ee173xed@sdcc3.UUCP ({|stu) (12/02/84)
Until comparatively recently Western music theorists
and the textbooks written for students of music theory have
concerned themselves almost exclusively with the interac-
tions of pitches. The study of harmony, chord progressions,
and melody, while very important to the structure of tradi-
tional music, neglects an important element of sound. Any-
one who listens to music can hear the difference between a
melody played by a flute and that played by a guitar or vio-
lin. Yet it has not been easy to articulate the details of
distinction, nor has the contrast always seemed to be impor-
tant to the composer.
The music that has survived from Medieval and Renais-
sance Europe rarely indicates the instruments for which the
music was intended. The theoretical treatises of the time,
particularly Praetorius' _S_y_n_t_a_g_m_a _M_u_s_i_c_u_m (16th century),
give detailed descriptions and drawings of many of the
ancient instruments, and historical interest has led to the
reconstruction of many of them. From these and the few
specimens that have survived as museum pieces, whoever
wishes can hear what early music might have sounded like.
Many musicologists believe that performances in the
Renaissance usually consisted of whatever instrumentalists
were available, and certainly vocal music predominated, if
only by virtue of the greater number of singers. In fact,
most polyphonic lines were written in a vocal style and
could be either sung or played. Not until the early Baroque
era did composers begin to specify instrumentations in their
scores. And not until the late Classic period, especially
with Beethoven, was music written that was truly dependent
upon the differences between instruments. Beethoven and
composers ever since the end of the Classic period have
included in their orchetsral writing specific instrucions as
to the number and type of instruments that were to be sound-
ing, as well as the dynamic level of each instrument. Pas-
sages of important melodic ideas would not be heard, nor
would the intended musical effect be achieved if the dynam-
ics were not observed or if the instrumentation were
changed. In effect, composers required control of the color
of the orchestra.
The word most commonly used by speakers of modern
English to denote the difference in the quality of sound
between instruments is defined concisely, but somewhat
vaguely, in its standard dictionary version (Webster):
"Timbre - the characteristic quality of a sound that distin-
guishes one voice or musical instrument from another: it is
determined by the harmonics of the sound and is dis-
tinguished from the intensity and pitch." Alternatives to
the word "timbre" (which means "postage stamp" in France and
means "doorbell" in Latin America and Spain) have been
offered, including color, tone quality, and Klangfarbe (from
the German). A direct translation of Klangfarbe has proved
unworkable, as Alexander Ellis, the translator of
Helmholtz's _O_n _t_h_e _S_e_n_s_a_t_i_o_n _o_f _T_o_n_e (1877), humorously sum-
marized:
Prof. Helmholtz uses the word Klang for a musical
tone, which generally, but not always, means a com-
pound tone. Prof. Tyndall therefore proposes to use
the English word clang in the same sense. . . Of
course, if clang could not be used, Prof. Tyndall's
suggestion to translate Prof. Helmholtz's Klangfarbe
by clangtint fell to the ground. I can find no
valid reason for supplanting the time-honored
expression quality of tone. Prof. Tyndall quotes
Dr. Young to the effect that "this quality of sound
is sometimes called its register, colour, or tim-
bre." Register has a distinct meaning in vocal
music which must not be disturbed. Timbre . . . is a
foreign word, often odiously mispronounced, and not
worth preserving.
"Clangtint" does sound a bit strange. However, Ellis lost
his fight to discredit the use of timbre, and current text-
books in physics and acoustics are specific, although not
complete enough for some, in their definition of it: "The
subjective measure of the number and relative strengths of
the overtones present, in addition to the fundamental. It
is represented as the shape of the wave, its waveform."
(Askill, 1979)
The idea of timbre being a function of the relative
strengths of the overtones in the complex waveform can be
credited principally to the work of the nineteenth-century
physicists and mathematicians Fourier, Ohm, and Helmholtz.
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Fourier states that every complex tone is made up of a
number of sinusoidal tones of different frequencies and
intensities, each sinusoid having a frequency that is an
integral multiple or harmonic of the lowest or fundamental
frequency. Stated simply, every musical tone is made up of a
fundamental sine wave accompanied by other sine waves which
are called overtones, harmonics, or partials, depending on
the context. Ohm maintains that each of the separate fre-
quencies in the complex sound is audible and can be per-
ceived separately.
Composers such as Stockhausen in this century have
shown, by use of the electronic medium, that pitch, rhythm,
and timbre are related. If a tone or a rhythm is repeated
quickly enough, so that the rate of the repetition falls
into the audible range (above about 20 repetitions per
second), that series of repetitions will be heard as a sin-
gle tone with its own entirely different timbre. Conversely,
if the pitch of a complex tone is steadily lowered, until
the fundamental is no longer audible, it will be perceived
as a periodic rhythm containing distinct pitches and tim-
bres. In a similar vein, present-day composers have written
music that is dependent primarily upon timbre as a unifying
compositional principle. Such pieces are often called
Klangfarbenmelodies, and are illustrations of what can be
done if timbre, instead of harmony, is paramount.
Helmholtz (1877), building on the theory of Ohm,
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developed what is called the Place Theory of pitch percep-
tion. This theory describes the cochlea of the inner ear as
a natural frequency decoder. Each of the 25 to 30 thousand
hair fibers along the basilar membrane resonates at a dif-
ferent frequency, the fibers near to the oval window where
the basilar membrane is thinnest corresponding to the higher
frequencies, and those fibers at the end of the basilar mem-
brane where it is thickest corresponding to the lower fre-
quencies. A complex waveform excites the fibers of the
basilar membrane in a one-to-one relationship with the over-
tones present. The relative strength of each of the over-
tones is expressed as a measure of the intensity of the
resonance of the hair fibers involved. This information is
then transmitted to the brain by the nervous system, where
the information is assimilated and interpreted. The timbre
of a sound in the Place Theory model is perceived directly
from the component sinusoidal frequencies that make up the
complex waveform.
Much of the present-day work in acoustics follows the
model given by Helmholtz. Plomp (1976) believes that the
ear performs a frequency analysis of the complex waveform,
but that the listener may not be aware of the existence of
the individual harmonics. Rather, the harmonics fuse into a
single percept. Plomp acknowledges the importance of the
change over time of the relative strengths of the individual
harmonics in the complex tone, but discusses only the steady
state and the dimensions present in the Fourier theorem.
- 6 -
The following equation demonstrates the multidimensionality
of the complex tone:
w(t) = a(k) sin(2 pi k f t + p(k))
where w(t) represents the complex waveform as a summation of
the sinusoidal components k as the wave develops through
time t, for k = 1 to the highest harmonic having any ampli-
tude. The variables, a - amplitude of the harmonic k, f -
fundamental frequency, and p - phase of the component k,
form the shape and timbre of the waveform.
It should be quite apparent that the phase of the indi-
vidual partials radically alters the shape of the waveform,
since the waveshape is a summation of the separate sine
waves that account for each partial. The perceived timbre,
on the other hand, is affected by phase, but not to a great
degree. Immensely differing waveshapes, distinct in their
spectral makeup only in the phase of the sinusoidal com-
ponents, sound very similar. Generally, the effect of vary-
ing the phase angle is noticeable only for the higher har-
monics of a complex tone.
Actual complex timbres, even in the steady state, do
not usually have overtones that are exact integer multiples
of the fundamental. For example, analysis has shown that
the harmonics of a bowed string closely approach the ideal,
but the harmonics of a plucked string do not form integral
- 7 -
ratios to the fundamental, as anyone who tries to tune a
guitar by using the harmonics will discover.
Musicians are generally not content to believe that the
steady state of a tone's spectrum is representative of its
timbre. The change of the spectral and amplitude envelopes
through the duration of the note, usually called the attack,
the initial decay, the sustain level, and the release, are
of great importance to the development of timbre. The spec-
trum and amplitude of a tone vary markedly, especially dur-
ing the attack and final release portions of its duration.
An interesting experiment (Grey and Moorer, 1977)
involves the analysis and resynthesis of the timbres of
various musical instruments using the additive Fourier
model, with the frequencies of the overtones and fundamental
as well as the amplitudes of the harmonics being time vari-
ant. To simplify the procedure, the tones chosen for
analysis purposely lack vibrato. The minute variations of
amplitude and frequency in the analyzed spectrum are
replaced in the resynthesis with straight line-segment
approximations. The resulting synthetic timbre is, in the
estimation of the researchers, virtually the same as the
original to the human perceiver, even though definite
changes in the waveform have been made.
When composers began writing for specific ensembles in
the Baroque era, they also began to include such timbre
- 8 -
notations in their scores as phrasing, legato or stacatto
tongueing or bowing, and dynamic markings. Between that
time and the present day the notation of attacks, releases,
phrasing, color combinations and transformations, and dynam-
ics has become more and more specific. Detailed instrucions
to a violin performer such as "am Steg," "col legno," and
"stop the A string with the left thumb at A# and with the
fourth finger of the left hand glissando from the octave to
the highest possible harmonic while bowing gently," have
become popular with modern composers.
Harmonic analysis has helped to categorize the differ-
ences between instruments. The flute, for example, produces
a tone with few strong overtones, most closely approaching a
pure sine wave of all the orchestral instruments. The clar-
inet, in contrast, has a prominent fundamental but also has
strong odd-numbered partials. The oboe and the bassoon
exhibit what are often called formant regions--regions in
the range of the instrument that are resonant and cause par-
tials that fall there to be much higher in amplitude. For
the oboe, the formant regions are from 1000-1500 hertz and
again at 3000-4000 hertz. For this reason, low notes on the
oboe (or bassoon) lack energy at the fundamental and appear
to be higher than the register in which they are notated.
Brass instrments are rich in partials in all but the highest
part of the range. A formant region covers approximately the
upper two-thirds of the range, causing the lower notes to
have weak fundamentals. Also, the louder a note is played
- 9 -
on a brass instrument, the stronger the upper partials
become.
The theories discussed so far do not adequately explain
how the mind is able to differentiate between the various
timbres present or how they are related to those timbres
previously encountered by an individual perceiver. One
popular theory of cognitive perception is the general notion
of "template matching" (Norman/Lindsay, 1977). It is pro-
posed that a record is kept and categorized by the brain in
the form of a template for each stimulus encountered by the
individual. When the brain is presented with a new
stimulus, the existing templates are searched for a match.
If a partial or near match can be found, the stimulus is
then identified and categorized, else a new category is
formed.
In the case of timbre, or any other multidimensional
stimulus, a host of parameters must be stored somewhere in
memory for retrieval in the template matching strategy.
Consider the clarinet, whose tone color is noticeably dif-
ferent in the low chalumeau register from its timbre in the
higher range. A listener will correctly identify all sounds
produced by the instrument as being typical of a clarinet,
yet a separate spectral template for each sound would have
to be stored in memory in order to make the positive iden-
tification possible. Within the template must be placed all
the spectral components obtained from the Place Theory
- 10 -
additive analysis of timbre. Clearly, the storage capacity
required of the mind by this process is enormous.
Other cognitive viewpoints exist and are sometimes at
variance with one another. For example, the Periodicity
Theory of pitch perception, in contrast to the Place Theory,
attributes an ability to the basilar membrane of being able
to decode frequency as a function of the resonance of the
membrane, where the nerve cells react to the velocity pat-
terns of the airwave and fire in synchrony with the regular
rise and fall of the beat frequency. However, with either
the Place or Periodicity theories the Fourier model is left
intact, the ear performing the spectral analysis and sending
the information to the brain for template matching. A favor-
ite technique, and a major problem with the development of
cognitive theories, is the use of illusions or other methods
of fooling the perceptual system in an effort to understand
and explain the mechanism. As interesting as illusions may
be, when undue emphasis is placed on their study, all they
really begin to show is how the perceptual system may be
fooled, not how it works.
The finding in another experiment by Grey (1975)
involves the relationship between timbres of various musical
instruments. Two instrument timbres are chosen, analyzed,
and resynthesized according to the additive technique
already described. An interpolation of the time-variant
amplitude/frequency spectra is made in several successive
- 11 -
steps between the two instrument timbres. The observers are
asked to determine the point at which the timbre of the
second instrument becomes noticeable, and to note the
abruptness or gradualness of the transition. It is
interesting that there are no naturally-occuring instrument
sounds between discreet musical instruments. Rather, the
range between the two instruments exhibits properties of
both timbres, more like a blend than a separate third tim-
bre. Also of note is that the moment chosen by the
observers as the point of crossover from one instrument to
the next invariably is closer to the second instrument than
to the first, exibiting a definite hysteresis in all transi-
tions.
Sound clearly has many dimensions including loudness,
intensity, pitch, and timbre. Many of these attibutes them-
selves are multidimensional. Pitch, for example, has at
least the two dimensions of pitch-class (C, C#, D, D# . . .)
and octave. An investigation into the facets of timbre
benefits from a device called multidimensional scaling. In
this system, perceptual data consisting of subjective simi-
larity judgements between pairs in a set of stimuli are
treated as measurements of subjective distance from which a
best-fitting geometric image, with the number of dimensions
specified by the investigator, is constructed. In one such
experiment (Grey, 1977), sixteen instrument tones were pro-
cessed according to the analysis/synthesis routine and
equalized for loudness, pitch, and duration, in order to
- 12 -
remove those dimensions from the analysis. It was found
that three spatial dimensions best represent the perceptual
relationships: (1) spectral energy distribution (wide band
versus narrow), (2) the presence of low-amplitude high-
frequency enery in the attack segment (most often inhar-
monic) as opposed to low frequency only, and (3) the degree
of synchronicity of the amplitudes of the harmonics in their
temporal progressions. As this was a test using specific
soundtypes, other dimensions are also possible in a dif-
ferent setting.
Using the time-variant amplitude synthesis technique to
represent various instruments involves the manipulation of a
large amount of data, and requires a substantial amount of
computational power (ie., a fast digital computer). Yet it
is plain that not all the variables or dimensions have been
taken into account. Each new dimension adds more complexity
to the information needed to be stored if the template
theory is to be satisfactorily implemented. The increasing
complexity as new parameters are identified and added to the
model suggests that there should be a simpler, more general
way of approaching timbre.
Harmonic analysis, using the Fourier theory, yields a
list of the frequencies present and their relative ampli-
tudes over a given time segment or window. If an accurate
picture of a timbre is desired which includes the continuous
- 13 -
time-variant information, many thousands of Fourier
transforms of sufficiently small window size are required.
To suggest that the ear works in this way, that it uses the
Fourier transform to obtain its harmonic data, is untenable
because of the impossibility of knowing where and when to
perform the Fourier analyses. However, the ear is able to
detect information specific to the dynamic events that pro-
duced the timbre, the sound source. Musical instruments and
all natural sound sources can be modeled as dynamic systems,
each with its own style of change over time. The source
would then be taken as an independent solution, with its own
set of boundary conditions and variables, to a wave equa-
tion, however complex it may need to be, such as that
described by Hiller and Ruiz (1971) for various plucked,
struck, and bowed strings. The method by which an object or
air column is excited into vibration is specific to that
system. The attack transients, steady state, release tran-
sients, the presence or absence of vibrato (controlled vari-
ations in frequency) or tremelo (variations in amplitude),
and the characteristic transients when moving between
notes--all of the perceptible timbre cues--are functions of
the source of the sound.
Gibson (1966) describes an alternative approach to the
cognitive model of visual perception which he calls ecologi-
cal optics. Ecological acoustics, after Gibson's model,
advocates an approach to the perception of sound that com-
bines the physical analysis of the source event and the
- 14 -
identification of the higher-order acoustic properties of
the event that are detectable by the listener. An analogy
can be drawn between the action of human perception mechan-
isms and operation of an interesting device called a polar
planimeter (Runeson, 1977). This device, while adept at
measuring the area of an irregular outline or shape, is
totally unsuited to the task of measuring the length of an
object, even though observers would remark that it should be
impossible to calculate area without being able to measure
length since length is part of the complex quantity--area.
If the senses can be modeled as direct receptors of complex
variables, then it becomes easier to believe that the ear
can detect timbre without resorting to the Fourier
transform. In fact, the human perceptual system ignores cer-
tain variables in the environment that other species con-
sider important--for example, the frequencies over about
20,000 hertz that dogs find useful, or the pattern of
skylight polarization that only insects can see. In support
of the Ecological view, Heyser (1976) suggests the thesis
that ". . . we reaize that a frequency domain expansion can
be completely accurate and yet have no meaning to a . . .
listener because it is in the wrong system of coordinates,"
because the frequency domain and the time domain are really
just two different ways of looking at the same thing, and
the ear is only perceptive of the latter. Balzano (1983)
maintains that if the basic form of the underlying dynamics
of a sound stimulus are invariant over its intensity range
- 15 -
and its frequency range, and if the ear can perceive infor-
mation of this sort directly, then the subconcious spectral
analyses and the encyclopedic memory for template matching
are not needed in the identification of timbres. With the
understanding that the method of tone generation is central
to the identification of timbre, it is easy to see why many
people group all of the synthesized sound that has been pro-
duced this century into a single "timbre-electronic."
The development of a workable definition of timbre has
been dominated for many years by the Helmholtz spectral
theory. Acousticians working more than a hundred years after
the proposal of Helmholtz have only expanded the original
theory, and are still primarily concerned with the steady
state timbre. Musicians recognize the importance of the
time-varying spectrum, and researchers can demonstrate com-
petent examples of analysis and resynthesis using multidi-
mensional time-variant parameters. Cognitive perceptualists
accept the Helmholtz Place Theory but offer the contrasting
Periodicity Theory, yet with either theory they rely on the
unwieldy system of template matching for information recog-
nition and retrieval. Ecological acousticians are interested
in the development of an alternative view to the
Helmholtz/Fourier model, suggesting that timbre and the per-
ception of timbre, instead of being the performance of
abstract analyses of the perceived sound structure, are a
matter of directly perceiving and interpreting the underly-
ing dynamics of the physical process that generates the
- 16 -
sound.
- 17 -
_B_i_b_l_i_o_g_r_a_p_h_y
[1] Balzano, Gerald J. "Changing Conceptions of Pitch
and Timbre: A Modest Proposal." Paper presented to
the 106th meeting of the Acoustical Society of Amer-
ica. Abstract in supplement to vol. 74 (November
1983): 518.
[2] ________. "Musical vs. Psychoacoustical Variables and
Their Influence on the Perception of Musical Inter-
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_M_u_s_i_c _E_d_u_c_a_t_i_o_n 70 (1982): 1-11.
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_C_o_n_t_e_m_p_o_r_a_r_y _M_u_s_i_c _T_h_e_o_r_y. New York: Norton, 1972.
Babbitt, M. "The Structure and Function of
Musical Theory."
Randall, J. K. "Two Lectures to Scientists."
[4] Clark, Melville and Luce, David. "Intensities of
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_S_o_c_i_e_t_y 13, 2 (1965): 151 - 57.
[5] Clark, M., Robertson, P., and Luce, D. "A Preliminary
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[6] Clark, Melville and Paul Milner. "Dependence of
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_S_o_c_i_e_t_y 12, 1 (1964): 28 - 31.
[7] Cogan, Robert and Pozzi Escot. _S_o_n_i_c _D_e_s_i_g_n.
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Chapter 4, "The Color of Sound."
- 18 -
[8] Erickson, Robert. _S_o_u_n_d _S_t_r_u_c_t_u_r_e _i_n _M_u_s_i_c.
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[9] Fechner, Gustav. _E_l_e_m_e_n_t_s _o_f _P_s_y_c_h_o_p_h_y_s_i_c_s.
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New York: Appleton-Century-Crofts, 1969. Chs. 1-5,
20.
[11] Gibson, J. J. _T_h_e _S_e_n_s_e_s _C_o_n_s_i_d_e_r_e_d _a_s _P_e_r_c_e_p_t_u_a_l
_S_y_s_t_e_m_s. Boston: Houghton, 1966. Introduction, Chs.
1-3, 5, 13-14.
[12] Grey, J. "An Exploration of Musical Timbre."
Dissertation. Department of Psycology, Stanford
University, 1975. pp. 75 - 86.
[13] ________. "Multidimensional Perceptual Scaling of
Musical Timbres." _J_o_u_r_n_a_l _o_f _t_h_e _A_c_o_u_s_t_i_c_a_l _S_o_c_i_e_t_y
_o_f _A_m_e_r_i_c_a 61, 5 (1977): 1270 - 1277.
[14] ________. "Timbre Discrimination in Musical Patterns."
_J_o_u_r_n_a_l _o_f _t_h_e _A_c_o_u_s_t_i_c_a_l _S_o_c_i_e_t_y _o_f _A_m_e_r_i_c_a 64, 2
(1978): 467 - 473.
[15] ________, and J. W. Gordon. "Perceptual Effects
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_J_o_u_r_n_a_l _o_f _t_h_e _A_c_o_u_s_t_i_c_a_l _S_o_c_i_e_t_y _o_f _A_m_e_r_i_c_a 63, 5
(1978): 1493 - 1500.
[16] ________, and J. A. Moorer. "Perceptual Evaluation of
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_t_h_e _A_c_o_u_s_t_i_c_a_l _S_o_c_i_e_t_y _o_f _A_m_e_r_i_c_a 62, 2 (1977): 454
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- 19 -
[17] Helmholtz, H. L. F. _D_i_e _L_e_h_r_e _v_o_n _d_e_n _T_o_n_e_m_p_f_i_n_d_u_n_g_e_n.
Brunswick: 1863. English trans., _O_n _t_h_e _S_e_n_s_a_t_i_o_n_s
_o_f _T_o_n_e. New York: Dover, 1954. Introduction,
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[18] Hiller, L. and Isaacson. "Experimental Music."
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Simon & Schuster, 1963.
[19] Heyser, R.C. "Perspectives in Audio Analysis: Changing
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667. II. _J_o_u_r_n_a_l _o_f _t_h_e _A_u_d_i_o _E_n_g_i_n_e_e_r_i_n_g _S_o_c_i_e_t_y
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_P_r_o_c_e_s_s_i_n_g. New York, 1972. Chapts. 1, 4, 5, 7.
[21] McAdams, Stephen and Albert Bregman. "Hearing Musical
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1979): 26-43, 60, 63.
[22] Meyer, Leonard. _E_m_o_t_i_o_n _a_n_d _M_e_a_n_i_n_g _i_n _M_u_s_i_c.
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