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MUSIC FROM MATHEMATICS
Played by IBM 7090 COMPUTER and DIGITAL TO SOUND TRANSDUCER
course of human development has always been marked by man's striving
for new techniques and tools in pursuance of a better life. This is
most dramatically manifested in the fields of science and technology.
But this dissatisfaction with available materials and methods and the
corresponding search for new ones is also evident in the arts, and artists
have continually sought to improve the tools of their trade. Today's
modern orchestral instruments, for example, hardly resemble their medieval
ancestors. On this recording, we illustrate another advancement in the
realm of tools available to the music-maker: the computer and the digital-to-sound
transducer. This new "instrument" combination is not merely a gadget
or a complicated bit of machinery capable of producing new sounds. It
opens the door to the exploration and discovery of many new and unique
sounds. However, its musical usefulness and validity go far beyond this.
With the development of this equipment carried out at the Bell Telephone
Laboratories, the composer will have the benefits of a notational system
so precise that future generations will know exactly how the composer
intended his music to sound. He will have at his command an "instrument"
which is itself directly involved in the creative process. In the words
of three of the composers whose works are heard on this recording:
Man's music has always been acoustically limited by the instruments
on which he plays. These are mechanisms which have physical restrictions.
We have made sound and music directly from numbers, surmounting conventional
limitations of instruments. Thus, the musical universe is now circumscribed
only by man's perceptions and creativity.
The process of composing music on and for the computer and transducer
is highly complex: we shall attempt here only a brief and simplified
description so that the listener may better understand what he is hearing.
At the very heart of this type of composition rests this fundamental
premise: "Any sound can he described mathematically by a sequence of
numbers." Our composer thus begins by determining what numbers specify
the particular sounds in which he is interested. These numbers are then
punched on IBM cards: the cards are fed into the computer and the digits
recorded in the memory of the machine. The computer is thus able to
generate limitless sounds, depending on the instructions given it by
the composer. The latter, instead of writing the score in notes, programs
his music by punching a second set of IBM cards, which when fed into
the computer cause it to register on tape certain sounds from its vast
storehouse. The composer may give the computer detailed instructions
for every "note," or he may allow it varying degrees of freedom by asking
it to select "notes" at random from a host of possibilities. The tape
which emerges from the computer contains music in the form of magnetic
impressions. To convert these impressions to actual sounds, the tape
is run on a digital-to-sound transducer, which translates the digital
indications to sounds, amplifies these sounds, and gives us the finished
As can be gleaned from the above description, the human element plays
a large role in computer music, as in any art medium. The sounds and
sound-producing methods are new; the composer's role is essentially
that which it has always been. History tells us that whenever a new
concept emerges, it is labeled revolutionary by either its proponents
or the public at large. The new techniques and tools of computer music
are not meant to replace the more traditional means of composition and
performance. Rather, they are designed to enhance and enlarge the range
of possibilities available to the searching imagination of musicians.
Science has provided the composer with new means to serve the same ends
- artistic excellence and communication.
1. FRERE JACQUES (Common round)
Here is an example of what imagination and the computer can do with
a familiar piece. The round is done with five voices, which have the
following attacks and timbres.
1. Modified square waves; modified rectangular attack.
2. Random noise, narrow band; percussive attack.
3. Modified square wave; percussive attack.
4. Timbre and attack taken from the trombone.
5. Same as voice 1.
2. FANTASIA (Orlando Gibbons)
This work (composed in the 16th century) is played to demonstrate the
performance of conventional music on the computer. The original score
is for three recorders. The wave shape is sinusoidal (the shape of a
tone which is a pure fundamental, without overtones or harmonics). The
attack was copied from that of a recorder, and includes an initial burst
simulating a tonguing noise.
3. BICYCLE BUILT FOR TWO (arranged by M. V. Mathews)
The computer can also be programmed to speak and sing, as is illustrated
by the last verse of this familiar ditty. The patterns of human speech
are analyzed in the same manner as the instrumental sounds. The computer
is then programmed to speak the desired words. On this band the computer
not only was programmed to sing but also to simulate a Honky Tonky type
of piano accompaniment that was popular during the era when this song
was a hit.
4. MOLTO AMOROSO (J. R. Pierce)
This piece features plucked tones. A squarish wave is modulated by a
jagged attack which rises rapidly in a sequence of peaks, then falls
slowly to zero at the end of the note, with decreasing jaggedness. Four
different attacks are used for the four voices. The contrasting effect
of similar attacks in long and short notes is of interest. This piece
displays exaggerated variations of loudness and duration.
5. VARIATIONS IN TIMBRE AND ATTACK (J. K. Pierce)
This composition consists of a melody and various accompanying parts,
making a total of five voices. Voice 2 was formed from selected notes
in the melody (Voice 1). The frequencies used in Voice 3 were obtained
as differences between the frequencies of the first two voices. The
difference between the frequencies of Voices 4 and 5 is approximately
the frequency of the melody. Thus a highly integrated musical texture
was obtained. The composition is divided into live sections.
Section I Voice 1
Section II Voices 1 and 2
Section III Voices 1, 2 and 3
Section IV Voices 4 and 5 only
Section V All Voices
6. STOCHATTA (J. R. Pierce)
This is a very simple work originally composed for recorder and piano.
The piece was programmed principally to show that the computer is capable
of producing a pleasant timbre and harmonious music. The recorder timbre
is that of a single sine-wave (sinusoidal), and the accompanying timbre
is a smooth saw-tooth shape.
7. FIVE AGAINST SEVEN - RANDOM CANON (J. R. Pierce)
The first work is a study of rhythmic patterns, in this case, five against
seven. The canon was composed by a random procedure. It is a two-voiced
canon at the octave, composed by selecting rests and notes of a whole-tone
scale. The notes and rests, of various lengths, were selected by chance.
After the first measure, the subsequent measures are, by chance, either
new music or repetitions of earlier bars.
8. BEAT CANON (J. K. Pierce)
In this work, certain timbre effects were studied. The first third of
each treble note was emphasized by playing it at an amplitude four times
that of the last two thirds. An envelope to the notes in the bass part
was formed by striking together neighboring frequencies.
9. MELODIE (J. R. Pierce)
This selection illustrates how the computer can be used to play rapidly
and flawlessly two rhythms at once. The coda shows how the computer
can execute an accent which would be difficult to achieve by conventional
10. NUMEROLOGY (M. V. Mathews)
This piece was produced as a demonstration of some of the special effects
which can be achieved with the computer. It has eight sections.
Section I A long crescendo in which loudness is increased by raising
vibrato frequency, vibrato amplitude, and pitch, as well as intensity.
Section II A duet in which the treble voice is gradually modulated in
timbre from a percussive, piano-like sound to a slow-attack, string-like
Section III The melody from the bass voice of the previous section is
moved to the treble. Between each note of the original melody, the voice
splits and divides continuously into three notes, which recombine on
the next note.
Section IV A melody is played three times; first in the treble, then
in the bass with slight alterations in rhythm, finally in the treble
with a frequency attack on each note.
Section V A bass line is played first with conventional notes and then
with notes in which the frequency has been blurred with a large, rapid
Section VI A sentimental, high-pitch section which demonstrates nuances
of vibrato and attack.
Section VII A canon in which the two identical voices are separated
by one rhythmic beat.
Section VIII A finale in which the tempo gradually increases to a point
beyond comprehension. Some of the notes have no single frequency; they
sweep over a range of as much as five octaves, and arc perhaps reminiscent
of a passing bullet.
1. THE SECOND LAW (M. V. Mathews)
This composition makes extensive use of random noise. The noise is used
with a variety of bandwidths (the frequency limits of a given sound
pattern) to achieve effects ranging from notes of definite pitch to
pitchless noise. Contrasted against the noise is a tone with vibrato.
2. MAY CAROL (M. V. Mathews)
This composition was produced to show the rhythmic possibilities of
playing three notes against four. The work has three sections: the first
is a study in rhythmic patterns, the second is a canon which is repeated
with varying time intervals between the two voices, and the third is
a development of the first section.
3. THEME AND VARIATIONS (S. D. Speeth)
This piece is a 12-tone composition which makes use of a tone row that
is the same when played upside down and backwards. The three note motive
E-F-D is the basic structural unit of the piece. The voice timbres employed
were taken from a piece by J. R. Pierce, but with one modification:
the rate and range of vibrato is made dependent on note intensity. The
four sections are:
Section I Theme statement (Row is played upside down and backwards.)
Section II Transition
Section III First variation (Row is played backwards)
Section IV Coda; this is a computer-processed recording of a seismogram,
sped up by a factor of 300.
4. STUDY NO. 1 (D. Lewin)
This 12-tone piece also exploits a durational series; augmentations,
inversions, and retrogrades of a basic series of six notes are used
throughout. The Study contains essentially only two types of notes,
which may he called "clink" and "hoot." All tones are presented with
the same amplitude, but their durations and registers are varied, affecting
the timbre and intensity. The piece is an attempt to determine how musically
effective a systematic control of the use of registers might be.
5. STUDY NO. 2 (D. Lewin)
This composition is also in the 12-tone idiom. The number of voices
has been increased to four. Simple instruments allowing control of attack
and waveforms only were used.
6. PITCH VARIATIONS (N. Guttman)
This piece illustrates how the content of a composition may be derived
from a particular psycho-acoustical phenomenon. An experiment, carried
out by the composer, showed that certain waveforms consisting of trains
of sharp pulses produce unusual pitch impressions which may not correspond
simply to the frequency of the waveform. At some frequencies, an impression
of more than one pitch is produced. In this composition, these waveforms
are reproduced at frequencies designed to make the pitch uncertainties
7. NOISE STUDY (J. Tenny)
Each of the "instruments" used in this piece includes a noise generator.
These "instruments" were designed in such a way that the amplitudes,
center frequencies, and bandwidths or the output signals were all continuously
variable within each "note" as well as from one "note" to another. The
large-scale structure of the piece is defined by the changes in average
amplitude, center frequency, bandwidth, and duration from "note" to
"note" and from section to section. In addition. each section is characterized
by a different range of variation in each of the four qualities mentioned
above. The particular qualities manifested by the sounds at any given
moment, however, were determined by chance methods (via random-number
tables and tossed coins). Thus, although the general shape of the piece
as a whole was predetermined by the composer, the way this general shape
was realized in detail was not, but rather, left to chance.
8. JOY TO THE WORLD (M. V. Mathews)
This well-known carol is performed to illustrate different timbres.
The first two playings differ only in the attack on each note; the first
features a slow attack and the second a fast one. The third playing
uses a vibrato which has a maximum intensity in the middle of each note.
FREQUENCY...The repetition rate which determines the pitch of a sound.
DECAY...The very end or "dying" of a tone.
AMPLITUDE...The magnitude which determines the loudness of a sound.
WAVEFORM...The pictorial representation of the shape of a sound wave.
TIMBRE...The quality of a tune which distinguishes voices or instruments;
INSTRUMENT... Pertaining to computer music, this term refers to a group
of computer instructions which generate the notes of a given type of
DR. J. R. PIERCE is Executive Director, Research, Communications Principles
Division, Bell Telephone Laboratories. This Division has contributed
a great deal to the fields of radio, television, electronics, acoustics
and vision, mathematics, and group behavior. Dr. Pierce's department
carried out the work on Bell Telephone's Project Echo, and Dr. Pierce
is credited with significant contributions to the success of the Telstar
DR. M. V. MATHEWS is a member of the staff of Bell Telephone Laboratories,
where he is doing important work in the area of visual and acoustical
research. His main activities involve the study of speech coding and
recognition methods by means of digital computer simulation. The programming
of the music on this recording is the result of Dr. Mathews' work.
DAVID LEWIN is a prominent composer presently teaching at the University
of California at Berkeley. Mr. Lewin also studied mathematics at Princeton.
He was a Junior Fellow in Music at Harvard.
JAMES TENNY is a recognized composer who studied with Dr. L. A. Hiller,
Jr. at the University of Illinois. He has since September 1961, worked
at the Bell Telephone Laboratories writing computer music.
DR. S. D. SPEETH is a psychologist with a musical background which includes
serious study on the violin.
DR. N. GUTTMAN is a psychologist at Bell Telephone Laboratories, specializing
in speech and phonetics. His special interest is the perception of the
pitches of unusual sounds.
Notes for this recording compiled and written by Dr. M. V. Mathews and
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