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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,
Amp = 2A{J0(I) + J1(I) + .......Jn(I)} .
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 and References
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 J.C., ?????Study of Trumpet Tones????Bell Labs
8. Corrington, op. cit.
9. Risset, op. cit.