

Musical Mathematics
on the art
and science of acoustic instruments
●
Table of
Contents
Buy now at
Amazon.com:
click
New or
Used.
Also available from the publisher at
Chronicle Books, San Francisco.
© 2000–2019 Cristiano M.L. Forster
All rights reserved.
www.chrysalisfoundation.org
Just Intonation: Two
Definitions
(Excerpts from Musical Mathematics,
Chapters 3 and 10.)
The term just intonation has two different definitions.
The first is ancient and strictly mathematical.
It states that just intonation is a method of tuning intervals
and scales based exclusively on rational numbers.
The second is modern and strictly acoustic.
It states that just intonation is a method of tuning based on
the intervals of the harmonic series.
In the West, some of the oldest extant sources that give detailed
accounts of rational or integer number ratios in music are the
Division of the Canon by
Euclid (fl. c. 300
b.c.), the
Manual of Harmonics by Nicomachus of Gerasa (b.
c.
a.d. 60), and the
Harmonics by Ptolemy (c.
a.d. 100 – c. 160).
Nicomachus chronicles the discoveries of the semilegendary
Greek philosopher and mathematician Pythagoras (c.
570 b.c. –
c. 500
b.c.) who reputedly
discovered the crucial nexus between sound and number.
According to Nicomachus, Pythagoras lacked an instrument that
would assist his ears in the same manner in which a compass aids the
eyes. Because human sense
perceptions are imperfect, we need a compass to draw a circle.
The act of drawing a circle with a compass produces the
following relation:
C/D
= π, where
C is the circumference of
the circle;
D is the diameter of the circle;
and π is the Greek
letter pi that represents
the ratio 3.1416... Nicomachus
states that since our ears are also incapable of precise numerical
analysis, Pythagoras contemplated the possibility of a device that
would verify the following intervalic relationship:
that the fourth E–A plus the fifth A–E produce the octave E–E.
To understand Pythagoras’ problem, imagine a world in which musicians
tune their instruments strictly by ear, without any knowledge of the
mathematical relationships between tones.
Before the time of Pythagoras, instrument builders and
musicians had no knowledge of musical ratios.
If one performer preferred sharp fourths, and another preferred
flat fourths, no method existed which enabled musicians to quantify
the magnitude of their tuning discrepancies. Lyre players must have
encountered serious difficulties in consistently tuning their
instruments to a given scale.
In those days, players of stringed instruments probably relied on wind
instruments to help remind them of difficult or long forgotten
tunings.
Nicomachus continues his discussion by observing that the frequency of
a string is inversely proportional to its length. Although musicians and mathematicians in the 1st century
a.d. had no method to
determine the exact frequencies of strings, Nicomachus’ conclusions
regarding a string’s rate of vibration are correct.
He accurately observes that onehalf of a string vibrates in a
2/1 relation, or twice as fast when compared to the vibrations of the
whole string; and twothirds vibrates in a 3/2 relation, or 1.5 times as fast
when compared to the vibrations of the whole string. In short, with this instrument Pythagoras confirmed the numeric
proportions of intervals, because a measured string section assists
the ears in the same manner in which a compass aids the eyes. The compass produces π,
or the ratio that defines the characteristic shape of a circle;
and a string section produces — in relation to the length of
the whole string — a ratio that
defines the characteristic sound of a musical interval.
Now, when we sound a onehalf string length, it produces frequency
ratio 2/1, or the interval of an octave;
and a twothirds string length, frequency ratio 3/2, or the
interval of a fifth. If we were
to write the conventional 12tone chromatic scale:
C, C#, D, D#, E, F, F#, G, G#, A, A#, B, C as a progression of
rational ratios, the following sequence would result: 1/1, 16/15, 9/8,
6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 16/9, 15/8, 2/1.
In response to Pythagoras’ problem, note that a fourth, ratio
4/3, times a fifth, ratio 3/2, equals 4/3 ×
3/2 = 2/1, or the octave.
Finally, in the West, most musicians trained on instruments
tuned in 12tone equal temperament have never heard this 12tone just
intoned scale. From a
historical perspective, the mathematical origins of just scales, which
consist of rational numbers, precede tempered scales, which consist of
irrational numbers, by more than two thousand years.
Marin Mersenne (1588–1648) was the first European to accurately
describe and mathematically define the first six harmonics — 1/1, 2/1,
3/1, 4/1, 5/1, 6/1 — of vibrating strings.
These discoveries forever changed Western music theory.
Suddenly, scientists and musicians realized that the rational
ratios of just intonation not only constitute a convention of man, but
also reflect a phenomenon of nature!
Consequently, integer number ratios, which comprised the core
of tuning and music theory since the time of the ancient Greeks, had a
physical reality in nature, and could, therefore, not be dismissed as
antiquated entities of longforgotten civilizations.
In Harmonie universelle: The
Books on Instruments, Mersenne dedicated Book 4, Prop. IX, to his
discoveries of string harmonics.
In this text, he used three different terms to describe
harmonics.
Petits sons delicats appears
in a general context, and literally means small, delicate sounds.
Sons differens du naturel
appears in a numerical context, and means sounds different from the
natural. Here
naturel connotes
son naturel, or natural sound, a term that Mersenne
consistently used to describe the lowest and most audible tone of a
string, or the fundamental, ratio 1/1.
Finally, sons
extraordinaires also appears in a numerical context, and means
extraordinary sounds. To simplify the following discussion, I will continue to use the
simple word harmonic to describe these special sounds
discovered by Mersenne. Even
though Mersenne accurately identified the first six harmonics, he had
no knowledge of the superposition of traveling waves in vibrating
systems; therefore, he
did
not have an exact
mathematical understanding of the harmonic series as a theoretically
infinite sequence of integers.
Consequently, the following infinite series of modern frequency
ratios: 1/1, 2/1, 3/1, 4/1,
5/1, 6/1, 7/1, . . . , etc. eluded him.
Despite these mysteries, difficulties, and frustrations,
Mersenne persevered and discovered important truths about the nature
of sound because he had phenomenal powers of hearing and observation,
and a tenacious curiosity. In
his own words, “Some new properties of strings will always be found if
one takes the trouble of examining them in all the ways possible.”
(All quotations from R.E. Chapman’s translation of
Harmonie universelle: The Books
on Instruments, Martinus Nijhoff, The Hague, Netherlands, 1957.)
To understand the following discussion based on Mersenne’s tone
numbering system, write three consecutive standard diatonic scales
beginning on the second C below middle C.
Then, the 1st tone is the second C below middle C, the 8th tone
is the first C below middle C, the 15th tone is middle C, and the 22nd
tone is high C.
In Harmonie universelle: The Books on Instruments, Book 4, Prop. IX,
Mersenne assures the reader that the harmonics he hears are not due to
the sympathetic vibrations of other strings.
He then assigns ratios to these harmonics, and ends by
comparing four string harmonics — 2/1, 3/1, 4/1, 5/1 — to the
“leaps,”
that is, to the harmonics of the natural trumpet:
* * ^{↓} * *
“Thus it is very certain that these different tones do not come from
other strings which are on the instruments and which tremble without
being played . . . since the single string of the monochords produce the same sounds.
“Now these sounds follow the ratio of these numbers, 1, 2, 3, 4, 5,
since one hears four sons differens du naturel, the first of which is at the upper octave
[2:1], the second at the twelfth [3:1], the third at the fifteenth
[4:1], and the fourth at the major seventeenth [5:1], as is seen by
the said numbers which contain the ratios of these consonances in
their lesser terms. At this
point, two things must be remarked, that is, [1] that no sound is ever
heard lower . . . than the son
naturel of the string [1:1], since they are all higher, and [2]
these tones follow the same progression as the leaps [i.e., harmonics]
of the trumpet . . .” (Ratios
and text in brackets mine.)
* * ^{↑} * *
Later in the text, Mersenne identifies the 19th tone of his numbering
system, ratio 6/1, as the sixth harmonic.
* * ^{↓} * *
“Nevertheless I add that these leaps and these points [on a trumpet
marine, an instrument with very long strings], which imitate the
sounds of the military trumpet, do nothing else but explain in great
volume what the string does being played open, that is, [the string
sounds] the octave [2:1], the twelfth [3:1], the fifteenth [4:1], the
seventeenth [5:1], the nineteenth [6:1], etc., one after the other . .
. which it produces all together at the same time . . .” (Ratios and
text in brackets mine.)
* * ^{↑} * *
This gives a total number of six harmonics:
1/1, 2/1, 3/1, 4/1, 5/1, 6/1.
In relation to the fundamental 1/1, frequency ratio 2/1 sounds
an octave, 3/1 sounds an octave and a fifth, 4/1 sounds a
doubleoctave, 5/1 sounds a doubleoctave and a major third, and 6/1
sounds a doubleoctave and a fifth.
Now, the interval ratio between 2/1 and 3/1 is 3/2 or a fifth,
between 3/1 and 4/1 is 4/3 or a fourth, between 4/1 and 5/1 is 5/4 or
a major third, and between 5/1 and 6/1 is 6/5 or a minor third.
In Western music, the only intervals classified as bona fide
consonances are interval ratios 2/1, 3/2, 4/3, 5/4, 6/5.
In this tuning system, a consonant interval may only include
prime factors 2, 3, or 5. Note
that in the previously mentioned 12tone scale: 1/1, 16/15, 9/8, 6/5,
5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 16/9, 15/8, 2/1, all integers include
no higher prime factors than 5.
That is, in Western music, interval ratios that include prime factors
7, 11, 13, . . . , etc. are not permitted, and therefore never heard.
Having never heard such intervals, most musicians and theorists
do not dismiss them for experiential reasons, but for academic
reasons.
From my book
Musical
Mathematics: On the Art and Science of Acoustic Instruments (see
M.M.
Pages > Musical Mathematics),
below please find Chapter 3, Figure 12, which illustrates the acoustic
basis of just intonation in the context of the harmonic series.
Many modern music theorists have attempted to intellectually degrade
the importance of just intoned scales by promulgating two serious
misconceptions. First, they
insist that in a musical context, the ratios of just intoned scales
must include only simple or small integers. This bogus rule is based on the premise that the frequency ratios
of just intonation are somehow synonymous with consonant tones and
intervals like 2/1, 3/2, 4/3, etc.
Almost four centuries ago, Mersenne observed, “. . . all the
vibrations of air which the consonances and dissonances make are
commensurable [rational] . . .”
(Text in brackets mine.)
Second, they insist that just intonation is a tuning system, or some
specific scale. Any scale that
includes interval ratios such as 7/5, 11/8, or 16/13 challenges both
of these highly opinionated assumptions.
First, such scales contradict the notion that interval ratios
with prime numbers larger than 5 are inherently dissonant and
musically unacceptable, and therefore serve no purpose in a just
intoned scale. Second, it refutes the erroneous conclusion that a specific scale
represents just intonation.
Just intonation is
not a tuning system. It is
a tuning principle, or a method of tuning.
Furthermore, a just intoned scale, like a tempered scale, may
have any number of tones. All
writers who applaud the “virtues” of tempered tunings while
criticizing the “faults” of just intoned tunings argue their case from
a very narrow perspective.
Ultimately, such criticisms yield irrelevant bits of information, like
the kind obtained from comparing apples and oranges.
Finally, with the exception of Western instruments influenced by
keyboard and fretted instruments tuned in 12tone equal temperament,
most musicians in the world tune their instruments in just intonation,
or based on intervals that have rational identities. 
