The score generation process can be considered as a scheme shown
below. Every note pfield pn,
i.e. instrument number p1,
start time p2, duration
p3 and all other parameters
p4, p5 ... will be calculated according to
its own realization of this scheme.
The blue moduls are number generators: constant, list, segment
function, random and oscillator. They can be used only alternatively,
either RND or OSC or SEG and so on.
The grey moduls are modifier: tendency mask, quantizer and
accumulator. QUANT and ACCUM are optional, they can be bypassed as
the red lines show.
The result of this process is a value for the note parameter.
A group of note parameters forms a field. A field is valid for a certain time span.
The description of one or more fields is contained in a text
file.
This file acts as input for CMask, similar to the .orc and .sco files
for Csound.
CMask's output is a score file that can be used directly for the
sound generation with Csound.
The next sections introduce some of the most important principles used in CMask.
The first stage in the score generation process is the generation
of "raw" numbers for a note statement pfield. "raw" means that they
are designated for a further processing as the scheme above
indicates. One of the possible number generators is the RND module.
Its random values follow a certain probability distribution. These
distributions are also implemented in Csound as unit generators (the
xlinrand family) and GEN
routines (GEN21). The
simplest of them is the uniform distribution. Every value in the
random generators range {0,1} has the same probability to appear,
like the six sides of a dice.
rnd uni
From a global point of view there is no preferred area in the
whole range from a certain minimum to a maximum ( 0.0 to 1.0. in case
of figure 1.1). Every normal random function in a programming
language should have a similar behavior as well as numbers in lotto
and roulette. Another word for random numbers in a succession is
noise. Uniform distribution corresponds to white noise. The rand unit generator in Csound produces
uniform distributed numbers and is therefore a white noise
generator.
A more important distribution for describing natural processes by
mathematical statistics is the gaussian or normal distribution. It
prefers numbers in the middle of a range. Two values mark this middle
area: the mean and the standard deviation. The mean is the number in
the range where we have the most frequent values. The standard
deviation ist the width around the mean. Figure 1.2 shows the
gaussian distribution (solid line) and the variation of both
parameters (dotted lines): a smaller mean and a smaller deviation.
A gaussian random generator with a mean of 0.5, a standard deviation of 0.2 and a range of 0.0 to 1.0 produces values shown in figure 1.3.
rnd gauss .2 .5
The numbers in the next picture was generated by the same gaussian generator except for the mean which was set down to 0.2.
rnd gauss .2 .2
Several probability distributions are available in Csound and CMask: uniform, linear, exponential, gaussian, beta and others. For a detailed desription of the various distributions refer in [1][2][3][a]. Some of them have parameters like the gaussian distribution. The exponential distribution for example, which is characterized by a dominance at the lower end of the range, is controlled by a value lambda. A higher lambda describes a stronger dominance of low values. A lambda near 0 results in a uniform-like distribution. If we change that lambda value during the generation of successive random values we can control the the relative appearence of lower values over the time. This is comparable to a low pass filter with a variable cut-off frequency. Below is shown the output of an exponential distribution generator (figure 1.5), whose lambda goes from 0.1 to 3.0 at the middle of the process and finally back to 0.1 (figure 1.6).
rnd exp (0 0.1 5 3.0 10 0.1)
Every random generator in CMask is limited to the range {0,1}. In the next score generation stage the numbers within this range will be mapped to a new range required for real Csound note pfields, for example frequencies between 400Hz - 1000 Hz or durations between 0.1 - 2.0 seconds. This is done by tendency masks.
A mask is a time variant range, which is described by the lower
and the higher limit. The limits itself can be constant or
time-variant. The mapping from the random range {0,1} to the mask's
range is a simple linear function: 0 goes to the lower limit of the
mask, 1 to the upper limit.
Let's take a uniform distribution and a mask with a constant lower
limit of 200 and a constant higher limit of 400. If we regard these
values as frequencies and use the numbers in audio oscillators, we
get random pitches approximately between g3 and g4:
mask 200 400
In the next example we have a real tendency in the mask. The lower limit goes from 200 to 600, the higher limit goes from 400 to 600 too. That means the range will be smaller and higher over time. At the end, where the limits have the same value, all the random values are mapped onto 600 - there is no other number possible than this number at this point!
mask [200 600] [400 600]
Figure 2.3 shows the same higher limit as in figure 2.2. But the lower limit now consists of three points: it starts at 200, goes to 10 at the middle and ends again at 600.
mask (0 200 5 10 10 600) [400 600]
Each boundary can be described by an unlimited number of points. Every point is a pair of a time value and a function value. The function values between the specified points will be computed by interpolation, like in the linseg and expseg unit generators. The normal case is a linear interpolation, but there are others. Here we have again the same mask with a negative exponential interpolation for the lower limit:
mask (0 200 5 10 10 600 ipl -1) [400 600]
More on tendency masks in [5][7][b][c].
We have seen that a tendency mask is a time-dependent random
range. Every note (i-statement) in Csound has its onset time
p2. This is the time at
which CMask evaluates the current range of each pfield p1...pn. The onset time p2 itself can also be generated by a
random function and a mask. The p2 value in a Csound score is normally
an absolut time measured in seconds (if the tempo is 60). But in
CMask it is necessary to handle p2 as the time difference or interval
between two successive events.
Figure 3.1 shows the first values of p2 generated by a uniform random function
in a constant range from 0.01 to 1 seconds (red diamonds). The next
onset time (black dots) is always the current time plus the random
value.
p2 rnd uni
mask .01 1The first time is 0 seconds. The generated value at this time is
by chance 0.75. Therefore, the next time is 0.75. At 0.75 the random
value is about 0.5, the new time is 0.75+0.5=1.25. And so on.
Now, we look at an almost complete example. Apart from random
functions and masks there are other possibilities for determining
pfield values: constants and segment functions.among other
things.
A group of events, that share the same masks, random functions etc.,
is called a field. The first line in the example below is the field
header with start and end time of the field. p1, the instrument number, is set to 1,
that is every note in this field belongs to instrument 1. The onset
time or the rhythm value comes from a uniform distribution between
0.01 and 1.0. p3 (duration)
and p4 (a frequency ?)
remain constant. p5 is in a
random range between 100 and 200. range is a shorthand for uniform
distribution and an unchanging mask range. p6 is generated by a segment or
break-point function with linear interpolation. One can regard
seg as a mask, whose lower and
upper bounds are the same. The list after seg contains 3 points: 150 at 0, 300 at 5
and 10 at 10 seconds.
f 0 10
p1 const 1
p2 rnd uni
mask .01 1
p3 const .1
p4 const 100
p5 range 100 200
p6 seg (0 150 5 300 10 10)
The diagram shows one possible output of this field for p4 (red diamonds), p5 (green dots) and p6 (blue triangles) over the 10
seconds. Note, that the three pfield values have always the same
time. This is the current onset time. You can see p2 as the gap between these times.
Another Example, now with a time-dependent range for p2. The brackets [] are a special notation for a segment
function that has only two points, one at the start time of the field
and one at the end. The number after the word ipl is the optional interpolation value.
The default is 0 - this is linear. 1 results in a slightly
exponential rise or decay. The mask for p4 has three points for both limits.
f 0 10
p1 const 1
p2 rnd uni
mask [.01 .5 ipl 1] [.02 1 ipl 1]
p3 const .1
p4 rnd uni
mask (0 100 2 500 6 300) (0 1000 2 500 6 1000)
Figure 3.3 depicts the mask for the rhythm values in red lines. These
values are very small at 0 seconds and rise to {0.5, 1.0} at the end.
The gray lines mark the mask of p4, the dots are the random values.
Note, that both last points of the mask are at the 6th second, their
values keep constant from now on. CMask generally uses the first
value before its time and the last value after its time - this is
different to the behavior of the line and expon modules in Csound were we have an
extrapolation of the funtion.
The next optional step in the score generation process is the
quantization. Three dynamical parameters determine this quantization:
the quantization interval, the offset and the strength. The interval
or the quantum can be regarded as the width of a grid. In figure 4.1
we see the output of a random generator in a constant mask between
100 and 400 after a subsequent quantization with an interval of 70
and a maximum strength of 100%. The multiples of 70 that fall in the
range {100,400} are 140, 210, 280 and 350. Figure 4.2. shows the same
interval but only a 70% strength.
range 100 400
quant 70 1
range 100 400
quant 70 .7
The strength is a kind of attraction. 0% means no quantization at
all. 50% means that every random number is attracted to the half
distance between this random value and the next grid value. 100%
means that all numbers go to their next grid point. The strength and
the other quantization parameters can be given as a constant or as a
segment function. Example 4.3 results from a dynamic strength that
goes from 0 to 1.
range 100 400
quant 70 [0 1]
The offset is a shift of the quantization grid. With an interval
of 70 the grid is ... -140 -70 0 70 140 210 ... . With an offset of
20 we have ... -120 -50 20 90 160 230 ... . The next picture shows a
linear rising offset between 0 at start and 70 at the end. The result
is a grid wrapped around the masks boundaries. If the values were
frequencies we would hear a shepard grain stream.
range 100 400
quant 70 .9 [0 70]
An elaborated concept of quantization - the sieves - can be found
in [8].
The last step in the score generation procedure is an optional accumulation of the up to now generated and modified numbers. This is done by a simple addition of all values to an inital value (0 by default). The example shown in the next figure is generated by the following code fragment:
p4 rnd uni
mask -50 50
accum on init 200The uniform random values was mapped into {-50,50} and then added to 200. The result looks like the walk of a drunken man/woman. Another name for this thing is brownian motion or 1/f^2 noise.
An accumulator can also have its own boundaries, like a mask, in order to prevent too small or too high values. Here we have constant limits at 100 and 400. One can see how the drunken dots knock against the upper wall.
accum limit 100 400 init 200
There are 2 other accumulation modes. The wrap mode handles the lower and the upper bounds as they were sticked together like a (cylindrical) tube. That means that a value above the upper limit, for example 420, comes out above the lower bound, in this case at 120.
accum wrap 100 400 init 200This section describes some applications of CMask and the instrument designs belonging to them.
The term texture stands for a musical object that has a
comparatively weak inner structure. Its counterpart is an object that
have a strong structure like a melody, a theme, a rhythm. Natural
sound textures can be found were many similar acoustic sources sound
together: applause, public chatter, crowds of birds, bees or
mosquitos, stones in an avalanche, the tuning of the orchestra ...
[7][8].
The first example uses a simple FM bell. Instrument 1 has 2
additional parameters: the base frequency and a panorama value
between 0 (left) and 1 (right).
;;; texture1.orc -----------------------
sr = 44100
kr = 4410
nchnls = 2
instr 1
;p4 frequency
;p5 pan (0...1)
ipanl table 1-p5,1,1
ipanr table p5,1,1
k1 expon 1,p3,.01
a1 foscil k1*4500,p4,1,2.41,k1*6,2
outs a1*ipanl, a1*ipanr
endin
;;; texture1.orc -----------------------The CMask parameter file texture1 contains the
necessary ftables for panorama crossfade and the sine wave. The
longer field (f 0 30)
consists of three processes: rhythms values gets smaller (p2), the frequency mask gets wider
(p4) - the frequencies
itself are bound to an overtone series build on 100 Hz - and the
stereo image gets broader (p5). The mapping value 1 in p4 guarantees more lower frequencies.
The shorter field (f 31 33)
describes a ritardando with random frequencies between 300 and 400 Hz
and a motion from left to right.
;;; texture1 ---------------------------
{
f1 0 8192 9 .25 1 0
f2 0 8193 10 1
}
f 0 30
p1 const 1
p2 rnd uni
mask [.01 .002 ipl 0] [.1 .01 ipl 0]
p3 range .5 1
p4 rnd uni
mask [860 80 ipl -1.2] [940 2000 ipl 1] map 1
quant 100 .9 0
p5 mask [.4 0] [.6 1]
f 31 33
p1 const 1
p2 seg [.08 .8 ipl 2]
p3 seg [.1 2]
p4 range 300 400
p5 seg [0 1]
;;; texture1 ---------------------------CMask produces with this parameter file a Csound score file
texture1.sco with about 1370 events in the first field
and 15 events in the second field.
;;; doors.orc -----------------------
sr = 44100
kr = 4410
nchnls = 2
garev init 0
instr 1
;p4 transposition (1=normal)
;p5 table number (1...6)
;p6 pan (0...1)
;p7 dry/wet (0...1)
ipanl table 1-p6,10,1
ipanr table p6,10,1
k1 expon .5,p3,.01
a1 loscil k1,p4,p5,1,0,0,2
a1 linen a1,0,p3,.05
garev = garev + a1*p7
a2 = a1*ipanr
a1 = a1*ipanl
outs a1*(1-p7*p7), a2*(1-p7*p7)
endin
instr 99
krev expon .03,p3-4,5
a1 reverb2 garev,krev,.2
a2 reverb2 garev,krev*1.1,.21
outs a1/2,a2/2
garev = 0
endin
;;; doors.orc -----------------------The doors parameter file begins with the GEN01 sound
file tables and the panorama function. Instrument 99, the
reverberator, sounds all the time. Rhythm values for instr 1 can be chosen from the range
{0.01, 0.1}. The beta distribution for p2 ensures that we get small groups or
rather gaps between events like stumbling. The durations increase
from 300 msecs to 2 seconds. p4 - the transposition factor - goes
from a range {3,5} to {0.1,0.2}. This means that we have values about
2 octaves higher at start and finally more than 2 octaves lower.
p5 selects the ftable from
a range between 1 and 6 with a precision of 0, i.e. as integer. The
reverb balance is controlled by p7.
;;; doors -------------------------- { f1 0 0 -1 "door1.aiff" 0 4 1 f2 0 0 -1 "door2.aiff" 0 4 1 f3 0 0 -1 "door3.aiff" 0 4 1 f4 0 0 -1 "door4.aiff" 0 4 1 f5 0 0 -1 "door5.aiff" 0 4 1 f6 0 0 -1 "door6.aiff" 0 4 1 f10 0 8192 9 .25 1 0 i99 0 23 } f 0 20 p1 const 1 p2 rnd beta .05 .1 mask (12 .01 18 .2) (12 .1 18 1) p3 seg [.3 2 ipl .4] p4 mask [3 .8 ipl .4] [5 1.2 ipl .4] p5 range 1 6 prec 0 p6 range 0 1 p7 seg (2 0 18 .5 ipl 1) ;;; doors --------------------------
Granular synthesis uses mostly sound files instead of synthetic wave forms and is therefore actually a mix of sampling and resynthesis. Granulation is the term for cutting a sound into small pieces - the grains. Their duration is normally very short: about 5 to 50 msecs. This time range marks also the transition area from low pitches to fast tempi. (It is not by chance that this is also the domain of frame rates in movies and vertical frequencies in monitors.) The sound quality of a single grain is among other things determinated by its envelope. There are many different envelopes or window functions for granular synthesis: real-time systems use often triangles, trapezoids or simple rectangles, but to have a cleaner sound it is advisable to use smoother functions like hamming or hanning windows, splines or phase shifted sine waves. Figure 6.1 shows a spline function.
The granulation might be considered as a meta-sampling process: every grain contains a small sample from the whole sound. First we take same grains - like snapshots with a camera - and then we rearrange them in any order and distance. For many applications it is necessary that a sequence of grains has a proper overlap. Depending on the envelope shape it is then possible to reconstruct the original soundfile. To do this, some conditions have to meet: the sum of the grain envelopes must have always the same amount at every time, and size, distance and order of the grains have to be the same as before the rearrangement.
Now, there are unlimited possibilities to transform a sound file
by granulation. We have many changeable parameters such as grain
duration, grain envelope and amplitude, grain distance or overlap and
grain order, as well as all the other traditional transformation
techniques like filtering, modulation, positioning in space (angle,
distance and reverberation) and so on.
A simple time stretching technique, for example, uses the overlap
factor as transformation parameter. If we rearrange the grains in the
same order but with a half overlap factor we will get a sound that
takes the double time. In order to prevent gaps in granular time
stretching it is better to have a higher overlap while scanning the
sound file and a normal overlap of 2 while rearranging. To shorten a
sound we have to multiply the overlap factor or rather shorten the
distance between the grains. The principle of stretching is shown in
figure 6.3. The first row is the original order, the second is the
new order.
For further studies on granular techniques see
[4][5][6][7][c].
In order to granulate a sound file with Csound and CMask we need some
preparation! Since we let CMask compute the grains as score notes it
is advisable to load the complete sound file in a GEN01 table instead of using soundin , in order to prevent thousands
of hard disk accesses during the sound generation process. Normally
we can use GEN01 ftables only in
conjunction with loscil if
the sound file size is unequal to a power of 2. But loscil is not very useful for our
purposes, so we have to lengthen the sound with silence or cut to the
nearest power of 2. For example:
f1 0 262144 1 "sound" 0 4 1If sr = 44100 this table
contains 5.94 seconds regardless of the original sound file
length.
Suppose we want to play a certain part of the ftable 1 at a certain
time p2 for a certain
duration p3. Then we must
have a time pointer or an index to this part. The following code
fragment shows a solution. It makes use of p4 as initial index measured in
seconds.
andx line p4,p3,p4+p3
asig table andx*sr, 1The index andx goes from
ftable time p4 during a
time p3 to ftable time
p3+p4. The multiplication
andx*sr converts seconds to
sample frames so that we have a raw index. ksig provides the envelope and a proper
amplitude if the GEN01 table is
normalized.
ksig oscil 20000,1/p3,2
out asig*ksigThe corresponding ftable 2 contains for example a phase shifted sine wave with a DC offset as envelope function:
f2 0 8192 19 1 1 270 1The macroscopic organization of the grains is the business of CMask. For a simple constant time stretching we need a constant time advance and duration:
p2 const .02
p3 const .04The overlap is set by the relation of p3 to p2 - in this case the factor is 2. The
grain pointer p4 goes over
the ftable's size with a constant speed:
p4 seg [0 5.9]
Now, let's have a look at a complete example.
;;; axa1.orc --------------------------
sr = 44100
kr = 4410
nchnls = 2
instr 1
;p4 grain pointer (in seconds)
;p5 pan (0...1)
ipanl table 1-p5 ,4,1
ipanr table p5 ,4,1
andx line p4,p3,p4+p3
asig table andx*sr,1
k1 oscil 30000,1/p3,2
asig = asig*k1
outs asig*ipanl, asig*ipanr
endin
;;; axa1.orc --------------------------The sound file used here is speech in a fictious language:
;;; axa1 ------------------------------
{
f1 0 65536 1 "axaxaxas.aiff" 0 4 1 ; 65536 samples @ 44.1 = 1.4861 sec
f2 0 8193 19 1 1 270 1 ; grain envelope
f4 0 8192 9 .25 1 0 ; pan function
}
f 0 5
p1 const 1
p2 const .02
p3 const .04
p4 seg [0 1.44]
p5 const .5
;;; axa1 ------------------------------
The interpolation exponent in the segment function for p4 is set to 0 by default. This means
that p4 follows a linear
function. The value of p4
begins at 0 and ends after 5 seconds (the field's duration) at 1.44.
Therefore we have a stretch factor of 5/1.44=3.47. But
if we change that value to -2, for example, we get a nonlinear
function: fast rising at the beginning and very slow in the end.
p4 seg [0 1.44 ipl -2] That is, the scanning overlap factor goes from small to large and
we get a dynamical time stretching: from fast to slow like a
ritardando.
The reverse principle - an accelerando effect - is shown in figure
6.4:
With a little change in the definition for p4 it is easy to get a texture of
grains instead of time stretching:
p4 range 0 1.44 This texture has a constant density because p2 and p3 are constant. To get a more random
texture we can write the following for example:
p2 mask .005 .1 map 1
p3 range .04 .2 Figure 6.5 shows the scheme of a random granular texture:
A mean between dynamic timestretching and a complete random structure can be found in random walks. The mask produces time steps between 2 and 50 msecs and the accumulator adds them together up to a maximum of 1.4. The wrap mode causes that the next steps starts again from 0.
p4 mask .002 .05 map 1
accum wrap 0 1.4
The last example uses an instrument similar to the previous
instr 1. Two small changes
happened here:
andx line p4,p3,p4+p3*p6
asig tablei andx*sr,1Parameter p6 is a factor for transposition. Supposing
p6 = 2 the index goes from
p4 over the time
p3 to a position that has
the double distance compared to p3. The grain comsumes now a part of
the sound file that is larger than its own duration. In case of
p6=2 this is the double
speed or a transposition to 1 octave higher. With this technique we
can transpose a sound without changing its length. If p6 will be very small it is
advisable to use an interpolating table to reduce the noise
level.
This example is a slightly random time stretching partly with
quantized transposition.
;;; whisper.orc --------------------------
sr = 44100
kr = 4410
nchnls = 2
instr 1
ipanl table 1-p5 ,4,1
ipanr table p5 ,4,1
andx line p4,p3,p4+p3*p6
asig table andx*sr,1
k1 oscil 8000,1/p3,2
asig = asig*k1
outs asig*ipanl, asig*ipanr
endin
;;; whisper.orc --------------------------
;;; whisper ------------------------------
{
f1 0 262144 1 "whisp.aiff" 0 4 1 ; = 5.94 sec
f2 0 8192 19 1 1 270 1 ; grain envelope
f4 0 8192 9 .25 1 0 ; pan function
}
f 0 60
p1 const 1
p2 mask (0 .0005 37 .007 60 .003) (0 .003 37 .15 60 .005)
p3 mask [.3 .02] [.7 .04]
p4 seg [0 5.9]
p5 range 0 1
p6 mask (0 .3 25 1 40 .7) (0 2 4 1 25 1.2)
quant .3 (0 0 25 .9 30 0 45 .9 55 0) (40 0 45 1.5 55 0)
;;; whisper ------------------------------CMask runs currently only on Macs, PowerMacs, SGI IRIX 5.3. and
WIN95
Program, manual and examples are available at:
For further information about CMask and other utilities as well as
computer music activities in Berlin check out my website.
http://www.kgw.tu-berlin.de/~abart/
[1] Ames, Ch. 1991. "A Catalog of Statistical
Distributions..." Leonardo Music Journal 1(1)
[2] Dodge, Ch. / Jerse, Th. A. 1985. Computer Music Collier
Macmillan, London
[3] Lorrain, D. 1980. "A Panoply of Stochastic 'Cannons'."
Computer Music Journal 4(1)
[4] Roads, C. 1991. "Asynchronous Granular Synthesis." in
Representations of Musical Signals, MIT Press
[5] Roads, C. 1985. "Granular Synthesis of Sounds." in
Foundations of Computer Music, MIT Press
[6] Truax, B. 1988. "Real-time Granular Synthesis with a Digital
Sound Processor." Computer Music Journal 12(2)
[7] Wishart, T. 1994. Audible Design Orpheus the Pantomime
[8] Xenakis, I. 1992. Formalized Music Pendragon Press
[a] Castine, P. Litter Package (Externals for MAX)
[b] Koenig, G.M. Project I, Project II
[c] Truax, B. POD programs
[d] Xenakis, I. Stochastic Music Program
March 1997, Andre
Bartetzki @ STEAM, HfM Berlin