Musical Mathematics

on the art and science of acoustic instruments

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CHAPTER 10: *WESTERN TUNING THEORY AND PRACTICE*

Part VI: Just Intonation

For background information on
the Greek prefix |

Section 10.38

Although Ramis managed to integrate 5-limit ratios into the 12-tone scale, he
did not develop a theory of consonance. In 1558, seventy-six years after the
publication of Ramis’ monochord, Gioseffo Zarlino (1517–1590) published a work
entitled *Istitutioni harmoniche* in which he proposed (1) that the *
numero Senario* (from the Latin: *senarius*, lit. *composed of six in
a group;* fig. *the number Series 1–6*) constitutes the source of all
possible musical consonances, and (2) that the harmonic division of strings
expresses the “nature
of Harmony” and produces
“the consonances which the composers call
perfect.” (See Section
10.44,
Quote V.) However, at the end of his life, in the *De tutte l’opere del R.M.
Gioseffo Zarlino da Chioggia*... edition of the *Istitutioni* (1589),[1] Zarlino
argued that the arithmetic division *and* the harmonic division of the
“fifth,” length ratio 3/2, are equally important to the development of music
theory. He thereby became the first European music theorist to mathematically
define what we now call the *minor tonality* and the *major tonality*,
respectively, of musical composition.

In the *Istitutioni* *
harmoniche *(1573), Zarlino acknowledges a theoretical contribution by the
Arabian physician, scientist, and music theorist Avicenna, or Ibn Sina
(980–1037), and plagiarizes a mathematical contribution by the German
mathematician Michael Stifel (1487–1567). Zarlino first quotes a crucial
sentence from a Latin translation of a compendium of Ibn Sina’s writings
entitled *Auicene perhypatetici philosophi:* *ac medicorum facile primi
opera in luce redacta*..., published in 1508.[2]
Later in the text, Zarlino utilizes a harmonic division of a “double-octave and
a fifth,” ratio 6/1, into *five interval ratios* as described in Stifel’s
*Arithmetica integra*, published in 1544.[3]
Although in the *Dimostrationi harmoniche* (1571),[4] Zarlino
severely criticizes Stifel’s geometric division of the “tone,” ratio 9/8, two
years later he fails to acknowledge Stifel’s stunning mathematical solution for
the division of musical intervals into two or more harmonic means.

To understand Zarlino’s theory
of musical consonance, the reader should thoroughly read Chapter 3, Sections
14–19, and carefully study Chapter 3, Figures 22 and 23. These two illustrations
show the notation of length ratios and interval ratios in the context of the
arithmetic division ( *Λ*A ),
and the harmonic division ( *Λ*H )
of the “octave” on canon strings. Throughout Zarlino’s treatises, *all*
ratios represent either (1) length ratios, or (2) interval ratios between length
ratios. This means that frequency ratios, or interval ratios between frequency
ratios, do *not* exist in his works.[5]
The great Arabian theorist, Al-Farabi (d. *c*. 950), deliberately set the
standard for notating musical ratios as expressions of string length
measurements when he stated

* * ^{↓} * *

The largest extreme of an
interval, that which corresponds to the largest number, is, for certain
mathematicians of another time, its lower extreme; for others, it is the upper {
extreme }. In our opinion, it matters little, either from the point of view of
theory or from the point of view of the ear, whether the large extreme is placed
at the lower note or at the upper note. But having regarded up to now the lower
note of an interval as being its large extreme, we will abide by this
convention; besides, it is suited to the principles we stated, and facilitates
our explanation of the rules of music; for it relates the measurement of the
notes to the lengths [of strings] from which they come. The longest has the
largest measurement and produces the lowest note; the shortest, which has the
smallest measurement, gives the highest note.[6]
(Text in braces mine. Text in brackets in *La Musique Arabe*.)

* * ^{↑} * *

Therefore, one finds that in the
musical treatises of the Arabian Renaissance,[7] *all*
ratios refer exclusively to length ratios, or to interval ratios between length
ratios.

During the 16th century, many
European artists, scientists, and intellectuals were inspired to observe the
physical processes and functions of nature. The Italian term *la scienza
naturale* (lit. *the science of natural things;* in other words: *
physics*), or the simplified expression *la naturale* (lit. *the
Natural*), signified this renewed interest in natural phenomena.
Consequently, there arose a heated debate between those who based their studies
on mathematics, or *abstract* truth, and those who based their studies on
physics, or *concrete* truth. The former philosophical approach is known as
*rationalism*, and the latter, as *empiricism*.[8]
In the *Istitutioni harmoniche*, Prima Parte, Cap. 19, Zarlino maintains
that numbers have the capacity to manifest themselves as parts of
“sounding bodies,” and
that only through the quantifications of
“sonorous numbers” are
human beings able to comprehend the tones and ratios produced by vibrating
strings. However, Zarlino resolves the debate between the science of numbers vs.
the science of nature in the following chapter heading and passage from Prima
Parte, Cap. 20:

* * ^{↓} * *

The reason why music is the subordinate of arithmetic, and the

intermediary between *la
Mathematica *[mathematics]

and *la Naturale*
[physics].

. . . I am so bold to assert, that music is not only the subordinate of mathematics, but also of physics, not with respect to numbers, but with respect to tone, which is something natural. From it arises every modulation, every consonance, every harmony, and every melodic song. Avicenna [Ibn Sina], who also advocates such an interpretation, says: “Music derives its principles from the science of nature [physics] and the science of numbers [mathematics].”[9] (Text in brackets mine. Italics in Zarlino’s original text.)

* * ^{↑} * *

Zarlino here quotes Ibn Sina
from a chapter (or book) entitled *Sufficientia* (lit. *Physics*, or
*Science of Nature*).[10]
In short, Ibn Sina and Zarlino refused to compromise and, thereby, focused their
attention on the best of both worlds. Given the personal nature and emphatic
tone of his resolve, it seems to me that Zarlino was greatly influenced not only
by Ibn Sina, but by other Arabian writers as well.

Section 10.39

To substantiate this assumption,
let us prepare an examination of the *Istitutioni harmoniche* by first
considering the division of musical intervals on canon strings in Ibn Sina’s *
Kitab al-shifa’* (Book of the cure) and in a treatise entitled *Risalat
al-Sharafiya fi*’*l-nisab al-ta*’*lifiya* (The Sharafian
treatise on musical conformities in composition)
by Safi Al-Din (d. 1294). (See Section
10.36.)
H.G. Farmer describes the former treatise by stating

* * ^{↓} * *

This great work by the famous
Avicenna — as he was known to the wide world — contained the entire sum of
knowledge in science and philosophy known in Islamic lands, if not western
Europe also. It includes a chapter (*fann*) on music which is divided into
six discourses (*maqalat*) dealing respectively with the physics of sound,
musical intervals, genres and species of melody, systems, and mutations, as well
as rhythm and composition.[11]

* * ^{↑} * *

Ibn Sina describes the arithmetic and harmonic division of the “octave” by noting

* * ^{↓} * *

The octave is called the
interval of *absolute consonance *[homophonic interval]; the fifth and the
fourth are called intervals of *similar notes* [symphonic intervals];
sometimes one attributes to them the ** characteristic of inversion**.
The extreme degrees of the octave are, as we have said, [

* * ^{↑} * *

Figure
10.34
shows that Ibn Sina’s ratios refer unequivocally to string length measurements.
Although so-called Western theorists such as Nicomachus,[13] Theon,[14] and
Boethius[15] (see
Section
10.4)
also described various arithmetic and harmonic divisions of the “octave,” Ibn
Sina was the first music theorist to define both of these divisions in *least
terms*. Notice that one cannot reduce Ibn Sina’s integers of the arithmetic
division of the “octave” expressed as interval ratios 4:3:2, or the harmonic
division of the “octave” expressed as interval ratios 6:4:3. As discussed below,
it is extremely important to notate arithmetic and harmonic divisions in least
terms when one attempts to divide a given length ratio into *three or more*
interval ratios.

Ibn Sina also observed that the intervals produced by an arithmetic and a harmonic division of a given interval are identical:

Quote I

* * ^{↓} * *

Now, we have already seen that
in giving a ratio an arithmetic mean, we obtain two ratios identical to those
that result from a harmonic mean; but ** their position has changed**.[16]
(Bold italics mine.)

* * ^{↑} * *

Therefore, since the calculation of a harmonic mean is more difficult than an arithmetic mean, Ibn Sina offers the following convenient solution:

Quote II

* * ^{↓} * *

When it comes to making this
division by way of the harmonic mean, if we do not find any number [integer]
that can serve this purpose, it will suffice to place in the ** lower
[position] the largest of the two ratios obtained by way of the arithmetic mean**.[17]
(Bold italics, and text in brackets mine.)

* * ^{↑} * *

Finally, consider this passage by Safi Al-Din, which constitutes the first known description of the arithmetic division of the “fifth”:

* * ^{↓} * *

If we are then asked which are
the two intervals whose ratios are made of consecutive numbers taken in the
natural order of numbers, and which together exactly complete the interval of
the ratio 1 + 1/2 [3:2], the fifth, the easiest way to solve the problem will be
this: We double the term of this ratio that represents ** the higher note,
that is 2**, and we will thereby know that of the two intervals required,
the one placed in the upper [position] will be in the ratio 1 + 1/4 [5:4], and
the one placed in the lower [position] in the ratio 1 + 1/5 [6:5].[18]
(Bold italics, and text and ratios in brackets mine.)

* * ^{↑} * *

Figure 10.35 shows that the arithmetic division of length ratio 3/2 produces a “minor third,” ratio 6/5, as the lower interval, and a “major third,” ratio 5/4, as the upper interval. In least terms, we may notate this division as interval ratios 6:5:4.

Section 10.40

Zarlino introduces the concept
of his *Senario* in Prima Parte, Cap. 15:

Quote I

* * ^{↓} * *

On the characteristics of the *numero Senario* and its parts,

and the relations between them, one finds the form

of every musical consonance.

Although the *numero Senario* possesses many special characteristics, I
will nevertheless, in order not to deviate, enumerate only those [properties]
that are suitable for our purpose. First, it represents the first of the perfect
numbers. It contains parts which stand in the following relation to one another:
if one picks out two arbitrary parts, they always indicate the relation or the
form of a proportion of a musical consonance — whether it concerns a simple or a
compound consonance — as one can see in the following figure [see Figure
10.36].[19]
(Text in brackets mine. Italics in Zarlino’s original text.)

* * ^{↑} * *

In the original engraving that
follows, Zarlino assigns traditional Latin names to the musical intervals
included in the *Senario*. To discuss the mathematical significance of this
illustration, Figure
10.36
expresses these interval names as ratios.

When viewed from a historical perspective, Quote I is of paramount importance to the development of European music because it shatters the 3-limit barrier of Pythagorean theory. As described in Section 10.5, the Pythagoreans recognized only five consonances: 4/1, 3/1, 2/1, 3/2, 4/3. These ratios represent all possible interval combinations in the series 1, 2, 3, 4. Figure 10.37 shows that one obtains these consonances through a division of a canon string into two, three, and four aliquot parts.

Dissatisfied with the
limitations of only five consonances, Zarlino proposed the number six, or the
first perfect number, as the underlying mathematical principle of all musical
consonances. By definition, a perfect number is a positive number that equals
the sum of its positive divisors. In this case, 1 + 2 + 3 = 6. To demonstrate
the musical potential of the *Senario*, or of the series 1, 2, 3, 4, 5, 6,
Zarlino continues his description with the arithmetic division of the “octave,”
expressed as interval ratios 4:3:2, and the arithmetic division of the “fifth,”
expressed as interval ratios 6:5:4. He then states that a
“harmonic divisor,” or a
harmonic division of these two ratios, would have placed the
“parts,” or these
interval ratios, in
“reverse order.”

Quote II

* * ^{↓} * *

Its parts [i.e., the six parts
of the *Senario*] are so sequenced and arranged, that the forms of either
of the two simplest and largest consonances [i.e., 2:1 and 3:2] — which the
musicians call perfect because they are contained in the parts of the number 3 —
may be divided by a middle number into two harmonically proportioned parts.
First, one finds the octave, without an inner term, in the form and the relation
of 2:1. Then the octave is divided by the number 3, which is situated between 4
and 2, into two consonant parts [4:3:2]; that is, into the fourth, which is
between 4 and 3, and into the fifth between 3 and 2. One finds the fifth in turn
between the numbers 6 and 4, which is divided by 5 into two consonant parts
[6:5:4]; that is, a major third between 5 and 4, and a minor third, which is
contained in the numbers 6 and 5.

I wrote
that the parts are arranged according to a harmonic proportion. However, this
does not concern the order of the proportions (because they are in reality in
arithmetic order), [i.e., 4:3:2 or interval ratios 4:3, 3:2; and 6:5:4 or
interval ratios 6:5, 5:4], but applies only to the relation of the parts as
determined by the middle number. Because these parts exist in such large
quantity and in as many relations as there are parts — which are fashioned [into
proportions] by a middle number or a ** harmonic divisor** — although
in a

* * ^{↑} * *

In Section
10.42,
we will discuss the mathematics of the latter paragraph in full detail. For now,
we conclude that while it may be impossible to prove a direct connection between
the musical treatises of Ibn Sina and Safi Al-Din on the one hand, and Quote II
from Zarlino’s *Istitutioni harmoniche* on the other, we should *not*
assume that the latter text states new or original ideas.

Section 10.41

Before we continue this
discussion on the arithmetic and harmonic divisions of length ratios, let us
first examine Latin expressions for ratios, which appear in countless European
treatises on music. Ordinarily, the Latin prefix *sesqui* means *one and
a half*, or *one-half more;* consequently, the Latin word *sesquiopus*
means the *work of a day and a half*, and *sesquicentennial* literally
equals (100 years ÷ 2) + 100 years = 150 years. However, this simple definition
of *sesqui* does not apply to mathematical descriptions of epimore or
superparticular ratios. (See Section
10.4.)
Although medieval and Renaissance theorists retained old Greek names of musical
intervals such as *diapente* for “fifth,” *diatessaron* for “fourth,”
*ditone* (*ditonon*) for “major third,” *tone* (*tonon*) for
“whole tone,” etc., they substituted new Latin mathematical expressions such as
*sesquialtera* for *hemiolios* [3/2], *sesquitertia* for *
epitritos* [4/3], *sesquiquarta* for *epitetartos* [5/4], *
sesquioctava* for *epogdoos* (*epiogdoos*) [9/8], etc. Except for
the first example, note that the Latin prefix *sesqui* replaces the Greek
prefix *epi*. To understand the derivations of these new constructions,
refer to Table
10.25,
which gives the Latin terms for the first ten ordinal numbers. Now, since *
sesqui* means *one-half more*, and since *alter* means *second*
— as in the *second part* of a whole, or *one-half* — *sesquialtera*
literally equals (1/2 ÷ 2) + 1/2 = 3/4. However, turn to Table
10.26
and note that *sesquialtera* actually means 1 + 1/2 = 3/2; similarly, since
*tertius* means *third* — as in the *third part* of a whole, or
*one-third* — *sesquitertia* literally equals (1/3 ÷ 2) + 1/3 = 1/2;
however, *sesquitertia* actually means 1 + 1/3 = 4/3. As a result, we
conclude that in the context of *musical ratios*, the Latin prefix *
sesqui* simply describes the operation of addition, which means that it has
the identical mathematical function as the Greek prefix *epi*. (See Chapter
3, Section 17.) Consequently, *sesquiquarta* denotes *one-fourth in
addition*, and connotes *one and one-fourth:* 1 + 1/4 = 5/4.

Finally, with respect to epimere
or superpartient ratios, Table
10.27
lists the Latin terms of seven ratios that appear in Table
10.3.
To identify a given ratio, extract the numerator and the denominator contained
in the term. First, to determine the denominator, simply identify the last word.
Second, to calculate the numerator, identify an inner value — *bi* for 2,
*tri* for 3, *quadri* for 4, or *quinque* for 5 — and add this
quantity to the denominator. Therefore, *super quadripartiens-quinta*
describes a ratio with 5 in the denominator, and 5 + 4 = 9 in the numerator, or
ancient length ratio 9/5.

Section 10.42

In preparation for a discussion on Zarlino’s arithmetic and harmonic division of length ratio 6/1 on canon strings, we must first review his methods of calculation. With respect to the first kind of division, Zarlino states in Prima Parte, Cap. 36,

* * ^{↓} * *

For example, we want to arithmetically divide a sesquialtera, which is formed by the basic numbers 3 and 2. The former includes consecutive prime numbers, which must first be doubled. Then we obtain the numbers 6 and 4. When they are added, the result is 10; when this result is divided into two equal parts, the result is 5. Therefore, I say that 5 is the divisor of our proportion. Because it not only produces the same differences in this proportionality, but also divides the proportion (as is the characteristic of an arithmetic proportionality) into two unequal relations, in such a manner, that one finds between the larger numbers the smaller proportion, and inversely, between the smaller { numbers }, the larger { proportion }. The sesquiquinta { exists } between the 6 and 5, and the sesquiquarta, between the 5 and 4 . . . It is true, that one will refer to this [sequence of numbers] as a progression rather than a proportionality. Because one begins with the smaller number, comes to a middle { number }, and from this, to the larger { number }. One progresses with equal distances. One always finds unity or two or three or another number which produces the mentioned distance.[21] (Text in braces mine. Text in parentheses and brackets in Fend’s German translation.)

* * ^{↑} * *

In short, after doubling the
outer terms, Zarlino utilizes Equation 3.37a
and calculates the *arithmetic division* of length ratio 3/2
in
the following manner:

This arithmetic progression expresses an ascending sequence of musical intervals: a low “minor third,” length ratio 6/5, followed by high “major third,” length ratio 5/4.

With respect to the second kind of division, Zarlino states in Prima Parte, Cap. 39,

* * ^{↓} * *

If we want to harmonically divide a sesquialtera, which is formed by the basic numbers 3 and 2, then we will first divide it arithmetically in the manner that I stated above. Then we obtain an arithmetic proportionality in the numbers 6:5:4. Second, to cause it to become a harmonic proportionality, we multiply the 6 and the 4 times the 5, and then [we multiply] the 6 times the 4. From the products we derive the desired [harmonic] division, which is formed by the numbers 30:24:20 . . . Because the relation between the numbers 6 and 4, which indicates the distance between the harmonic numbers, corresponds to that between the numbers 30 and 20. They are the outer terms of the sesquialtera, which are divided into a sesquiquarta between 30:24 [5:4], and into a sesquiquinta with the relation 24:20 [6:5]. Thus, one finds between the larger numbers the larger relation, and between the smaller [numbers], the smaller [relation], and this is the characteristic of this proportionality.[22] (Text and ratios in brackets mine.)

* * ^{↑} * *

To summarize, Zarlino does not
utilize Equation 3.38a, but
calculates the *harmonic division*** **of length ratio 3/2
based on the three terms of the arithmetic division of 3/2.

This harmonic progression expresses the former ascending sequence of musical intervals in “reverse order”: a low “major third,” length ratio 15/12 = 5/4, followed by high “minor third,” length ratio 12/10 = 6/5.

Section 10.43

Next, to prepare for Zarlino’s
arithmetic and harmonic divisions, we must also consider the work of Michael
Stifel. In commentaries to his German translation of the Prima Parte and Seconda
Parte of the *Istitutioni harmoniche*, Michael Fend gives the following
description and translated excerpt from Stifel’s *Arithmetica integra*
(1544):

* * ^{↓} * *

Zarlino owes the realization of
the reciprocity of both sequences of ratios indirectly to *Arithmetica integra*,
by Michael Stifel who . . . first taught how harmonic sequences can be
constructed beyond three terms provided that one derives them from arithmetic
sequences. Stifel began with the observation that a cube, which possesses 6
faces, 8 vertices, 12 edges, and 24 face-angles, represents a harmonic
proportion, and he transferred this sequence of proportions (6:8:12:24) to a **
descending sequence** of tones: dd, aa, d, D. He compared it to the
[descending] sequences cc, gg, c, C; aa, e, a, A; g, d, G,
Γut
[lit.

*
“can produce — from any given
arithmetic progression — a harmonic [progression], which includes as many terms
as the arithmetic [progression]. One proceeds in the following manner: Multiply
the terms of your arithmetic proportion in sequence with one another. Then
divide the product by the individual terms of your arithmetic progression,
beginning with its largest term. From the arithmetic sequence 1, 2, 3, 4, 5, 6
comes the harmonic [sequence] 10, 12, 15, 20, 30, 60. In the same manner, out of
a given harmonic sequence comes an arithmetic [sequence]!”**
*[23]

(Bold italics and text in brackets mine. The harmonic progression in parentheses in Fend’s German commentary.)

* * ^{↑} * *

Regarding
Stifel’s cube analysis, refer to Equation 3.38a,
and for a given string where *a* = 6 units and *c* = 12 units,
calculate *Λ*H =
8 units; and where *a* = 8 units and *c* = 24 units, *
Λ*H =
12 units. Consequently, Stifel’s harmonic progression demonstrates that the
“double-octave,” ratio 24/6 = 4/1, has *two harmonic means*, namely, 8 and
12. Now examine Figure
10.38,
which shows Stifel’s descending tones: dd, aa, d, D, on a canon string with a
length of 6 units, 8 units, 12 units, and 24 units, respectively.

Because Stifel interprets the harmonic progression 6:8:12:24 as a descending sequence of tones, we must, in turn, interpret his series as a descending sequence of interval ratios: 8/6, 12/8, 24/12, or simply 4/3, 3/2, 2/1, respectively. Now, turn to Chapter 3, Figure 12, and observe that if we play these three interval ratios in ascending order: 2/1, 3/2, 4/3, Stifel’s harmonic division of the “double-octave” produces the first three intervals of the harmonic series, namely, the “octave,” “fifth,” and “fourth.”

With respect to the transformation of the arithmetic progression 1, 2, 3, 4, 5, 6 into the harmonic progression 10, 12, 15, 20, 30, 60, Stifel calculated the integers of the latter sequence in three steps:

Step 3 reduces the quotients of
Step 2 to least terms.
Although Stifel did not explicitly interpret the arithmetic and harmonic
division of the “double-octave and a fifth,” ratio 6/1, as a descending sequence
of tones, it is highly likely that he *also* experienced these two
divisions on canon strings. Figure
10.39
shows the arithmetic progression 1:2:3:4:5:6 as a descending sequence of tones:
C6, C5, F4, C4, Ab3, F3, or a descending sequence of interval ratios: 2/1, 3/2,
4/3, 5/4, 6/5. If we express each tone as a length ratio, and play them in
ascending order: F3–1/1, Ab3–6/5, C4–3/2, F4–2/1, C5–3/1, C6–6/1, we find that
the first three tones produced by Strings VI–IV sound the F-minor triad: F–Ab–C.
Finally, examine Figure
10.39
to see that this so-called *arithmetic division* actually consists of a
sequence of *increasing* string lengths, where each succeeding length is an
integer multiple of the first length. That is, String I has a length of 1 unit,
String II has a length of 1 unit × 2 = 2 units, etc. Hence, Stifel’s descending
division of ancient length ratio 6/1 into *four arithmetic means*: 1,
2, 3, 4, 5,
6.

In contrast, Figure
10.40
shows Stifel’s harmonic progression 10:12:15:20:30:60 as a descending sequence
of tones: G5, E5, C5, G4, C4, C3, or a descending sequence of interval ratios:
12/10, 15/12, 20/15, 30/20, 60/30, or simply 6/5, 5/4, 4/3, 3/2, 2/1,
respectively. If we express these tones as length ratios, and play them in
ascending order: C3–1/1, C4–2/1, G4–3/1, C5–4/1, E5–5/1, G5–6/1, we find that
the last three tones produced by Strings III–I sound the C-major triad: C–E–G.
Now, turn back to Chapter 3, Figures 10, 13, and 15, and observe that the
standing waves in Figure
10.40
represent the first six modes of vibration of a flexible string. The first
complete analysis of the mode shapes and mode frequencies of vibrating strings
exists in a work entitled *Système général des intervalles des sons*... by
Joseph Sauveur (1653–1716), published in 1701. (See Section
10.56.)
We conclude, therefore, that Stifel comprehended the mathematical and musical
significance of the division of ancient length ratio 6/1 into *four harmonic
means*: 10, 12, 15, 20, 30, 60,
approximately 150 years before Sauveur discovered the *harmonic progression *1,
1/2, 1/3, 1/4, 1/5, 1/6,*
. . . as a natural phenomenon of subdividing strings.*

Section 10.44

In the *Istitutioni harmoniche*,
Prima Parte, Cap. 40, Zarlino begins his formal demonstration of the *Senario*
on canon strings by first summarizing the fundamental differences between the
“compound unities” of the arithmetic division, and the “sonorous quantities” of
the harmonic division.

* * ^{↓} * *

The harmonic proportionality
possesses the same proportions as the arithmetic [proportionality] because the
forms of the consonances are contained (as we saw) in the parts of the *numero
Senario;* however, in the case of the arithmetic proportionality, among the
smaller numbers exist the larger proportions, and among the larger [numbers],
the smaller [proportions]; while one finds the opposite in the case of the
harmonic proportionality, that is, we have among the larger numbers the larger
proportions, and among the smaller [numbers], the smaller [proportions]. This
difference stems from the fact that the one [the former] is associated with pure
numbers, and the other [the latter], with sonorous quantities. They progress in
opposite directions, that is, one [the former] increases, the other [the latter]
decreases in relation to their respective starting point, as I showed. None of
them deviates from the natural progression, which one finds in the order of the
proportions. This [order] is formed by the numbers in the following manner: in
the arithmetic proportionality, the numbers form ** compound unities**,
while in the harmonic proportionality, they are parts of

* * ^{↑} * *

As discussed in Section 10.43, the “compound unities” represent increasing string lengths that are integer multiples of the first string length, and the “sonorous quantities” represent decreasing string lengths of a manually subdivided string. Zarlino then continues with the arithmetic division of an “octave and a fifth,” or length ratio 3/1.

Quote III

* * ^{↓} * *

In order to better understand these things, we will give an example. We draw a line AB, which for an arithmetician represents unity, and for a musician, a sonorous body, hence a string. Its length is one foot. If we want to give it an arithmetic progression, then we must leave it whole and undivided, because one may not divide unity of an arithmetic progression. Thus an [arithmetic] proportion, consisting of three numbers, is given in such a manner, that the proportion of a tripla [3:1] is divided by a mean into two parts.

We must proceed in the following manner: First, the mentioned line (if possible) is to be doubled, so that unity is doubled [to form] a duality, which follows unity directly. After we doubled it, we have the line AC of a two-foot length. If we compare the doubled line AC with the line AB, then we discover between them the proportion of the dupla [2:1], which is first in the natural order of the proportion, as one also finds between the numbers two and one. When we want to find the third term in this kind of progression, we must extend the line AC to a three-foot length, so that it reaches the point D, because three directly follows two. Then we will have the proportion of the tripla between DA and BA, because AD is measured exactly three times by AB, or AD contains AB three times, as in the case of numbers the three contains the one three times. And the proportion from AC can be divided into two parts in the following manner: in a dupla CA and BA, and in a sesquialtera [3:2] DA and CA, indeed an arithmetic proportionality . . .[25] (Ratios in brackets mine.)

* * ^{↑} * *

At this point in the text, Zarlino includes a simple figure to illustrate his division on a canon string, but he neither describes nor demonstrates the arithmetic division of number six, length ratio 6/1, in full detail. As an alternative, refer to Figure 10.41, which takes Zarlino’s arithmetic method to its logical conclusion. Here Strings I–III illustrate the first three steps described by Zarlino in Quote III, and String III shows the complete arithmetic division of length ratio 3/1, interval ratios 3:2:1. The column to the left of the string gives Zarlino’s method for calculating the arithmetic mean, and the staff to the right of the string shows C5 as the arithmetic mean between C6 and F4. Finally, Strings IV–VI demonstrate three succeeding constructions that result in the arithmetic division of length ratio 4/2, interval ratios 4:3:2, of ratio 5/3, interval ratios 5:4:3, and of ratio 6/4, interval ratios 6:5:4.

Zarlino then continues with the harmonic division of length ratio 6/2 [3/1].

Quote IV

* * ^{↓} * *

However, if we want to construct a harmonic progression, we will proceed in the following manner: First, we divide the mentioned line AB at its center, the point C, because the half comes before every other part. I now say that one finds between the given string AB and its half CB . . . the proportion of the dupla [2:1], which is the first in the natural order of the proportions. Then we will decrease the mentioned line AB by 2/3 at the point D, and we will thus obtain the proportion of the sesquialtera [3:2], which takes the second place in the order of the proportions. I say that the sesquialtera exists between CB and DB, furthermore the tripla [3:1] [exists] between AB and DB, which are [both] divided by CB into two proportions according to the harmonic proportionality . . .[26]

* * ^{↑} * *

Here again, Zarlino includes a simple figure to illustrate his division on a canon string, but he neither describes nor demonstrates the harmonic division of length ratio 6/1 in full detail. So, refer to Figure 10.42, which takes Zarlino’s harmonic method to its logical conclusion. Strings I–III illustrate the first three steps described by Zarlino in Quote IV, and String III shows the complete harmonic division of length ratio 6/2 [3/1], interval ratios 6:3:2. In conformity with Zarlino’s method of calculating a harmonic division based on the three terms of a corresponding arithmetic division, the column to the left of String III gives the latter three terms in a rectangular frame. Also, the staff to the right of the string shows C4 as the harmonic mean between C3 and G4. Finally, Strings IV–VI demonstrate three succeeding constructions that result in the harmonic division of length ratio 6/3 [2/1], interval ratios 6:4:3, of ratio 20/12 [5/3], interval ratios 20:15:12, and of ratio 30/20 [3/2], interval ratios 15:12:10. Note carefully the transition in string length units between Strings IV and V. Although it is possible to extend the fractional string length notation 3.0 : 2.0 : 1.5 of String IV, to 2.0 : 1.5 : 1.2 for String V, and to 1.5 : 1.2 : 1.0 for String VI, in Quote V below Zarlino explicitly gives Stifel’s harmonic progression 60:30:20:15:12:10 as the harmonic division of number six. Consequently, the mathematical complication that requires a transition from 6 units for the overall length of Strings I–III, to 60 units for the overall length of Strings IV–VI, probably explains why Zarlino neither described nor illustrated the complete harmonic division of length ratio 6/1.

Zarlino continues with the following comparison of the arithmetic and the harmonic division of length ratio 3/1:

* * ^{↓} * *

And as the numbers of the arithmetic progression are multiplied unities, so those [numbers] of the harmonic [progression] represent the number of parts that can be determined from a sonorous body, [and] which originate from the subdivision of this body. Therefore, in the former one regards the multiplication of unity, as in the following sequence: 3:2:1. And in the latter, one regards the multiplication of parts on a divided object, which is formed by the numbers 6:3:2. Because if we regard the whole divided into its parts, then we discover that the line CD is the smallest part of the line AB, and that it measures AB altogether six times, the line CB, three times, and the line DB, two times.[27] (Text in brackets mine.)

* * ^{↑} * *

With respect to these two different kinds of divisions on canon strings, Zarlino concludes

Quote V

* * ^{↓} * *

Now it can be seen, that in the
harmonic progression [6:3:2], the larger numbers [6:3] contain the larger
proportions and the lower sounds [i.e., the “octave”], while the smaller numbers
[3:2] correspond to the smaller proportions and the higher sounds [i.e., the
“fifth”]. Because they [the higher sounds] are brought forth on strings with
smaller dimensions, while in the case of the lower tones, the strings have
larger dimensions. Furthermore, we can see: As one progresses in the arithmetic
proportionality (provided that one would realize it in the manner shown) from
the high to the low sound by multiplying the string length, so one proceeds in
the harmonic [proportionality] in reverse from low to high by shortening the
string. In the arithmetic progression [3:2:1] the intervals of the smaller
proportion [3:2] have their position in the lower [sounds], [i.e., the “fifth”],
contrary to the *natura dell’Harmonia* [lit. *nature of Harmony*],
whose characteristic it is that the deep sounds possess a larger interval than
the high [sounds], and these [the high sounds], in turn, [possess] a smaller
[interval].

However, since all the proportions that belong to the arithmetic progression —
because they follow the natural order of the proportions — also exist in the
same order in the harmonic progression [that is, the “octave,”
“fifth,” “fourth,”. . . descend in the arithmetic progression, and the “octave,”
“fifth,” “fourth,”. . . ascend in the harmonic progression],
we can now understand in which manner one should take the meaning of the words
in Chapter 15, which state that in the terms of the *numero Senario* are
contained all the forms of the simple musical consonances that can be produced,
and that ** the consonances, which the composers call perfect**, are
fashioned after the harmonic division of this number. Because when the
consonances are transferred to a sounding body with the aid of the consonant
ratios 60:30:20:15:12:10, then one recognizes that these consonances are so
divided as the parts of the number 6, although they are now arranged in another
manner. Likewise, it is comprehensible in which sense the words of the very
learned Jacobus Faber Stapulensis in his “Musica” (Prop. III, 34) are to be
understood: that the harmonic proportionality is completely indispensable and
that, although the magnitudes of its proportions agree with those of the
arithmetic proportionality, the sequence and the place [position] of the ratios
are different.[28]
(Bold italics, and text and ratios in brackets mine. Italics in Zarlino’s
original text. Text in parentheses in Fend’s German translation.)

* * ^{↑} * *

In the first paragraph of Quote
V, Zarlino identifies the “nature of Harmony” — or the very essence of musical
harmony — with the harmonic division of strings. His preference for the harmonic
division is based on the performance of choral music, where one places large
intervals in the bass, or in the lower position of a chord, and small intervals
in the treble, or in the upper position of a chord. In the second paragraph,
Zarlino establishes an irrefutable nexus between his *numero Senario* and
Stifel’s harmonic division of length ratio 6/1, notated here as the harmonic
progression 60:30:20:15:12:10, or an ascending sequence of interval ratios:
60/30, 30/20, 20/15, 15/12, 12/10, or simply 2/1, 3/2, 4/3, 5/4, 6/5. To
strengthen his argument, in the next sentence Zarlino paraphrases a passage from
a famous mathematical treatise entitled *Musica libris quatuor demonstrata*,
by Jacobus Faber Stapulensis (Jacques Le Febvre), (*c.* 1455 – d. 1536),
first published in 1496.

Zarlino’s unrelenting
determination to shift the focus from the arithmetic division to the harmonic
division reveals how deeply entrenched the practice of direct canon string
division had become. As discussed in Sections
10.4
and
10.8, the latter method
always produces an arithmetic division, where the smaller interval appears in
the lower position, and the larger interval, in the upper position. With respect
to the “octave,” it matters little if in playing a chord one utilizes the
arithmetic or the harmonic division. The former places the “fourth,” ratio 4/3,
in the lower position, and the “fifth,” ratio 3/2, in the upper position; and
the latter places the “fifth” in the lower position, and the “fourth” in the
upper position. However, with respect to the “fifth,” and what would later be
called triadic harmony, the difference between the arithmetic division of length
ratio 3/2, and the harmonic division of length ratio 3/2, literally defines the
emotional polarity of Western music. Figure
10.43
shows that the arithmetic division of the “fifth” places the “minor third,”
ratio 6/5, in the lower position, and the “major third,” ratio 5/4, in the upper
position; we call the chord C–Eb–G a *minor triad*, or *minor tonality*.

In contrast, Figure
10.44
shows that the harmonic division of the “fifth” places the “major third” in the
lower position, and the “minor third” in the upper position; we call the chord
C–E–G a *major triad*, or *major tonality*.

Now, turn back to Figure 10.39, and note that the minor triad F–Ab–C occurs in Stifel’s arithmetic progression when realized on a string with 6, 5, 4, aliquot parts; and, in Figure 10.40, the major triad C–E–G occurs in Stifel’s harmonic progression when realized on a string with 15, 12, 10, aliquot parts.

Section 10.45

Before we examine Zarlino’s
final analysis of the arithmetic and harmonic division of the “fifth,” let us
first evaluate two important mathematical aspects of the *Senario*. In his
paper, Robert W. Wienpahl states that “. . . the major and minor sixth are not
considered by Zarlino to be basic consonances . . .”[29] because
they are not superparticular or epimore ratios. In Section
10.4,
we noted that a superparticular ratio has a numerator that exceeds the
denominator by one. Therefore, since the integers of the “major sixth,” ratio
5/3, and the “minor sixth,” ratio 8/5, do not differ by unity, Zarlino concludes
that they are Composite or imperfect consonances. Zarlino distinguishes between
Simple and Composite consonances in the *Istitutioni harmoniche*, Prima
Parte, Cap. 16. Near the end of this chapter, he concludes

Quote VI

* * ^{↓} * *

In the *Senario*, that is,
in its parts, one finds every Simple musical consonance *in atto* [lit. *
in actuality*], and beyond that, the Compound [musical consonance] *in
potenza* [lit. *in potentiality*].[30]
(Text in brackets mine. Italics in Zarlino’s original text.)

* * ^{↑} * *

In the middle of this chapter, Zarlino contends

* * ^{↓} * *

To the . . . [compound consonances] belongs the mentioned [major] sixth, which consists of the fourth and the major third. It is recognized by the simplest terms of its proportion, 5 and 3, which is divided by 4 into 5:4:3.[31] (Text in brackets mine.)

* * ^{↑} * *

In other words, because the “major sixth,” expressed as interval ratios 5:4:3, consists of two smaller consonances, Zarlino considers it a composite consonance: 5/4 × 4/3 = 5/3. He then continues by applying the same argument to the “minor sixth”:

Quote VII

* * ^{↓} * *

Next to it, I will place the
minor sixth, which arises from a union of the fourth and the minor third. Its
simplest terms are contained in the genus superpartiens as the ratio
supertripartiensquinta, and they can be joined by a middle term. Since one finds
this proportion between 8 and 5, a middle harmonic number is included between
them, namely, the 6. It divides the proportion 8:5 into two smaller ratios
8:6:5, that is, a sesquitertia and a sesquiquinta. For this reason we can
characterize this consonance as a Compound [consonance]. Until now, it has
received a friendly reception by musicians, and it is counted among the other
consonances. If, under the parts of the *Senario*, one does not come across
its form *in atto* [in actuality], one finds it there nevertheless *in
potenza* [in potentiality]. Because, it builds its form in truth from the
parts that are contained in the number 6, that is, from the fourth and the minor
third.[32]
(Text in brackets mine. Italics in Zarlino’s original text.)

* * ^{↑} * *

That is, because the “minor sixth,” expressed as interval ratios 8:6:5, also consists of two smaller consonances, it too is a composite consonance: 8/6 × 6/5 = 8/5. Therefore, both the “major sixth” and “minor sixth” are considered less than perfect consonances.

Although the “major sixth” and
the “minor sixth” are both superpartient ratios, note that the *Senario*
includes both integers of ratio 5:3, but only one integer of ratio 8:5. Because
of this, Zarlino’s rationalization with respect to ratio 5/3 seems unnecessary
and contradictory. To resolve this confusion, we may conjecture that if the *
Senario* had contained both integers of ratio 8/5, Zarlino would probably not
have given an inconsistent description of ratio 5/3. In other words, the
necessity to rationalize the inclusion of superpartient ratio 8/5 forced him to
rationalize superpartient ratio 5/3 as well. The inscription in the inner circle
of Figure
10.36
is evidence enough that Zarlino primarily regarded ratio 5/3 as a Simple or
basic consonance.

As if to acknowledge his
inconsistent treatment of ratio 5/3, Zarlino draws a hard distinction between
5/3 and 8/5 by directly stating in Quote VI, and by indirectly stating in Quote
VII, that the former is found “in actuality” in the *Senario*, but the
latter is only found “in potentiality.” When viewed from this perspective, ratio
5/3 is an *actual *consonance among all the other consonances in Figure
10.36,
but ratio 8/5 is only a *potential *consonance that stands apart from the
ratios in this figure. To understand Zarlino’s apparent reluctance to classify
ratio 8/5 as a bona fide dissonance, consider this sequence of ratios: 6/5, 4/3,
3/2, 8/5. Now turn back to Section
10.20,
and observe that in the Catalog of Scales of the *Harmonics* by Claudius
Ptolemy (*c*. a.d.
100 – *c*. 165), only
Ptolemy’s Tense Diatonic includes these four ratios. In Section
10.21,
Figure
10.15
shows that in the Lydian Mode, the latter sequence transforms to ratios: 5/4,
4/3, 3/2, 5/3.[33]
Given Zarlino’s fascination with Ptolemy’s scale, his artistic predilections led
him to regard ratios 5/3 and 8/5 as unequal but musically acceptable
consonances.

Section 10.46

Toward the end of his life,
Zarlino abandoned the needlessly conflicted rhetoric of his early writings. In
the *De tutte l’opere*... edition of the *Istitutioni harmoniche* he
gives equal consideration to the harmonic *and* the arithmetic division of
the “fifth.” With the exception of the modern G-clef, Figure
10.45(a)
is an exact copy of Zarlino’s illustration as it appears in *De tutte l’opere*...,
Terza Parte, Cap. 31, p. 222. Figure
10.45(b)
gives a detailed ratio analysis of Figure
10.45(a).
For the harmonic division of the “pure fifth,” expressed as length ratio 180/120
= 3/2, Zarlino describes the lower interval, ratio 180/144 = 5/4 [386.3 ¢], as a
*ditono* (lit. *two-tones*) and *sesquiquarta;* and he describes
the upper interval, ratio 144/120 = 6/5 [315.6 ¢], as a *semiditono* (lit.
*flat two-tones*) and *sesquiquinta*. To verify his *harmonic mean*
calculation, substitute the outer two terms — 180 units and 120 units — into
Equation 3.38a to obtain the
units:

Unfortunately, the arithmetic
division of the “fifth” is not so simple. In Figure
10.45(b),
observe that the length ratios above the staff represent the first six tones of
Ptolemy’s Tense Diatonic in the ancient Lydian Mode: 1/1, 9/8, 5/4, 4/3, 3/2,
5/3; hence, string lengths 180 ÷ 9/8 = 160, 180 ÷ 5/4 = 144, . . . , etc. Given
this sequence of tones, the interval between the “whole tone,” ratio 9/8, and
the “major sixth,” ratio 5/3, is a “flat fifth”: 5/3 ÷ 9/8 = 40/27 [680.4 ¢], or
a “fifth” tuned 1 syntonic comma flat: 3/2 ÷ 81/80 = 40/27. A substitution
of the outer two terms — 160 units and 108 units — into Equation 3.37a
gives the following *arithmetic mean**:*

Now, if Zarlino had
given this exact result in Figure
10.45(a),
it would have produced two very complex interval ratios with prime number 67;
that is, a lower “flat minor third,” ratio 160/134 = 80/67 [307.0 ¢], and an
upper “flat major third,” ratio 134/108 = 67/54 [373.4 ¢]. He avoids this
difficulty by increasing 134 units to 135 units, but also incurs a small
mathematical error. The lower interval is now a Pythagorean “minor third,” ratio
160/135 = 32/27 [294.1 ¢] — or a 5-limit “minor third” tuned 1 syntonic comma
flat: 6/5 ÷ 81/80 = 32/27 — and the upper interval, the desired 5-limit “major
third,” ratio 135/108 = 5/4, or a true *sesquiquarta*. However, in the
final analysis, the “flat fifth” and the Pythagorean “minor third”
approximations do not contradict Zarlino’s original intent, namely, to
demonstrate the major and minor tonalities in the context of a single musical
scale.

To clarify the musical distinction between the harmonic division of the “fifth” and major tonality on the one hand, and the arithmetic division of the “fifth” and minor tonality on the other, Zarlino describes Figure 10.45(a) by stating

Quote VIII

* * ^{↓} * *

. . . the variety of
the harmony . . . consists not only in the variety of the consonances which
occur between the parts, but *also* in the variety of the harmonies, which
arises from the position of the sound forming the third or tenth above the
lowest part of the composition. Either this is minor and the resulting harmony
is ordered by or resembles the arithmetical proportion or [arithmetic] mean, or
it is major and the harmony is ordered by or resembles the harmonic
[proportion].

** On this variety depend the whole diversity and perfection of the
harmonies.** For . . . in the perfect composition the fifth and third, or
their extensions [or “octave” equivalents; i.e., the “twelfth” and “tenth,”
respectively], must always be actively present, seeing that apart from these two
consonances the ear can desire no sound that falls between their extremes or
beyond them and yet is wholly distinct and different from those that lie within
the extremes of these two consonances combined. For in this combination occur
all the different sounds that can form different harmonies.[34]
(Italics, bold italics, and text in brackets mine.)

* * ^{↑}
* *

He then continues by attributing a “joyful” sensibility to the major tonality, and a “mournful” sensibility to the minor tonality.

Quote IX

* * ^{↓} * *

But since the
extremes of the fifth are invariable and always placed subject to the same
proportion, apart from certain cases in which the fifth is used imperfectly, the
extremes of the thirds are given different positions. I do not say different in
proportion; ** I say different in position**, for . . . when the major
third is placed below, the harmony is made joyful, and when it is placed above,
the harmony is made mournful. Thus, from the different positions of the thirds
which are placed in counterpoint between the extremes of the fifth or above the
octave, the variety of harmony arises.[35]
(Bold italics mine.)

* * ^{↑}
* *

We conclude that the
*Senario* not only enabled Zarlino to define a theory of consonance, but
also provided him with two mathematical means to describe the polar emotions of
human existence.

Stifel, who also recognized the musical importance of both means, was not swayed
by the rhetorical arguments of his day. He wrote in the *Arithmetica integra
*:

* * ^{↓} * *

But I do not see what the Harmonic [progression] may explain about musical concords that the Arithmetic [progression] does not explain in equal proportion [i.e., just as well].[36] (Text in brackets mine.)

* * ^{↑}
* *

Zarlino’s contributions to Western music are truly monumental. By integrating
the mathematical principles of Ptolemy’s scale, Ramis’ monochord, Stifel’s
arithmetic and harmonic divisions of length ratio 6/1, and his theory of
consonance as defined by the *Senario*, Zarlino gave Western music its
modern roots. Although Zarlino favored 1/4-comma meantone temperament for the
tuning of keyboard instruments,[37] his
theory of consonance was exclusively based on rational or just intoned ratios.
Irrational or tempered ratios do not play any part in the formulation of his
musical ideas. Four hundred years later, Western music theory still agrees with
the basic premise of the *Senario*, and teaches that only these ratios
constitute desirable consonances.

[1]*De
tutte l’opere del R.M. Gioseffo Zarlino da Chioggia*... is available in a
facsimile edition from Georg Olms Verlag, Hildesheim, Germany.

[2]Ibn
Sina (Avicenna): *Auicene perhypatetici philosophi: ac medicorum facile
primi opera in luce redacta*... This Latin translation was published in
1508. Facsimile Edition: Minerva, Frankfurt am Main, Germany, 1961.

The second chapter (or book) of this compendium is entitled *Sufficientia*.

[3]Facsimile
editions or translations of *Arithmetica integra *are not available.
See Sections 10.43 and 10.46, and Chapter 9, Footnote 6, for translated
excerpts from this work. Stifel was the first mathematician to use the term
*exponent*, and to state the *four laws of exponents*.

[4](**a**)
Zarlino, R.M.G. (1571). *Dimostrationi harmoniche*,
Ragionamento Terzo, Proposta
VIIII (i.e., IX), pp. 158–160.
Facsimile Edition, The Gregg Press Incorporated, Ridgewood, New Jersey,
1966.

(**b**)
Kelleher, J.E. (1993). *Zarlino’s “Dimostrationi harmoniche” and
Demonstrative Methodologies in the Sixteenth Century*,
pp. 265–268.
Ph.D. dissertation printed and distributed by University Microfilms, Inc.,
Ann Arbor, Michigan.

This dissertation includes many translated excerpts from Zarlino’s *
Dimostrationi.*
Here Kelleher describes not
only Zarlino’s criticism, but also quotes several translated passages from
Stifel’s *Arithmetica integra*.

[5]See Chapter 3, Section 12.

[6]D’Erlanger,
R., Bakkouch,
‘A.‘A.,
and Al-Sanusi,
M., Translators (Vol. 1, 1930; Vol. 2, 1935; Vol. 3, 1938; Vol. 4, 1939;
Vol. 5, 1949; Vol. 6, 1959). *La Musique Arabe*, Librairie Orientaliste
Paul Geuthner, Paris, France.

Forster Translation: in *La Musique Arabe, Volume 1*, pp. 100–101.

[7]See Chapter 11, Part IV.

[8]Reichenbach,
H. (1951). *The Rise of Scientific Philosophy*. The University of
California Press, Berkeley and Los Angeles, California, 1958.

[9](**a**)
Fend, M., Translator (1989). *Theorie des
Tonsystems*: Das erste und zweite Buch der *Istitutioni harmoniche*
(1573), von Gioseffo Zarlino. Peter Lang, Frankfurt am Main, Germany.

This German translation includes the Prima and Seconda Parte of the *
Istitutioni harmoniche*.

Forster Translation: in *Theorie des Tonsystems*, pp. 104–105.

(**b**)
Zarlino R.M.G.
(1573). *Istitutioni harmoniche.* Facsimile Edition, The Gregg Press
Limited, Farnborough, Hants., England, 1966.

Forster
Translation:
in *Istitutioni harmoniche*, pp. 37–38.

[10]*Auicene
perhypatetici philosophi . . .*
I, Cap. 8, p. 18.

[11]Farmer,
H.G. (1965). *The Sources of Arabian Music*, p. 36. E.J. Brill, Leiden,
Netherlands.

[12]Forster
Translation: in *La Musique Arabe*, *Volume 2*, p. 124.

With respect to the “puissance” translation error, see Al-Farabi's 'Uds, Footnote 22.

[13]Levin,
F.R., Translator (1994). *The Manual of Harmonics, of Nicomachus the
Pythagorean*,
pp. 107–108.
Phanes Press, Grand Rapids, Michigan.

[14]Lawlor,
R. and D., Translators (1978). *Mathematics Useful for Understanding Plato*,
by Theon of Smyrna, pp. 76–79. Wizards Bookshelf, San Diego, California,
1979.

[15]Bower,
C.M., Translator, (1989). *Fundamentals of Music*, by A.M.S. Boethius,
pp. 65–72. Yale University Press, New Haven, Connecticut.

[16]Forster
Translation: in *La Musique Arabe*, *Volume 2*, p. 136.

[17]Forster
Translation: in *La Musique Arabe*, *Volume 2*, pp. 136–137.

[18]Forster
Translation: in *La Musique Arabe*, *Volume 3*, pp. 36–37.

[19]Forster
Translation: in *Theorie des Tonsystems*, p. 87; in *Istitutioni
harmoniche*, p. 31.

[20]Forster
Translation: in *Theorie des Tonsystems*, pp. 87–88; in *Istitutioni
harmoniche*, pp. 31–32.

[21]Forster
Translation: in *Theorie des Tonsystems*, pp. 148–149; in *
Istitutioni harmoniche*, pp. 54–55.

[22]Forster
Translation: in *Theorie des Tonsystems*, p. 156; in *Istitutioni
harmoniche*, p. 60.

[23]Forster
Translation: in *Theorie des Tonsystems*, p. 167.

[24]Forster
Translation: in *Theorie des Tonsystems*, p. 161; in *Istitutioni
harmoniche*, p. 61.

[25]Forster
Translation: in *Theorie des Tonsystems*, pp. 161–162; in *
Istitutioni harmoniche*, pp. 61–62.

[26]Forster
Translation: in *Theorie des Tonsystems*, p. 162; in *Istitutioni
harmoniche*, p. 62.

[27]Forster
Translation: in *Theorie des Tonsystems*, pp. 162–163; in *
Istitutioni harmoniche*, p. 62.

[28]Forster
Translation: in *Theorie des Tonsystems*, pp. 163–164; in *
Istitutioni harmoniche*, pp. 62–63.

[29]Wienpahl,
R.W. (1959). Zarlino, the *Senario*, and tonality, p. 31. *Journal of
the American Musicological Society* **XII**, No. 1, pp. 27–41.

Wienpahl gives a detailed analysis of the “major sixth” and “minor sixth” in
Zarlino’s *Senario*. This paper also includes many translated excerpts
from Zarlino’s *Istitutioni harmoniche*, and from Salinas’ *De musica
libri VII*.

[30]Forster
Translation: in *Theorie des Tonsystems*, p. 93; in *Istitutioni
harmoniche*, p. 34.

[31]Forster
Translation: in *Theorie des Tonsystems*, p. 91; in *Istitutioni
harmoniche*, p. 33.

[32]Forster
Translation: in *Theorie des Tonsystems*, pp. 91–92; in *Istitutioni
harmoniche*, pp. 33–34.

[33]Zarlino
recognized the musical importance of Ptolemy’s Tense Diatonic in the Lydian
Mode and passionately advocated its implementation.
In the *De tutte l’opere*... edition of the
*Istitutioni
harmoniche**,*
Seconda Parte, Cap. 39, Zarlino described Ptolemy’s Tense Diatonic
Tetrachord as the most natural of all diatonic scales. Today we know it as
the just-intoned version of the Western major scale.

[34]
(**a**) Strunk, O., Editor
(1950). *Source Readings in Music History*, p. 242.
W. W. Norton & Company,
Inc., New York.

(**b**) *Journal of the
American Musicological Society* **XII**, No. 1, p. 27.

(**c**)
Shirlaw, M. (1917). *
The Theory of Harmony*, p. 50. Da Capo Press Reprint Edition. Da Capo
Press, New York, 1969.

[35](**a**)
*Source Readings in Music History*, pp. 242–243.

(**b**)
*Journal of the American Musicological Society* **XII**, No. 1, p.
28.

[36]Forster
Translation: in *Theorie des Tonsystems*, p. 168.

In his commentaries, Fend gives Stifel’s original Latin text: “Non enim video quod Harmonica habeat quod ad concentus Musicos pertineat, quod Arithmetica non habeat aequali commoditate.”

[37]See Section 10.27.