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--| Paradoxal Music |--- 

The endless staircase stands as a classic visual paradox, continuously
tricking the eye into a geometrically impossible journey. First devised in
the 1950’s by l.S. Penrose and roger penrose of the university of london
and later made famous by the dutch artist m.C. Escher, this paradox has a
rich set of acoustic counterparts. In the early 1960’s roger n. Shepard of
bell telephone laboratories produced a rather remarkable example. He
repeatedly played the same sequence of computer generated tones that moved
up in an octave. Instead of hearing the pattern stop and then start again,
listeners heard the pattern descend endlessly.

...The physical basis of musical pitch has fascinated scientists since
antiquity. Pythagoras established that the pitch of a vibrating string
varies inversely with its length: the shorter the string, the higher the
pitch. He also demonstrated that certain musical intervals – the pitch
relation between two tones – correspond to ratios formed by different
lengths of string. In the 17th century galileo and the french
mathematician and theologian marin mersenne showed that the basis of these
associations lay in the relation between string length and frequency of

Mersennse also demonstrated the existence of overtones, or harmonics, in
vibrating bodies. That is, a vibration occurs both at the frequency
corresponding to the perceived pitch (the fundamental frequency) and at
frequencies that are whole number multiples of the fundamental
(harmonics). In other words, a tone whose fundamental frequency is 100
hertz contains components at 200 hertz, 300 hertz, 400 hertz and so on. In
the 1930s jan schouten of philips laboratory in eindhoven showed that the
auditory system exploits this phenomenon. When presented with a harmonic
series, we can perceive a pitch taht corresponds to the fundamental
frequency, even if the fundamental itself is missing.

The relations between pitches enable us to hear musical patterns. When two
tones are presented simultaneously or in succession, we perceive a musical
interval. Intervals are heard to be the same in size when the fundamental
frequencies of their component tones stand in the same ratio.
(Technically, the tones within each pair are separated by the same
distance in log frequency.)

This principle forms one of the bases of the traditional musical scale.
The smallest unit of this scale is the semi-tone, which is the pitch
relation formed by two adjacent notes on a keyboard. The semitone
corresponds to a frequency ratio of approximiately 18:17. Intervals
composed of the same number of semitones are given the same name. For
example, the interval corresponding to a ratio of 2:1 (12 semitones) is
termed an octave, the interval corresponding to a ratio of 3:2 (seven
semitones) is termed a fifth and the ratio 4:3 (five semitones) is called
a fourth.

Tones related by octaves are in a sense perceptually equivalent. Each of
the 12 semitones in an octave is assigned a name (c, c#, d and so on). The
entire scale (called the chormatic scale) consists of note names across
octaves. The note names are identified by subscripts for example, middle c
can be written as c4. The c one octave lower is c3, and the one above is

The pitch of a tone can thus be regarded as varying along two dimensions.
The first, known as pitch height, extends from low to high, which can
experience by sweeping a hand all the way up a keyboard. The second is the
circular dimension of pitch class, which defines a tone’s position within
the octave. Researchers call this dimension as the pitch class circle. The
circle leads to an immediate assumption: it is nonsensical to ask whether
one tone, say, C, is higher than another, such as f#. To clarify the
question, one would need to give the octaves in which the two tones occur.

In the absence of such information the human brain still tries to organise
tones so that it can judge relative pitch. Shepard demonstrated this
phenomenon in 1964. Using a music synthesis programme developed by his
colleague max mathews, he generated a series of tones that were clearly
defined in terms of pitch class but in which the octave containing the
tones was unclear. Each tone consisted of sinusoidal components (smoothly
oscillating waves) separated by octaves, so that the tones were composed
only of harmonics in the same pitch class.

Shepard found that when two such tones were played, one after the other,
subjects heard either an ascending pattern or a descending one. The
perceived direction depended on the distance seperating the two tones
along the pitch class circle: listeners followed the shorter distance
between the tones. For example, subjects heard the pair c#-d as ascending,
because the shorter distance here is clockwise. Analoguously, the pair
a-g# was always heard as descending.

This finding enabled shepard to produce the striking demonstration
described at the beginning of this article. A series of tones repeatedly
traverses the pitch class circle in clockwise steps appears to ascend
endlessly in pitch. If the series of tones traverses the circle in
counterclockwise steps, it appears to descend infinitely.

Jean-claude risset, now at the laboratory for mechanics and acoustics at
the cnrs in marseilles [france], produced an intriguing variant. He
created a single tone that glided around the pitch class circle in a
clockwise direction. The tone appeared to ascend endlessly [just like i
heard in that *slowdive* song :) ] …risset used such a gliding tone when
he composed incidental music to pierre halet’s [play, the] *little boy*…

…in my laboratory i recently created pitch circularity effects using a set
of tones, each of which constituted a full harmonic series but in which
the relative amplitudes (loudnesses) of the harmonics were such as to
generate ambiguities of perceived pitch height. Listeners obtained an
impression of a series that ascended infinitely in pitch.

These demonstrations of pitch circularity illustrate that the human mind
tends to form linkages between elements that are close together rather
than those that are far apart. Analogous phenomena can be found in vision.
For example, we tend to group together dots that are next to one another
and to perceive movement between neighbouring lights turned on and off in
succession. [Like the movement perceived in the lights in a marquee.]

[To envision the pitch class circle, imagine the notes of the chromatic
scale arranged around the face of a clock, with the notes resident at the
following designations: 12:00-c, 1:00-c#, 2:00-d, 3:00-d#, 4:00-e, 5:00-f,
6:00-f#, 7:00-g, 8:00-g#, 9:00-a, 10:00-a#, 11:00-b, and back to 12:00-c.]

Musical paradoxes occur when sequences of tones appear to rise or fall
even though the tones lack the physical cues normally used to judge pitch
height [this is done by electronically eliminating the fundamental
frequency which is ussually heard when a string or conventional instrument
is agitated. (?) ] Such paradoxes can be understood in terms of the pitch
class circle, which represents the tones in an octave. The tones played
are opposite one another along the circle. In an example of this
phenomenon, called the tritone paradox, d is played at time t1 followed by
g# at t2. Some listeners heard the sequence ascend; others heard it
descend. In a variation called the semitone paradox, d# and g# are
presented simultaneously, follwed by d and a. Another version, the melodic
paradox, uses three pairs of tones. In these cases, some subjects heard
the ascending sequence as higher than the descending one, and others heard
it as lower. The results show that the subjects must have preferred
orientations of the pitch class [depending on locality and dialect of the
language they speak] circle with respect to it’s pitch height [for
example, british and americans will hear the tri-tone paradox differently.
The british typically hear the same notes ascend that the californians
hear as descending.].

(Scientific American, August 1992, Volume 267, Number 2, 
 Paradoxes of Musical Pitch, Diana Deutsch, pp. 88-90, ISSN: 0036-8733.)


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SUBMIT AN ARTICLE posted: january 1, 1995 updated: december 16 2004