This is the 2nd chapter of the book ***"Music by the Numbers: Fr...
### TL;DR Pythagoras of Samos was intrigued by the sounds emitte...
An Octave corresponds to the fundamenta interval between one musica...
### The Monochord The monochord is a musical instrument that is ...
Illustration of the Pythagorean intervals on the strings of a guita...
#### Physics of a vibrating string The wavelength of the fundame...
> Pythagoras called them perfect consonances and used them to const...
This sequence: \frac{9}{8},\frac{9}{8},\frac{256}{243},\frac{9}...
Joseph Sauveur was a French mathematician and physicist and a membe...
It would be nice to listen the music. A mp3 file, perharps?
How did Johannes Kepler come to his assignment of a celestial melod...
CHAPTER 2
String Theory,
500 BCE
IT IS A STRANGE TRUISM: the earliest experimental sci-
ence to establish quantitative relations between ob-
servable entities was acoustics. Pythagoras of Samos
(ca. 585500 BCE), the legendary philosopher who will
forever be associated with the right- triangle theorem
named after him, began his scientic career by investi-
gating the vibrations of sound- emitting objects. Accord-
ing to legend, while walking down a street one day he
heard sonorous sounds coming from a blacksmith’s shop.
Stopping by to investigate, he noticed that the sound had
originated from the craftsman’s hammer hitting a metal
sheet; the heavier the sheet, the lower the pitch of the
sound it emitted.
Not being satised with just a qualitative observation,
Pythagoras went on to experiment with all kinds of vi-
brating bodies—taut strings, water- lled glasses, bells,
and pipes (gure 2.1). He is said to have built a primitive
musical instrument, the monochord—a single string at-
tached to a sound board with a numerical scale along it
(gure 2.2). The strings effective length could be varied
by inserting a small bridge between the string and the
board. Pythagoras found that, when the string was al-
lowed to vibrate rst at its full length and then stopped
at half its length, the two sounds bore a pleasant, har-
monious afnity to one another: they were separated by
an octave. A melody played in different octaves sounds
14 CHAPTER 2
essentially identical, like walking down the hallway on
different oors of a hotel. The octave, Pythagoras had
found, corresponds to the ratio 1:2.
Having established the octave as a fundamental musi-
cal interval, Pythagoras next attempted to subdivide this
FIGURE 2.1. Pythagoras experimenting with sound-emitting objects. From
Franchino Gaffurio, Theorica Musicae (Milan, 1492).
STRING THEORY, 500 BCE 15
rather large interval into smaller parts. He experimented
with other ratios of string length, leading him to a dis-
covery that left a deep impression on him: ratios of small
numbers produced harmonious, pleasant combinations of
soundsconsonances—whereas ratios of larger numbers
produced dissonances. Foremost among the former were the
octave (1:2), the fth (2:3), and the fourth (3:4) (the names
derive from the position of these intervals in the musical
scale; see gure 2.3). Pythagoras saw in this a sign that
nature itself—indeed, the entire universe—is governed by
simple numerical ratios. Number rules the universe became
the Pythagorean motto, and it would dominate scientic
thought for the next two thousand years.
FIGURE 2.3. The octave, perfect fth, and perfect fourth.
FIGURE 2.2. Monochord.
16 CHAPTER 2
𝄓
We must digress here for a moment and mention that be-
ginning around 1600, it became the practice to describe
musical intervals in terms of their frequency ratios, rather
than ratios of string length. For any given string, the fre-
quency is inversely proportional to the strings length, so
the octave, the fth, and the fourth correspond to the ra-
tios 2:1, 3:2, and 4:3, respectively. We will adhere to this
practice in what follows.
𝄓
The three intervals just mentioned were to play a funda-
mental role in music. Pythagoras called them perfect con-
sonances and used them to construct a musical scale
the rst known attempt to organize musical sounds into
an orderly numerical system. He found that, starting
with any note, going up a fth and then another fourth
brings us to a note exactly one octave above the starting
note. Translated into ratios, the relation can be expressed
as
2
3
3
4
1
2
# =
. This is true in general: to add two intervals,
multiply their frequency ratios. Unbeknownst to him, Py-
thagoras had discovered the rst logarithmic relation in
history.
Next, he took each perfect consonance and raised its
ratio to successive powers. Powers of 2:1 merely carry us
to higher octaves, while powers of 4:3 result in inversions
of 3:2 (an interval is said to be inverted if its lower note
is moved up by one octave or its higher note down by one
octave). This left him with powers of 3:2, starting with
()
1
3
2
2
3
=
and leading to the following sequence:
1
.
2
3
3
2
2
3
2
3
2
3
2
3
4
9
2
3
8
27
2
3
16
81
2
3
32
243
1 012345
=
a aaaaaa
kkkkkkk
STRING THEORY, 500 BCE 17
Of the seven ratios in this sequence, only the second
and third lie within one octave. To bring the remaining
ratios into the range of an octave, we multiply or divide
them by powers of 2:
.
3
4
1
2
3
8
9
16
27
64
81
128
243
When this new sequence is arranged in ascending
order and augmented by the ratio 2:1 to complete it to a
full octave, we get the following array:
.
1
This sequence is known as a diatonic scale. It gives the
ratio of each note to the fundamental (lowest) note. But in
music, what matters most is the ratio between two notes,
that is, the interval separating them. By taking the ratio
of each note to the one preceding it, we get the sequence
,
8
9
8
9
8
9
8
9
8
9
243
256
243
256
which represents the intervals of the Pythagorean dia-
tonic scale. It consists of just two distinct intervals, a
large one of 9:8 (= 1.125), called a whole tone, and a small
one of 256:243 (~ 1.053), called a semitone or half tone.
𝄓
At rst thought, the Pythagorean scale seems like a
great invention; it stands out for its simplicity, employing
powers of just one ratio, 3:2. But this simplicity is deceiv-
ing, and for a number of reasons. First, as every music
student learns early on, there is a scheme called a circle
of fths: start with any note and go up in a succession of
fths. After doing this twelve times (and in the process
going through a series of sharps and ats, notes that are
18 CHAPTER 2
a half tone above or below those of the diatonic scale), you
should arrive back at the base note, albeit seven octaves
higher (see gure 2.4). Alas, this is impossible to do with
the Pythagorean scale: no positive integer values of m
and n can ever satisfy the equation
() 2
mn
2
3
=
.
1
But even more troubling is the fact that the Pythago-
rean scale was out of tune with the natural sequence of
harmonics, or overtones, generated by practically all mu-
sical instruments. When a string is vibrating, it emits a
note with a denite pitch that can be placed on the musi-
cal staff, but there are also other, higher notes that come
along with it. This mix of overtones gives the sound its
characteristic color, or timbre—the quality that distin-
guishes the sound of a violin from that of a clarinet, even
when they play the same note.
As we will see in the next chapter, the frequencies of
these overtones are always whole multiples of the strings
lowest, fundamental frequency, so they follow the se-
quence 1, 2, 3, . . . (relative to the fundamental). In theory
FIGURE 2.4. The circle of fths.
C
A
E
D
G
G
F
B
Major Keys
Minor Keys
G
(1 sharp)(1 at)
F
D (2 sharps)
E (4 sharps)
A (3 sharps)
B (5 sharps)
STRING THEORY, 500 BCE 19
this series can go on forever, producing an innite blend
of ever higher notes. Usually, however, the amplitudes of
these overtones, and therefore their intensities, quickly
diminish as we go up the sequence, making them increas-
ingly feeble and difcult to hear. Indeed, for nearly two
thousand years they remained hidden behind the funda-
mental tone, barely noticed until the eighteenth century,
when a little- known French scientist by the name Joseph
Sauveur conrmed their existence (see chapter 3). Never-
theless, these harmonics play a crucial role in music, for
they are the raw material from which the natural musi-
cal intervals are derived. The Pythagorean scale, being
based solely on the ratio 3:2 while leaving out the remain-
ing harmonics—including such important ratios as 5:4
and 6:5 (a major third and a minor third, respectively)
was therefore out of sync with the laws of acoustics; it was
a purely mathematical creation, divorced from physical
reality. This was the rst known attempt to impose math-
ematical rules on music, but it would not be the last.
𝄓
The Pythagorean scale was typical of the Pythagorean
philosophy in general. Obsession with musical numerology
led Pythagorass followers to believe that everything in
the universe, from the laws of musical harmony to the mo-
tion of the celestial bodies, was governed by simple ratios
of whole numbers. To understand this giant leap of faith,
we must remember that in Greek tradition music ranked
equal in status to arithmetic, geometry, and spherics (as-
tronomy)—the quadrivium comprising the four disciplines
every learned person was expected to master, the equiva-
lent of the core curriculum of todays university.
2
Signicantly, to the Pythagoreans the word “arithme-
tic” had a different meaning than it has today; it meant
number theory, the study of the properties of integers,
20 CHAPTER 2
rather than the practical skills needed to compute with
them. Likewise, they regarded the music component of
the quadrivium as referring to music theory, the study of
scales and harmony, not the actual art of playing music.
This was typical of the aloof attitude of the Pythagore-
ans to all things practical. Theirs was a perfect universe,
governed by notions of beauty, symmetry, and harmony
but removed from daily, mundane considerations. It may
have been one reason why they kept all their discussions
secret, fearing they would be ridiculed by their fellow
citizens, the vast majority of whom had to toil daily to
eke out a living. None of the Pythagorean writings—if
they left any writings at all—survived. All that we know
about them came from later writers, who lived hundreds
of years after Pythagoras and often outdid each other in
extolling the virtues of their revered master.
But if their writings did not survive, the Pythagorean
legacy lasted well over two thousand years. Number rules
the universe became a rallying motto to generations of sci-
entists and philosophers, who sought to explain the mys-
teries of the cosmos on the basis of musical ratios or in
terms of simple, elegant geometric shapes. The planets,
for example, had to move around the Earth in perfect
circular orbits; it was inconceivable that any shape other
than the perfectly symmetric circle could rule the uni-
verse. Thus, by subjugating the laws of nature to their
ideals of beauty, harmony, and symmetry, the Pytha-
goreans may have actually impeded the progress of sci-
ence for the next two millennia.
One of the last Pythagoreans was the eminent German
astronomer Johannes Kepler (1571–1630), considered the
father of modern astronomy. Kepler, at once a devout mys-
tic and an ardent believer in the Copernican heliocentric
system, spent more than half his life trying to derive the
laws of planetary orbits from those of musical harmony.
STRING THEORY, 500 BCE 21
He believed that each planet, in its orbit around the Sun,
plays a tune that our ears are unable to hear, being below
the range of audible frequencies (not to mention that it
was produced in the vacuum of outer space, where sound
cannot propagate). He actually assigned a celestial mel-
ody, written down in musical notation, for each of the ve
then- known planets (gure 2.5)—the celebrated music of
the spheres. It was only after decades, during which he fol-
lowed this blind path, that Kepler nally abandoned the
Greek