Welsh's Synthesizer Cookbook

129 Pages·44.873 MB·English

by  Fred Welsh

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OSCILLATORS
OSCILLATORS
Harmonics andoscillatorwaveforms
The most fundamental sound isthe sine wave. Every sound whether it isnatural or
synthetic is made up ofsine waves.
Figure1.Allsounds, natural and
synthetic,are made up ofsinewaves
The sound ofa sine wave could be described as extremely mellow and not particularly
interesting, but when numerous sine waves ofdifferent pitches and amplitudes are mixed
together the sonic possibilities become endless.
It ispossible to create any waveform using nothing but sine waves. Let's begin by
creating a sawtooth wave. Each sine wave isgoing to have a different pitch and as they
go higher inpitch they will decrease in amplitude. The first one will havethe lowest
pitch and itwill in fact determine the overall pitch ofthe sawtooth. This is called the
fundamental frequency. For the case ofthis examplewe will give it apitch of440 Hz
correspondingto the A4 key.
Hz '1000Hz eoooHz
440Hz i
·20dB 1.- --.- -- +---.- _ __ , _... __ _._ __ .
•'10dB .............__ __...... .+..__.................. .......................................................................... . -t- .
.eodB
·80dB
·100d
Figure2. A440 Hz sine wave and a graph representing its frequency and amplitude
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The bottom portion ofthe figure is what iscalled a harmonic diagram. The horizontal
axis isfrequency/pitch and the vertical axis isamplitude. The spike represents the
frequencyand amplitude ofthe 440 sine wave. Let's add a second sine wave with a
frequencyof880 Hz (440Hzx 2).
+
440Hz 880 Hz
2000Hz 4000Hz 6000Hz
440fQ
aeofQ
·20dB
-·-40 dB ~._.
·50dB
~---
-80dB
~ .
--100d .. l...;'",ti::l SF<A:H RhJC) SEF',N fB
Figure3. A440Hz sine waveadded to a880Hz sine wave createa newwaveform
Notice there isa second spike inthe diagram representing the880 Hz sine wave. Now
add athird sine wave at a frequencyof1320Hz (440Hz x 3)
+ +
440 Hz 880Hz 1320Hz
2000Hz 4000Hz "000Hz
440Hz
880Il"
-20dB 13:'0Hz
-40dB
·60dB
1
j
·80dB
.~~.
·10od~ ........•••......l1;:;1~ "J SRR!-4 RND SERN ta .e '.
Figure4.Threesine wavesoffrequency 440Hz,880Hz, and 1320Hz added
togethercreateawaveform thatbeginsto look like a sawtoothwave.
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Alreadythe new waveform is beginning to lean to one side like a sawtooth wave. Keep
adding sine wavesthat are frequency multiples of440 Hz and eventually a sawtooth
wave is produced.
12000H= f1000H= 5000 H=
__
-20dEt ........... .....
•4<)dB ...~..- 1- ~... -~~-..- ~._.. _.~_.
_
·50dB _. ..•~-...~..- .._._.. ...._-~ ~-_...... ~~...- _..~..~...•~.f-----•..~.... ....-.... ......•.•.•_-..........•
_
·80dB .- ._..__... ._. ._~ -..- .. .-
- ~D.
·"tOOd .-----'~.~.r"". ) a,· IJ::i:1.3.1::1hJ .5 6: :u,j 4:..51<Is.S:I~LS-~.j..L: d..Nlt
Figure5. Asawtoothwaveand its harmonicdiagram
Each one ofthe harmonics(spikes) in the diagrams above are called partials. The first
partial at the far left, as stated earlier, is known asthe fundamental and itdetermines the
pitch ofthe waveform. The other partials are all frequency multiples ofthe fundamental
and they are known as overtones. Also notice that the fundamental is the loudest sine
wave ofall the harmonics and that asthe overtones get higher inpitch their amplitudes
decrease. There is ofcourse nothing preventing us from mixing sine waves that all have
the same amplitude it's justthat inthis particular case it is a requirement ofcreating a
sawtooth that the amplitudes diminish with higher frequency. Just as apoint ofinterest if
we were to mix these harmonics all at the same amplitude it would produce something
that as successively higher-pitched sine waves are added would at first look like a
sawtoothwave with a bad complexion and as more harmonics are added would look
nothing like a sawtooth at all. Harmonic amplitudes are important!
Figure6. Waveform produced bysetting
the first 16harmonicsto equal amplitudes.
A square wave can be createdusing the same process. The difference isthat a square
wave is only made up ofodd-numbered partials.
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:::0:00Hz AoOOHz 5000Hz
1
,
-20dB
5
7
9
-40dB 11
13
'15
._-_ __
-eodB ...__. ....._._- - _ . .......
-80dB
~D
-100d f~!!,'cjCI SE i'10.1 G?
Figure7.Asquarewave has only odd partials. Thissquarewave was produced by an actual
oscillator. Noticetheslightimperfectionsin thewaveform. Nooscillatorproduces perfect
waveforms nor do any two oscillatorsproduceidenticalwaveforms.
Triangle waves are also made up ofodd-numberedpartials but the higher harmonics are
weaker than in a square wave.
roo
aHz
-20dB
I
I
·40dB
-60dB
i
·80 dB
·100 d9 M'::lS P8~Ak AND SE ~N ca SM Ft~TELEf-TI=tClNI)
Figure8. Atrianglewave has odd harmonicsthatareweakerthan those ofa squarewave.
The process ofadding sine waves togetherto create sounds is known as additive
synthesis. This method isused on afew digital synthesizers and soft synths and has also
been used bypipe organs for hundreds ofyears. In apipe organ each pipe produces a
sinewave ofa different pitch and by controllingthe amount ofairto each pipe it is
possible to control the individual amplitudes ofeach sine wave which in tum makes it
possible to produce sounds that are harmonically similar to other instruments. It could be
saidthat pipe organs were the first synthesizers.
Analog synthesizers use aprocess called subtractive synthesis which is simply additive
synthesis in reverse. Here's some synth terminonlogy for you: Sounds created by
synthesizers are referred to aspatches. This goes backto the early days ofmodular
synthesizers when patch cables were used to route signals from module to module to
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reate a sound. Patches created using subtractive synthesis start with waveforms that are
already rich in harmonics such as sawtooth, square,and triangle waves. These
waveformsare then passed to a filter which removes harmonics from the waveforms in
ordertoproduce the desired sound. The harmonics are subtractedout hence the process
is known as subtractive synthesis.
The most common waveforms on analog synthesizers are the sawtooth, square/pulse, and
triangle wave. There are three main reasons these are used almost universally rather than
otherwaveforms. First isthe fact that they all have lots ofharmonics which can be
chiseled away by the filter. Secondly they are relatively easy to produce using analog
circuits. Thirdly they are each harmonically similar to broad families ofacoustic sounds
even without any filtering. This almost sounds as ifto imply that acoustic imitation isthe
goalofsynthesis which itmost certainly is not. It is apractical criteria to start with if
nothing else. Sawtooth waves are similar to brass and string instruments, square waves
are similar to woodwinds,and triangle waves with their diminished higher harmonics are
good for mixing together to produce inharmonic sounds (bells,chimes,etc) as well as
adding the occasional rogue harmonic to saw and pulse waveforms. This may be
surprising but even though these waveforms have similar harmonics and sound similar to
certain families ofinstruments the waveforms ofacoustic instruments look nothing like
squares/pulses, sawtooths, ortriangles. A waveform's usefulness lies with the
harmonics that make itup and isNOT inherent to the shape ofthe waveform. Two
sounds can have very similar harmonics yet have waveforms that look nothing alike.
Adding waveforms together in an attempt to match the waveshape ofan acoustic
instrument won't produce desirable results most ofthe time.
While some synthesizers only have one oscillator and a few others have many,the
majority have two oscillators that can be independently setto different waveforms. Two
oscillator configurations are so common because they are cheaperto produce than multi
oscillator synths yetthey are much more capable ofproducing rich sounds than single
oscillator instruments. They are a good balance ofeconomy and ability. By using two
oscillators it ispossible to not only create more interesting strings, brass,and woodwinds
but italso becomes possible to create vocals, inharmonic sounds such as bells,and
countless other acoustic and synthetic sounds. One simple fact is that a single oscillator
will have harmonics where each successivepartial has less amplitude. While the higher
harmonics ofacoustic instruments do diminish in amplitude they do not follow this
progression sorigidly. Below are harmonic diagrams ofsome acoustic instruments.
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2000Hz 4000Hz Hz
·20dB
·40dB
Figure9. Violin harmonics
pOOOHz Hz Hz
I
+_--+
·20dB \
-·40 d-B --I- --1l--+-+-,.-------- --;-- --------+---
·60dB
Figure 10.Flute harmonics
Hz Hz 00Hz
-20dB
·40 dB
60dB
Figure 11.Trumpet harmonics
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The noise atthe bottom ofthe graphs is from resonance,reverberation, and recording
noise,
Notice that while the higher harmonics do die away they do not do so in a uniform
fashion like synthetic waveforms. These slightharmonic deviations help separate one
acoustic instrument from another in the same family and give what could be considered a
"natural" sound. Again, imitation is a possibilityofsynthesis but is not necessarily the
goal,however lessons learned from studying and recreating acoustic sounds can help to
produce more interesting synthetic patches that have character and life. By detuning
oscillators it is possibleto come up with much more interesting sounds. To produce the
harmonics ofa marimba for example take atriangle wave and add a second triangle wave
at slightly lower amplitude and tuned up two octaves.
Hz 00Hz 15000Hz 8000Hz
d,
·20 -I i ........................... ; _..................................................•..................
·40d
·150d
·80dB
/
-'
·100dB
Figure 12. Using two oscillators to synthesizean instrumentsuch as the
marimba allows for a morecomplex harmonicstructurethan usingsingle
waveforms alone.
By simply mixing two humble triangle waves with different tunings a more harmonically
sophisticated sound has been produced than could be done withjust a single waveform.
The harmonics ofa singletriangle may die out uniformly but the harmonics oftwo
detuned triangles do not. While this sound is literallythe sum oftwo triangle waves,to
the ear it sounds far richer than the sum ofits parts. The importantpointhere is that using
two oscillators gives more control overthe harmonic make up ofthe final sound. As we
shall see there are many othertools at our disposal that will allow us to further fine tune
harmonic structure.
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PW=50%
-20dB
(square wave)
...,d.
_r- L
-eodB
-80dB
'~V_
-100d M:lS BRA ~NLl SE RN Ii! S'" I'lTELErTRLlNI
PW=40% -20dB
.4(1dB
-50dB
-80ee
-~D_
A~
-100d ~:lSI B Nr1 C::;E N S I'lT LE T'"bNI>
jOOH%
PW=30% -20dB
-4(1dB
-50ae
,
-80dB ;
~o.Dt-_
-100d It-,s B A l=lNn S"" Nii SM "'T L'" Tl:: NI>
PW=20%
-20dB
-40dB
J L
-00d•
•80dB '
I
, I ~r
-100d :lS S~ R -t ANl:i SE Nii SMtll'lTELE TI=bNI
PW=10%
I
-20ee
I
I
-40dB
II
I
d.
-00
d. I
-80
! E:\_
I
-100d l:JS SFR4 RNCISE Nii SM RT ELE TI=bNl I
Figure13.Pulsewidths. Narrowing thepulsewidth createsa brighter,edgiersound similar to a
sawtooth.