Description
BACKGROUND OF INVENTION
Field of Invention
This invention relates generally to a technique for
manipulating the nominal notation values of a musical score with
respect to the amplitude contour of individual tones, the
reactive loudness of different tones, slight changes in tone
duration and other deviations from the nominal values which
together constitute the expressive microstructure of music. More
particularly, the invention deals with a computerized system
capable of manual or automatic operation for impressing an
expressive microstructure on a musical score inputted therein in
terms of nominal notational values, the system being usable for
composing music that includes microstructure.
Music has been defined as the art of incorporating intelligible
combinations of tones into a composition having structure and
continuity. A melody is constituted by a rhythmic succession of
single tones organized as an aesthetic whole. The standard
system of notation employs characters to indicate tone, the
duration of a tone (whole, half, quarter, etc.) being
represented by the shape of the character and the pitch of each
tone by the position of the character and the pitch of each
notation, a melody is a musical line as it appears on the staff
when viewed horizontally.
While the notation of a musical score gives the nominal values
of the tones, in order for a performer to breathe expressive
life into the composition, he must read into the score many
subtilties or nuances that are altogether lacking in standard
notation. Some expressive subtilties are introduced as a matter
of accepted convention, but most departures from the nominal
values appearing in the score depend on the interpretive power
of the performer.
Thus, a musical score, while it may indicate whether a section
of the score is to be played loudly (forte) or softly (piano),
does not generally specify the relative loudness of component
tones either of a melody or of a chord with anything approaching
the degree of discrimination required by the performer. The
performer decides for himself how loudly specific notes are to
be played to render the music expressive.
Even more important to an effective performance is the
amplitude contour of each tone in the succession thereof. To
satisfy musical requirements, the amplitudes of the tones must
be individually shaped. Though, in general, amplitude contours
are completely unspecified in standard notation, each performer,
such as a singer or violinist who has the freedom to shape
tones, does so in actual performance to impart expressivity
thereto. Indeed, with those instruments that lend themselves to
tone shaping, variations in the amplitude shapes of the tones
constitute a principal means of expression in the hands of an
expert performer.
Another factor which comes into play in the microstructure of
music are subtle deviations from the temporal values prescribed
in the score. Thus, in actual performance, to avoid temporal
rigidity which dehumanizes music, the performer will in actual
practice amend the nominal duration values indicated by standard
notation. These nominal values are arithmetic ratios of simple
whole numbers such as 1/2, 1/3, 1/4, 1/16, etc.
Yet another expressive component of music which is unspecified
in the score is the timbre to be imparted to each tone; that is,
the harmonic content thereof. A performer of a string
instrument, by varying the pressure and velocity of the bow on
the string, can give rise, not only to variations in the
loudness of the tone, but also variations in its tonal timbre
independently of loudness. On a wind instrument, the performer
can achieve similar effects by changes in lip pressure and wind
velocity.
In short, the macrostructure of a musical composition is
defined in the score by standard notation. If, therefore, one
executes this score by being assiduously faithful to its
macrostructure, the resultant performance, however expertly
executed, will be bereft of vitality and expression. The term
"microstructure" as used herein encompasses all subtle
deviations from the nominal values of the macrostructure in
terms of amplitude shaping, timing, timbre and all other factors
which impart expressiveness to music.
It has been found that the measure of expressivity that can be
attained using only sinusoidally-generated tones whose amplitude
is shaped is quite surprising, despite the absence of harmonics
which enrich the tones. Our ears appear to be highly sensitive
to changing amplitudes and shapes, and our memories can
effortlessly detect relative amplitudes and amplitude-shapes
sounded in sequence in a musical context, even when the
corresponding tones are sounded up to 10 or 15 seconds apart
with many other tones in between.
This faculty of short term memory for comparing tone amplitudes
and tone shapes makes it possible for the typical listener to
distinguish between identical forms that are mostly perceived as
mechanical and monotonous, and slightly varying forms; for the
latter, played even a few seconds apart, are perceived in
relationship to one another and can produce varied meaning and
vitality. Thus, the relationship which constitute the
microstructure in music, though not explicit in the score, are
vital to its appreciation.
Essential also to an understanding of controllable
microstructure in a system in accordance with the invention are
A essentic forms; that is, the dynamic expressive forms of
specific emotions; and B, the inner pulse of composers. These
will now be separately considered.
As explained in an article by Clynes and Nattheim (pp 47-82)
included in Music, Mind and Brain: The Neuropsychology of Music,
M. Clynes (ed.), Plenum Press, New York (1982), touch
expressions of specific emotions such as love, grief and hate,
can be transformed into sound expressions of like emotions;
i.e., the nature of the transforms was found so that the sound
expresses the same emotional quality as the touch expression
from which it is transformed.
The touch expressions are measured by recording the transient
forms of finger pressure when these are voluntarily expressed.
The instrument enabling this measurement to be made is called
the Sentograph; it measures both the vertical and horizontal
components of finger pressure independently as vector components
varying with time. The sentographic forms obtained are stored in
a computer memory and can be reproduced at will. (See Clynes
U.S. Pat. No. 3,755,922, "System for Producing Personalized
Sentograms" which discloses in greater detail the nature of
essentic forms and how sentographs are produced.)
In transforming the sentograms for touch expression of specific
emotions into corresponding sound expressions it was found that
the dynamic form (essentic form) of the touch was preserved to
become the frequency contour of the sound. The sound is a
frequency and amplitude modulated sinusoid. The amplitude
modulation also was related to the dynamic touch form but needed
to be passed through an imperfect differential network.
These expressive sound shaped of "continuous" frequency
modulation are related to melodies which represent a form of
"discrete" frequency modulation. As spelled out in the
above-identified Clynes and Nattheim article, it is possible to
create musical melodies which express an emotional quality
similar to dynamic, expressive sound forms in such a way that
the melodic steps of the created melody act to outline the
frequency contour of the "continuous" form, and the amplitude
contour of the expressive sound is preserved. As different
melodic steps are chosen, the durations would be constrained so
as to conform to the "continuous" frequency modulation wave.
(essentic form).
The inner pulse of specific composers such as Beethoven, Mozart
and Schubert, are expressed by sentographic forms obtained by
having an individual think the music of a selected composer in
his mind and by concurrently expressing the pulse by "conducting
the music on a sentograph with finger pressure. The resultant
sentograms indicate that major composers, such as those
previously identified, impart individual pulse forms to their
music which characterize their creativity identity or personal
idiom. This inner pulse characteristic of each composer is, to a
degree, analogous to individualistic brush strokes which
distinguish one painter from another, regardless of the subject
matter of their paintings. (See M. Clynes, "Sentics, The Touch
of Emotions"- published by Doubleday - 1977.)
SUMMARY OF INVENTION
In view of the foregoing, the main object of this invention is
to provide a computerized system for processing the nominal
values of a musical score to impart an expressive microstructure
thereto.
More particularly, an object of the invention is to provide a
system of the above type which is operable in a manual mode in
which the values representing the microstructure are entered by
the user, or in the automatic mode wherein microstructure values
are calculated from a shaping function responsive to the melodic
contour and by means of pulse matrices having certain values
relating to the amplitude and duration of pulse component tones
stored in the system.
Among the significant advantages of a system in accordance with
the invention is that it makes it possible to enliven a musical
score and render it highly expressive, the system thereby acting
to deepen the user's understanding and appreciation of music and
its structure.
Briefly stated, these objects are attained in a computerized
system into which is fed the nominal values of a musical score,
the system acting to process these values with respect to the
amplitude contour of individual tones, the relative loudness of
different tones in a succession thereof, changes in the duration
of the tones and other deviations from the nominal values which
together constitute the microstructure of the music notated by
the score. The system performs the specified tones in the score
as modified by the microstructure, thereby imparting
expressivity to the music that is lacking in the absence of the
microstructure. The microstructure may include changes in pitch
or vibrato.
OUTLINE OF DRAWINGS
For a better understanding of the invention as well as other
objects and further features thereof, reference is made to the
following detailed description to be read in conjunction with
the accompanying drawings, wherein:
FIG. 1 illustrates a family of curves generated from one
set of beta function values;
FIG. 2 illustrates a family of curves generated from a
second set of beta function values;
FIG. 3 illustrates a family of curves generated from a
third set of beta function values;
FIG. 4 illustrates a family of curves generated from a
fourth set of beta function values.
FIG. 5 illustrates a family of curves generated from a
fifth set of beta function values.
FIG. 6 illustrates a family of curves generated from a
sixth set of beta function values;
FIG. 7 illustrates the notes of a Mozart theme below
which are three graphs representing different microstructural
aspects of the theme;
FIG. 8 similarly illustrates the notes of a Chopin
theme;
FIG. 9 similarly illustrates the notes of a Beethoven
theme;
FIG. 10 is a block diagram of a computerized system in
accordance with the invention which is operable in the manual
mode;
FIG. 11 is a graph illustrating how an amplitude shaped
tone is obtained from a succession of digital values yielded by
the calculator included in the system;
FIG. 12 shows both a sinusoidal wave and a
differentiated square wave which, when combined, produce a
non-sinusoidal wave rich in harmonics; and
FIG. 13 illustrates in block form a portion of a system
operating in the automatic mode.
DESCRIPTION OF INVENTION
Microstructure and Expressiveness
Consideration will first be given to the extent to which
detailed expressiveness is possible without vibrato and timbre,
using only sinusoidal tones with individually-shaped amplitude
envelopes, representing the notes of the music, with duration
modification. Such amplitude modulation appears to be the most
basic mode of dynamic expression and thus the understanding of
this should precede the analysis of the expressive effects of
vibrato, timbre and of timbre variation. It has been found that
a significant degree of expressiveness is indeed possible with
these very simple means, and that many of the subtlest nuances
can be realized.
In doing this, we shall also point out the relationship
existing between the amplitude shapes of individual tones and
the shape of the melody (i.e. essentic form). We shall show that
a simple relationship holds between the shapes of individual
tones of a melody and the form of its time course. Its shape
within a continuum of upright ("assertive") and forward sloping
("plaintive") form is seen to be a function of the slope of the
essentic form. (Assertive will hereafter be represented by the
letter A and plaintive by the letter P) This means in practice
that the shape of a tone is strongly and predictably influenced
by the pitch of the next tone and by the time when it occurs.
The shape of individual tones has a musically predictive
function contributing to continuity, and thus, we may say, to
musical logic.
We shall also demonstrate how the individual pulse
characteristics of a number of composers, in particular Mozart,
Beethoven and Schubert, systematically modulate the pattern of
tones both in amplitude and temporally, in characteristic ways
which will be precisely given. In the course of these
considerations we shall also outline how certain microstructure
functions, involving durations, silences, and amplitude shapes,
relate to various aspects of phrase and larger structure.
Beta Function for Calculating Amplitude Shapes
In order to produce convenient shapes for amplitude-modulating
individual tones, we have used a mathematical means, briefly
called the Beta Function. This term derives from a similar-named
function in mathematic statistics. The Beta Function permits us
to create a wide variety of shapes with the aid of only two
parameters (P.sub.1 & P.sub.2). It has a considerably wider
applicable scope than the use of two exponential would have, for
example.
In electronic generation of musical sounds, it has heretofore
been conventional to specify tones using parameters of rise
time, decay time, sustain time, release time and final decay, or
some subset of these. These parameters, natural to the
electronic engineer, do not really have a musical function of
like aptness. Amplitude shapes of musical tones often need to be
convex rather than concave (or vice versa) in particular
portions of their course (e.g., convex in their termination),
and hardly ever have sustained plateaux. Moreover, separation of
the termination of a tone into a decay and a release is
generally the result of the mechanical properties of keyboard
instruments and not a musical requirement.
We have found that the varied rounded forms available through
the Beta Function allow a simple and time-economical realization
of the multitude of nuances of musical tone amplitude forms.
Beta Function is defined as:
x.sup.p.sbsp. (1-x) .sup.p.sbsp.2 for 0.ltoreq.x.ltoreq.1,
Equation 1
and is normalized for a maximum amplitude of 1 by dividing by a
constant. ##EQU1## for a particular set of values of p.sub.1 and
p.sub.2.
p.sub.1, p.sub.2 have values .gtoreq.0.
The resulting shape is multiplied by a parameter G to give the
amplitude size of the particular tone. The shape stretches over
a number of points determined by the duration of the tone.
By choosing suitable values of p.sub.1 and p.sub.2, a shape may
be selected from families of shapes such as the ones shown in
FIGS. 1 to 6. Choosing the value 1 for both parameters gives
rise to a symmetrical, rounded form, and 0.89 for both
parameters produces a form very close to a sine half wave.
Smaller values of p.sub.1 result in steeper rises; zero being a
step function. Larger values than 1 for either p.sub.1 or
p.sub.2 make the curve concave, at the corresponding regions. A
combination of zero and 1 results in a sawtooth.
Most commonly used p values for musical tones generally lie
within the region of 0.5 to 5, and most frequently in the region
of 0.7 to 2. Where required, a second or several more Beta
Functions may be added to produce the desired shape--this is
seldom necessary, however.
FIGS. 1 to 4 illustrate different families of Beta Function
shapes, showing some of the kinds of shapes readily obtained by
choosing appropriate p.sub.1 and p.sub.2 values. In each group
of curves, pairs of p.sub.1, p.sub.2 values are given starting
from the leftmost curve. Maximum amplitude is normalized as 1.
FIGS. 5 and 6 are Beta Function shapes illustrating degrees of
skewness, starting from a symmetrical (1, 1) form, with pairs of
p values forming a series as shown. These are types of shapes
used for the amplitude of many musical tones (called A [left
group] and P [right group] types).
Computer Program for Shaping and Playing Melodies
The Beta Function is used in a computer program that calculates
individual tone shapes. In this program, the amplitude character
of a tone is specified by three numbers, respectively denoting
the amplitude magnitude G, and parameters p.sub.1 and p.sub.2.
We use a linear scale for the magnitude with a resolution of 1
part in 4096. This in accordance with our findings that such
transient sound phenomena may often be better understood by
changes in amplitude on a linear rather than logarithmic scale.
It also presents a visually more tractable aspect. Silences of
various durations in the millisecond and centisecond range can
also be readily inserted between tones.
The duration of each tone is specified by the number of points
it occupies over which the Beta Function is calculated. The
calculation for each tone is done without affecting the duration
and number of points of other tones. The temporal resolution of
tones is usually better than 1 millisecond. In practice, the
amplitude contour may be constituted by a 12 bit DA Converter
and modulates a voltage-controlled amplifier (VCA) of linearity
better than 0.1% over a dynamic range of 1 to 4096. The
frequency of the tones is set by another channel of a DA
converter which modulates a voltage-controlled oscillator (VCO).
In practice, the invention may be realized with D to A
converters having 8 to 16 bits. One may also use a
digitally-controlled VCA and VCO which can, in effect, be
integrated within a digital synthesizer, in which event there is
no need for a D to A converter to operate the VCA and VCO. In
that case, the ultimate sampling rate per channel must be 44 kHz
or higher to obtain good quality results.
The FORTRAN IV program was run on a PDP 11-23 computer. The
tempo can be varied over a very wide range. Parameters of any
tone can be readily varied and the changed result listened to in
a few seconds, typically 2-10 seconds. Any desired portion of
the music can be listened to. The maximum length of the piece to
be played is only limited by the length of the microscore than
can be stored on the disc. In practice, many tens of thousands
of tones can be stored.
Manually-Created Microstructure
Before considering examples of manually-produced
microstructures, some general comments may be appropriate.
(i) It may take a musical person a day or two to perfect such a
theme by gradually improving the values of the parameters, and
repeatedly listening until it is refined to a degree where he no
longer is sure what change would be an improvement.
(ii) As one listens repeatedly and changes parameter values for
a time, one eventually experiences satiation or lessening of
sensitivity to particular aspects and/or the totality of the
expression. It appears that one's sensitivities in this regard
are dulled with repeated exposure, and need some time to
recover, to regain their spontaneous freshness. A few minutes is
sometimes sufficient, but after many hours of work, several
hours or overnight may be required to revitalize one's
listening.
(iii) Working on a theme in this manner in fact sharpens one's
hearing and attention to detail in regard to expressive
qualities, and can be considered invaluable for that reason
alone.
(iv) A further phenomenon occurs as one continues working with
a theme in this manner: the theme teaches one its own nature. As
one repeatedly interacts, many new aspects relationships and
meanings become clear. One becomes more and more involved with
the theme, and it gains more vitality and clarity in one's mind.
The goal becomes clearer; yet its form also develops as one
works towards it. The process becomes self-refining, a
systematic interaction with a stable limit; assymptotic and not
seemingly finite, but stable. When one finally reaches the point
of not being able to improve it any further, one feels this is
not because it could not be improved, but rather that the
required changes are so subtle, that greater understanding than
can be summoned at the time for A-B comparison would be required
in order to continue. But still, there is joy and satisfaction
in having created something vital, particularly on hearing it
freshly at a later time.
This method of sculpturing tones and melodies allows a musical
artist, then, to perfect the expression in much the same manner
as a painter or a sculptor can, working with a painting for long
periods of time, gradually perfecting the forms so that they
correspond to his inner vision, a vision which itself becomes
more perfect as the interaction grows. At what stage to say "it
is enough" depends on a high level of integration where another
vision and its realization interact in a like manner.
It is also in many ways similar to the process of practicing
for a musical performance, in the course of which the artist,
enamoured with the piece, refines his performance and
understanding through repeated reciprocal
interaction--feedforward and feedback.
Resolution of Parameter Values
Sensitivity in discriminating different shapes of tones is
typically of the order of 0.01-0.02 in the p values in the range
of 0.5 to 2 (most commonly used). For larger values it is
correspondingly greater. The limen of discrimination of the
magnitude of amplitude peaks within a melody is of the order of
2% or about 1/4dB. This means that the ear is more sensitive to
the shape than to peak amplitude. For example, a difference in
shape resulting in 2% deviation of critical portions of the
shape (referred to peak amplitude) will be considerably more
noticeable than a 2% change in overall amplitude. Concerning
relatively critical portions of the shape of a tone with respect
to sensitivity, see Clynes and Walker, supra.
Mozart Quintet
An example of a theme embodied by this method is the first
eight bars of the Mozart Quintet in G minor, as shown in FIG. 7.
In this figure, the stave on top shows the notes of the melody.
Graph 1 below this represents the pitch of the sinusoidal tone.
Small markers on this graph indicate repetition of the same note
and micropauses.
Graph 2 is the amplitude contour (in linear scale from 0 to
4096, the 200 point being thus equivalent to -26 dB referred to
as the loudest level; 1000 being played at about 50 dB above
threshold normally).
Graph 3 represents the temporal deviations from the nominal
values for each tone, in percent, upward deflection being
slower. (Micropauses are often included in the representation.)
The timer marker at the bottom of all figures represents 1
second.
The digital printout prints only every sixth point of the
functions--actual resolution is this six times greater than that
shown in the illustration.
The table shown below for the Mozart Quintet and similar ones
gives a list of all the tones and rests of the computer
realization, specifying for each;
1. the duration of the tone, or rest, in points
2. amplitude size
3. amplitude shape (p.sub.1, p.sub.2)
In some tables, where Beta functions are used to span only part
of the duration of the tone, as may occur for staccato tones,
the sound duration of staccato tones is given in parenthesis
next to the total tone duration. Micropauses are indicated as
"P", Rests as "R". When more than one Beta function is used for
a tone amplitude, they are listed in vertical sequence for that
tone. Metronome mark for the nominal unit used (often a
subdivision of a quarter note) is also given.
______________________________________
MICROSCORE
MOZART: QUINTET IN G MINOR, 1ST MOVEMENT
Duration (sec): 8.73
MM (100 pt tones/min): 230.00
DUR
T# NOTE (PTS) AMPL P1 P2
______________________________________
1 D4 100 800 1.00 1.00
2 G4 123 1580 1.10 0.88
3 B4.music-flat.
88 480 1.20 0.80
4 D5 100 950 0.85 3.40
5 D5 112 430 1.20 4.60
6 D5 223 660 0.96 0.83
7 R 80 0 0.00 0.00
8 G5 105 920 1.10 1.35
9 G5 133 2100 1.15 1.70
10 F5.music-sharp.
102 600 0.97 0.80
11 F5 101 850 0.80 3.20
12 F5 110 400 1.00 3.60
13 E5 231 390 1.45 1.40
14 R 83 0 0.00 0.00
15 E5.music-flat.
105 580 1.20 0.65
16 E5.music-flat.
139 1550 1.28 1.65
17 D5 103 430 0.85 1.00
18 D5 105 380 0.85 0.90
19 E5.music-flat.
108 620 0.80 1.10
20 E5.music-flat.
135 800 1.80 1.20
21 D5 95 290 0.80 1.30
22 D5 113 400 0.85 1.90
23 C5 98 240 0.80 1.30
24 C5 215 530 1.55 0.90
25 B4.music-flat.,C,B.music-flat.
62,40,38
245 0.83 0.80
26 B4,B4.music-flat.
31,39 210 0.80 0.70
27 A4 227 225 1.30 1.20
______________________________________
The chosen unit of time in the melody (in this case, an eighth
note) is assigned 100 points duration nominally, so that a
quarter note becomes 200 points nominally, a half note 400
points, and so forth. The actual duration of each tone is
modified from these so that a particular eighth tone may have a
duration of 96, say, a half tone 220; and so on, depending on
its position and expressive requirements.
In this example, all parameters (4 for each tone: duration,
peak amplitude, p.sub.1 and p.sub.2) were chosen by trial and
error; that is, by repeated listening and gradually improving
the values as dictated by "the ear."
The program allows us to play any portion or the entire theme,
and will repeat as many times as desired (with a short pause
between repetitions). The metronome mark entered (i.e. 230)
refers to the nominal chosen unit of duration (in the present
example, 100 points for an eighth note). If an actual tone has
more or fewer number of points than 100, it will have a
correspondingly different duration. Minute tempo variations
within the theme arise from the differences from nominal values
in the number of points for each of the tones.
In this example, we may note the following:
1. Amplitude Relationship within a Four-tone Group (One Pulse)
The amplitude relationship between the four eighth notes of the
first bar shows that the second and fourth tones are much
smaller, the fourth one being a little less than the second. The
third tone is considerably larger in amplitude than the second
and fourth but less than the first. A similar pattern is
repeated in the third bar, but the accentuation of the first
tone is even greater. Throughout the theme, the first tone of
each bar is considerably larger in amplitude.
2. Peak Amplitudes Outline Essentic Form
The peak amplitude form a descending curve from bar 3 to bar 4.
The form of this descending curve combines with the frequency
contour to produce an essentic form related to grief (this form
may well be considered to be a mixed emotion: predominantly sad,
with aspects of loneliness, anxiety, and perhaps regret). Bars 1
and 2 provide similar forms of diminishing amplitude, but in bar
1 combined with rising frequency. Pain and sadness are implicit
in bar 2. Bar 1 suggests a resigned view, accepting fate,
without the quality of hope; "this is how it is; there is
nothing that can be done about it." The combined effect is a
combination of grief with a stoical, strong acceptance of what
is; without defiance or rebellion.
3. Individual Tone Shapes
The shapes of each individual tone are governed by their place
in the melodic context. The shorter tones may seem similar in
shape on the graph, but in fact they are varied, as can be seen
from the p values in Table 3. (Small changes in the p values
noticeably affect the quality of the sound.) In the longer
tones, such as the fifth tone of bars 1 and 2, first and fifth
tones of bar 4, the shape of the termination of the tone is as
important as its rise, for appropriate expression. For the
shorter tones, the termination phase of the tone relates to the
degree of legato that is achieved. Smaller p values result in
greater legato. It is not generally necessary to include a DC
component to maintain a legato between succssive tones. The
momentary drop in amplitude between tones shaped by the Beta
Function is not perceived by the ear if it is quite short, as is
the case for appropriately low p values. Staccato tones are
produced in the first few examples by choosing appropriate Beta
Functions with high p.sub.2 values.
4. Duration Deviations
We may note systematic time deviations from the given note
values. First notes in each bar are lengthened, the second
shortened. Hardly any tones correspond in duration to the actual
note value. Some tones are lengthened by as much as 39% (first
tone bar 3). Specially lengthened are first notes of beats 5, 9,
that correspond to accentuated dissonant tones, which like
suspensions are resolved in the following, second tone of the
bar. Such prolongations induce a lamenting quality in the
expression.
Further Observations
(a) When working with such a theme, one notices fairly quickly
that when one changes the amplitude, or duration, of one note,
it affects the balance of all notes: that is, the theme is an
organic entity. Increasing the loudness of the fifth tone of bar
3, for example, even by only, say, 5%, or one dB, will affect
its relationship with first tones of bars 3 and 4. Also, it
affects its prominence compared to its adjacent tones, the sixth
tone of bar 3 and the fourth tone of bar 3. But further, the
altered amplitude contour now constituted by the peaks of the
first and fifth tones of bar 3, and the first tone of bar 4
affects how bars 1 to 2 balance with bars 3 to 4.
Similar organic behavior is evident in changing the duration of
a tone. Additionally, changing the duration of a tone also
affects its relative emphasis. For example, lengthening the
upbeat duration will tend to give greater prominence and energy
to the following main beat, which then in turn will affect its
relation to the other main beats.
(b) The appropriate way to realize the trill, in bar 4, seemed
to be to group the trill in terms of 3 and 2 notes (the latter
being its termination), each group under a single amplitude
contour.
(c) When a phrase is repeated such as in bar 3, the amplitude
relationships may be similar, but never the same. Whether they
tend to be augmented or diminished in successive repetitions
depends on the specific piece, the structural design, and the
nature of the composer. In the present theme, the second
presentation is less emphatic than the first.
Chopin Ballade
As illustrated in FIG. 8, the opening theme of Chopin's Ballade
Number 3 in A flat shows how a melody written for piano, an
instrument of limited ability to vary amplitude shapes, can be
expressed by varying amplitude shapes according to its character
and not in violation of it. In this example, amplitude tone
shapes area realized that are implicit in the melody, and are
heard inwardly even when they are not actually produced. One
thinks this melody with these shapes. Notable in this example is
the strong rubato in the second part of the first bar. This
quickening reaches its maximum extent on the fourth eighth note
and is counterweighed by a slowing down in much of the second
bar.
The microscore for this Chopin piece is as follows:
______________________________________
CHOPIN: BALLADE NO. 3 IN A FLAT, OP 47
Duration (sec): 5.81
MM (100 pt tones/min): 145.00
DUR
T# NOTE (PTS) AMPL P1 P2
______________________________________
1 E4.music-flat.
224 840 0.80 0.73
2 F4 130 800 1.20 0.67
3 G4 95 800 1.00 1.50
4 A4.music-flat.
80 1150 1.00 0.80
5 B4.music-flat.
98 1680 1.00 0.90
6 C5 276 2170 1.70 2.53
P -- 9 0 0.00 0.00
7 F5 151 810 0.95 2.36
8 E5.music-flat.
340 660 0.85 1.55
______________________________________
Beethoven Piano Concerto
FIG. 9 illustrates the first movement, second subject of the
Beethoven piano concerto no. 1. The microscore for this piece
which illustrates the automatic mode of operation, using both
pulse matrix and automatic amplitude and shape calculations
(P.sub.1 & P.sub.2 values) is as follows:
______________________________________
BEETHOVEN: PIANO CONCERTO NO. 1 1ST MOVT
Duration (sec): 6.44
MM (100 pt tones/min): 261.00
Base P1, P2: 1.18 0.82
DUR
T# NOTE (PTS) AMPL P1 P2
______________________________________
1 D5 400 1000 1.16 0.83
2 C5.music-sharp.
107 1000 1.30 0.75
3 D5 89 399 1.46 0.66
4 E5 96 799 0.96 1.00
5 D5 108 799 1.18 0.82
6 D5 107 1000 0.98 0.99
7 C5 89 399 1.06 0.91
8 B4 96 799 0.96 1.00
9 A4 108 799 0.98 0.99
10 G4 107 850 1.07 0.90
11 F4.music-sharp.
89 340 0.95 1.01
12 E4 96 680 0.96 1.00
13 D4 108(99) 650 1.18 0.82
P -- 17 0 0.00 0.00
14 G4 107 1000 1.18 0.82
15 R 89 0 0.00 0.00
16 G4 96 800 1.18 0.82
17 R 108 0 0.00 0.00
18 A4,G4 24,83 1000 1.07 0.90
19 F4.music-sharp.
89 400 1.31 0.74
20 G4 96 800 1.77 0.55
21 B4 107 800 0.89 1.09
P -- 7 0 0.00 0.00
22 G4.music-sharp.
210 1000 1.24 0.78
23 A4 170 280 1.00 1.00
______________________________________
1 pulse considered to be 1 half note
Automatic Linking of Amplitude Shapes of Individual tones to
Melodic Form
Having observed and worked with a large number of examples in
the manner illustrated in the previous section in connection
wiht a Mozart Qunitet and considered the interaction between
amplitude shapes of individual tones and the music, we were
enabled to posit a certain relationship between the amplitude
shapes and the melodic course of the music.
A and P Classes of Amplitude Shapes
Let us consider these shapes to belong to a continuum between
two classes and with intermediate shapes between the two
extremes. We may conveniently call these classes (A) assertive
and (P) plaintive or pleading, respectively--without wishing at
all thereby to tie the musical expression to such categories, of
course.)
We can then consider an actual tone shape as placed somewhere
along this continuum--and consider the nature of the influence
that displaced it from a neutral position (the base shape) to
the place on this continuum where it needs to be.
We can describe forms along this continuum by pairs of p
values, starting from a base shape (say 1,1) such that as the
values of p.sub.1 increase those of p.sub.2 decrease in
proportion, and vice versa. Thus, say, for example (0.9, 1.11),
(0.8, 1.25), (0.7, 1.42) etc. will give a series of shapes
shifting gradually towards class (A) which has a relatively
sharp rise time). The inverse series (1.11, 0.9), (1.25, 0.8),
(1.42, 0.7) etc. will tend more and more towards class (P) (of
gradual rise time).
We then can consider the influence which causes the shift to be
the slope of the essentic form, as measured by the slope of the
pitch contour.
More particularly, the deviation from a base shape for a
particular tone is seen to be a function of the slope of the
pitch contour (essentic form) at that tone: both pitch and
duration determine the deviation in such a way that:
(a) Downward (-ve) steps in pitch deviate the shape towards A,
upward steps toward P, in proportion to the number of semitones
between the tone and the next tone.
(b) Deviations are affected by the duration of the tone so that
the longer the tone, the smaller the deviation (since the slope
is correspondingly smaller).
Further, in practice, the slope is measured from the beginning
of the tone considered to the following tone. In measuring the
slope between the tone and the next tone, rather than the
previous tone, the amplitude shape acquires a predictive
function. The first derivative of a function has a predictive
property (lead in phase). The amplitude shape associated with a
particular slope leads us to expect a melodic step in accordance
with it. The movement of the melodic line is thus prepared.
This appears to match well with its actual function in music,
relating the present tone to what is to follow, and may be
considered to be in accordance with musical logic. It gives a
sense of continuity both musically and in terms of feeling.
Experience shows that the proportionality constant needs to be
an approximately 10% change in the p values peer semitone, for
tones of 250 msec duration. The shift is, of course, to be
expected to be linear only over a limited range; a degree of
nonlinearity for both the duration and pitch factors existing
over a broader range.
In order to see how the duration factor and the pitch factor
may obey different power laws, the equation is put into the
form: ##EQU2## where s=number of semitones to next tone
b=const. of p.sub.1,2 by frequency
a=const. of modulation p.sub.1,2 by duration
T=duration of tone in milliseconds
P.sub.1(i), P.sub.2(i) =base (initial values of p.sub.1 and
p.sub.2)
Experience has shown that preferred values are in the region of
______________________________________
a = 0.00269
b = 0.20
______________________________________
for such music.
It is to be understood that a simplified or linear equation may
be substituted for equation (3) having approximately the same
behavior within the range considered.
Choice of Base Values
There remains to consider the choice of the base form, which we
have nominally put at 1,1.
In music of different composers, and of different types, it
would seem that certain preferred base values apply. Values in
the vicinity of (1.2, 0.8) appear as an appropriate choice for
much of Beethoven-giving a greater legato and more gradual
attack. For Mozart, base values around (0.9, 1.1) give a more
rapit decaying sound, and a somewhat sharper attack. For
Schubert, base values in the vicinity of (1.15, 0.9) are seen to
be appropriate.
These values are influenced by the type of instrumental sound
that the style of the composer appears to require, and may also
be linked to historical consideration of the instruments in use
at the time. They may also be related to how the inner pulse
affects the microstructure, which we will consider in the next
section.
Equation 3 relates to the use of Beta Functions to obtain the
desired shift in amplitude shape. In practice, when one chooses
not to employ Beta Functions, the amplitude shape may be
modified as a function of the slope of melodic form using
traditional expedients for individually shaping the component
tones of a melody; that is, such expedients as attaack, decay,
sustain and release, which are appropriately varied.
Other Implications
Silence before Downward Leaps
The rule implies that within a "legato" melody for a tone of
given duration, the larger the upward leap in pitch to the tone
that follows, the more should the tone have a (P) form, and the
larger the downward leap, the more marked the (A) form, and the
larger the downward leap in moderate and fast tempo this can
(depending on the size of the leap and the duration of the tone
starting the leap) in a momentary silence--a microsilence (due
to the tail of the (A) form) before the low tone. This is felt
as appropriate in larger downward leaps. In upward leaps,
silence will tend to occur before the tone starting the leap.
Expressiveness of Scales
Also implied is that tones of scales have somewhat more (P)
shape going up, and going down more (A) (the exact deviation
depending on the tempo as given by equation (3)). In fact,
scales become considerably more musical when this rule is
appropriately observed.
(We may also relate the amplitude shape deviations to the type
of gestures with which one would conduct the music; often a
downward gesture goes with greater (A)--assertive; an upward
gesture with more (P)--plaintive, pleading).
Limitations of the Equation
By the nature of the rule, it cannot predict the amplitude
shape of the last tone of a melody (since it measures slope to
the next note)--this must be entered manually. (It is possible,
however, that a rule referring to last notes of phrases may be
formulated which would also take into account aspects of larger
structure.) Further, it does not apply to tones which require
more than one Beta function.
The basic thought behind this formulation is: if we can
consider the available shapes to lie along a continuum between
(A) and (P) forms, then the slope of the essentic form at that
tone determines where in that continuum a particular tone lies.
The particular merit of this formulation may be seen in that it
harmoniously integrates the behavior of single tones to that of
the larger whole, for different kinds of music. In this, it
seems to show a surprising elegance and power--and musicality.
The Composer's Specific Pulse Expressed in Microstructure
In the previous section, we have been how the amplitude shape
of individual tones can be derived from the form of the melody.
We can now proceed and ask, how much of the remaining
unaccounted microstructure can be attributed to the function of
the inner pulse? It seems an unexpectedly large amount.
The inner pulse of a given composer is not the same as the
rhythm or meter of a piece; it is found in slow movements, in
fast movements; in duple time, in triple time, or compound time.
The tempo of the pulse has generally been considered to be in
the range of 50-80 per minute. In slow movements, one pulse may
correspond to an eighth note, even a sixteenth in a very slow
movement; in a fast movement a half note; and in a moderate
movement a quarter note.
The inner pluse as a specific signature of a composer became
established in Western music around the middle of the eighteenth
century and continued until the advent of music in whose
rhythmic motion there no longer was interfused and intimate
revelation of the personality of the composer. In the music of
Mozart, Haydn, Beethoven, Schubert, Schumann, Chopin, Brahms,
for example, we find a clear and unique personal pulse which the
composer has impregnated successfully into his music (the
knowledge of which we ultimately acquire from the score).
Indeed, because of the time course of the inner pulse (0.7-1.2
sec. approx. per cycle), the matrix of its wave form is most
prominently expressed in microstructure.
In considering the nature of the beat, and of the inner pulse,
the property of the mervous system called Time Form Printing is
very relevant. This function enables the human organism to
decide on a particular form of movement to be repeated, and the
rate, and then to repeat the movement at that rate without
further attention--until a separate decision is made to stop or
to alter the form or the rate of the movement. Once conceived as
a repetitive movement and initiated, the movement will continue
in "automatic" fashion until stopped. This process takes place
mentally when thinking the beat, and the inner pulse.
In practice, this means that once initiated, the inner pulse
can carry through the musical piece without need for further
specific initiation of form, although the rate will be caused to
vary to a degree. Small pauses can momentarily suspend the
pulse, and act as punctuation, as it were, in the musical
phrasing.
In any composition, certain parts of the score embody the pulse
more clearly and obviously than others. However, through Time
Form Printing, the pulse will tend to carry through all parts,
if it has been well initiated at the beginning of the piece.
Scale passages, for example, can become characteristic of the
composer through the pulse: the same scale passage can be played
with a Mozartian pulse or with a Beethovenian pulse, for
example, and will sound appropriate correspondingly. (This does
in fact happen in satisfying performances.) Thus "neutral"
passages acquire from the pulse the characteristic "flavor" of
the composer. This, of course, is precisely the province of
microstructure, and we shall show how this occurs.
By experimenting with such neutral structures, and combining
the results with knowledge of the inner pulse forms as expressed
through touch, we can arrive empirically at an answer to the
question: What does the inner pulse do to the specific tones in
a piece of music?
Concerning the Derivation of Pulse Microstructure Values
The following considerations have aided the derivation of these
values.
Characteristics of the pulse matrix may be in part connected to
the kinesthetic "feel" of the pulse, observed in recording it
sentographically. We have seen that different degrees of
inertia, damping, can be experienced for pulses of various
composers, and for some pulses different kinds of tensions
occurring at specific phases.
These show up indirectly in the recorded sentographic form; but
appear to be significant clues to the deviations in amplitude
and duration that constitute the matrix. Different degrees of
inertia, or sluggishness, are brought about by different modes
of tensions by agonist and antagonist muscle groups (as a result
of which greater or lesser massiveness is displayed by the
arm--e.g., more of the upperarm mass is involved). This
massiveness, a mental program, translates to the degree of
flexibility of modulation within a pulse. The Beethoven pulse
has high inertia, the Mozart pulse low inertia. Accordingly, we
would expect to see much greater differences bewteen the
amplitude of component tones for Mozart than for Beethoven.
Indeed, there is a relative effortlessness in changing the
loudness level in Mozart within a pulse, that is not present in
Beethoven.
The massiveness has an influence also on the duration
deviations, since each pulse cycle contains an initiating point
at a particular phrase (near the time of the upbeat). The
deviation values can also be looked at the light of rhythm
studies that relate the energy of a beat to the duration and
amplitude of the upbeat, in relation to a given downbeat. High
inertia would accordingly tend to be accompanied by a longer
duration upbeat.
Pulse Matrix of Mozart, Beethoven and Schubert
A two-fold effect may be observed: The inner pulse affects
1. Relative amplitude sizes of its component tones.
2. Duration deviations of its component tones.
Both 1 and 2 must occur. Either alone is insufficient.
Accordingly, the influence of a composer's pulse is stated for a
particular meter by a matrix that specifies (1) amplitude ratio,
(2) duration deviations.
The following matrices specify the influence of the inner pulse
for Mozart, Beethoven and Schubert, respectively, for the 4/4/
meter. For each tone, two numbers are given. One specifies the
amplitude size ratio, referred to the first tone as 1. The other
gives the duration referred to as 100 as a mean duration for the
4 components.
______________________________________
Tone No.
Amp. Ratio Duration
##STR1## Beethoven
Amp. Ratio Duration
##STR2## Mozart
Amp. Ratio Duration
##STR3## Schubert
______________________________________
Salient Features of the Pulse Matrices
The Mozart Pulse
Large amplitude difference between first tone and the
others--archness in articulation, second tone very slightly
softer than the fourth tone, the third tone having a subsidiary
accent, subdividing the four tones.
Duration deviations are rather symmetrical, first and third
tones moderately longer.
The Beethoven Pulse
Amplitude ratios generally more even, fourth tone not softer
than third. Second considerably softer than first.
Durations
First tone considerably extended
Fourth tone considerably extended.
The extended duration of the fourth tone combined with its
relatively high amplitude is a cardinal feature of the Beethoven
pulse. The third tone is not extended in duration, and is more
nearly equal to the first tone in amplitude than for Mozart. A
more even, less arched articulation, with a special aspect to
the fourth tone. (A resistance is displayed against "excessive"
amplitude modulation--experienced often as a kind of "ethical
restraint.") First and second tone taken together are contracted
compared with the third and fourth tone together.
The Schubert Pulse
Low amplitude of the third tone--no subdivision; higher
amplitude and short duration of the fourth tone. No duration
extension of the first tone (or third tone) but a considerably
extended duration of the second tone. Third and fourth tones,
taken together, are contracted compared with first and second
tones together.
The unusually extended duration, without accent, of the second
tone of the Schubert pulse can be linked to the following:
1. The special tension that occurs in the corresponding part of
the Schubert pulse, a pulling upward, uniquely characteristic of
the Schubert pulse. Its initial downward phase acts on low
inertia. During the course of a rebound, however, a tension is
added, increasing the resistance and slowing down what otherwise
would be a carefree bounce, into an entirely different
character, an impression of being pulled upwards.
2. It can be seen in the scores of Schubert's music that there
is a predilection for special treatment melodically, or in other
ways, of the weak tone following the first tone of the bar. In
performance and in thought, these deviations are not experienced
as unevenness, but appear as an appropriate flow.
Pulse matrix values for triple meters are as follows for the
three composers illustrated:
______________________________________
##STR4##
##STR5##
##STR6##
______________________________________
Values for duple meter are derived simply from the quadruple
values by adding the duration of tones 1 and 2, and of tones 3
and 4, respectively, to obtain the duration proportions, and
keeping the amplitude values for tones 1 and 3, which now become
1 and 2.
Thus the matrix values for duple meter are:
______________________________________
##STR7##
##STR8##
##STR9##
______________________________________
Pulse matrices for compound meters can be derived from the above
as follows:
The two matrices are combined so that ##EQU3## where A.sub.c
(i,j) is the compound pulse amplitude
D.sub.c (i,j) is the compound pulse duration
A.sub.1 (i), A.sub.2 (j) are the simple pulse amplitudes and
D.sub.1 (i), D.sub.2 (j) are thesimple pulse durations
respectively for the i.sup.th and j.sup.th tone of the simple
pulses.
To allow for different degrees of hierarchical dependence
attenuation factors are introduced that allow the effectiveness
of the subordinate pulse structure to be de-emphasised or
emphasised, with the parameters n and m, so that when n=1 and
m=1, a full hierarchical effect is obtained, and for smaller
values the duration and/or amplitude effects of the subordinate
pulse structure are relatively more attenuated. Thus ##EQU4##
Values of n and m in the range of 0.7 to 0.8 are found to be
often appropriate.
For example, the 6/8 pulse for Beethoven is
______________________________________
##STR10##
##STR11##
______________________________________
The effect is that each group of 3 tones constitutes a small
3-pulse, and the two groups of 3 tones form a 2-pulse.
Applying the Pulse to a Melody Having Various Note Values
When a melody has a combination of notes of different values,
as is generally the case (some larger and some smaller than the
component values), the following appears to apply:
1. Duration deviations are proportioned according to the
component tones of the pulses; e.g., a dotted quarter has the
duration deviation of one tone plus half that of the next.
2. The amplitude is taken as that prevalent at the beginning of
the tone; i.e., is not averaged; e.g., the dotted quarter has
the same amplitude as the quarter would have had without the
dot.
While it appears that the range of 50-80 per minute is an
approximately useful guideline in applying the pulse, some
pieces may present alternate possibilities of a larger or
smaller frame for the pulse.
An example of automatic operation of the system using a pulse
matrix and using automatic calculations of amplitude shape (p
values) is given in FIG. 9.
In general, it should be emphasized that the pulse and its
effects in microstructure as described herein is in no way to be
considered a binding Procrustes bed, but rather as a level from
where fine artistic realization of the music can be more readily
attempted, taking into account the individual concept of the
piece, and personal interpretive perferences.
Additional Expressive Functions And Properties Considered
Further properties of melodic relationships relevant to manual
and computer realizations are:
1. Microstructural
In working with the various examples manually, we may readily
note that:
(a) A tone of smaller amplitude following a larger amplitude
tone will sound more legato than in the reverse order, for given
amplitude shapes (p values).
(b) A tone will sound softer after a tone of greater amplitude
than before it (a masking effect, the degree depends on the
tempo).
(c) Micropauses often need to be inserted between phrases, and
longer Luftpauses between major sections. They are appropriate
for all composers. The placement of these is important, and not
automatic. These pauses tend to be placed at the end of pulses
(before the next upbeat tone, where there is one). They also
occur immediately before a subito piano.
At times the required pauses are actually provided for by the p
values, as calculated by the rule given, by means of the tail of
the amplitude shape--this means that in such a place, the
composer's indication of phrasing indicates the amplitude shape
required by the music. Also, they may be appropriate between
separate bowing marks (as distinguished from phrasing signs--an
often difficult distinction), but not always.
2. Structural
(a) Pitch Crescendo
It can be noted that there is a general tendency within a
melody to increased loudness with higher pitch. This tendency is
not found to the same degree, however, for different music and
different composers. For Beethoven, it is appropriate to keep
the amplitudes in terms of voltage similar. This insures a small
degree of crescendo, since a given voltage amplitude is
perceived as louder with increasing pitch. For Mozart, however,
this crescendo tends to be greater--approximately an additional
4 db/octave. For Schubert, it can be even more than 6 db/octave.
If the same melodic crescendo is applied to Beethoven, the
effect tends to sound exaggerated. There is a resistance to such
crescendo in Beethoven, a restraint that (as in the modern
modulation of tone amplitude heights by the Beethoven pulse)
appears to translate into the strength of an ethical constraint.
(b) Alternation of "heavy" and "light" bars
This is a general structural property (not explicit in the
score) which can be produced on the computer by scaling down the
"light" bars. For Mozart, a sutiable amplitude proportion may be
0.72, and bars often (but of course not always) alternate as
Heavy-Light, Heavy-Light. This ads considerably to the sense of
musical balance and logic, where appropriate; it presents larger
units to the mind, allowing a greater overview.
In Beethoven, other patterns tend to dominate and often special
dramatic devices, such as crescendo and subito piano. Crescendi
and Diminuendi specially noted by the composer in the score
have, of course, to be included in the computer realizations.
A Preferred Procedure for Shaping A Melody
We may concisely summarize the steps involved in controlling
microstructure for the artistic shaping of a melody. Steps 7 to
9 are not always required. Some of these steps can be combined
into single, automatic operations:
1. Enter tones and nominal durations according to score.
2. Choose tempo.
3. Apply pulse matrix for the particular composer, and
particular meter (choose length of pulse) (B, M, or S for
Beethoven, Mozart and Schubert, respectively).
4. Insert micropauses of appropriate duration at appropriate
places (end of phrases and other special points).
5. Choose base p.sub.1, p.sub.2 values and let the program
calculate p. values for amplitude shapes of individual tones.
6. Adjust last tones of phrases by "ear" (both amplitude and p
values).
7. Apply pitch cres-dim function as appropriate, and any cres
or dim or special accents directed by the score.
8. As required, modify amplitude and/or duration of one or
several key tones to shape a particular essentic form.
9. If phrases repeat, subtly change the repetition as desired.
The first tone only of the theme is generally to be lengthened
in duration by about 5%, at the beginning of a piece (but not on
subsequent reintroductions of the theme).
A few minutes are generally sufficient to carry out these
steps. It is, of course, useful to listen to the result after
each step is taken. Micropauses p.sub.1, p.sub.2 base values or
other parameters can be adjusted at any time to improve the
result, as desired.
The Stability of Pulse Matrices
The pulse matrix values given previously are subject to further
refinement. The salient features of the specific pulse matrices
hold over a wide range; but their degree of prominence is likely
to be influenced to some degree by the tempo (with the range
given; 50-80 per minute), and by pitch height of the entire
theme (i.e., transposition). These factors may modify the
elements within the pulse matrices as a second order effect.
Other second order changes in the pulse forms may occur with
variables such as variables related to specific pieces and the
composer's age.
Relation of Timbre to Microstructure
Timbre is essential for melodic expressiveness when there is
more than one melodic line. Several sinusoids tend to coalesce
and fuse--for distinctness of voice leading and contrast sounds
with different dynamic proportions of harmonics are required.
Secondly, we can now, in the light of the present studies with
sinusoids, investigate how timbre variations are of help in
improving expressiveness of a single melogic line.
Within each tone it is possible to add timbre (and also
vibrato) various dynamic ways to the expressive forms already
determined, and to see how such individually shaped time varying
functions of timbre for each tone (time-shaped harmonic
content--not only in relation to the attack) may augment or
interfere with expressiveness. One must take into account the
complication that addint timbre (and/or vibrato) can modify the
perception of amplitude shapes, in part because the overtones
require their own amplitude shaping (not according to the
amplitude shapes of the fundamental), as well as for
psychophysiologic reasons related to persistence of hearing and
dynamic masking.
Summary of Findings
We have found that:
1. Expressiveness and Microstructure can be fruitfully studied
with amplitude-modulated sinusoidal sounds. Musical meaning in
its subtlety can be largely expressed by this means for single
melogic lines.
2. A systematic realtion appears to exist between individual
amplitude shapes of tones of a melody and its course, so that
the shape can be predicted from the slope; and thus is heard to
presage the next tone of a melody--forging an organic link, and
making it possible for the living qualities inherent in the
melodic shape to cast a presence within each tone.
3. In music which incorporates a personal pulse, this is shown
to systematically affect both amplitude and duration of
component tones in a way characteristic to that personal pulse,
as also an individual's signature bears the continuing stamp of
his person. The realization of the pulse and its effects is seen
to be necessary for the life, power and beauty of such music.
Among the advantages of imparting microstructure to a musical
score in a computerized system in accordance with the invention
are the following:
1. It can improve one's artistic understanding and output.
2. Far from being "mechanical," it can infuse life and
livingness into music.
3. It tends to give us a degree of understanding of the very
nature of that livingness, as bound to iconic and unique form.
4. It allows us to use imagination and creative insight from a
higher hierarchic point of view, using as units of though
entities that before needed to be specially constructed every
time from constituent parts.
The practical applications of a computerized system in
accordance with the invention to music education are obvious and
therefore need not be detailed. In an age of personal computers,
the programs developed therefor can give access to creative
interpretation to all so inclined without the need to acquire
physical musical skills or manual dexterity.
The invention is also useful to a serious or amateur composer,
for it allows the composer to incorporate his own realization of
microstructure into the macrostructure of his own composition,
and it also allows him to experiment with inner pulse forms. The
final proudct thereby reflects the imagination, feeling and
discernment of the individual who shapes a musical composition.
The computerized system serves as a tool which assists its user
in thinking musically in a manner somewhat analogous to the
relationship of an electronic calculator to a mathematical
concept.
The Computerized System
Referring now to FIG. 10, there is shown a manually-operated
computerized system in accordance with the invention based on
the technique disclosed hereinabove for processing the nominal
tones of a raw score entered therein to impart an expressive
microstructure thereto.
The system includes a beta function calculator 10. In the
calculator is entered by way of a computer keyboard or means
performing a like function represented by entry station 11, the
successive tones of a music score in terms of their nominal
pitch and duration values expressed in alpha-numeric terms. Thus
the pitch of a given note, which depends on its position on the
staff, is represented by an appropriate value, as is the
duration of the same note. In practice, an electronic piano
keyboard may be used. By depressing a selected key, there is
produced an appropriate value for entry into the calculator. In
that case, the tone duration will have to be normalized.
Also entered manually into calculator 10 is the desired
microscore of each nominal note. By "microscore" is meant
digital values representing desired deviations from the nominal
values of the note necessary to its processing to impart a
microstructure thereto.
Thus the p.sub.1 and p.sub.2 values required by the beta
function to shape the amplitude contour of each note is entered
as well as digital values representing changes in the duration
of each note and values representing the relative amplitude of
successive notes in the score. Also entered are micropauses and
whatever other variables are to be processed by the system, such
as timbre.
From entry station 11, there are two channels C.sub.1 and
C.sub.2 leading into calculator 10, channel C.sub.1 conveying
the amplitude and timing data, and channel C.sub.2 the pitch and
timbre data for each note.
Calculator 10 is programmed to process the data supplied
thereto and to yield a series of equi-spaced digital values V
during a specified interval, as shown in FIG. 11, that represent
the successive amplitude levels in the contoured tone T whose
microstructure duration is interval P. In the absence of the
microstructure impressed on the nominal tone, its form would be
represented by a square wave of constant amplitude and
predetermined duration, which depends on whether it is a whole
note, a half note or whatever else is notated.
The series of digital values V, which outline the amplitude
shape, are applied to a D-to-A converter 12 to yield an analog
voltage A.sub.1 reflecting the amplitude contour or envelope of
the processed note. The digital data derived from entry station
11, which represents the frequency of each note, is also fed to
D/A converter 12 to yield an analog voltage A.sub.2, reflecting
the pitch of the tone to be played.
Analog voltage A.sub.1 representing the amplitude contour is
applied to a voltage-controlled amplifier 13 (VCA) whose output
is applied to a loudspeaker 14. Analog voltage A.sub.2
representing the pitch is applied to a voltage-controlled
oscillator 15 whose output frequency is in accordance with the
pitch of the tone. The sinusoidal output of this oscialltor may
be applied directly to amplifier 13, in which event the
reproduced structure impressed thereon. Alternatively, the
oscillator output may be applied to the amplifier through a
timbre network 16 which changes the sinisoidal wave shape so
that the resultant tone is rich in harmonics and therefore has a
timbre depending on its harmonic content.
Any known means may be used for introducing a varying harmonic
content. One approach is to combine a sinusoidal wave SW, as
shown in FIG. 12, with the differentiated form DW of a square
wave having the same period, the resultant sharp pulses being
adjustably clipped and rectified to provide sharp peaks which,
when summed with the sinusoidal wave, produce a nonsinusoidal
wave having a desired harmonic content that depends on the
adjustment of clipping and rectification. The resultant sum may
be further variably rectified to provide preponderantly even or
odd harmonics.
With each tone, the timbre is varied through a number of D to A
control channels, typically up to 4 channels, each output of
which is shaped by beta functions or equivalent means. All of
these functions can also be carried out in an entirely digital
manner in a digital synthesizer.
It is to be understood that among the means usable for
producing a varying harmonic content are additive or subtractive
synthesizers, wave shaping and other means, realized either
digitally or through analog means.
It is to be understood that while calculator 10 is
advantageously operated in accordance with the beta function
disclosed herein requiring only two parameters (p.sub.1 and
p.sub.2), in order to produce a desired amplitude contour, any
known electronic means to effect amplitude shaping in response
to applied digital parameters may be used for the same purpose.
In operating a system in accordance with the invention in an
automatic mode, all of the digital values with respect to
amplitude and duration necessary to impart a microstructure to
the nominal note values of the raw musical score entered therein
may be stored, as shown in FIG. 13, in a pulse matrix 17 which
in one output channel A.sub.3 yields the amplitude and timing
data required to process each note, and in another output
channel A.sub.4 yields the necessary pitch data for each note.
The digital data from channel A.sub.3 is applied to an
amplitude-shape calculator 18, while digital data from channel
A.sub.4 is applied to a timbre calculator 19.
The amplitude envelope from amplitude shape calculator 18 is
applied to a tone generator 20, while tone shape data from
timbre calculator 19 is also applied to the tone generator which
generates tones having the desired microstructure. In the
amplitude-shape calculator, the shaping function related to the
melodic and essentic form is calculated as a deviation from a
base shape in accordance with Equation 3, as explained
previously.
It is to be understood that when two or more tones are to be
sounded simultaneously, separate channels will be required for
each tone and their outputs summed. Tones may also be directed
to different loudspeakers to crease stereo and spatial sound
effects.
While there has been shown and described a preferred embodiment
of a computerized system for imparting an expressive
microstructure to a musical score in accordance with the
invention, it will be appreciated that many changes and
modifications may be made therein without, however, departing
from the essential spirit thereof. Thus, while the invention has
been illustrated as it applied to classical music, the invention
is by no means limited to this application, for it is fully
applicable to all other forms of music, including popular and
ethnic.
It must also be noted that the microstructure elements set
forth herein may be used to modulate visual presentations so
that by employing video graphics, the shape, brightness and
color of visually displayed forms may be variously modified to
express visual counterparts to the expressiveness imparted to
music by the microstructure. Simultaneous presentations of such
sound and visual forms may further enhance their expressive
quality.
The visual presentations may assume free or abstract forms
which, by reason of the microstructural modulation, become more
expressive and appear to move or dance and thereby take on a
more animated character. For example, the visual presentation
may be in the form of a conductor's baton whose movement is
related to a musical score and its microstructure. Human body or
facial expressions may be made responsive to microstructural
modulations. Such microstructure can also be used to supplement
or refine existing dance notation, such as Laban notation.
