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.
Manual of Harmonics by Nicomachus of Gerasa (b.
a.d. 60), and the
Harmonics by Ptolemy (c.
a.d. 100 – c. 160). Nicomachus chronicles the
discoveries of the semi-legendary 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
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 one-half of a string vibrates in a 2/1 relation, or twice as fast
when compared to the vibrations of the whole string; and two-thirds 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 one-half string length, it produces frequency ratio 2/1, or
the interval of an octave; and a
two-thirds string length, frequency ratio 3/2, or the interval of a fifth. If we were to write the conventional
12-tone 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 12-tone equal temperament have never heard this
12-tone 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 long-forgotten 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
different from the natural.
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,
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,  that no sound is ever heard
lower . . . than the son naturel of the
string [1:1], since they are all higher, and  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 double-octave, 5/1 sounds a
double-octave and a major third, and 6/1 sounds a double-octave 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 12-tone 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.
Musical Mathematics: On the Art and Science of Acoustic
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
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 12-tone equal temperament, most musicians in the world
tune their instruments in just intonation, or based on intervals that have rational