .05 * ifreqaenv linseg 0, .1, 1, idur-.3, 1, .2, 0avib oscili kvibwidth, ivibrate, iwave1asig oscili iamp, ifreq+avib, iwave1

out asig * aenvendin

Vibrato -----> FMVibrato -----> FMVibrato -----> FMVibrato -----> FM

• A sine wave with vibrato becomes a full A sine wave with vibrato becomes a full spectrum when vibspectrum when vibraterate is in the audio is in the audio

range (above 20 Hz), especially as vibrange (above 20 Hz), especially as vibraterate

approaches freqapproaches freq11..

• Since it is no longer vibrato, we use the Since it is no longer vibrato, we use the term term modulation frequencymodulation frequency instead of instead of vibrato rate.vibrato rate.

• When vibWhen vibraterate is above 20 Hz: is above 20 Hz:

freqfreqmodmod = vib = vibraterate

Vibrato -----> FMVibrato -----> FMVibrato -----> FMVibrato -----> FM• With FM, we may not get the frequency With FM, we may not get the frequency

out that we put in.out that we put in.• We call the base frequency of the outer sine We call the base frequency of the outer sine

wave the carrier frequency freqwave the carrier frequency freqcarcar::

sample(time) = sin(2 freq(time)time)sample(time) = sin(2 freq(time)time)

with:with:

freq(time) = freqfreq(time) = freqcarcar + vib + vibwidwidsin(2 freqsin(2 freqmodmodtime)time)

Vibrato -----> FMVibrato -----> FMVibrato -----> FMVibrato -----> FM• FM uses a FM uses a modulation indexmodulation index as well as as well as

the vibrato width to describe the the vibrato width to describe the amount of modulation. The relationship amount of modulation. The relationship between them is:between them is:

vibvibwidwid = Index * freq = Index * freqmodmod

or:or:

Index = vibIndex = vibwidwid / freq / freqmodmod

• Typical values for modulation index:Typical values for modulation index:0 <= Index <= 10 0 <= Index <= 10

[iii:6][iii:6] Vibrato -----> FM Vibrato -----> FM[iii:6][iii:6] Vibrato -----> FM Vibrato -----> FM• freqfreqmodmod = 1 ---> 6 Hz ---> 261.6 Hz (= vib = 1 ---> 6 Hz ---> 261.6 Hz (= vibraterate))

(vib) (FM)(vib) (FM)• Index = .02Index = .02

• (vib(vibwidwid = .02 * freq = .02 * freqmodmod))

• When the frequency of When the frequency of the modulator reaches the modulator reaches 6 Hertz (at 5 seconds), 6 Hertz (at 5 seconds), the effect changes the effect changes from vibrato to FM. from vibrato to FM.

Vibrato -----> FMVibrato -----> FMVibrato -----> FMVibrato -----> FM• The amplitude of the first harmonic The amplitude of the first harmonic

remains the same ...remains the same ...

Harmonic 1 Amplitude

Vibrato -----> FMVibrato -----> FMVibrato -----> FMVibrato -----> FM• … … but the amplitudes of the other but the amplitudes of the other

harmonics change.harmonics change.

Amplitudes of the Other Harmonics

Vibrato -----> FMVibrato -----> FMVibrato -----> FMVibrato -----> FM• The frequency changes of all the The frequency changes of all the

harmonics get much faster during FM.harmonics get much faster during FM.

Harmonic 1 Frequency

Vibrato -----> FMVibrato -----> FMVibrato -----> FMVibrato -----> FM• The frequency changes of all the The frequency changes of all the

harmonics get much faster during FM.harmonics get much faster during FM.

Harmonic 2 Frequency

Vibrato -----> FMVibrato -----> FMVibrato -----> FMVibrato -----> FM• orchestra:orchestra:;fmbasic.orc - use with fmbasic.scoinstr 3 ; vibrato --> fmidur = p3 ; (10)iamp = p4ifreq = p5ivibwidth = .02 * ifreq ; (~5.2 Hertz)iwave1 = 1

kvibratelinseg 1, idur * .5, 6, idur * .5, ifreqaenv linseg 0, .1, 1, idur-.3, 1, .2, 0amod oscili ivibwidth, kvibrate, iwave1acar oscili iamp, ifreq+amod, iwave1

out acar * aenvendin

Modulation IndexModulation IndexModulation IndexModulation Index

• Index = 0 (no modulation)Index = 0 (no modulation)• Index = .01 - .05 (vibrato range)Index = .01 - .05 (vibrato range)

• As modulation index increases, the As modulation index increases, the spectrum bandwidth increases (and spectrum bandwidth increases (and gets brighter).gets brighter).

[iii:7][iii:7] Modulation Index Modulation Index[iii:7][iii:7] Modulation Index Modulation Index• freqfreqmodmod = 261.6 Hertz = 261.6 Hertz

• index = 0 -----> 20 (over 0:10) index = 0 -----> 20 (over 0:10)

Harmonic Amplitude vs. Time

Bessel Function,

orders 0 - 3

Modulation IndexModulation IndexModulation IndexModulation Index• freqfreqmodmod = 261.6 Hertz = 261.6 Hertz

• index = 0 -----> 20 (over 10 seconds) index = 0 -----> 20 (over 10 seconds)

Frequency vs. Time

Modulation IndexModulation IndexModulation IndexModulation Index• orchestra:orchestra:;fmbasic.orc - use with fmbasic.scoinstr 4 ; vibrato --> fmidur = p3 ; (10)iamp = p4ifreq = p5imodfr = ifreqicarfr = ifreqiwave1 = 1

kindex linseg 0, idur, 20aenv linseg 0, .1, 1, idur-.3, 1, .2, 0amod oscili kindex*imodfr, imodfr, iwave1acar oscili iamp, icarfreq+amod, iwave1

out acar * aenvendin

Frequency Frequency Modulation Modulation

Block DiagramBlock Diagram

Frequency Frequency Modulation Modulation

Block DiagramBlock Diagram

Add NoiseAdd NoiseAdd NoiseAdd Noise

• Random Noise VariationRandom Noise Variationanoiseanoise randirandi .3, 15, giseed.3, 15, giseed

giseedgiseed == frac(giseed*105.947)frac(giseed*105.947)

anoiseanoise == anoise + 1anoise + 1

avrateavrate linseglinseg 1, idur, 101, idur, 10

avrateavrate == avrate avrate * anoise* anoise

avibwidavibwid == ivibwid ivibwid * anoise* anoise

• Try different combinations of the parameters to Try different combinations of the parameters to find one that really suits your soundfind one that really suits your sound