Aesthetics No.15 (2011)

The Japanese Society for Aesthetics / 125

Pietro Mengoli’s Theory of Perception of Musical Intervals:
A Mathematical Approach to the Sense of Hearing in the
Scientific Revolution
OAI Takaharu
University of Tokyo, Tokyo

Introduction
What is the agent of recognition of musical intervals -reason or the sense of hearing? Thiswas one of the most important questions in Western music theory from antiquity to the earlymodern period. On the one hand, music was thought to be basically mathematical, becausemusic theory was one of the four mathematical sciences (Quadrivium) in medieval times and it
used ratios to define musical intervals. In this sense, musical intervals were objects of rationalrecognition. On the other hand, they were also associated with the sense of hearing because
they could be heard as sounds in reality.
For musical theorists of that period, music was primarily a mathematical science because
they firmly believed in the Pythagorean view of music. This view maintained that music should
be judged not by the fallible sense of hearing but by reason using numbers as the subject ofmusic. That is, musical intervals were to be understood through mathematical equations thatassured the certainty of recognition.
We can find an example of this attitude in the works of Giovanni Maria Artusi (ca.15401613), an Italian musical theorist known as a critic of Claudio Monteverdi. In the first part of
his L’Artusi overo delle imperfettioni della moderna musica (1600), Vario, Artusi’s spokesman,says:
I admit you that the sense of hearing is the first that rejoices at that part to which it
belongs: that is sound, its proper object and the matter of the songs. But I also tell you thatthe sense without intelligence cannot judge anything by itself, because it is subject to many
deficiencies and many imperfections, and it does not receive things unless [those received
things are] somewhat vague, even if [they are] close to their true [states].[1]
According to Artusi, the sense of hearing could receive sounds as the matter of musicalintervals and songs, but it could not judge how they relate with each other. He believed thatusing only hearing to perceive musical intervals would only result in confusion. Therefore,
Artusi insisted on the necessity of intelligence for the clear and distinct judgment of music.
Thus, the primacy of intelligence/reason over the sense of hearing was maintained in the
tradition of music as a mathematical science.
However, in the latter half of the seventeenth century, a theoretical work appeared that
focused on the function of the sense of hearing: the Speculationi di musica (1670) by Pietro

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Mengoli (1625-86), an Italian mathematician and professor at the University of Bologna. In thisstudy I examine Mengoli’s theory of interval perception and the functions of the sense ofhearing. Specifically I focus on Mengoli’s discussion of the recognition of errors in intervals,
which was a subject that was discussed very rarely at that time[2].

1. Mengoli and Speculationi
Mengoli was active as a mathematician of the Galilean school, which promoted newnatural sciences. However, when he was ordained in 1660, he began to waver between thereligious doctrines of the Roman Catholic Church and the new scientific trends of the scientificrevolution. After all, he represented Catholicism and opposed the Galilean doctrines (forinstance, he supported the geocentric theory). Due to the fact that he developed his original
metaphysics while attempting to reconcile the Catholic faith with the new scientific view ofnature, the terms and logic in his works were often difficult to understand. Mengoli’s
inclination toward speculation prevented his works from having much influence on future
generations. It was not until the beginning of the twentieth century that his mathematicalachievements were reevaluated[3].
In addition to his works of pure mathematics (arithmetic and geometry), Mengoli wrote
about “mixed mathematics” (astronomy, optics, and music, etc.), in which natural phenomena
were examined by means of pure mathematics. Speculationi is one of his works on “mixed
mathematics” and was written in Italian. It consists of the following sections: a dedication,
preface, “Natural History of Music” (Historia Naturale della Musica; a summary of the wholework), a “Protestation of the author” (Protesta dell’autore; a declaration that the standpoint of
the work is not against the doctrines of the Roman Catholic Church), and twenty-five
speculations (Speculationi) - the main parts of the work.
In 1673, three years after its publication, Speculationi was discussed in articles in
Philosophical Transactions of the Royal Society, and Giornale de’ letterati, a book review
journal issued in Rome[4]. Both articles mentioned that Speculationi contained new knowledge
about the auditory organ, which reflected the prevalence of empiricism in those days. In theformer article, after a short introduction of the work, the “Natural History of Music” was whollytranslated into English. Henry Oldenburg (1619-77), the first secretary-general of the Royal
Society and the supposed author of this article, pointed out that Speculationi was “somewhat
obscurely written”, indicating his difficulty understanding the work[5]. In the latter article, thereviewer stated that Mengoli’s main purpose was to argue the nature of the soul (anima) rather
than to teach music theory[6]. This remark is particularly astute, because Speculationi deals
exclusively with the recognition of musical intervals by the human soul, not with the practice ofmusic.

2. The Physiological Structure of Hearing
Mengoli’s theory of perception of musical intervals is based on the anatomical knowledgeof the auditory organ offered by Giovanni Galeazzo Manzi, an anatomist at the University ofPietro Mengoli’s Theory of Perception of Musical Intervals

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Bologna. The anatomy described in the first Speculatione is not far from the current knowledgeof the ear for the most part, with exception that there are two eardrums -one inside and one
outside of the middle ear. According to Mengoli, the inner eardrum vibrates two times while
the outer eardrum vibrates once; in other words, the ratio of the vibrational frequencies of thetwo eardrums is 2:1[7]. This is because the weight and velocity of the air confined in the inner
ear is twice that of free air in the external ear. Consequently, the load added to the inner
eardrum is twice that of the outer, and the tension of the inner eardrum is also twice that of theouter[8]. This assumption, which is naturally false according to the current knowledge ofanatomy, played an important role in Mengoli’s theory of musical perception.
Additionally, sound is considered to be the wavy motion of air, which is believed to beparticulate matter, and the space between the particles is filled with “aura”, which is subtle
matter like ether. This assumption must have been taken from the scholastic conception ofhorror vacui. Aura is matter that reacts passively to the motions of the objects that surround it
and it reflects their motions exactly; therefore, it is a medium that can accurately transmit the
motions of air particles perceived as sound. Furthermore, it is so subtle that it can pass throughthe eardrum (the middle and inner ear where air particles cannot reach) and transmit the
movement of air particles from the outer eardrum to the soul. According to Mengoli, “Aura is
the immediate instrument of the soul, in which it receives and senses, and for which it operates
and moves”[9]. That is, aura is a medium through which the soul works both passively andactively. It should be noted that aura was conceived to be matter with functions similar to both
ether (transmitting the external phenomena of nature) and animal spirits (mediating between
the mind and body inside a human being). Aura bridged the gap between the physicalphenomena of the motions of air particles and the subjective perception of sound.
Next, Mengoli developed his ideas about the soul’s function in hearing. The rational
human soul spiritualizes all corporeal things that vanish away in process of time and keeps
them in intelligence eternally. Sound is also abstracted as an intellectual object, which is,according to Mengoli, the logarithm of the ratio of the durations that the outer and inner
eardrums vibrate (the two eardrums are linked to each other through the aura that fills thespace between them)[10]. Then, the soul perfects the knowledge about the received things byreflecting on such abstracts as logarithms. However, Mengoli declared that he focused only on
naturally executed abstractions, not the soul’s reflection on the results of those abstractions[11].
This suggests that he intended to focus on the way the sense of hearing perceived sounds, not
the rational judgments on what was perceived.
In addition, Mengoli distinguished the real motion of the eardrum and the “inclination” tothat motion. On the basis of Galileo’s theory that the ratio between the tensions of two chords
emitting an octave is not 2:1 but 4:1 (the square of 2:1), he asserted that if the ratio between
each vibrational frequency of the two eardrums was 2:1, the ratio between the each inclination
to the motion was √2:1 (dimidiata della doppia) and the soul abstracted the logarithms ofboth ratios. This assumption makes his argument rather confusing. However, because the“inclination” Mengoli referred to can be interpreted as the impression the real motion leaves on
the soul, it seems that the ratio perceived by the soul is in fact only that of the inclination.
About the interval of the ratio √2:1, half-octave or tritone, Mengoli says:

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OAI Takaharu[...] this interval is the first ratio, according to which the soul is made to perceive what isdone by sound in the eardrum of the ear and in the ethereal aura between the [two]eardrums. Therefore it is right that it should be the first of all the ratios to be found out
and also the gauge of all the other [intervals], [...] [12]Because the ratio between the vibrational frequencies of the inner and the outer eardrums isfixed at 2:1, the sound perceived by the outer eardrum ought to be perceived as being an octave
higher by the inner eardrum. However, according to his idea of inclination, the soul perceives it
as a half-octave; whenever the ear receives a certain sound, the soul abstracts the logarithm of√2:1, the ratio of a half-octave. Therefore Mengoli believed that a half-octave was the firstinterval perceived when hearing any sound, and he believed this to be the basis of every
auditory experience at the physiological level. He called this interval “musical measure (misuramusica)” and used it to gauge the errors of intervals.

3. The Mechanism of Interval Perception and the Logarithm
When hearing two differently pitched sounds, how is the interval between these pitches
recognized? This was the main problem addressed in Speculationi; Mengoli provided amechanical interpretation by defining the interval as the difference between the frequencies ofcontact the air particles have with the eardrum. When sounds are emitted at a certain pitch,
they cause air particles to make contact with the outer eardrum, making it vibrate. If there are
two tones, one high-pitched and the other low-pitched, the air particles from the high-pitchedtone moves faster than those of the low-pitched tone, so they make contact with the eardrum
more frequently, causing more frequent vibrations; the opposite is true for the low-pitched
tone. Even when those two tones are in harmony, the air particles emitted from each tone
strike the eardrum alternately, never at the same time. This theory is particular to Mengoli andis decidedly different from Galileo’s theory, which states that the more frequently the pulses of
air from two tones coincide, the more consonant the interval between the two tones is[13].
Mengoli may have designed this theory to accommodate the idea that the soul could recognize
the frequencies of both high and low tones and therefore understand the ratio of the interval
between those tones with mathematical accuracy, even when hearing sounds that wereperceived to be harmonious. He explains this idea:
Now, listening to two sounds is, in my opinion, none other than that the soul is attentive tothe two strokes that are alternately hitting the eardrum of the ear, and perceives thealternation of the strokes, the period of the alternation, and the return to the same period,and is counting how many times a sound gives a stroke and how many times in the same
period of alternation another gives a stroke at intervals. [14]Mengoli showed the relationship between alternating strokes of air and intervals in Fig.1[15].
The lines of numbers on the left represent the periodic patterns of the numbers of strokes by129

Pietro Mengoli’s Theory of Perception of Musical Intervals

i

ii

iii

● ● ○

iv

v

● ●

vi

vii

○ ●

viii

○ ・・・

● :Strokes by the high tone
○ :Strokes by the low tone
Fig. 2 The pattern of the alternation of
the strokes for a major sixth chord (5:3)

Fig. 1 Periodic patterns of the numbers ofstrokes by high tones and the ratios ofeach stroke by high and low tones

high tones; the ratios on the right represent those of each stroke by high and low tones.
For example, the number of strokes produced by high tone 2. 2. 1 corresponds to the ratioof a major sixth 5:3. This means (see Fig.2) that since air particles from the high tone reach theeardrum faster than those from the low tone, the high tone strikes the eardrum twice (i, ii),
then the low tone strikes once (iii), followed by the high tone twice (iv, v), the low tone once (vi),the high tone once (vii), the low tone once (viii), and so on. This alternating pattern of strokes
will be repeated. In this example, we can see that five strokes by the high tone correspond to
three strokes by the low tone. Thus, in the perception of two contemporaneous sounds, the soulcounts the strokes that each sound transmits to the eardrum alternately and then grasps theperiodic pattern of their regular alternation[16]. Finally, the soul obtains the logarithm fromthe ratio of the strokes and recognizes the size of the interval.
Now, we must consider the significance of Mengoli’s idea that the soul gets the logarithm
from the ratio as a cognitive function. The logarithm, invented in England around 1600, was arather new mathematical tool in Mengoli’s time and was useful calculating large figures, suchas those used in astronomy, because it could replace multiplication or division with addition or
subtraction. Therefore, its use spread rapidly in various mathematical fields, including music
theory. In addition to its utility, introducing logarithms into the music theory was significant
for another reason: it radically changed the view of musical intervals. While an interval wasformerly understood to be the “relationship” between two pitches shown by the ratio of
integers in the Pythagorean tradition of music, the logarithm, which replaced the ratio of two
numbers with one real number, played a large role in establishing the new view of intervals as
the “distance” between two pitches. This concept of an interval as a continuum became thepremise for the division of the “musical measure” in Mengoli’s arguments concerning theerrors of intervals.

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4. Counting through the Sense and Errors of Intervals
Although we can find novelty in his understanding of musical intervals, Mengoli’s idea ofthe mechanism of their perception is not far from the traditional doctrine that reason should
judge the size of an interval. That is, he treated intervals as a mathematical object to beunderstood by reason. Nevertheless, he attempted to explain the perception of intervals as afunction of the sense that receives sound directly, not of reason. In the “Protestation of theauthor”, Mengoli expressed his basic attitude:
I talk about some operations of the soul: count, division, and about the errors whichhappen to the soul in these operations, and about its active attention, [...] [those are]always [done] according to the sense and without abstraction by mind, as far as all thesethings belong to the inferior part of human being, which is sensitive, never to the superiorpart, which is rational. For this I don’t divide human being in two forms or natures, but
ignoring only that human being is rational, I consider it as sensitive being, and no more.
[17]
The soul perceived an interval by counting the strokes on the eardrum and Mengoli attributed
this function not to reason, which is the superior cognitional faculty of human beings, but tothe sense of hearing, which is its inferior cognitional faculty. He believed that since sound was
the proper object of the sense of hearing, there must be a system of musical perception that
relied on that sense, not on reason. Therefore, since interval perception is executed by counting
strokes of air particles, it is necessary that the soul count them only through the sense ofhearing without reason. Mengoli had the following to say regarding this idea:
Now I say that the rational soul can count according to the sense and make sure of thenumber without any effort until it finds how many counts have been done in itself,
independently of its every operation. And this [count] is one, two, three, up to three, andno more. Because the rational soul has one in itself for its natural faculty of learning one
thing, and no more, at one time; it has two for its natural faculty of putting two things incomparison with each other and judging them; it has three for the natural faculty of
discussing three things by putting the first in comparison with the third through thesecond to which the both are compared. I don’t refer to the very acts of judgment anddiscussion, but to its natural faculties of having two and three without any expertness.
[18](Emphasis added.)
It should be noted that, according to Mengoli, the soul can count from one to three through thesense “without any effort” or “without any expertness” and that this function is carried out by
the “natural faculty” of the human soul. Mengoli emphasized that the function is based on the
innate faculties of the soul and that it is clearly distinguished from rational acts such as
judgment and discussion as if it were carried out unconsciously[19]. According to this principle,
Mengoli limited the numbers the soul can count using the sense of hearing to 1 and multiples

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of 2 and 3 up to 36: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36. He added 5 to them for the seemingly naïve
reason that there are five fingers on one hand. The limitation of numbers is indicative of thelimitation of intervals the sense of hearing can perceive. Therefore the perceptible intervals arelimited to those of multiple ratios (n:1, where n is natural number) and superparticular ratios(n+1:n) as shown in Fig.3. All of the antecedents of these ratios are the numbers that arecountable using the sense of hearing; the smaller the number is, the easier the perception of theinterval (unison (1:1) and octave (2:1) are equally easy to perceive).
Multiple Ratios
Superparticular Ratios
Fig.3

1:1(=)2:1, 3:1, 4:1, 6:1, 9:1, 5:1
3:2, 4:3, 6:5, 9:8, 5:4, 12:11, 18:17, 27:26, 36:35

The Ratios of Intervals Perceptible to the Sense of Hearing

Mengoli adapted this theory of counting using the sense of hearing to his thoughts on theerrors of intervals. He distinguished intervals that contained errors from pure intervals (i.e. theintervals given in Fig.3) into three categories according to tolerance the sense of hearing musthave to identify the intervals: 1. Interval with an error that is unrecognizable to the sense ofhearing (Apparente), 2. Interval with an error that is recognizable but acceptable to the senseof hearing (Aggradevole), 3. Interval with an error that is recognizable and barely tolerable tothe sense of hearing (Tollerabile). The larger the error becomes from Category 1 to Category 3,
the more difficulty the soul has in identifying the interval, and the less pleasure it gets fromunderstanding it.
Mengoli carried out the categorization of errors in two ways. The first was through theequal division of a half-octave he called “musical measure”. The second was through the ratiosof the strokes created by two sounds.
In the first method of error categorization, the sizes of errors are defined by the parts of“musical measure” that can be divided equally. Since the biggest countable number using thesense of hearing was 36, he used 1/36 of a half-octave as the smallest perceptible error. If theerror is smaller than this interval, it is “unrecognizable” (Category 1). If the error is between
1/36 and 1/18 of a half-octave, it is “recognizable but acceptable” (Category 2). If the error is
between 1/18 and 1/12 of a half-tone, it is “tolerable” (Category 3). In a case where the error isbigger than 1/12 of a half-octave, that is a quarter of a tone, it is no longer tolerable. Needlessto say, 18 and 12 are numbers that the soul can count only through the sense of hearing.
On the other hand, in the second method of error categorization, errors are related toconfusions in the periodic regular pattern of each interval according to which air particles fromtwo tones alternate. The confusions are caused by mixtures of foreign patterns from otherintervals that break the regular pattern of the alternation. That is, when a certain interval has
an error, the ratio of that interval is mixed with its proximate ratios. For example, an error in a
fifth chord, whose ratio is 3:2, is produced by mixtures with the ratios 2:1 and 4:3. The sizes oferrors are defined by the frequencies of those mixtures in the strokes by the low tone. If there isa mixture in more than 36 strokes by the low tone, the error is “unrecognizable” (Category 1). If
there is a mixture in more than 24 strokes, the error is “recognizable but acceptable” (Category2). If there is a mixture in more than 12 strokes, the error is “tolerable” (Category 3). In the case

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of a fifth, intervals with an error of Category 1 are within 56:37 ((54 + 2):(36 + 1); 54:36=3:2)
and 58:39 ((54 + 4):(36 + 3)); intervals with an error of Category 2 are within 38:25 ((36 + 2):(24+ 1); 36:24=3:2) and 40:27 ((36 + 4):(24 + 3)); and intervals with an error of Category 3 arewithin 20:13 ((18 + 2):(12 + 1); 18:12=3:2) and 22:15 ((18 + 4):(12 + 3)). If there is a mixture inless than 12 strokes by the low tones, the error is no longer tolerable. Errors caused by a
mixture in more than 36 strokes are not perceptible, because the soul cannot count numbers
over 36 through the sense of hearing. 24 and 12 are two thirds and one third of 36, respectively,
and both are countable numbers only through the sense of hearing. Thus, the more frequent
confusions in the patterns of alternations are, the less tolerant the errors are. Fig.4 sums up theabove-mentioned argument.
Category of Errors

1. Unrecognizable

2. Recognizable but acceptable

3. Tolerable

Parts of“musical measure”

Less than 1/36

1/36 ~ 1/18

1/18 ~ 1/12

Frequency of mixture offoreign intervals in the
strokes by the low tone

Once in more than
36 strokes

Once in more than
24 strokes

Once in more than
12 strokes

Fig.4

Categorization of the Tolerance of Errors

Why did Mengoli approach interval errors in these two ways? He did not provide ananswer to this question, but the very fact that he set up such double standards might show his
ambivalent attitude toward the understanding of musical intervals. From the viewpoint of
temperament, the division of “musical measure” corresponds to the method of creating anequal temperament. The ratio of strokes corresponds to just intonation in which the intervals
are defined by simple ratios of integers. The latter method for categorizing errors in intervals
is based on the Pythagorean concept that a musical interval should be considered as the“relationship” between two pitches. This relationship is indicated by a ratio that compares twoquantities, whether they are lengths of monochord or frequencies of the strokes of air particles.
On the other hand, the former method is in accordance with the idea that a musical interval
should be considered as the “distance” (continuum) between two pitches. In order to equallydivide a half-octave, indicated by the ratio √2:1, it is necessary to treat it as a continuum. Thisis because √2:1 is an irrational ratio, which is impossible to deal with according to the
Pythagorean theory of intervals because it can only handle rational numbers. Although the ideaof dividing an interval into equal parts in the manner of geometry had already appeared inmusic theory by the late sixteenth century[20], Mengoli’s originality is indicated by the fact that
he related his view of musical intervals as continua to the physiological structures of hearing
and to the functions of the sense of hearing. In other words, Mengoli considered the nature ofmusical intervals from the way human beings perceive them.
In this sense, the examination of interval errors through the divisions of “musicalmeasure” seems to be more suitable to Mengoli’s purpose. Nevertheless, when considering a
musical interval as a physical phenomenon external to humans, he had no choice but to definePietro Mengoli’s Theory of Perception of Musical Intervals

133

it using the ratio of strokes of air particles, and he also had to believe that the soul recognized
the size of the interval errors through the frequency of confusions in the alternation pattern ofstrokes. Consequently, we suppose that Mengoli must have considered interval errors from theboth points of view, and this gap in his view of intervals indicates his failure to connect theirexternal factors and the physiological structures of hearing.
As we have seen so far, Mengoli’s argument about counting through the sense of hearing
developed in relation to the problem of determining the errors of interval. We should focus lesson the seemingly arbitrary numbers that he used and more on his assumption that the soul canonly count the numbers through the sense of hearing, which is very limited. The confusion ofsense perception is caused by this limitation. Nevertheless, the rational human soul requires
mathematical evidences based on the limited faculty of the sense of hearing when perceivingmusical intervals. We will examine this aspect of sense perception in the next two sections.

5. The Active Attention of the Soul
In the nineteenth Speculatione, Mengoli emphasized the activeness of the soul’s reactionin hearing. He asserted that when perceiving a certain interval, due to its natural desire for theinterval to be understandable through the sense of hearing, the soul actively affected the bodyand changed the tension of the eardrum; that is, it changes the real ratio of the interval.
He used the case of perceiving the interval of 36:35, which is the smallest and most difficultinterval to recognize using the sense of hearing as shown in Fig.3 above. First, when receiving
the interval of 39:38, the soul relaxes the eardrum and decreases the vibrational frequencies
caused by this interval to those caused by the interval of 36:35. Then, it changes the interval of
39:38 to the interval of 36:35. In other words, it perceives the interval of 39:38 as the interval of
36:35 in order to satisfy the soul’s desire for clear understanding. On the other hand, when
receiving the interval of 34:33, the soul strains the eardrum and increases the vibrationalfrequencies caused by this interval to those caused by the interval of 36:35. In this case, it also
perceives the interval of 34:33 as the interval of 36:35.
With regard to this function of the soul, Mengoli said, “It is necessary to say that therational soul, by its active attention, gives the true number to the first sound offered to theear”[21]. The “true number” mentioned here must be one of the numbers that constitute theratios of pure intervals; consequently, it becomes one of the numbers the soul can understandonly through the sense of hearing. Therefore, we can conclude that the soul’s active attention is
the function of perceiving the provided intervals as intervals of the ratios that can beunderstood clearly through the sense of hearing. If the provided interval has a ratio that cannot
be understood through the sense of hearing, the soul corrects that ratio by affecting theappropriate part of the body - the eardrum. Thus, Mengoli believed that the essential elementof interval perception was the mathematical evidence based on the limited counting faculty ofthe sense of hearing. Therefore, the natural desire for mathematical evidence caused the soul todirectly affect the structures of the body. This means that he considered sense perception not
as a passive function where the soul received external stimuli one-sidedly, but as an activefunction where the soul understood the stimuli according to its own inner principles[22].

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However, it is difficult to imagine that such phenomena really occur in the eardrum. Can
we suppose that Mengoli’s intention was to assert that the human sense of hearing mustcorrect the ratios of intervals because of the imperfections of its cognitional faculty? A
reexamination of his classification of intervals according to the sizes of their errors should
provide an answer to this question.

6. The Tolerance of the Sense of Hearing concerning the Errors of Intervals
In the sixteenth Speculatione entitled “Definitions of the intervals”, Mengoli divided allintervals into two categories, “natural intervals” and “artificial intervals”. Then, he appliedthem to the three genera of tetrachords in Ancient Greek music theory. “Natural intervals” areequivalent to intervals in the diatonic genus, and “artificial intervals” are all the other intervals,
including those in the chromatic and enharmonic genera. Diatonic intervals, which aresupposed to be “natural” and the most easily perceived, are characterized by unrecognizableerrors (that is, they are intervals in Category 1: Apparente). They are unrecognizable withregard to the alternation pattern of the strokes and the divisions of “musical measure”. Mengolireferred to “natural intervals” as follows:[I call these intervals “natural”] because they are made easily and recognized easily by thesense [...], and because the errors of these intervals from their true forms are withincertain confines where the sense cannot discern them. Therefore [the soul] considers theintervals with unrecognizable errors (gli apparenti) as true intervals without any
discrepancy. And like true intervals it understands the intervals with unrecognizable errors
completely both in terms of the number [of the strokes] and the [musical] measure they
have, in which consists the complete satisfaction of the soul, according to the sense. [23]Since “natural intervals” contain errors that fall within the “unrecognizable” category, the soulconsiders them to be pure intervals and is satisfied with the completion of its recognition; thissatisfaction comes from the mathematical evidence based on the limited counting faculty of thesense of hearing.
On the other hand, “artificial intervals” contain clearly perceptible errors. According to
Mengoli, errors must fall within the confines of the “unrecognizable” category, both in terms of
the ratio of strokes and the divisions of “musical measure” in order for an interval to beunderstood completely using the sense of hearing only. However, it is necessary to use both thesense of hearing and reason to recognize “artificial intervals”, because they contain perceptible
errors in the ratio of strokes and/or the divisions of “musical measure”[24].
In his argument concerning which intervals are suitable to melodies, Mengoli includes
intervals from the chromatic and enharmonic genera in the category of intervals that the soul
can understand through the sense of hearing. According to Mengoli, intervals from thechromatic and enharmonic genera cannot be understood fully through the cognitive faculty ofthe sense of hearing alone; however, the soul imagines understanding them and it satisfies its
desire to recognize them (he compared this satisfaction of the soul with that of the person who

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imagined understanding fully the contents of a book only by glancing at its table ofcontents)[25]. This means that intervals from these genera belong to intervals that the soul canunderstand through the sense of hearing in the same way as it understands intervals from thediatonic genus. He considered “artificial intervals” except those from the chromatic andenharmonic genera to be unsuitable to melodies, because they contained “recognizable but
acceptable” errors (Category 2) both in the ratio of strokes and the divisions of “musicalmeasure” and consequently their errors were clearly recognized[26]. Therefore, the errors in
the “artificial intervals” suitable to melodies must be “recognizable but acceptable” for only oneof two criteria. Thus, Mengoli concluded that “artificial intervals” from the chromatic andenharmonic genera as well as “natural intervals” from the diatonic genus were suitable for
melodies.
It should be noted that the intervals suitable for melodies are allowed to have certain
errors that are tolerable to the sense of hearing; in other words, a note in a melody does notnecessarily have a single allowable pitch, it can represent a certain range of pitches. We
perceive each interval that constitutes a melody as a category containing a certain error. If eachinterval in a melody has an error within this category, we perceive it in its pure form and the
melody is not spoiled.
From these observations, we can infer that the correction of the interval ratios by the soul’s
active attention indicates the categorical perception of intervals. When the soul recognizes (orwhen it imagines recognizing) an interval through the sense of hearing, the interval is within
the category where it is identifiable. At the same time it is perceived in its pure form under thecorrection of the soul. Mengoli must have intended to explain this way of perception by
reducing it into the physical function of changing tension in the eardrum. Moreover, thecategorical perception indicates the limitation of the capacity of human sense perception, andthe correction by the soul’s active attention might be a form of mind-body relationship that wasinvented in order to make up for this limitation.

Conclusion
Mengoli’s Speculationi is an unusual work on music. Although it deals with musicalintervals and melodies, they are argued exclusively from the physiological aspect of musicalhearing without notes. Its main purpose was to explicate how the soul recognizes the size of aninterval when it is heard. Therefore, the work is a kind of epistemology that considers musicalperception rather than music theory, as was pointed out by the article in Giornale de’ letterati.
Certainly many of Mengoli’s theories appear very strange to us because they are deduced fromhis own speculations and do not involve any actual proof, however, his focus on the cognitive
functions of the sense of hearing itself was significant. In this sense, we can regard Speculationi
as a major milestone in the birth of aesthetics advocated by Alexander Gottlieb Baumgarten(1714-62) as a study of sensuous recognition and its proper objects: beauty and art[27].

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OAI Takaharu

Notes
[1]

[2]

[3]

[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]

[13]

[14]

[15]
[16]

Giovanni Maria Artusi, L’Artusi overo delle imperfettioni della moderna musica (prima parte),
Venezia: Giacomo Vicenti, 1600 (fac., Arnaldo Forni Editore, 2007), fol.11v.: “Vi confermo, chel’udito è il primo, che gode, & si diletta di quella parte che à lui s’appartiene; cioè del suono, suo
proprio oggetto, & è la materia delle Cantilene; ma vi dico ancora, che il senso senza l’intelletto,
non può da se solo iudicare cosa alcuna, perche è sottoposto à molti difetti, & à molte
imperfettioni; nè riceue le cose, se non in vn certo modo confuse, se bene vicine al vero:”
Gozza pioneered the study of Mengoli’s Speculationi by putting it in the historical context of thescience of his time: Paolo Gozza, “A Mechanical Account of Hearing from the ‘Galilean School’:
Pietro Mengoli’s Speculationi di Musica of 1670”, C. Burnett, M. Fend, P. Gouk (eds.), The Second
Sense: Studies in Hearing and Musical Judgement from Antiquity to the Seventeenth Century,
The Warburg Institute: University of London, 1991, pp.115-136. However, he hardly deals with themathematics that Mengoli used to explicate musical perception. Wardhaugh’s paper is rather
satisfying in this respect: Benjamin Wardhaugh, “The Logarithmic Ear: Pietro Mengoli’s
Mathematics of Music”, Annals of Science, Vol.64, No.3 (2007), pp.327-348.
Studies of the history of mathematics in the 1910s demonstrated that Mengoli was the first to
state the concept of “convergence” and “divergence” in infinite series, and he was the first to
demonstrate that the harmonic series is divergent. The achievement concerning harmonic series
was attributed to Jakob Bernoulli (1654-1705), a Swiss mathematician, but in reality Mengoli was
the one who discovered it. For Mengoli’s bibliographical informations, see Marta Cavazza“Introduzione”, La Corrispondenza di Pietro Mengoli, a cura di G. Baroncini e M. Cavazza,
Firenze: Olschki, 1986, pp.1-22.
Philosophical Transactions, Royal Society of London, Vol.8, 1673, pp.6194-7000; Giornale de’letterati, Roma: Nicol’Angelo Tinassi, 1673, pp.131-133.
Philosophical Transactions (note 4), p.6195.
Giornale de’ letterati (note 4), p.131.
Pietro Mengoli, Speculationi di musica, Bologna: l’Herede del Benacci, 1670, p.4.
ibid., p.16.
ibid., p.14: “l’aura è l’instromento immediato dell’anima, nel quale riceuendo, sente; e per lo quale
operando, moue.”
ibid., pp.22-24.
ibid., p.22.
ibid., p.51: “[...] questo interuallo è la prima ragione, secondo la quale l’anima è costretta ad
auuertire quel che ne’ timpani dell’orecchio, e nell’aura eteria trà gli timpani per il suono si fà.
Onde hà ragione per essere il primo di tutte le ragioni à scoprirsi, di essere anche la misura di tutti
gli altri, [...]”
Galileo’s “coincidence theory of consonance” appears at the end of the First Day in his Discourses
and Mathematical Demonstrations Concerning Two New Sciences published in 1638. For moredetails on this theory, see H. F. Cohen, Quantifying Music, Reidel, 1984, pp.85-97.
Speculationi (note 7), p.40: “Hor l’vdire due suoni, non è altro, per mio credere, se non chel’anima stà attenta a i due percutienti, che vanno alternatamente battendo il timpanodell’orecchio; ed auuerte l’alternatione de’ battimenti, e il periodo dell’alternatione, e il ritorno
allo stesso periodo; e và contando, quante volte vn suono percuote, e quante volte nel tempo dello
stesso periodo d’alternatione, interuallatamente percuote l’altro.”
ibid., p.37.
As Wardhaugh pointed out, this explanation has a number of problems. First, unless air particles
reach the eardrum at a fixed distance from one another, they cannot strike the eardrum at adeterminate frequency, so they cannot keep a determinate pitch. However, that is only under idealPietro Mengoli’s Theory of Perception of Musical Intervals

[17]

[18]

[19]

[20]

[21]
[22]
[23]

[24]
[25]
[26]
[27]

137

circumstances. Also, even if air particles reach the eardrum at a fixed distance from one another,in order for the soul to distinguish between the two kinds of particles, there should be somedifferences in their shapes or sizes, which is not addressed by Mengoli. See Wardhaugh, op.cit.
(note 2), p.336.
Speculationi (note 7), n.pag. [xix-xx]: “Parlo di alcune operationi dell’anima, numerare, partire, ede gli errori, che in queste operationi gli occorrono, e del suo attendere attiuo, [...] sempre
secondo il senso, e senza astrattione di mente, e in quanto tutte queste cose appartengono alla
parte inferiore dell’huomo, come che è sensitiuo, e non mai alla superiore, come che è
ragioneuole: non perciò io diuido l’huomo in due forme, ò nature, ma prescindendo solo dall’eßer
suo ragioneuole, lo considero come sensitiuo, e niente più.”ibid., p.54: “Hor’io dico, che può l’anima rationale numerare secondo il senso, ed accertare il
numero senza punto d’applicatione, sino à quanto troua in se la numeratione già fatta,
independente da ogni sua operatione. E questa è, vno, due, tre, sino al tre, e non più oltre.
Impercioche l’anima ragioneuole hà l’vno in se, per la facoltà naturale, che hà d’apprendere vna
cosa alla volta, e non più: hà il due, per la facoltà, che hà naturale di giudicare, e mettere due cosel’vna à paragone dell’altra: hà il tre, per la facoltà naturale di discorrere per tre cose, mettendo àparagone la prima della terza, mediante la seconda, à cui si paragonano tutte due, non dico per gli
stessi atti di giudicare, e discorrere, ma per le facoltà, che hà naturali, nelle quali hà il due, e il tre,senz’alcun magistero.”Mengoli’s idea of counting through the sense reminds us of the famous definition of music byLeibniz: “Music is a secret exercise of arithmetic by the soul not knowing that it counts (Musica
est exercitium arithmeticae occultum nescientis se numerare animi).” Leibniz regarded musicalperception as the result of the soul’s unconscious counting. Unconscious perception, which wascalled petite perceptions by Leibniz, is the basis of sense perception that is clear but not distinct(i.e. its cause is unknown). This “inferior cognition” was addressed by Christian Wolff (1679-1754),
which led to the birth of aesthetics in the middle of the eighteenth century.
Gioseffo Zarlino (1517-89) and the other theorists of his time attempted to find a way to divide aline into equal parts geometrically in order to construct equal temperament. See J. Murray
Barbour, Tuning and Temperaments: A Historical Survey, New York: Dover, 2004 (originallypublished in 1950), p.49 ff.
Speculationi (note 7), p.162: “[...] bisogna dire, che l’anima rationale, con l’attentione attiua, dia ilvero numero al primo suono, che gli si rappresenta nell’orecchio.”
The influence of St. Augustine is evident here. According to him, perception is the soul’s acting on
the passive state of the body to adjust it to the soul itself. De musica, Book VI, Chap.V.
Speculationi (note 7), pp.105-106: “perche facilmente si fanno, e facilmente col senso si riconoscono,[...] e perche gli errori di questi dalle vere loro forme, sono dentro à certi confini, che non può ilsenso discernerli; onde prende gli apparenti per veri, senz’alcuna discrepanza; e come i veri, cosìcomprende gli apparenti del tutto, e quanto al numero, e quanto alla misura, che hanno: nel che
consiste tutto l’intiero gusto dell’anima, secondo il senso.”
ibid., p.106.
ibid.
ibid., pp.106-107.
See also note 19.