

Musical Mathematics
on the art
and science of acoustic instruments
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© 2000–2014 Cristiano M.L. Forster
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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. 
