Musical Mathematics

on the art and science of acoustic instruments

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CHAPTER 3:* FLEXIBLE STRINGS*

Part V: Musical, Mathematical, and Linguistic Origins of Length Ratios

Two definitions at
Just Intonation
provide background information |

For descriptions of the arithmetic and harmonic divisions of musical intervals that determine the 'minor' and 'major' tonalities in Western music, see |

Section 3.16

We may use the preceding example, which describes the tuning process of two
unfamiliar frequency ratios, 7/5 and 7/2, to analyze the musical, mathematical,
and linguistic origins of length ratios. Although aural experiences of sound
depend on frequencies, our anatomy does not enable us to quantify rates of
vibration from oscillating strings. As such, the frequencies produced *while
playing musical intervals *have always and will always remain
incomprehensible quantities. Therefore, if we spontaneously tune two canons
strings to an unfamiliar musical interval, we cannot identify the new interval
by counting cycles per second and constructing a frequency ratio. To comprehend
such an experience, we must *measure* the corresponding string lengths, and
through the construction of a length ratio, identify the unfamiliar interval.
Conversely, to intentionally tune two strings to a new musical interval, we
cannot locate the required position of a canon bridge by simply listening to
frequencies. Instead, we must divide the strings according to a length ratio
that determines the frequency ratio we have not yet experienced. In both of
these cases, our intellectual ability to comprehend aurally perceived frequency
ratios depends on our mathematical ability to measure and compare string
lengths.

We may summarize the preceding discussion by referring to Equation 3.3,
which states that frequency is a *function* of wavelength. Before we
analyze this statement further, let us first examine modern mathematical
notation with respect to the term “function.” Typically, a function constitutes
a cause-and-effect relation between two variables *x* and *y*, where
the *x*-number, chosen randomly or at will, is called the *independent
variable*, and the *y*-number, determined only after the *x*-number
has been selected, is called the *dependent variable*. For example, if we
state that the area of circle ( *y* ) equals *π* (3.1416) times the
radius ( *x* ) squared, then the expression *y* =* πx*2 defines
*y as a function of x*, or the area as a function of the radius. To specify
this functional relation, mathematicians replace *y* with *ƒ*(*x*),
which reads, “*ƒ* of *x,*” or stated in full, “the value of the
function ƒ at *x*.” Therefore, the previous equation in functional notation
states *ƒ*(*x*) = *πx*2.
This notation informs us that in the functional relation between variables *x*
and *y*, values of *x* determine values of *y*. Now, substitute
the radius *x* = 2 into these two equations, and calculate the area *y:*

In the left column, a
sequence of equations simply terminates in a solution for the area *y*,
whereas in the right column, a sequence of equations in functional notation
terminates not only in a solution, but indicates the value of radius *x*
that determines area *y*.

If we solve this equation for *x*, the expression
defines *x as a function of y*, or the radius as a function of the area.
Therefore, in functional notation:
.
Substitute the area *y* = 4 into these two equations, and calculate the
radius *x:*

Now, recall Equation 3.3,

and express *F
(frequency) as a function of λ (wavelength):*

Next,
solve Equation 3.3 for *λ,*

and express *λ as a
function of F:*

Equation 3.36 expresses a functional relation that is
algebraically and scientifically correct: as frequencies increase, wavelengths
decrease; *vice versa*. However, since we cannot count rates of vibration
from oscillating strings, we cannot substitute values for the independent
variable *F* into this equation. That is, our aural experience of an *
unfamiliar* interval does not enable us to quantify a frequency ratio, which
we could then invert to identify the corresponding length ratio.

In contrast, Equation 3.35 essentially describes the musical and mathematical
process by which musicians create ascending sequences of tones that result in
tuned scales. By shortening the lengths of strings in precise increments, as
wavelengths decrease, frequencies increase. This tuning process requires two
steps. (1) Our anatomy enables us to measure, quantify, and comprehend all
changes in string lengths. (2) The inverse proportionality between length ratios
and frequency ratios — l.r.
f.r
—
permits us to interpret these length ratios as frequency ratios, which we then
identify as musical intervals. This second step is crucial simply because we **
do not** hear wavelengths or string lengths. However, note carefully
that we also do not hear f.r. 3/2 as an auditory phenomenon where one frequency
vibrates 1.5 times faster than another frequency. Instead, to give meaning to
such an aural experience, we define f.r. 3/2 as a musical interval called the
“fifth.” Although for known musical intervals, we may notate the inverse
proportionality between frequency ratios and length ratios — f.r.
l.r.
— music as an exploratory science and as a meaningful art principally depends on
the knowledge acquired through comprehensible string length measurements and on
the construction of length ratios.

Furthermore, in the context of the harmonic series, recall that the superposition of traveling waves causes nodes in strings. These internal obstructions cause decreases in loop lengths, which in turn cause decreases in wavelengths: as wavelengths decrease, mode frequencies increase. From this perspective, Equation 3.35 expresses the natural process by which a string, through a harmonic progression of length ratios 1, 1/2, 1/3, 1/4, 1/5, 1/6, . . . , shortens itself![1] We conclude, therefore, that in the scientific relationship between vibrating strings and the internal processes of subdivision, mode frequencies are a function of mode wavelengths; and in the historical relationship between musicians and the mathematics of canon string divisions, frequency ratios are a function of length ratios. Stated differently, for a constant transverse wave speed, length ratios constitute the primary factor in identifying, determining, and controlling frequency ratios.

Section 3.17

The Greeks primarily used verbal terms to describe mathematical ratios.
Furthermore, due to linguistic complications, the Greeks never employed
fractions such as 1/2, 2/3, 3/4, 8/9, etc., to identify musical intervals.
Instead, the ratios of Greek music always express quotients greater than one.
For example, examine Figure
3.21
and observe that the term for the ratio of the “octave” (or *diapason*) is
*diplasios*. Since the prefix *di* means *two*, and *plasma*
means *something formed*, *diplasios* means *twofold*, or *
double*, as in 1 + 1 = 1/1 x 2 = 2/1. Therefore, the “octave” is a musical
interval produced by a string that is twice as long as another string; hence,
l.r. 2/1. Similarly, the term for the ratio of the “fifth” (or *diapente*)
is *hemiolios*. Since the prefix *hemi* means *half*, and *olos*
means *whole*, *hemiolios* means *half and whole:* 1/2 + 1 = 1/2
+ 2/2 = 3/2. Therefore, the “fifth” is a musical interval produced by a string
that is by a half longer than another string; hence, l.r. 3/2. The Greek term
for the ratio of the “fourth” (or *diatessaron*) is *epitritos*.
Since, in a mathematical context, the prefix *epi* describes the operation
of addition,[2]
and *tritos* means *third*, the term *epitritos* denotes *
one-third in addition*, and connotes *one and one-third:* 1 + 1/3 = 3/3
+ 1/3 = 4/3. Therefore, the “fourth” is a musical interval produced by a string
that is by a third longer than another string; hence, l.r. 4/3. Similarly, the
term for the ratio of the “whole tone” (or *tonon*) is *epogdoos* (or
*epiogdoos*) which connotes *one and one-eighth:* 1 + 1/8 = 8/8 + 1/8
= 9/8. Therefore, the “whole tone” is a musical interval produced by a string
that is by an eighth longer than another string; hence, l.r. 9/8.

In
contrast, to notate common ratios whose quotients are less than one, the Greeks
inverted these ratios by attaching a prefix *hyp* or *hyph*, from *
hypo*, which means *under*, as in the reciprocal of a number, or 1/*x*.
Therefore, *hypodiplasios* means *half*, or 1 ÷ 2 = 1/2; *
hyphhemiolios* means *two-thirds*, or 1 ÷ (1 + 1/2) = 2/3; *
hypepitritos* means *three-fourths*, or 1 ÷ (1 + 1/3) = 3/4; and *
hypepogdoos* means *eight-ninths*, or 1 ÷ (1 + 1/8) = 8/9.[3]
These kinds of ratios occurred very rarely in the context of canon string
divisions, and were never used to identify musical intervals.

In his translation of Euclid’s *Division of the Canon*, André
Barbera observes

* *
^{↓} * *

If the long version of the *Division of the Canon*
originated in the fourth century b.c.,
it contains the earliest musical division of a canon in Western history.[4]

* * ^{↑} * *

Because this book was written by a geometrician, all letter
ratios and number ratios refer exclusively and unequivocally to the lengths of
straight lines, which in turn represent the lengths of canon strings. Indeed,
all ancient Greek[5]
and Arabian[6]
theorists who described the art of tuning from the
perspective of practicing musicians, tabulated their tuning data in either
string length units, or in length ratios. The author of the *Division of the
Canon* specifies *diplasios*, *hemiolios*, *epitritos*, and *
epogdoos* in Propositions 6–9, 12–14, and 19–20; the latter two propositions
describe the process of tuning a complete diatonic “double-octave” on canon
strings.

The oldest extant departures from this practice occurred approximately
400 years later in the *Manual of Harmonics *by the mathematician
Nicomachus of Gerasa (b. *c*.
a.d.
60) and in a dialog entitled *On Music*, contained in the *Moralia*
of the biographer Plutarch (*c.*
a.d.
46 – *c.* 120). Before we discuss these exceptions, note that both
texts are devoid of the mathematical techniques and procedural methods required
to describe a *complete* tuning on a musical instrument. The writers only
assign numeric values to the “octave,” “fifth,” “fourth,” and “whole tone,”
without integrating these intervals into a mathematical description of a
complete scale. Such accounts typically stem from theoreticians who are not
practicing musicians, which probably explains why neither Nicomachus nor
Plutarch contributed a single original tetrachord division to the history of
music.

Nicomachus abandons the concept of length ratios in Chapter 6 of his *
Manual of Harmonics*. He begins the discussion by telling a fallacious story
in which Pythagoras (*c*. 570
b.c.
– *c*. 500
b.c.) compares various weights suspended from the ends of
strings and, thereby, discovers that the “octave,” “fifth,” and “fourth” vary
directly with tension:

Quote I

* *
^{↓} * *

{ Pythagoras } found that the string stretched by the
greatest weight produced, when compared with that stretched by the smallest, an
octave. The weight on one string was **twelve pounds**, while that on the
other was **six pounds**. Being therefore in a double ratio [2:1], it
produced the octave, the ratio being evidenced by the weights themselves. Again,
he found that the string under the greatest tension compared with that next to
the string under the least tension (the string stretched by a weight of **eight
pounds**), produced a fifth. Hence he discovered that this string was in a
hemiolic ratio [3:2] with the string under the greatest tension, the ratio in
which the weights also stood to one another. Then he found that the string
stretched by the greatest weight, when compared with that which was next to it
in weight, being under a tension of **nine pounds**, produced a fourth,
analogous to the weights. He concluded, therefore, that this string was
undoubtedly in an epitritic ratio [4:3] with the string under the greatest
tension . . .[7]
(Bold text, and text in braces mine. Text in parentheses and ratios in brackets
in Levin’s translation.)

* * ^{
↑} * *

In short, Nicomachus claims that one may produce these three cardinal consonances of music by tensioning strings according to the following “weight ratios”:

Nicomachus then assigns the following unspecified quantities to the
“tonic,” “fourth,” “fifth,” and “octave” of the standard Greek Dorian Mode
between E and E':

Quote II

* *
^{↓} * *

{ Pythagoras } called the note partaking of the number 6, *
hypate* { or E }, that of the number 8, *mese* { or A }, this number
being in an epitritic proportion with the number 6; that of the number 9, he
called *paramese* { or B }, which is higher than *mese* by a whole
tone and what is more, stands in a sesquioctave { Latin for 9:8 } proportion
with it; that of 12, he called *nete* { or E' }. Filling out the
intervening intervals in the diatonic genus with analogous notes, he thus
subordinated the octachord to the consonant numbers, the double ratio [2:1], the
hemiolic [3:2], the epitritic [4:3], and the difference between them, the
sesquioctave [9:8].[8]
(Italics, and text in braces mine. Ratios in brackets in Levin’s translation.)

* * ^{
↑} * *

If we assume that Nicomachus is conjuring “weight ratios,” then

Contrary to observable fact, these weight measurements are false because
Nicomachus assumed that “the notes,” or the aurally perceived frequencies of
strings, are directly proportional to tension.

As discussed in Chapter 2, Section 6, the frequency of a string is
directly proportional to the square root of tension ( *T* ), in
pounds-force (lbf). Therefore, with all other variables held constant, to
increase the fundamental frequency (*
F*1
) by an “octave,” we must increase Nicomachus’
tension of 6.0 lbf by the factor 2/1 squared. Similarly, to increase
*F*1
by a “fifth”
and by a “fourth,”
we must increase the tension by the factors 3/2 squared and 4/3 squared,
respectively:

Therefore, the correct weights for these frequencies are

Nicomachus continues his numeric analysis of vibrating strings in Chapter
10, where he correctly concludes that frequency is inversely proportional to
length. He begins the discussion by comparing functional differences in
magnitude between the numeric values assigned to “weight ratios” and those
assigned to length ratios:

* *
^{↓} * *

. . . measurements based on the lengths . . . of strings . . . are seen to be inverse to measurements that are based on tension, [since] in . . . [the latter] case [1] the smaller the term, the lower the pitch, and the greater the term, the higher the pitch. [However], in the former case, there is an inverse proportion in that [2] the smaller the term, the higher is the pitch, while the greater the term, the lower is the pitch.[9] (Text and numbers in brackets mine.)

* * ^{
↑} * *

Observation 1 is true because lesser tension or a smaller term produces a lower tone, and greater tension or a larger term produces a higher tone. Furthermore, Observation 2 is also true because a shorter length or a smaller term produces a higher tone, and a longer length or a greater term produces a lower tone. Nicomachus then gives the following highly accurate description:

Quote III

* *
^{↓} * *

If, therefore, one takes a long string that is kept under one
and the same tension . . . — the string having been stopped by a bridge . . . at
its very center so that the vibration caused by the plucking of the string may
not progress beyond the half-way point — he will find the interval of an octave,
the sound of half { *hemiseias* } the string compared with that of the
whole string being in a greater proportion, that is, in a duple { 2:1 }
proportion, a result ** exactly inverse** to the reciprocal data of the
length. And if one keeps the vibration down to a third of the string . . . the
sound from two thirds {

* * ^{
↑} * *

Refer to Equation 2.9 and observe that the frequency of a
string is inversely proportional to length. Therefore, with all other variables
held constant, to increase
*F*1 by
an “octave,” we must decrease a string length by the factor 2/1. Similarly, to
increase* *
*F*1
by a “fifth” and by a “fourth,” we must decrease the length by the factors 3/2
and 4/3, respectively. If we utilize Nicomachus’ numeric values to quantify
string lengths, then

Although mathematicians in the 1st century
a.d. had no scientific methods to
determine the exact frequencies of strings, Nicomachus’ conclusions regarding
rates of vibration are correct. He accurately observes that “the sound” from 1/2
of a string vibrates 2/1 times faster, from 2/3 of a string vibrates 3/2 times
faster, and from 3/4 of a string vibrates 4/3 times faster than the whole length
of the string. Therefore, if we utilize Nicomachus’ numeric values to quantify
string vibrations (or modern frequencies, in cycles per second), then

Note that for these musical intervals, “weight ratios” and “vibration
ratios” are identical with respect to the numeric values of the numerators and
denominators. Since “weight ratios” represent *incorrect* quantities, and
“vibration ratios” represent *correct* quantities, we will avoid “weight
ratios” in all future calculations. In other words, when Nicomachus and Plutarch
refer to unspecified quantities such as 6, 8, 9, and 12, we will assume they
mean vibration numbers that are equivalent to modern frequencies, such as 6.0
cps, 8.0 cps, 9.0 cps, and 12.0 cps. However, because “vibration ratios” are
also problematic, we will only work with frequency ratios.

Finally, if we express these musical intervals according to the previously calculated string lengths, then

Of the Greek
treatises on music that have survived, Quotes I and II represent the first
occurrences where the terms *diplasios*, *hemiolios*, *epitritos*,
and *epogdoos* do not refer to ancient length ratios, but to “weight
ratios,” or to numerically identical “vibration ratios.” Consequently, whereas
in the former case music theorists assign larger numbers to lower tones, in the
latter case, they assign larger numbers to higher tones. In Sections
3.18–3.19
we will discuss how these two different quantifications of musical intervals
affect the division of the “octave”; and in Section
3.20,
we will examine several reasons why — throughout the history of music — creative
theorists and musicians never mention weight numbers or vibration numbers in
their descriptions and discussions of tetrachord divisions and scale
constructions.

Section 3.18

Before we compare crucial differences between ancient length ratios and
frequency ratios, let us first discuss several basic concepts of formal scale
construction. A scale consists of a progression of tones expressed as a sequence
of numbers in a definite order. The numbers contained in such a sequence are
called *terms*. The Greeks defined three fundamental number sequences
through the identification of three different *means:* the arithmetic mean,
the harmonic mean, and the geometric mean. A mean represents an intermediate
value, or several intermediate values, between two outer terms called the *
extremes*. Consequently, in an arithmetic progression, in a harmonic
progression, or in a geometric progression, a mean expresses a value that is
larger than the small extreme, and smaller than the large extreme. In a musical
context, a mean represents the division of a large interval into two or more
smaller intervals. For the rest of this chapter, we will focus exclusively on
Greek definitions of the arithmetic and harmonic means. See Chapter 9, Part I,
and Chapter 10, Part III, for detailed discussions on geometric mean
calculations.

The oldest extant Greek text that defines the three means of music is
preserved in a fragment by Archytas (fl. *c*. 400
b.c.). Like Euclid
(fl. *c*. 300
b.c.) after him, Archytas
was a famous geometrician, who also wrote extensively on sound and music.
Unfortunately, none of his works have survived intact. According to Ptolemy
(*c*.
a.d.
100 –*
c*. 165),
Archytas calculated three different tetrachord divisions in string length units,
which means that Archytas used length ratios to identify the musical intervals
of his scales.[11]
Again, this is perfectly consistent with the work of a geometrician. We may
calculate the three means with the following modern equations:

When there are only two variables (*a*, *
c* ), these equations simplify to

Archytas describes these three means in the following manner:

* *
^{↓} * *

There are three means in music. One is arithmetic, the second geometric, the third subcontrary, which they call ‘harmonic’.

[1] There is an **arithmetic mean** when there are three
terms, proportional in that they exceed one another in the following way: **
the second exceeds the third** by the same amount as that by which the
first exceeds the second. In this proportion it turns out that the interval
between the

[2] There is a **geometric mean** when they are such that
as the first is to the second, so is the second to the third. With these the
interval made by the greater terms is equal to that made by the lesser.

[3] There is a subcontrary mean, which we call ‘**harmonic**’,
when they are such that the part of the third by which ** the middle term
exceeds the third** is the same as the part of the first by which the
first exceeds the second. In this proportion the interval between the

* * ^{
↑} * *

Because Archytas did not explicitly state whether the “greater terms” and “lesser terms” represent length numbers or vibration numbers, these definitions are completely open to interpretation. However, note carefully that in Definitions 1 and 3, the second or middle term “exceeds the third” term, which means that for the arithmetic mean and the harmonic mean, respectively, Archytas describes two descending sequences of numbers. As we shall see, in the context of musical interval calculations, music theorists primarily associate such progressions with string length measurements, or with the construction of length ratios.

The arithmetic mean refers to the familiar average value of two numbers. On the other hand, the harmonic mean is not so simple. Recall that Archytas’ definition specifies, “. . . that the part of the third [term] by which the middle term exceeds the third is the same as the part of the first [term] by which the first exceeds the second.” (Text in brackets mine.) Therefore, if the second term exceeds the third term by a given fractional value of the third term, the first term must exceed the second term by the same fractional value of the first term. Or, stated another way, the harmonic means is smaller than the large extreme by the same fraction that it is larger than the small extreme. For example, since in the progression 12:8:6, the harmonic mean (8) is smaller that the first term (12) by one third (1/3) of 12, the harmonic mean (8) must be larger than the third term (6) by one third (1/3) of 6:

To understand how
differences in length numbers and vibration numbers yield diametrically opposed
interval divisions, begin by considering the arithmetic division of the “octave”
according to length numbers. Refer to Equation 3.37a,
substitute the values *a* = 12.0 in. and *c* = 6.0 in. into this
equation, and calculate the arithmetic mean ( *Λ*A )
of the “octave”:

To compute the musical interval of the bridged canon string at 9.0 in.
with respect to the fundamental at 12.0 in., divide the length ratio of the
upper interval 12/9 by the length ratio of the fundamental 12/12:

And to calculate the musical interval of the bridged string at 9.0 in.
with respect to the “octave” at 6.0 in., again divide the length ratio of the
upper interval 12/6 by the length ratio of the lower interval 12/9:

Refer to Figure
3.22,
and observe that we may interpret these interval ratios as the quantities
described by Archytas’ first definition: the greater terms [12/9] identify the
lesser arithmetic interval [4/3], and the lesser terms [9/6] identify the
greater arithmetic interval [3/2].

However, before we continue, consider a harmonic division of
the “octave” according to vibration numbers. Refer to Equation 3.38a,
substitute the values *a* = 6.0 cps and *c* = 12.0 cps into this
equation, and calculate the harmonic mean ( *F*H )
of the “octave”:

Again, examine Figure
3.22
and note that length numbers in arithmetic progression ( 12.0 in., 9.0 in., and
6.0 in.) produce vibration numbers in harmonic progression (6.0 cps, 8.0 cps,
and 12.0 cps). Consequently, with respect to the latter vibration numbers, we
may equally well interpret these frequencies as producing interval ratios
described by Archytas’ third definition: the greater terms [12/8] identify the
greater harmonic interval [3/2], and the lesser terms [8/6] identify the lesser
harmonic interval [4/3].

So, although I describe Figure 3.22 as the arithmetic division of the “octave” expressed as length ratio 2/1, most scholars of ancient Greek music interpret Figure 3.22 as the harmonic division of the “octave” expressed as frequency ratio 2/1.

Next, consider a harmonic division of the “octave” according to length
numbers. Refer to Equation 3.38a,
substitute the values *a* = 12.0 in. and *c* = 6.0 in. into this
equation, and calculate the harmonic mean ( *Λ*H )
of the “octave,”

Now, to compute the musical interval of the bridged canon string at 8.0
in. with respect to the fundamental frequency divide

Similarly, to compute the musical interval of the bridged string at 8.0
in. with respect to the “octave” at 6.0 in., divide

Refer to Figure
3.23,
and observe that we may interpret these interval ratios as the quantities
described by Archytas’ third definition: the greater terms [12/8] identify the
greater harmonic interval [3/2], and the lesser terms [8/6] identify the lesser
harmonic interval [4/3].

However, before we continue, consider an arithmetic division
of the “octave” according to vibration numbers. Refer to Equation 3.37a,
substitute the values *a* = 6.0 cps and *c* = 12.0 cps into this
equation, and calculate the arithmetic mean ( *F*A )
of the “octave”:

Again, turn to Figure
3.23
and observe that length numbers in harmonic progression ( 12.0 in., 8.0 in., and
6.0 in.) produce vibration numbers in arithmetic progression (6.0 cps, 9.0 cps,
12.0 cps). Consequently, with respect to the latter vibration numbers, we may
equally well interpret these frequency ratios as producing interval ratios
described by Archytas’ first definition: the greater terms [12/9] identify the
lesser arithmetic interval [4/3], and the lesser terms [9/6] identify the
greater arithmetic interval [3/2].

So, although I describe Figure 3.23 as the harmonic division of the “octave” expressed as length ratio 2/1, most scholars of ancient Greek music interpret Figure 3.23 as the arithmetic division of the “octave” expressed as frequency ratio 2/1.

However, to confirm the historic authenticity of my interpretations as shown in Figures 3.22 and 3.23, see Chapter 10, Section 8, Quote I, and Chapter 10, Section 16, Quote II, for Ptolemy’s descriptions of the arithmetic and harmonic divisions of the “octave” on canon strings.

In Figure 3.22, notice that we may express interval ratios 4/3 and 3/2 with respect to string lengths, and interval ratios 4/3 and 6/4 [3/2] with respect to frequencies, in shorthand notation 4:3:2. Similarly, in Figure 3.23, we may express interval ratios 6/4 [3/2] and 4/3 with respect to string lengths, and interval ratios 3/2 and 4/3 with respect to frequencies, in shorthand notation 2:3:4. Now, observe carefully that the integers of the latter two sequences are identical. To distinguish between intervals based on string lengths on the one hand, and frequencies on the other, music theorists (who understand this crucial difference) use two different methods of notation. Length ratios appear in descending numerical order; here the first integer represents the overall length of an open string. Consequently, one writes interval ratios according to string length calculations as 4:3:2, 6:4:3, etc. In contrast, frequency ratios appear in ascending numerical order; here the first integer represents the fundamental frequency of an open string. Consequently, one writes interval ratios according to frequency calculations as 2:3:4, 3:4:6, etc.

Finally, as discussed in Chapter 10, Part VI, the arithmetic division of
strings plays an extremely important role in determining the *minor tonality*,
and the harmonic division, in determining the *major tonality* of Western
music. Furthermore, although the musical intervals in Figures
3.22
and 3.23
are notated in ascending order, music theorists have also described the
arithmetic and harmonic divisions of musical intervals in descending order. For
example, in Chapter 10, Sections 62–64, we will examine the work of
Jean-Philippe Rameau (1683–1764), who investigated a so-called *subharmonic
series*, where musical intervals appear in *reverse order* from the
harmonic series. Even though Rameau eventually conceded that such a phenomenon
does not exist in nature, he nevertheless used a purely mathematical
construction of the subharmonic series to define the minor tonality. Note,
therefore, that the subject of interval notation is extremely complex, and that
we must be constantly aware of the mathematical context in which a musical
description of interval ratios occurs.

Section 3.19

As dimensionless entities, ancient length ratios and modern frequency ratios are indistinguishable. For example, it makes no difference if a theorist specifies a.l.r. 4/3 or f.r. 4/3 because both ratios represent the “fourth.” Crucial differences occur only in the context of interval divisions! Figure 3.24 shows that the arithmetic division of the “octave” expressed as a length ratio yields a “fourth” as the lower interval, and a “fifth” as the upper interval; whereas the harmonic division yields a “fifth” as the lower interval, and a “fourth” as the upper interval. In contrast, an arithmetic division of the “octave” expressed as a frequency ratio yields a “fifth” as the lower interval, and a “fourth” as the upper interval; whereas the harmonic division yields a “fourth” as the lower interval, and a “fifth” as the upper interval.

We
may also find evidence of the use of length ratios in Plato (427–347
b.c.).
Although Plato mentions Philolaus (fl. *c*. 420
b.c.)
on two occasions in a dialog entitled *Phaedo*,[13]
he does not acknowledge the scientific and musical accomplishments of this
renowned philosopher. (See Chapter 10, Section 10.) Consequently, he plagiarizes
Philolaus’ famous diatonic tetrachord: 256/243, 9/8, 9/8 in a dialog entitled *
Timaeus*. In commentaries by Nicomachus and Plutarch, both writers discuss
the following passage from the *Timaeus*, which describes how God created
the soul of the universe by dividing its divine “mixture” into several sequences
of numerically defined musical intervals:[14]

* *
^{↓} * *

And { God } proceeded to divide after this manner. First of
all, he ** took away one part** of the whole [1], and then he separated
a second part which was double the first [2], and then he took away a third part
which was half as much again as the second and three times as much as the first
[3], and then he took a fourth part which was twice as much as the second [4],
and a fifth part which was three times the third [9], and a sixth part which was
eight times the first [8], and a seventh part which was twenty-seven times the
first [27]. After this he filled up the double intervals [that is, between 1, 2,
4, 8] and the triple [that is, between 1, 3, 9, 27],

* * ^{
↑} * *

If it were not for the cosmic “mixture,” it would be
difficult to imagine how the process of taking parts, cutting portions, and
quantifying mathematical intervals depicts anything but the work of a master
musician at a canon. I am particularly intrigued by the verbs *to take* and
*to cut*, which also occur in Euclid’s canon tuning description.[16]
However, because Plato does not explicitly state whether his ratios express
length numbers or vibration numbers, this passage (like Archytas’ fragment) is
completely open to interpretation. Of course, neither Nicomachus nor Plutarch
allude to Plato’s obvious references to canon tuning techniques. So, it is not
surprising that both authors suppressed the *mathematical possibility* that
the “intervals of 3/2 and 4/3” could have originated from harmonic and
arithmetic divisions of the “octave” expressed as length ratio 12/6.
Approximately 500 years after Plato wrote the *Timaeus*, Nicomachus and
Plutarch attempted to eliminate length ratios in favor of frequency ratios
because they undoubtedly believed that the concrete reality of geometry
interfered with developments in “pure” mathematics. In Chapter 8, Nicomachus
gives the following interpretation:

*
* ^{↓} * *

**Explanation of the references to harmonics in the
Timaeus**

. . . Plato expressed himself as follows: “so that within each interval there are two means, the one superior and inferior to the extremes by the same fraction, the other by the same number. He (the Demiurge) filled up the distance between the hemiolic interval [3:2] and the epitritic [4:3] with the remaining interval of the sesquioctave [9:8].”

For the double interval is as 12 is to 6, but there are two means, 9 and 8. The number 8, however, in the harmonic proportion is midway between 6 and 12, being greater than 6 by one third of 6 (that is, 2), and being less than 12 by one third of 12 (that is, 4). ...

The other mean, 9, which is fixed at
the *paramese* degree { or B }, is observed to be at the arithmetic mean
between the extremes, being less than 12 and greater than 6 by the same number
(3).[17]
(Italics, and text in braces mine. Text in parentheses and ratios in brackets in
Levin’s translation.)

* * ^{
↑} * *

Refer to Figure 3.24, and note that because Nicomachus identifies abstract number 9 as an arithmetic mean that represents the tone B, he interprets Plato’s division of the “octave” expressed as frequency ratio 12/6.

Finally, here is Plutarch’s explanation:

* *
^{↓} * *

So Plato, wishing to show in terms of the science of
harmonics the harmony of the four elements in the soul [represented by the
numbers 6, 8, 9, and 12] . . . presents in each interval two means of the soul,
in accordance with the ratio of music. For it so happens that in music the
consonance of the octave has two mean intervals. ... Now the consonance of the
octave is seen to be in the duple ratio; and this ratio, expressed in numbers,
is illustrated by six and twelve . . . Six then and twelve being the extremes,
the *hypate* { or E } . . . is represented by the number six, the *nete*
{ or E' } . . . by the number twelve. ... it is evident that the *mese* {
or A } will be represented by the number 8, the *paramese* { or B } by the
number nine.[18]
(Italics, and text in braces mine. Text and numbers in brackets in Einarson’s
translation.)8

* * ^{
↑} * *

Again, since Plutarch equates the tone A with abstract number 8, and tone B with abstract number 9, the former number represents the harmonic division, and the latter number represents the arithmetic division of the “octave” expressed as frequency ratio 12/6.

Section 3.20

Many ancient writers avoid dimensional descriptions of quantities such as
6, 8, 9, and 12, because a specific reference to a “weight ratio” or “vibration
ratio” would have compelled them to define a new kind of complicated “length
ratio.” For example, in Section
3.17,
Quote III, Nicomachus states that the *sound* of the “fifth” — or the
frequency ratio of the “fifth” — is in a *hemiolic* [3/2] relation with the
whole string; consequently, the *length* of the “fifth” must be in a *
dimoiros* [2/3] relation with the whole string.[19]
In his next example, the *sound* of the “fourth” — or the
frequency ratio of the “fourth” — is in an *epitritic* [4/3] relation with
the whole string; consequently, the *length* of the “fourth” must be in a
*hypepitritic* [3/4] relation with the whole string. Here already,
Nicomachus fails to identify the complicated fraction that describes
three-fourths of the string. Instead, he simply compares four parts to three
parts and, thereby, equates ancient length ratio 4/3 with frequency ratio 4/3.

The introduction of weight numbers and vibration numbers instigated a *
double standard* whereby music theorists silently interpreted ancient length
ratios as “weight ratios” or “vibration ratios.” Since ancient length ratio
is indistinguishable from Nicomachus’ “vibration ratio,” a switch from
a.l.r.
to f.r.
incurs no numeric changes, and a switch in the opposite direction from
f.r.
to a.l.r.
also incurs no numeric changes. However, all readers of ancient texts,
and of most interpretations of ancient texts, must remain vigilant with respect
to interval divisions. Confusion, mistakes, and anachronisms will inevitably
occur if a reader fails to recognize whether a particular writer is describing
the division of a.l.r.
,
or of f.r.
.

Creative theorist-musicians like Euclid (fl. *c*. 300
b.c.), Ssu-ma Ch’ien (*c*.
145
b.c. –
*c*. 87
b.c.), Ptolemy (*c*.
a.d.
100 –
*c*. 165),
Al-Kindi (d. *c*. 874), Al-Farabi (d. *c*. 950), Ibn Sina (980–1037),
Safi Al-Din (d. 1294), Ramis (*c*. 1440 –*
c*. 1500), Ramamatya (fl. *c.* 1550), Zarlino (1517–1590), and
Narayana and Ahobala (fl. *c.* 1670) reject or simply ignore “weight
ratios” and “vibration ratios.” The inclusion of such ratios would have
hopelessly confused their monumental achievements. In Chapters 10 and 11, we
will examine detailed tuning descriptions by all these writers who — within the
realm of human experience — based their work on the premise that frequency
ratios are a function of length ratios.

[1]See Chapter 10, Section 58.

[2]Liddell,
H.G., and Scott, R. (1843). *A Greek-English Lexicon*, p. 623. The
Clarendon Press, Oxford, England, 1992.

[3](**a**)
D’Ooge, M.L., Translator (1926). *Nicomachus of Gerasa: Introduction to
Arithmetic*,* *p. 299. The Macmillan Company, New York.

(**b**)
*A Greek-English Lexicon*, p. 1857.

[4](**a**)
Barbera, A., Translator (1991). *The Euclidean Division of the Canon:
Greek and Latin Sources*, p. 60. University of Nebraska Press, Lincoln,
Nebraska.

The long version includes the text that spans pp. 170–185, which describes the tuning of a diatonic “double-octave.” See Chapter 10, Section 11, for a complete analysis and discussion of this tuning.

(**b**)
Barker, A., Translator (1989). G*reek Musical
Writings, Volume 2*,
pp. 205–207.
Cambridge University Press, Cambridge, Massachusetts.

In Barker’s translation, the long version includes Propositions 19 and 20.

[5]See Chapter 10, Part II.

[6]See Chapter 11, Part IV.

[7]Levin,
F.R., Translator (1994). *The Manual of Harmonics, of Nicomachus the
Pythagorean*, p. 84. Phanes Press, Grand Rapids, Michigan.

[8]*Ibid*.,
p. 85.

See Chapter 10, Figure 10, which shows
the Greek note names *hypate*, *mese*, *paramese*, and *
nete* in the context of a “double-octave” scale known as the Greater
Perfect System.

[9]*Ibid*.,
p. 141.

[10](**a**)
*The Manual of Harmonics*, p. 141.

(**b**)
Jan, Karl von, Editor (1895). *Musici Scriptores Graeci*, p. 254. This
work includes the Greek text of Nicomachus’ *Manual of Harmonics*.
Lipsiae, in aedibus B.G. Teubneri.

In contrast to the discussion in this
section on Greek terms for ratios whose quotients are less than one,
Nicomachus uses *hemiseias* from *hemisus* which means *half*,
and *dimoirou* from *dimoiros* which means *two-thirds*.

(**c**)
Zanoncelli, L., Translator
(1990). *La Manualistica Musicale Greca*, p. 164, 166. Angelo Guerini e
Associati, Milan, Italy.

This book gives the Greek text of
Nicomachus’ *Manual of Harmonics* as it appears in Jan’s *Musici
Scriptores Graeci*, and it includes a modern Italian translation.

[11](**a**)
Diels, H. (1903). *Die Fragmente der Vorsokratiker*, *Griechisch und
Deutsch*, *Volume 1*, p. 428. Weidmannsche Verlagsbuchhandlung,
Berlin, Germany, 1951.

Diels identifies Archytas’ description of three tetrachord divisions as authentic fragment A16. This page gives Archytas’ fragment as it appears in Ptolemy’s original Greek text.

(**b**)
Burkert, W. (1962). *Lore and Science in Ancient Pythagoreanism*, pp.
379–380, Footnote 47. Translated by E.L. Minar, Jr. Harvard University
Press, Cambridge, Massachusetts, 1972.

In Footnote 47, Burkert agrees with Diels that A16 is an authentic fragment. On p. 380, Burkert states, “. . . Archytas apparently assigned the smaller number to the high tone and the larger to the low.”

(**c**)
Düring, I., Translator (1934). *
Ptolemaios und Porphyrios über die Musik*, pp. 46–47. Georg Olms Verlag,
Hildesheim, Germany, 1987.

(**d**)
*Greek Musical Writings, Volume
2*, pp. 43–44,
and
p. 304.

[12]*Greek
Musical Writings, Volume 2*,
p. 42.

[13]Hamilton,
E., and Cairns, H., Editors (1963). *The Collected Dialogues of Plato*,
p. 44. The *Phaedo* translated by H. Tredennick. Random House, Inc.,
New York, 1966.

[14]*Greek
Musical Writings, Volume 2*,
pp. 59–60, Footnote 17.

Barker gives a complete analysis of these sequences.

[15](**a**)
*The Collected Dialogues of
Plato*, pp. 1165–1166. Quotation from the *Timaeus* translated by B.
Jowett.

(**b**)
*Greek Musical Writings, Volume 2*, pp. 59–60.

[16]See Chapter 10, Section 11.

[17]*The
Manual of Harmonics*, p. 107.

[18]Einarson,
B., Translator (1967). *On Music*, by Plutarch, pp. 401–403. In *
Plutarch’s Moralia, Volume 14*. Harvard University Press, Cambridge,
Massachusetts.

[19]See
Footnote 10(**b**).