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The Synthesis of Complex Audio Spectra
by Means of Frequency Modulation
John M. Chowning
Stanford University
One of the monkeys sitting at his typewriter finally produced a
work by Shakespeare, he was then asked to explain how.
Although frequency modulation is well understood as applied
in radio transmission, the relevant equations have not been applied
in any significant way to the generation of spectra where both the
carrier and modulating frequencies are in the audio band and the
sidebands are used directly to form the spectrum. Given an audio band
oscillator capable of producing the proper wave when
the sign of its frequency input becomes negative,
it will be shown that a startling variety of spectra
can be produced which are altogether predictable, and with an extraordinarily
simple system.
The research described here was done on a Digital Equipment Corporation
PDP-10 computer for which there is a special sound synthesis program
which is designed to make optimum use of the
time-shared capability of the machine.
Implementation of this research, however,
will be described for mUSIC V, a sound synthesis program which is both
well documented and generally available (1).
MUSIC V is a program which generates samples or a numerical representation of
a sound pressure wave according to data which specify the physical
characteristics of the desired sound. The samples are stored on a
memory device such as a digital magnetic tape or disk as they are
computed. On completion of the computation the samples are passed
at a fixed rate, the sampling rate
(typically 10000 to 30000 samples/sec), to a digital to analog converter,
which generates a sequence of voltage pulses whose amplitudes are
proportional to the samples. The pulses are smoothed by a filter and
the resulting signal is passed to an audio system.
The program is designed such that the computation of the samples is done
by program blocks, called unit generators. A typical unit generator
is the oscillator which has two inputs, an output, and a function. The
first input specifies the amplitude of the output, the second the
frequency of the output, and the function determines the shape of
the output. The value of an input can be either a constant or the output
of another unit generator, thereby allowing higher level operations on
waves. A collection of interconnected unit generators is called an "instrument",
which is supplied data through a set of parameters (P1 to Pn) set by the
user. P1 and P3 are reserved by the program for begin time and duration,
P2 is reserved for the "instrument" number of a set of
instruments, and the remaining
parameters are assigned their function by the user. Shown in Figure 1,
is an instrument diagram which consists of three unit generators, two
oscillators and an adder. This instrument is designed to produce a
tone which has a "vibrato", or a periodic variation of a frequency
around some average. The function for each oscillator in this case
is a sinusoid. In order to produce a tone of 440 Hz which has a
vibrato width of 4 Hz above and below the average 440 at a rate of
5 Hz, the following assignments would be required.
P4 = 1000 (arbitrary loudness scaling)
P5 = 440
P6 = 4
P7 = 5
Oscillator 1 produces a sinusoidal output whose amplitude is
scaled by P6 to be + and - 4, at a frequency of 5 periods/second.
This output is then added to the constant 440 in the adder 2, whose
output is the frequency input of the other oscillator 3. This oscillator
produces an output wave whose amplitude is scaled by P4 to be + and -
1000 amplitude units and whose frequency varies 4 Hz + and - 440 Hz
at the rate of 5 Hz. Figure 2 is a graph of the instantaneous frequency
vs. time for this case.
In order to generate this sound, the "instrument" would be encoded
and then the paramenters passed to it with the addition of P1,P2, and
P3 as defined above. The program would then compute the numbers and
store them for conversion.
"Vibrato" is a special case of frequency modulation and will serve
to define three basic parameters.
In frequency modulation the instantaneous frequency of a carrier
wave is varied according to a modulating wave such that the rate
at which the carrier varies around its average is at the frequency of
the modulating wave while the amount the carrier varies away from
the average is the deviation which is proportional to the amplitude of the
modulating wave.
The parameters in frequency modulation are referred to
as carrier frequency c (average),
frequency deviation d (peak deviation of carrier from its average),
and modulation frequency m (rate at which the deviation occurs).
The parameters of Figure 1 would be defined to be then:
P4 = A = amplitude of carrier wave
P5 = c = carrier frequency
P6 = d = frequency deviation
P7 = m = modulating frequency
As long as the modulating frequency is well below the audio band (Figure 2),
the ear has little difficulty tracking the instantaneous frequency of the
carrier which results from the modulation.
For example, by changing the frequency deviation to 220 Hz the
deviation will range between 220 and 660, as in Figure 3, which is perceived as
a periodic frequency sweep over an interval of a twelfth at the rate of
5 times per second.(2)
If the modulating frequency were changed to 440 Hz, where the modulating
frequency equals the carrier frequency, the ear can
no longer track the instantaneous change in frequency as a sweep, but rather
perceives a complex spectrum.
In order to determine the spectral components
it is necessary to consider in some
detail the theory of frequency modulation.
When a sinusoid is frequency modulated by a sinusoid, side frequencies are
produced above and below the carrier frequency
at intervals of the modulating frequency.
The number of side frequencies produced is related to the frequency
deviation, in such a way that as the deviation is increased from 0, energy is
"stolen" from the carrier frequency and distributed between more and
side frequencies. The frequency band which contains the side frequencies
above the carrier is called the upper sideband and that containing the
side frequencies below the carrier is called the lower sideband.
The amplitudes of the carrier and side frequencies in the upper and lower
sidebands are determined by Bessel functions (3) of the first kind and nth
order, Figure 4, the argument to which is the ratio of the frequency
deviation to the modulating frequency and called the modulation index.
modulation index = frequency deviation / modulating frequency
I = d/m
The 0th order Bessel function and some modulation index I, J0(I), yields an
amplitude scaling coefficient for the carrier frequency, the 1st order
Bessel function, J1(I), the coefficient for the first upper and lower
side frequencies, the 2nd order Bessel function, J2(I), the coefficient for the
second upper and lower side frequencies, etc. In
general, there are frequencies of significance on either side of the
carrier frequency within a band which is slightly larger than
the frequency deviation .(4)
The equation for a frequency modulated wave of peak amplitude A is
e = A sin (αt + I sin βt) Eq. 1
where
e = the instantaneous amplitude of the modulated carrier
α = 2π times the carrier frequency c
β = 2π times the modulating frequency m
I = the modulation index (d/m).
The trigonometric expansion of the above equation is in a form which
allows the determination of all sideband frequencies and amplitudes for a
carrier and modulating wave which are both sinusoidal (5).
e = A{J0(I) sin αt
+ J1(I)[sin(α+β)t - sin(α-β)t]
+ J2(I)[sin(α+2β)t + sin(α-2β)t] Eq. 2
+ J3(I)[sin(α+3β)t - sin(α-3β)t]
+ J4(I)[sin(α+4β)t + sin(α-4β)t]
+ . . . . . . . . . . . . . . . . }.
It should be noted that the lower side frequencies have alternate signs
and therefore inverting phase relationships.
The frequency potential is, then,
c amplitude = J0(I)A
+(c+m) -(c-m) amplitude = J1(I)A
+(c+2m) +(c-2m) amplitude = J2(I)A
+(c+3m) -(c-3m) amplitude = J3(I)A
+(c+4m) +(c-4m) amplitude = J4(I)A
. . Table 1
. .
. .
+(c+nm) +(c-nm) amplitude = Jn(I)A
-
upper lower
sideband sideband
carrier
for all cases where the carrier and modulating frequencies are
invariant. The amplitudes are proportional to the scaling coefficient
of the nth order Bessel function at modulaion index i. The energy is
distributed between the carrier and the side frequencies according to
the relation,
A2 = (J0(I)A)2 + (J1(I)A)2 + .......(Jn(I)A)2 .
From Table 1 and Figure 4, it is now possible to determine the spectrum in
the case above where the carrier and modulating frequencies are both
440 Hz and the frequency deviation is 220 Hz. The modulation index is
I = d/m
=220/440
=.5 .
The carrier and sideband components are then,
440 Hz J0(.5) = .95A
+(+880) Hz -(0) Hz J1(.5) = .24A
+(+1320) Hz +(-440) Hz J2(.5) = .02A
+(+1760) Hz -(-880) Hz j3(.5) = .0A
As mentioned above, the sign for the odd order lower
side frequencies is negative, which means that these components will
have an inverted phase.
It can be seen in Figure 4 that any higher order
side band component, Jn(.5) where n > 3,
will have a Bessel coefficient which is near 0 and therefore is of
decreasing significance. Figure 5a represents the spectrum for this
case.
The two lower side frequencies need some
explanation. The component at 0 Hz can be ignored since it
is a constant in the wave. The component at -440 Hz is reflected
into the positive domain at +440 Hz and since
sin(-α) = -sin(α)
its phase will be inverted and subtract from the carrier component,
Figure 5b. It is important, therefore, that one consider the
sign of the Bessel coefficient and the sign in the equation in
plotting the spectrum since the reflected side frequencies will
in some cases add and other cases subtract.
By increasing the frequency deviation to 440 Hz, the
modulation index increases to 1, and from Figure 4 it can be seen that
the amplitude of the carrier component will be reduced, while the
amplitude of the 1st, 2nd, and 3rd side frequencies will be increased.
should be noted that in this case the instantaneous frequency of
the carrier will range between 0 Hz and 880 Hz. If the frequency
deviation were increased still more, yet higher order side frequency
components should appear.
However, as can be seen in Figure 6, for
a modulation index of 4, where
d = I * m or
d = 4 * 440
d = 1760 ,
the instantaneous frequency of the carrier will become less than 0 Hz!
"Negative frequency" may be a conceptual paradox, but it does have
mathematical meaning in this case.
In order to understand the application of "negative frequency",
we will first consider the case above
where c = m = d = 440 Hz and, therefore, I = 1. Figure 7 is a graph
of the instantaneous frequency. For Eq. 1
e = A sin (αt + I sin βt) (Eq. 1)
we substitute
e = A sin (αt - I cos βt + I) Eq. 3
which includes the phase information and will allow us to see the
exact relationship between the instantaneous frequency, as in Figure 7,
and the modulated wave resulting from Eq. 3.
We have for
αt
the graph in Figure 8, where αt reaches the value 2π at the time
t0 = 1/440.
We have for
- I cos βt + I
the graph in Figure 9, having the period 1/440, since c = m.
Figure 10 is a graph of the sum of the two or,
αt - I cos βt + I
where the slope of the curve represents the rate of
change in frequency for a duration
of 1/440 sec. When the slope of the curve is 0 at k, there is no change
in angle and, therefore, the frequency is 0. Figure 11 is the sine
of the curve in Figure 10, or
A sin (αt - I cos βt + I) ,
and represents the instantaneous amplitude of the modulated
carrier wave.
If the modulation index is increased at all, the deviation
becomes greater than the carrier, the instantaneous
frequency becomes negative, and the slope in
Figure 10 becomes negative. Such a case can be seen in the graphs,
Figures 12 - 16, where I = 1.5.
As the frequency becomes negative between k and l,
Figure 12, the slope becomes negative between k and l, Figure 15, and the
angle DECREASES with time causing the mirror-like deformation of the
carrier, Figure 16. Figure 17 is 4 periods of the wave in Figure 16.
A negative slope will occur whenever the modulation index times the
modulating frequency is greater than the carrier frequency, or
Im > c.
In order to realize this condition where the angle has negative slope,
it necessary to make a small change to the algorithm of the computer
oscillator.
In MUSIC V, the oscillator is a block of instructions which operates
on a function stored in an array of n elements.
A variable, S,
is initialized such that it indexes the first element of a stored
function, F. The contents of F(S) is multiplied by an amplitude
scale factor, A, and the result is stored away.
The index is then added to an increment, I, which is proportional
to the frequency input to
the oscillator, where the higher the frequency the larger the
increment. Another value is then taken from the stored function, then scaled and
stored. When the cumulative sum of increments stored in
the index reaches a value that is larger than the function length, FL,
a "wrap around" occurs by subtracting from the index the
function length. The algorithm is, then
Oi = Ai * F(Si mod FL)
Si+1 = Si +Ii
where
Oi = the ith output sample
Ai = the ith amplitude input
Ii = the ith increment input
F = a stored function
Si = the ith sum of increments
FL = the length of the function in samples
Normally, the frequency is positive
and therefore the increment is positive, but in the case of interest
where the frequency deviation is greater than the carrier as a result
of extreme modulation,
"negative frequencies" and
therefore negative increments will occur. The
oscillator must be altered such that when the index is less than 0 the
array length will be added to the index thereby allowing a "wrap around"
in either direction. The algorithm is
IF Si ≥ 0
Oi = Ai * F(Si mod FL)
ELSE
Oi = Ai * F((Si mod FL) + FL) .(6)
With this alteration we have a rather exceptional
oscillator, for it is now capable of reversing the sampling
direction of its stored function when the frequency becomes
"negative". The reversal of direction is equivalent
to DECREASING the angle, thereby meeting the requirements of the equation.
With this form of the oscillator, then, the higher order sideband
components can be used in forming spectra.
The spectrum which results from an instantaneous change in
frequency as in Figure 6, where the modulation index is 4,
is shown in Figure 18a. The adjusted spectrum, with the "negative"
frequencies reflected in the positive domain, is shown in
Figure 18b. It can be seen that the 1st order side frequencies
are very near 0 amplitude since J1(3.8) is a zero crossing of
the function (Figure 4).
Normally, in radio applications, one thinks of the side frequencies
as being symmetrical about the carrier,
however, in this case, where the frequency deviation is
greater than the carrier, the lower side frequencies add algebraically
to the upper side frequencies, eliminating the symmetry and
causing all of the components to fall at or above the carrier
except for the component at 0 Hz.
HARMONIC SPECTRA
The significance of the case above, where the ratio of the carrier
to the modulating frequency is 1/1, is that it is a
member of the class of ratios of integers, all of whose sideband
components are a subset of the harmonic series; for example, a ratio of
c/m = 1/2 will produce sideband components which are all odd
numbered partials.
The algorithm for determining the partial numbers of the sideband
components is,
partial k = |c + nm| , Eq. 3
-
where
n = 0,1,2,3......j
and the ratio c/m is a rational number.
Tables 2 and 3 give the partial of the carrier and sideband components in
the harmonic series through
four orders for simple ratios of c/m. For orders 1 through 4, the left
number of each pair is the partial in the harmonic series of the upper
side frequency,
while the right number is the partial in the series of the lower side
frequency. A "0" means that the component is at 0 Hz. In cases where
the carrier is greater than the modulating frequency the lower sideband
may span both the positive and negative frequency domain. As an example,
in Table 3, column 2, where c/m = 3/1, the first two low order side
frequencies are not reflected (3 - 1 and 3 -2, from Eq. 3),
the third is at 0, and the fourth is reflected (3 - 4).
c/m where c ≤ m
1/1 1/2 1/3 1/4 1/5 2/3 2/5 3/4 3/5
_____________________________________________________
nth 0 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3
order 1 |2 0|3 1|4 2|5 3|6 4|5 1|7 3|7 1|8 2
side 2 |3 1|5 3|7 5|9 7|11 9|8 4|12 8|11 5|13 7 Table 2
freq. 3 |4 2|7 5|10 8|13 11|16 14|11 7|17 13|15 9|18 12
4 |5 3|9 7|13 11|17 15|21 19|14 10|22 18|19 13|23 17
c/m where c > m
2/1 3/1 4/1 5/1 3/2 4/3 5/2 5/3 5/4
_____________________________________________________
nth 0 | 2 | 3 | 4 | 5 | 3 | 4 | 5 | 5 | 5
order 1 |3 1|4 2|5 3|6 4|5 1|7 1|7 3|8 2|9 1
side 2 |4 0|5 1|6 2|7 3|7 1|10 2|9 1|11 1|13 3 Table 3
freq. 3 |5 1|6 0|7 1|8 2|9 3|13 5|11 1|14 4|17 7
4 |6 2|7 1|8 0|9 1|11 5|16 8|13 3|17 7|21 11
1 2 3 4 5 6 7 8 9
Column
Reading down a column of the tables coresponds to an increasing
modulation index which, therefore, indicates the order of entry of
the side frequencies. For example, if the modulation index is in
a range where the 4th order side frequencies become significant,
the frequency potential includes all lower order side frequencies
as well. The term "potential" has special significance. The
amplitudes of the side frequencies are determined by
a specific index in reference to the Bessel functions. Since
the functions are quasi periodic around 0, there is a possibility within
a range of values for the index, where a side frequency is of "significance",
that the coefficient will be at or near 0. As an example, an index
that is in a range between 4 and 6 will produce components including
the 2nd order side frequencies; howver, a specific index of 5.2 will
yield a coefficient near 0 for that order.
Where a frequency component is redundant within a column, it should
be noted that the redundancy is always between different orders.
The components will not cancel then, since the coefficients
for all orders are different at a given index.
There is one case where there is
cancellation of side frequencies. Figure 19 is the spectrum for
the case where,
c = 0
m = 1
I = 3 .
Because the side frequencies are symmetrical about 0, each order has
the same coefficient and magnitude, and as a result of the alternating
sign in the lower sideband, the odd orders add and the even orders
cancel. The spectrum is composed of the odd partials where m is
the fundamental.
Other spectra composed of the odd
partials occur when the ratio c/m is such that c is odd and m is even
( 1/2, 1/4, 3/4, 3/2, 5/2, 5/4).
Spectra composed of all the partials occur
when the ratio is such that m = 1 ( 1/1, 2/1, 3/1, 4/1, etc.). See
Tables 2 and 3.
INHARMONIC SPECTRA
Inharmonic spectra will result from ratios of large integers or
real numbers.
Although the spectrum may be composed of components which are
technically a subset of the harmonic series, the spread and/or
location in the series causes the spectrum to be heard as inharmonic.
Ratios in a form where the c is unity and m is a real number express
the frequency relations more directly.
Table 4 gives examples of large integer and real number ratios.
c/m
2/11 11/2 9/11 11/9 1/1.4 1/.7 1/2.1
_____________________________________________________
nth 0 | 2 | 11 | 9 | 11 | 1 | 1 | 1
order 1 |13 9|13 9|20 2|20 2|2.4 .4|1.7 .3|3.1 1.1
side 2 |24 20|15 7|31 13|29 7|3.8 1.8|2.4 .4|5.2 3.2 Table 4
freq. 3 |35 31|17 5|42 24|38 16|5.2 3.2|3.1 1.1|7.3 5.3
4 |46 42|19 3|53 35|47 25|6.6 4.6|3.8 1.8|9.4 7.4
1 2 3 4 5 6 7
Column
If m > 2c as in column 7, Table 4, all of the side frequency components
will be greater than the carrier frequency.
Figure 20 is the spectrum for the ratio of 1/1.4 of the carrier to
the modulating frequency and a modulation index
of 5. The lower sideband components are
interleaved with the upper components producing an inharmonic
spectrum.
In summary, the ratio c/m determines the distribution of the side
frequencies in the spectrum and the degree to which they are
harmonic or inharmonic, while the ratio d/m, or the modulation
index, determines the amount
of energy of the side frequencies as an index to the Bessel
coefficients. It should be understood that a particular spectrum
can be tranposed in frequency as long as the above ratios are unchanged.
For example, the shape of the spectrum defined by the relations
c = 220
m = 440
d = 1540
where
c/m = .5
d/m = 3.5 = I ,
is identical to the shape of the spectrum defined by
c = 440
m = 880
d = 3080
where
c/m = .5
d/m = 3.5 = I
since the ratios are the same. The deviation d, then, is the dependent
variable and should be computed according to
d = Im
DYNAMIC SPECTRA
A characteristic of most natural sounds is that the amplitudes of
the frequency components of the spectrum are time-dependent or dynamic.
The energy proportions of the components often evolve in complicated
ways, in particular during the attack and decay portions of the sound(7).
The evolution of the spectrum is in some cases easily followed as
with bells. In other cases the evolution is not easily followed
because it occurs in a very short time period,
but, it is nevertheless an important cue in our perception of timbre.
Many natural sounds have characteristic spectral evolutions
which in addition to providing cues to their timbral uniqueness, are
largly responsible for what we judge to be their "lifelike" quality.
In contrast, it is largly due to a fixed proportion spectrum in some
synthesized sounds that so readily imparts to the listener the
"electronic cue" and "lifeless" quality.
The special application of frequency modulation described above,
has an inherent and desirable characteristic; the complexity of
the spectrum is directly related to the modulation index.
If, then, the modulation index were made to be a function of time,
the evolution of the spectrum could be generally described by the
shape of the function.
In order to specify the modulation index as a function of time
and control the attack and decay of the output wave, it
is necessary to alter the instrument, Figure 1, by adding three more
unit generators. In Figure 21, oscillator 1 is used to impose an
amplitude envelope on the modulated wave, oscillator 2 and adder 3
together form a dynamic control of the modulation index, and oscillators
4 and 6 and adder 5 have the same functions as 1,2, and 3 in Figure 1.
The parameters for this instrument with which the user is
concerned are:
(User Parameters)
P1 = Begin time of instrument
P2 = Instrument number of a set of instruments
P3 = Duration of the note
P4 = Amplitude of output wave
P5 = Carrier frequency
P6 = modulating frequency
P7 = modulation index 1
P8 = modulation index 2
A special routine is used to convert the parametric information
specified by the user into suitable data for the computation
of the samples. In MUSIC V, this routine is usually supplied
by the user, and in this case it would perform the following
functons, where V = functionlength / sampling rate to convert
frequency into increments:
P5 = P5 * V
P6 = P6 * V
P7 = P7 * P6 , d = Im for deviation (P6 is already scaled by V)
P8 = (P8 * P6) - P7 , for deviation, (the relationship of P7 and P8
will be explained below)
P9 = 1 /P3 * V , or V/P3 , the frequency input to oscillators 1 and 2,
where the relation 1 / note duration, scaled by V to
increments, causes the functions associated with these
oscillators to be sampled at a rate such that one period
is completed in the duration P3.
Oscillator 2 and adder 3 are related in such a way that P7 becomes the
effective value of the function F2 at x,y = 0,0, while P8 becomes the effective
value of the function at x,y = 0,1. For example, if Figure 22 represents
the stored function F2, and
P7 = 2
P8 = 8
P6 = 100 Hz
P3 = .6 sec
then the output of the adder 3 would be a deviation increasing from
200 to 800 Hz in the first 1/6 sec, 800 to 450 in the next 1/6 sec, etc.
On the other hand, if
P7 = 8
P8 = 2
P6 = 100 Hz
the output of the adder would be a deviation decreasing from 800 to
200 Hz in the first 1/6 sec, etc. Having the capability of scaling
the deviation, in direct or inverse proportion to the function,
between any two values as a function of index 1 and index 2, will be shown
to be very useful in generating a variety of dynamic spectra.
In the follwing section, techniques for simulating three classes
of timbres will be defined, where the system will be that shown in
Figure 21 controlled by the user parameters listed above.
In visualizing the effect of sweeping between modulation indexes,
a careful study of Figure 23 will be helpful.(8) This is a representation
of the orders J(0) to J(15) for indexes 0 to 20, and is a sufficient
range of orders and indexes for most audio spectra discussed below.
Contour lines A, B, and C are for constant values of the Bessel
function Jn(I) = .01, .001, and .0001 respectively. Line A, then,
indicates which order side frequency is just becoming significant for
a given index. Line D represents the order of the function equal
to the argument or Jn(I) where I = n. This relation, easily
remembered, indicates that any order side frequencies greater than
the value of the index I decrease rapidly in significance. Line E
represents the absolute maximum amplitude value for each order, for
0 ≤ I ≤ ∞ . Lines F,G,H,I,J, and K show the zero crossings,(0 amplitude),
of the functions. Because the ear is not sensitive to small changes
in amplitude, this representation of the Bessel functions has proven
to provide sufficient information for most of the synthesis presented
below.
Brass-like Sounds
Risset demonstrated in his revealing analysis of brass sounds(9), that
there is a fundamental characteristic in this class of timbres; the
amount of energy in the spectrum is distributed over an increasing
band, in proportion to the increase of intensity. A simulation of
this class of timbres would be developed on the premises:
1. The frequencies in the spectrum are in the harmonic series,
2. Both odd even numbered harmonics are at some time present,
3. The higher harmonics should increase in significance with
intensity,
4. The rise-time of the amplitude is rapid for a typical attack
and "overshoots" the steady state.
Oscillators 1 and 2, Figure 21, controlling amplitude and modulation
index (deviation indirectly), will both use the function shown in Figure 22.
The parameter values for a brass-like sound are: (see user parameters)
P3 = .6
P4 = 1000 (arbitraary amplitude scaling)
P5 = 440 Hz
P6 = 440 Hz (ratio of c/m = 1/1)
P7 = 0
P8 = 5
The modulation index (therefore deviation) changes in direct proportion to
the amplitude of the carrier wave, with the result being an increase or
decrease in significance of the side frequencies in direct proportion
to the amplitude function. The ratio c/m = 1 produces components that
are in the harmonic series (see Table 2). By changing the values and
function shapes a large number of variations can be achieved. One
particularly useful variation is the addition of a small constant
to the modulating frequency. If the value .5 Hz were added,
for example, the reflected
lower side frequencies would not fall directly on the upper side
frequencies, producing a beat frequency or tremulant of 1 cps.
The brass-like quality is preserved in octave transpositions as long as
the ratios are maintained.
Woodwind-like Sounds
It is sometimes the case with woodwinds and organ pipes that the first
frequencies to become prominant with the attack are the higher harmonics
followed by the lower harmonics as the amplitude of the wave settles.
This type of spectral evolution can be achieved in several ways.
By making the carrier frequency an integral multiple of the modulating
frequency, any partial can be given emphasis during the attack.
For example, a ratio of c/m = 3/1 will give emphasis to the 3rd partial
(see Table 2). The parameter values might be as follows:
P5 = 900 Hz
P6 = 300 Hz
p7 = 0
P8 = 1 ,
and the amplitude and index function as in Figure 24. The perceived
fundamental frequency is the modulating frequency, in this case.
A ratio of c/m = 5/1 will produce bassoon-like quality in the lower
octaves. The functions remain as above.
P5 = 500 Hz
P6 = 100 Hz
P7 = 0
P8 = 1 .
The fundamental is 100 Hz. Another reed quality
can be produced by choosing a ratio of c/m which is a subset of the
odd harmonics. For example, the parameters
P5 = 900 Hz
P6 = 600 Hz
P7 = 4
P8 = 2 ,
will produce a clarinet-like quality where 300 Hz is the fundamental
(see Table 3, c/m = 3/2), and the index is inversely proportional to
amplitude function.
In all of the above examples, the realism can be improved considerably
by making the function controlling the index, the same as the amplitude
function only through the attack and steady state portions. For the
decay portion the index function remains constant. If Figure 24
is the shape of the amplitude function, then, Figure 25 would be
the shape of the index function. The evolution of the spectrum
during the attack is apparently not reversed during the decay.
Percussive Sounds
A general characteristic of percussive sounds, is that the decay
shape of the amplitude envelope is roughly exponential as shown
in Figure 26. A simulation of this class of timbres would be
developed around the following premises:
1. The spectral components are not nescessarily in the
harmonic series,
2. The evolution of the spectrum is from the complex
to the simple.
Bell-like sounds can be produced by making the change in the index
directly proportional to the amplitude function. Figure 26 is the
function for amplitude and index. The parameters are set to the
following:
P3 = 15
P4 = 1000
P5 = 200 Hz
P6 = 280 Hz
P7 = 0
P8 = 10 .
The ratio c/m = 1/1.4, Table 4. With the large initial index, the
spectrum is dense and as the amplitude decreases the spectrum becomes
ever more simple. At the time the amplitude function reaches 0, the
spectrum is composed of a sinusoid at 200 Hz. By changing the amplitude
function to that shown in Figure 27, and with the following parameter
values, a drum-like sound can be produced.
P3 = .2
P4 = 1000
P5 = 200 Hz
P6 = 280 Hz
P7 = 0
P8 = 2
A wood drum-like sound can be produced by maintaining the previous
amplitude function, but modulating the index according to the
function shown in Figure 28. The parameters are:
P3 = .2
P4 = 1000
P5 = 80 Hz
P6 = 56 Hz
P7 = 0
P8 = 25 .
These relations produce a burst of energy through a wide frequency
band for a duration of 20ms followed by a sinusoid which has the
perceptual effect of a resonance. It should be noted that an additional
amplitude modulation occurs when there is a rapid sweep of the index
over a wide range. Because the Bessel functions are quasi-periodic
around 0, each of the frequency components undergoes an asynchronous amplitude
modulation as the index, in this case, decreases.
The above examples are intended to give some feeling for the power
of this technique of synthesis, although they by no means exhaust
the potential of this instrument. With the addition of five
more unit generators as shown in Figure 29, additional control
can be gained over the spectrum. Oscillator 10 provides an additional
carrier wave whose modulation frequency is the same as the other
carrier, but whose index function can be scaled up or down by the
multiplier 8. Since the new carrier frequency is independent,
it can be set to be a multiple of the first
carrier frequency, thus providing components in another
region of the spectrum. The proportion of the two modulated carriers
is determined by the multiplier 7, which scales the amplitude function
before it is applied to the second carrier. The outputs are "mixed"
by the adder 11. With these parameter values:
P4 = 1000
P5 = 300
P6 = 300
P7 = 1
P8 = 3
P10 = .2
P11 = .5
P12 = 2100 ,
the second carrier will add components centered around the 7th
partial ( c/m = 7/1), where the index will range between .5 and 1.5, and at
an amplitude ratio of 1/5. The effect is that of an additional resonant
region added to the spectrum.
Conclusion
This means of synthesis provides a very simple temporal control over a variety
of spectra whose component frequencies can have a variety of
relationships. Because "nature" is doing most of the "work", the system
is far simpler than additive or subtractive systems which produce similar
spectra. Perhaps the most surprising aspect of this technique, is that
the seemingly limited control over the spectral components, a function
of the Bessel coefficients, proves to be no limitation at all in most
cases. This suggests that, the precise amplitude curve for each frequency
component in a complex spectrum, is not nearly as important, perceptually,
as the general character of evolution of the components as a group.
Foot Notes
1. Mathews, M.V., The Technology of Computer Music, MIT Press, 1968.
A complete description of computer synthesis and the MUSIC V
program.
2. At this point it would seem that any useful
application of frequency modulation should account for the
logrithmic perception of
frequency since in Figure 3. it can be seen that for 1/2 of
the modulating period the instantaneous frequency encompasses the octave
below the carrier while in the other 1/2 it encompasses only a fifth
above the carrier;
however, for the application described below, the frequency deviation
must be linear.
3. Tables and equations for Bessel functions are available in most of
the standard books of tables.
4. Corrington, Murlan S.,Variation of Bandwidth with Modulation Index
in Frequency Modulation, Selected Papers on Frequency Modulation,
edited by Klapper, Dover Publications, 1970.
5. Terman, Frederick E., Radio Engineering, McGraw Hill, 1947, pp 483-489.
6. The change in the code for the oscillator
in MUSIC V follows.
for
290 IF(SUM-XNFUN)288,287,287
287 SUM=SUM-XNFUN
substitute
290 IF(SUM.GE.XNFUN)GO TO 287
IF(SUM.LT.0.0)GO TO 289
and
for
GO TO 293
292 J6=L1+J3-1
substitute
GO TO 293
287 SUM=SUM-XNFUN
GO TO 288
289 SUM=SUM+XNFUN
GO TO 288
292 J6=L1+J3-1
7. Risset************
9. Corrington
8. Risset