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------=_NextPart_01C9565C.F8CD64C0 Content-Location: file:///C:/3A815E52/Synthesize_v12_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C9565C.F8CD64C0 Content-Location: file:///C:/3A815E52/Synthesize_v12_files/slide0001.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Digital Music: Musical Instrument Synthesis
12/4/2008
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Music and Engineering:
Musical Instrument Synthesis
Tim Hoerning
Fall 2008
(last edited on 11/19/08)
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quoted-printable Content-Type: text/html; charset="us-ascii" Digital Music: Musical Instrument Synthesis
12/4/2008
DRAFT
2
Outline
•<= /span>Early Electronic & Electro-mechanical Instruments
–<= /span>Hammond Organ, Mellotron, Theremin, etc
•<= /span>Fundamentals (Building Blocks) <= /div>
•<= /span>Synthesis techniques
–<= /span>Additive Synthesis
–<= /span>Subtractive Synthesis
–<= /span>Distortion Synthesis
–<= /span>Synthesis from analysis
–<= /span>Granular Synthesis
–<= /span>Physical Modeling
•<= /span>Representations for Musicians
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Electromechanical Instrument
•<= /span>Several Famous instrument were created with using = coils similar to electric guitar pickups and a ton= e generators
–<= /span>The Fender Rhodes electric piano used a piano like action to strike metal tines (small bars) to generate a pitch
–<= /span>The Hohner Clavinet used a tangent connected direc= tly to a key to strike a string= which was generated a pitch for a pickup.
•<= /span>Musical Example: Superstition by Stevie Wonder = ;
–<= /span>The Hammond B3 used a rotating varying reluctance = tone wheel positioned above a pickup to generate the smooth organ sounds.
–<= /span>The Mellotron actually used loops of tapes to prod= uce the notes
•<= /span>Musical Example: Sgt. Peppers album by The Beatles=
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Fully Electric Instruments
•<= /span>Some older Organs used large banks of vacuum tube oscillators connected to a conventional organ keyboard
–<= /span>Hammond NovaChord
–<= /span>Allen Organ
•<= /span>One of the first completely electronic instruments= was the Theremin
–<= /span>Invented less than 20 years after the invention of vacuum tubes
–<= /span>Unique interface required musicians to play without touching the instru= ment
•<= /span>Two antennas were used
–<= /span>The upright antenna controlled the pitch.  The closer to the antenna, the hig= her the pitch
–<= /span>The horizontal loop antenna controlled the output volume.  The closer to the antenna the quieter.  This allowed notes to be plucked.=   
•<= /span>Very difficult to play
–<= /span>The extreme sensitivity required the user to hold = their body steady while playing so as not to affect the pitch
–<= /span>Clara Rockmore was the only person to tour exclusi= vely as a Theremin player
•<= /span>Mostly used for sound effects
•<= /span>Other instruments were created around non-standard interfaces
–<= /span>Ribbon controller
–<= /span>Electro-Theremin, (Tannerin) – sounds like a Theremin, but easy to contro= l.
Musical Examples: = Edison’s Medicine - Tesla
Musical Examples:
Good Vibrations – Beach Boys
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The Theremin
•<= /span>The Theremin utilizes two RF devices (typically ~ 300kHz)
–<= /span>One has a fixed frequency
–<= /span>The other has a variable frequency determined by the antenna
•<= /span>These are beat against each other (heterodyned) to generate an audio output.
•<= /span>Another variable oscillator can be used to create a volume control (not always present on simpler modern Theremins)
*
Var. RF Osc
= Fixed Rf Osc.
Variable RF Osc.
Power Detector
*
LPF
Pitch Antenna
Audio
Output
Volume Antenna
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12/4/2008
DRAFT
6
Comput= er Synthesis Building Blocks
Instrument= s are implemented as algorithms typically using a specialty software package
C= ould be in a rack mount synthesizer
O= r a general purpose computer
Synthetic Instruments are often built up from Unit generators.
S= implifies the technical details for musicians
U= Gs are interconnected to form instruments
U= Gs are often modeled graphically so than an instrument flowchart
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12/4/2008
DRAFT
7
Signal Flowchart
•<= /span>Behaves like a simplified “digital circuit”
–<= /span>Output can be tied to more than one input
–<= /span>Outputs can never be tied together
–<= /span>Can combine outputs through mathematical operations
•<= /span>Addition (+) is used for mixing audio signals
Subtraction (+ with the negative input labeled with a – sign) <= span style=3D'position:absolute;top:60.5%;left:18.53%;width:29.77%;height:3.25%= '>Combining two signals while inverting one.
•<= /span>Multiplication (*) is typically used for amplification of a constructed <= span style=3D'position:absolute;top:71.0%;left:18.53%;width:29.77%;height:3.25%= '>signal
•<= /span>Division (a/b) is typically used for attenuation of a constructed signal
•<= /span>Output is defined a small empty = circle
= Unit = Gen
+
= Unit = Gen
3D"Trapezoid:
Envelope Gen
*
a/b
1
-
Amplitud= e
Duratio= n
Frequenc= y
Amplitud= e
= All Input Parameters
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12/4/2008
DRAFT
8
Oscillator
Mo= st fundamental UG is the oscillator
Symbol insi= de generator describes type (s= ine, square, general waveform)
Inputs gen= erally given short represen= tative input names
A= MP =3D peak amplitude
F= REQ =3D frequency
N= umber of Hertz
S= ampling Increment (SI)
PH= ASE =3D starting point in the cycle
WF
AMP
FREQ
PHASE
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DRAFT
9
Implementation
Di= rect Evaluation
C= ompute while generating
V= ery slow for most synthesizers
Wa= vetable
S= tored waveform (buffer in ROM)
Contains one period of the waveform
Later Romplers may contain more = complete samples of actual = musical instruments
Starting Sample is determined by the Phase input
-1
-0.5
0
0.5
1
0
128
256
384
511
0.0123
255
0.0
256
-0.0123<= /span>
511
-0.0245<= /span>
510
̷= 0;
-0.9999<= /span>
385
-1.0
384
-0.9999<= /span>
383
̷= 0;
-0.0123<= /span>
257
̷= 0;
0.9999
129
1.0
128
0.9999
127
̷= 0;
0.0123
1
0=
0
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Sampling Increment
•<= /span>To generate the “fundamental” of the w= ave = shape, read out at the sampling rate
•<= /span>Harmonics can be generating ready every other = sample (octave) or other multiples (i.e. every 3rd=  = sample =3D fifth above octave) <= /div>
•<= /span>Other frequencies can be created by specifying = a Sampling Increment (SI)
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Fractional Indexes
T= he SI is likely not to be an integer
Three meth= ods exist for using  fractional SI’s while reading out = the waveform from the wavetab= le.  The complexity increases in this list
T= runcation – round down to the nearest integer
R= ounding – round to the nearest integer (up or down
Interpolation – Estimate the value at this t= ime via a linear interpolat= ion (or more complex interpolation)
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SNR Effects of 3 methods
•<= /span>Consider the following table where N is the number= of elements in the table
•<= /span>SNR is approximated by the following equations = ;
–<= /span>Truncation =3D 6k – 11dB
–<= /span>Rounding =3D 6k – 5 dB
–<= /span>Interpolation =3D 12(k-1)dB
•<= /span>For the 512 element example table this yields 43, = 49 & 96 dB respectivel= y
•<= /span>This SNR would need to be combined with D/A SNR to= get a true estimate of the effect on the quality&#= 13;
•<= /span>This illustrates a implementation between computat= ional power and memory usage.
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Other Methods of Defining Waveform= s
•<= /span>Besides direct evaluation or stored wavetable, The= waveform can be described with a piece wise linear= evaluation
–<= /span>This is defined as a set of breakpoints
•<= /span>Points in time and amplitude that dictate where the waveform changes slops =
•<= /span>A line is drawn between the breakpoints to determi= ne the waveform
•<= /span>All points are described as a phase and the amplit= ude at that phase.
•<= /span>After generation these functions are usually store= d in a RAM wavetable.
–<= /span>Problems can arise from a harmonically complex wav= eform being generated with a high frequency fundamental&= #13;
•<= /span>The upper harmonics may exceed the Nyquist rate (f= s/2) and create images in the freq= uency domain.
•<= /span>This would generate an in harmonious instrument
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Define in the Frequency domain
To combat = the possible introduction of upper harmonics that will create images, one can specify the waveform in the frequency domain
W= aveforms are defined a series of data structures where each structure element includes the
A= mplitude
P= artial Number
P= hase
Partials above the Nyquist rate can be eliminated = by not adding that partial to the rest during the syn= thesis phase of the proc= ess.
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Functions of Time
•<= /span>It is often desired to make = an oscillator vary it’s = amplitude with time
•<= /span>This will modify the = “envelope” of the signal, = hence their name of envelope generators
The Envelope generator is connected to the AMP input of the UG<= /i> to modify the amplitude
Attack
Sustain
Decay
Decay
Time=
Rise
Time=
Envelope Gen
WF
AMP
FREQ
PHASE
DUR
DECAY TIME
RISE = TIME
Connections can also be reversed with the WF function feeding the Envelope Generators
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•<= /span>The function describing the segments of the envelope can be linear or exponential
•<= /span>Both are useful for different modeling purposes
–<= /span>Exponential is the method by which <= span style=3D'position:absolute;top:45.0%;left:14.04%;width:44.94%;height:4.5%'= >natural instruments die away.
–<= /span>Linear is useful for the sustain region and slow attack times
•<= /span>The envelope can have a great effect on the timbre of the sound
–<= /span>Short attacks are more common in percussion
–<= /span>Long attacks are more commonly found in acoustic instruments such <= span style=3D'position:absolute;top:86.25%;left:14.04%;width:44.94%;height:4.5%= '>as a pipe organ.
Envelope Gen
WF
AMP
FREQ
PHASE
DUR
DECAY TIME
RISE = TIME
More Envelopes
Lin Slope
Lin View
Exp Slope
Lin View
Lin Slope
Exp View
Exp Slope
Exp View
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12/4/2008
DRAFT
17
Additional complexity
•<= /span>Can add another segment to the envelope to better match more instruments
–<= /span>Section added after attack to simulate the fast die out of a struck note before the sustain portion – This section steals the name Decay
–<= /span>Decay section at the end of the wavefrom is renamed “Release”
Attack
Sustain
Release
Decay
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12/4/2008
DRAFT
18
Programming languages
•<= /span>Before GUIs and HW synthesizers, there were software languages for generating computer music
•<= /span>Csound and Cmusic are the two descendants of the first packages designed to create sound on = workstations
•<= /span>Like any good programming environment, the tasks are build = up in stages.&nbs= p; The sound definit= ion is used in parallel with the mus= ic definitions.  This keeps the code cleaner
•<= /span>
Instrument Definition
Score Editor
Score
Instrument Algorithms to generate sound
Performance Program
Sound
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12/4/2008
DRAFT
19
Csound vs. Cmusic
LINEN
F2
P4
P3
P8
P6
P5
F1
F2
P7
P10
P11
P9
P5
instr<= span style=3D'mso-tab-count:1;width:16.26%'> 1
k1 linen p5,p6,p3,p8
a2 oscil k1,p4,2
out a2
endin<= /span>
ins 0 SIMPLE;
osc b2= p5 p10 f3 d;
osc b1= b2 p6 f1 d;
out b1= ;
k1 =3D env gen output, p5=3Damplitude of note, p6 =3D rise = time, p3=3Dduration,p8=3Ddecay time,p4=3Dfrequency,2=3Dtype of waveform
b2 =3D 1st oscillation output, = p5=3Damplitude of note, p10 =3D dur, f3=3Dfunction to control envelope shape, d=3Dphase of oscillator, p6=3Dfrequency, f1=3Dwaveform pattern to generate
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12/4/2008
DRAFT
20
Additive Synthesis
•<= /span>Previous diagrams were fine for describing steady = state tones, but couldn’t match transients
–<= /span>Harmonics all arrived and departed at the same tim= e
–<= /span>Higher frequencies were perfect – no adjustm= ent for out of tune
•<= /span>New Model (shown below) represents every component= with its own set of sine wave UGs
–<= /span>Adding all the outputs gives the desired sound = 211; Additive Synthesis
–<= /span>Often called = Fourier recomposition – uses synthe= sis by analysis
–<= /span>Can combine multiple instruments, but care should = be taken to align temporal peaks
–<= /span>Requires significant computational resources to ge= nerate one sound
–<= /span>Required multiple configurations to support differ= ent intensity levels (instruments sound different depending on the force of the physical attack
AMP 1
FREQ 1
=
AMP 2
FREQ 2
AMP N
FREQ N
+
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Modulation
M= odulation is alteration of the following
A= mplitude
A= mplitude modulation
Basically tremolo.  A signal source is connect to the Amplitude input of the audio generator
R= ing modulation
Moves result to a different frequency center (same process as in the ring modula= tor effect from the last lecture)
S= ingle-sideband modulation
N= ot discussed – a radio method with little use in music
F= requency
v= ibrato
------=_NextPart_01C9565C.F8CD64C0 Content-Location: file:///C:/3A815E52/Synthesize_v12_files/slide0024.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Digital Music: Musical Instrument Synthesis
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22
Amplitude Modulation
•<= /span>Basically tremolo.  A signal source= is connect to the Amplit= ude input of the audio genera= tor
•<= /span>Generates side bands
•<= /span>Perception
–<= /span>< 10Hz – ear tracks amplitude variations
–<= /span>10Hz < x < critical band boundaries – user hears amplitude of the average of the output
–<= /span>> 1/2 critical band – perceived as additional tones
»
WF
AMP
fc<= /span>
+
WF
fm<= /span>
m*AMP
AMP
m/2*AMP<= /div> 3D"Text
fc
3D"Text
fc + fm
3D"Text
fc - fm
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Ring Modulation
•<= /span>Multiplies two waveforms together to spectrally dense signal also called
–<= /span>Balanced Modulation
–<= /span>Double Sideband Modulation
–<= /span>Called mixing in the RF field
Produces outputs at fc + f<= /span>m<= /span> and = fc<= /span> - fm 
•<= /span>Can use multiply to generate RO instead of 2 oscillators
•<= /span>If either oscillators are zero – no <= /span>output
•<= /span>If both waveforms have p and q harmonics respectively, the output <= span style=3D'position:absolute;top:71.5%;left:13.85%;width:43.63%;height:4.5%'= >contains 2*p*q harmonics (all possible products of the harmonics
»
WF2
fc<= /span>
WF1
fm<= /span>
AMP
3D"Text
fc
WF 1
3D"Text
fc
Out
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12/4/2008
DRAFT
24
Frequency Modulation
•<= /span>Applies a small shift to the frequency center
–<= /span>Average is still center frequency, but pitch varies around it
–<= /span>Modulation usually at most a few percent of the center frequency
–<= /span>Modulation rate is below the audio range
–<= /span>Higher rates lead to frequency modulation synthesis
WF
+
WF
= VIB = Rate
VIB Width
fc<= /span>
AMP
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12/4/2008
DRAFT
25
Noise Generators
&#= 8226;Generate a Distributed Spectrum
&= #8211;Fills many bands
&= #8211;White noise is flat across all bands
Generated by a random (or pseudo random) number generator
When random samples are picked at a rate < the sampling frequency, the high end is rolled off
------=_NextPart_01C9565C.F8CD64C0 Content-Location: file:///C:/3A815E52/Synthesize_v12_files/slide0007.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Digital Music: Musical Instrument Synthesis
12/4/2008
DRAFT
26
Spectral Interpolation
•<= /span>Implemented by using a mixer to gradually <= /span>= switch between two sounds
–<= /span>With mix value set to 0 all of sound 1 =
–<= /span>With mix value set to 1 all of sound 2 =
–<= /span>With mix value set to 0.5 – 50% of sound 1 a= nd 50% of sound2
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Distortion Synthesis
•<= /span>Additive Synthesis required too much = computational complexity
•<= /span>Non Linear methods were introduced to allow a = wide range of sounds while keeping complexity = down
–<= /span>The spectral complexity increases with distortion.=
•<= /span>Several Methods are commonly used
–<= /span>Frequency Modulation
–<= /span>Nonlinear Wave-shaping
–<= /span>Discrete Summation Formulas (not covered)
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FM Synthesis
Early FM synthesis research was lead by J. Chowning in the mid to late 1970s
FM synthes= is saw widespread use in PC sound cards before the falling price of memory made wave table based cards more affordable =
Unlike the vibrato example on a previous slide, now the modulation is in the audible range.
T= he can yield non-harmonic results caused by the modulation process.
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FM Synthesis
•<= /span>Typically only used Sinusoids for oscillators since more complex signals produce more complex spectra
•<= /span>d=3Ddeviation =3D max (fm) – min (fm<= /span>)
–<= /span>Instantaneous frequencies are fc<= /span>-d to fc+d
–<= /span>When d=3D0, the output is = sinusoidal
–<= /span>If d>f, negative frequencies result =
•<= /span>Requires processor to output = sample in reverse to show phase change
•<= /span>Frequency is folded over to positive axis with a phase change.
WF
+
WF
d
fc<= /span>
AMP
fm<= /span>
Modulating Oscillator
Carrier Oscillator
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12/4/2008
DRAFT
30
FM Synthesis Spectra
•<= /span>Using Sinusoids, the output spectrum will look similar to the one at left
Frequencies present are       where k is a natural number.
–<= /span>Power division depends on d
•<= /span>d=3D0 = means all power is in fc
•<= /span>As d increases, k increases and more power is added to the sidebands
–<= /span>Define the Index of Modulation <= /div>
3D"Text
fc
Out
3D"Text
fc+fm
3D"Text
fc+2fm
3D"Text
fc+3fm
3D"Text
Fc-fm
3D"Text
Fc-2fm
3D"Text
Fc-3fm
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Sp3gsHgoPqXKTDSUrJZ0p+10nDZ/1s93vEQgGBAKYBsqBQYEBwQDgS4EAgeEcwcACAAJExEAOx== ------=_NextPart_01C9565C.F8CD64C0 Content-Location: file:///C:/3A815E52/Synthesize_v12_files/slide0038.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Digital Music: Musical Instrument Synthesis
12/4/2008
DRAFT
31
Bessel Functions
•= The index of modulation = determines the amplitude = of each of the side bands according to the Bessel <= span style=3D'font-size:56%'>functions listed in the = chart
•= The sign (phase) of each = component is not audibly <= span style=3D'font-size:56%'>significant unless there is = spectral folding and a <= span style=3D'font-size:56%'>wrapped negative component cancels a <= span style=3D'font-size:56%'>positive component.
–= Then the two components must be added. =
–= Remember that folding negative components to the positive frequency also flips their sign
Jk<= /span>(I)
Fc<= /span>+kfm
(-1)k<= /span>Jk(I)
fc<= /span>-kfm
etc
J5<= /span>(I)
fc<= /span>+5fm
-J5<= /span>(I)
fc<= /span>-5fm
5
J4<= /span>(I)
fc<= /span>+4fm
J4<= /span>(I)
fc<= /span>-4fm
4
J3<= /span>(I)
fc<= /span>+3fm
-J3<= /span>(I)
fc<= /span>-3fm
3
J2<= /span>(I)
fc<= /span>+2fm
J2<= /span>(I)
fc<= /span>-2fm
2
J1<= /span>(I)
fc<= /span>+fm
-J1<= /span>(I)
fc<= /span>-fm
1
J0<= /span>(I)
fc<= /span>
0
Amp
Freq
Amp
Freq
k
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12/4/2008
DRAFT
32
Bessel Functions
Plots of the first 8 Bessel functions are shown below.
•<= /span>Note that for I=3D0, the only frequency present is= the carrier.
•<= /span>A Rule of Thumb: Only sidebands up to k=3DI+1 cont= ain significant power (from Jerse)
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DRAFT
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12/4/2008
DRAFT
34
Dynamic Spectra
•<= /span>In order to have the spectrum = evolve as a function of time, provide a envelope control to = the d parameter.
Two different envelope = generators are used
–<= /span>One for the overall envelope of the sound
–<= /span>One for the evolution of the spectrum
•<= /span>IMAX is the maximum deviation
•<= /span>Does not allow a specification of a specific spectral evolution, but a varying amount of richness
WF
+
WF
IMAX*f<= span style=3D'font-size:67%;position:relative;top:.37em;mso-text-raise:-25%'>m<= /span>
fc<= /span>
AMP
fm<= /span>
Modulating Oscillator
Carrier Oscillator
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Example Instruments
= 226;See Section 5.1D of Jerse.
&= #8211;Bell
&= #8211;Wood Drum
&= #8211;Brass
&= #8211;Clarinet
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3D"Flowchart:
Double Carrier
•<= /span>Useful in mimicking the formant = (fixed resonant frequency) present in acoustic instruments = that isn’t captured with Single Carrier FM synthesis.
•<= /span>Two carriers are at fundamental and first formant frequency.
•<= /span>IMAX is maximum modulation
–<= /span>I2 is the ratio of the 2nd carr= ier to the first.  Usually pretty small
A2 is usually less than unity too
Fc2 is usually chosen as the <= span style=3D'font-size:56%'>harmonic of the fundamental closest to the formant.
•<= /span>Used by Morrill in synthesis of = trumpet tones.
WF
*
WF
I*fm<= /span>
fm<= /span>
Carrier 1
3D"Flowchart:
WF
Carrier 2
+
+
fc1=
+
fc2=
*
A2
Amp
I2/I1
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Double Carrier Example Instrume= nts
= 226;See Section 5.1F of Jerse.
&= #8211;Trumpet w/ Vibrato
&= #8211;Soprano Voice
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Complex Waveforms
Example sh= ows sine modulated by wavef= orm with 2 spectral components
F= requencies in the output are&#= 13;
Amplitude of the resulting sidebands are determined as the product of Bessel functions
WF
+
WF
I2* fm1=
fc<= /span>
AMP
fm1=
Carrier Oscillator
WF
I2* fm2=
fm2=
+
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Complex Modulation Example Instrume= nts
= 226;See Section 5.1H of Jerse.
&= #8211;Violin
------=_NextPart_01C9565C.F8CD64C0 Content-Location: file:///C:/3A815E52/Synthesize_v12_files/slide0049.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Digital Music: Musical Instrument Synthesis
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Synthesis by Waveshaping
= 226;A different type of non-linear processing
&= #8211;Similar to FM
&= #8226;is more efficient than additive
&#= 8226;Dynamic evolution in spectral complexity
&= #8211;Unlike FM
&#= 8226;Can generate a band-limited spectrum
WF
a
fc<= /span>
= Input Oscillator
Waveshaper
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Waveshaping through non-linear transfer functions
•<= /span>Waveshaping uses the same concept of a transfer function that we saw when considering distortion effects.
The output shape will depend on the input amplitude
–<= /span>The shape of the transfer function <= span style=3D'position:absolute;top:53.0%;left:14.04%;width:48.68%;height:4.5%'= >will determine the richness of the <= span style=3D'position:absolute;top:56.49%;left:14.04%;width:44.19%;height:4.5%= '>output
•<= /span>Discontinuities add high frequency <= span style=3D'position:absolute;top:64.25%;left:18.53%;width:39.88%;height:4.0%= '>components.
•<= /span>Standard Symmetry rules apply
Odd functions only contain odd harmonics
–<= /span>Even functions only contain even harmonics
Max input amplitude =3D1
Max input amplitude =3D 0.5
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Content-Location: file:///C:/3A815E52/Synthesize_v12_files/slide0050_image168.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhaQAhAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABn ACAAgQAAAAAAAP//zAECAwKDhBGpy+0Po5yUHZOE3rz7D4biSJZdcATmyrbuiyLvTNduZuf6ruH8 D1z5gsTiSWVMGofK5o7pjNKg0iqLas2OsNquh+sNC8DiLrmcPaOr6nW07W7C48k5vWi/B/P6H7// hASYJjjIVmj4hpgoJ8P45vioFIOwKMmTcFFZwdnp+VlxUAAAOw== ------=_NextPart_01C9565C.F8CD64C0 Content-Location: file:///C:/3A815E52/Synthesize_v12_files/slide0051.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Digital Music: Musical Instrument Synthesis
12/4/2008
DRAFT
42
Polynomials
In order to keep the waveshaping problem tractable, limit the transfer functi= ons to polynomials
•<= /span>
This guarantees that t= he output spectrum will not have frequencies greater that N*f0&= #13;
For any given single t= erm polynomial, xN = the ratio of power in the harmonics is given in the table on the following slide.
•<= /span>
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12/4/2008
DRAFT
43
Harmonic levels
Example:
F(x)=3Dx5
h1<= /span>=3D 0.625
h3<= /span> =3D0= .3125
h5<= /span> =3D0= .0625
Check on adding to 1
h10=
h11=
h9<= /span>
h8<= /span>
h7<= /span>
h= 6<= /span>
h5<= /span>
h4<= /span>
h= 3<= /span>
h2<= /span>
h1<= /span>
h0<= /span>
x11=
x10=
x9<= /span>