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6 Variants of Pythagorean Orthodoxy: Gioseffo Zarlino, Giovan Battista Benedetti

6 Variants of Pythagorean Orthodoxy: Gioseffo Zarlino, Giovan Battista Benedetti

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6 With the Latin Alphabet, Above All



Sebastian Bach to Arnold Schoenberg, the musical scale was to be considered as

decomposed into so many semitones.95 But we shall see that these semitones were

to be different from the Pythagorean ones, because they were to be tempered, that is

to say, made all the same as one another, at least approximately.

The variant of Zarlino was different from the combinations of Maurolico, and

was destined to catch on among theoreticians of music; even if it, too, was born

within Pythagorean theory, like that of Ptolemy. Instead of working on the interval

of the fourth, as the Greeks and the Arabs had done, he decomposed the interval of

the fifth as follows: in order to divide it, he calculated, between 3 and 2, both the

arithmetic mean:

1

5

.2 C 3/ D

2

2

and the harmonic mean:

2



.2 3/

D2

.2 C 3/



6

5



which he introduced into the interval of the fifth, shifting them by an octave. The

ratio of the fifth 3:2 was thus decomposed into two new ratios, 5:4, the major third,

and 6:5, a minor third. These latter went outside the Pythagorean tetractis, because

they also used the numbers 5 and 6. The innovation of Zarlino was thus called a

senarius, and the two thirds were included among the consonances, whereas the

Greeks had not admitted the ditone.

Thus, also the moderate innovators now appeared on the scene of the sixteenth century: though conserving the general mathematical approach determined

by numerical proportions, they introduced other ratios, and consequently, other

intervals. Therefore the classification of the intervals was enriched by the major

third, which took the place of the Pythagorean ditone, the minor third and the sixth

5:3.

It should be underlined that the first serious variant to Pythagoreanism, partly

inspired by Ptolemy, came from professional musicians. But for the sake of brevity,

we cannot, unfortunately, digress on music. We will only recall that polyphony

had reached the heights of Giovanni Pierluigi da Palestrina (c.1525–1594), Roland

De Lassus (c.1530–1594) and other Flemings: like Josquin Després (1440–1521),

Henricus Isaac (c.1450–1517), Adrien Willaert (c.1480–1562) or Cyprien De Rore

(1516–1565), who was at home between Rome, Florence, Venice and Parma. We

could not keep silent either about the Venetians Andrea Gabrieli (c.1510–1586)

and Giovanni Gabrieli (1557–1612), or the Neapolitan Carlo Gesualdo, Prince of

Venosa (c.1560–c.1613), not to mention the madrigalists, Luca Marenzio (1553–

1599) and Adriano Banchieri (1568–1634). Pythagorean theories no longer seemed

to be sufficient to discipline all this creativeness. In this way, the simple melodic



95



See Part II, Sect. 12.4. Tonietti 2004.



6.6 Variants of Pythagorean Orthodoxy: Gioseffo Zarlino, Giovan Battista Benedetti



305



linearity of the Greek conception, or of Gregorian chant, might perhaps have been

handled, with their relatively few meetings between the notes. But now, with those

complex polyphonic Masses with 4, 8, 16, or 32 voices, the meetings and clashes

between notes were multiplied, with effects that invited reflection on consonances

or not.

In view of these problems, the Franciscan monk and priest Zarlino, a pupil

of Willaert and the successor of De Rore as master of the chapel at St. Mark’s,

constructed his proposal for the new consonances of the third. His declared aim was

allowing variety. “La varietà dell’harmonia : : : non consiste solamente nella varietà

delle consonanze che si fa tra due parti. Ma nella varietà anco delle harmonie : : :”.

[“The variety of harmony : : : does not consist only of the variety of consonances

between two parts. But also in the variety of the harmonies : : :”]. Although the

Venetian musician continued essentially to deal with vocal polyphony, the term

‘harmony’ in time entered into use with a precise technical meaning, starting

precisely from his major and minor thirds, but going much further than he had

imagined. In the end, ‘harmony’ was even to be proposed as an alternative to

polyphony. However, it is clear that our musician from Venice desired to maintain

the general Pythagorean-Boethian picture, introducing only that amount of change

that was sufficient to take into account the current music of the period. However,

other musicians were to propose more radical changes than his. We shall now also

write about these, because they maintained more or less close relationships, but all

clearly with the evolution of the sciences.

Giovanni Battista Benedetti (1530–1590) was another Venetian, and was thus

a member of that cultural context which has always, up to the present day, been

outstanding particularly for music. His knowledge on the subject, and of the people,

are certified by the letter to Cyprien De Rore that he published in the book

Diversarum Speculationum Mathematicarum et Physicarum Liber [Book of various

mathematical and physical speculations]. It also contains a chapter (XXXIII) in

which this natural philosopher raised objections against the music of the spheres,

which went back to the Pythagoreans. He took his inspiration also from Aristotle,

and he tried to base his criticism mainly on terrestrial observations. If there were

no air in the sky, how could the stars emit sounds? If the orbits in contact were

“politas ac lenas” [“smooth and soft”], how could they generate sounds, seeing that

rubbing two smooth mirrors together, no sound can be heard. Only if there were

some element of “asperitatis, aut inaequalitatis” [“asperity or inequality”] and not

perfectly smooth (as many believed the stars to be), a rotating sphere would generate

sounds. “: : : ut etiam experientia à corpore aliquo fluido, quod in alio velocissime

moveretur desumpta fretus : : :” [“: : : as also basing myself on the experience taken

from a body that moves with great speed into some other fluid, : : :”].

Benedetti even noted that the music of the spheres lacked consistency, because

the relationships between the musical intervals, as fixed by the ratios of the

Pythagoreans, did not correspond exactly to the aspects of astrology: such as

sextiles, trines, quadratures and oppositions. In spite of Ptolemy, difficulties would

arise. “Quod autem attinet ad motus, ad magnitudines, ad distantias, & ad influxus,

nihil est, quod hisce proportionibus conveniat, sed quia haec omnia dependent



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6 With the Latin Alphabet, Above All



ab infinita, & divina providentia Dei, necessario fit ut istae velocitates, eae magnitudines, distantiae, & influxus, talem ordinem, & respectum inter se ipsa, &

universum habeant, qualis perfectissimus sit.” [“As regards movement, dimensions,

distances and the influences [of stars], nothing of this conforms to these ratios. But

as all these things depend on the infinite and divine wisdom of God, then necessarily

the speeds, dimensions, distances and influences will have that certain order and

relationship between themselves and the universe, in such a way to be the most

perfect.”]

Among the consonant musical ratios, our Venetian patrician placed also the

sesquifourth (5:4) and the sesquififth (6:5), which Zarlino had introduced.96 In his

letter97 to his beloved friend, Cyprien De Rore “musico celeberrimo” [“most

renowned musician”], Benedetti denied “: : : quod aliquis recte possit intelligere

rationes consonantiarum musicae, absque cognitione illarum mediante ipso sensu,

: : :” [“: : : that anyone can understand correctly the relationships of musical consonances without a knowledge of those [obtained] through the senses themselves,

: : :”]. And by “the senses”, our natural philosopher from Venice intended also musical practice. Vice versa, however, the “pratico puro” [“pure practical man”] would

not be able to understand the intervals properly, and thus “: : : ad comparandam

perfectionem musicae necessarium sit, & theoriam & praxim addiscere.” [“: : : to

prepare the perfection of music, it is necessary to devote oneself both to theory and

to practice.”]

In order to explain to a practical musician the numerical ratios of intervals, he

even knew how to write notes on a stave. He could thus face up to the central

problem that remained also in the variant of Zarlino: that of tuning organs and

harpsichords, avoiding undesirable dissonances. For this reason, the fifths needed to

be suitably diminished and the fourths increased slightly. If they left the theoretical

3:2, with 27:8 every three fifths, diminished by an octave 27:16 (do sol C sol

reCre la D do la), the la would be found not to be tuned in the same way as the

consonant major sixth 5:3. The major thirteenth do la was considered: “: : : valde

odiosa : : : sensui auditus : : : auribus valde inimica : : :” [“: : : rather distasteful to the

sense of hearing : : : rather hostile to the ears : : :”]. They used the term comma, like

the Pythagorean one, for this other fastidious difference to be eliminated between

27

5

27 5

81

16 and 3 , that is to say, 16 W 3 D 80 , called the comma of Didymus, or syntonic,

since the times of Ptolemy. Zarlino had brought the matter up, because the new

ratios of the third generated, together with the Pythagorean tone 9:8, a second minor

tone 54 W 98 D 10

9 . The difference between the major tone and the minor one equals

9 10

81

W

D

.

A

similar reasoning should also be carried out also for the fourths,

8 9

80

to avoid three of them (do fa C fa si C si mi D do mi ) differing from

the major third do mi by the same comma. In order not to offend the ear, then,



96



Benedetti 1585, pp. 190–191.

Written between 1558, when Zarlino published the book Benedetti quoted, and 1565, the year

De Rore died.



97



6.6 Variants of Pythagorean Orthodoxy: Gioseffo Zarlino, Giovan Battista Benedetti



307



musicians had to tune their instruments with suitably dropping fifths and increasing

fourths.

Benedetti still reasoned with proportions, e.g. “81 to 80” and “10 to 9” or with

the Latin words “sesquioctuagesima” [9:8] and “sesquinona” [10:9], without using

fractions as we have done. Perhaps also for this reason, there are some errors in the

text. He did not make any mistake, on the contrary, when he calculated how many

9

commas 81:80 there were in the major and minor tones. He calculated that . 81

80 / D

150094635296999121

81 10

12157665459056928801

134217728000000000 and . 80 / D 10737418240000000000 , concluding that 9 commas exceed

10:9, and 10 commas 9:8. Less motivated, perhaps, by minor calculations, he did not

repeat the mistake of Boethius, or the brilliant demonstration of Maurolico either,

however.

He quoted the “Excellentissimus Zarlinus in secunda parte Istitutionum

Harmonicarum” [“Most excellent Zarlino in the second part of the Institutioni

Harmoniche”], adding: “Sed quia sensus auditus non potest exacte cognoscere

debitam quantitatem excessus, vel defectus, intendendo vel remittendo chordas

instrumentorum, ideo hanc viam sequutus sum.” [“But as the sense of hearing

cannot know exactly the due quantity of the excess or the defect, in tightening

or relaxing the strings of instruments, for this reason I have followed this way.”]

Helped more or less by the calculations on the commas, the ear of Benedetti seemed

to tolerate dropping fifths and increasing fourths quite well. With him, the ‘perfect’

Pythagorean ratios now came down to the earth, where they had to be content with

approximations.

And yet here, the Venetian patrician also sought other justifications for the

way of generating consonances: “: : : qui quidem modus fit ex quadam aequatione

percussionum, seu aequali concursu undarum aeris, vel conterminatione earum.”

[“: : : which becomes, indeed, the measure by means of a certain equality of strokes,

or by means of the equal convergence of waves of the air, or the joint termination of

these.”] Therefore, he carried out experiments “nella mente” [in his mind] with the

monochord, shifting the ponticello so as to divide the string according to the ratios

desired: one half, one third, two fifths. He thus obtained unison, the octave and the

fifth. He described the movements of the vibrating strings well. “: : : quo longior est

chorda, etiam tardius moveatur, quare cum longior dupla sit breviori, & eiusdem

intensionis tam una quam altera, tunc eo tempore, quo longior unum intervallum

tremoris perfecerit, brevior duo interValla conficiet.” [“: : : the longer the string is,

the more slowly it moves; thus, when the longer is twice as long as the shorter, and

both the one and the other have the same tension, then in the same time that the

longer completes one interval of vibration, the shorter completes two.”]

When the vibration of the string becomes percussion in the ear, “: : : in qualibet

secunda percussione minoris portionis ipsius chordae, maior percutiet, seu concurret

cum minori, eodem temporis instanti, : : :” [“: : : the greater strokes or converges

with the lesser at the same instant of time, at every second stroke of the lesser

portion of string, : : :”], in proportion to the lengths. Thus the ratio of the lengths

gave the ratio of the strokes, which corresponded better, the better the consonance

was: all in unison, one every two for the octave, etc. But with these “percussiones”



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6 With the Latin Alphabet, Above All



[“percussions, strokes”], what would prove to be in proportion? “: : : hoc est tempus

maioris intervalli ad tempus minoris erit sesquialtera : : :[: : :] : : : eadem proportio

erit numeri intervallorum minoris portionis ad interValla maioris, quae longitudinis

maioris portionis ad longitudinem minoris : : :” [“: : : that is to say, the time of the

greater interval will stand in a sesquialtera ratio [3:2] to the time of the lesser : : :[: : :]

: : : the same ratio as the number of intervals of the lesser portion compared with

the intervals of the greater portion will be that of the length of the greater portion

compared with the length of the lesser : : :”].

For Benedetti, sound was a wave in the air, though he was not clear enough about

how to measure it. He wrote about “time”, “numbers of intervals”, “percussions”;

if he had calculated the ratio, he would have obtained the wave frequency. Instead,

he left us other numbers, with the idea of seeking what consonances maintained

in common, when varying the length of the string that generated them. “: : : unde

productum numeri portionis minoris ipsius chordae in numerum intervallorum

motus ipsius portionis, aequale erit producto numeri portionis maioris in numerum

intervallorum ipsius maioris portionis; : : :” [“: : : hence, the product of the number

for the lesser portion of the same string with the number of intervals for the

movement of the same portion will be equal to the product of the number for the

greater portion with the number of intervals for the same greater portion; : : :”].

Thus, the diapason (octave) was characterised by the number 2 (2 1), the diapente

(fifth) by 6 (3 2), the diatessaron (fourth) by 12 (4 3), the major hexachord (major

sixth) by 15 (5 3), the ditone (major third) by 20 (5 4), the semiditone (minor

third) by 30 (6 5), the minor hexachord (minor sixth) by 40 (8 5). The better the

consonance, the lower the number would be.

In the end, the patrician from the Veneto region, a music lover, concluded his

open letter to his friend, the renowned polyphonist, as he had started it: with a hymn

to sensuality. “Voluptas autem, quam auditui afferunt consonantiae fit, quia leniuntur

sensus, quemadmodum contra, dolor qui a dissonantiis oritur, ab asperitate nascitur,

id quod facile videre poteris cum conchordantur organorum fistulae.” [“On the other

hand, the pleasure that consonances procure for the hearing originates because the

senses are sweetened by them, just as, on the contrary, the pain that stems from

dissonances is born from roughness; you will easily be able to appreciate this when

the pipes of organs are tuned properly.”]98

Historians have generally judged Benedetti comparing him with the future

Galileo Galilei. The former described the fall of heavy objects through the air

as independent of their weight, in contrast with Aristotle, but without giving the

mathematical law in a void like the latter. For music, we have preferred here to

narrate the results in relation to Pythagoreanism. Our Venetian natural philosopher,

who had stayed at Parma, and ended up at Turin, moved away from that tradition,

because he criticised the music of the spheres and declaredly used his ear with

an empirical spirit foreign to that tradition. However, in an attempt to combine

together a mathematical theory based on numbers with the practice of musicians,



98



Benedetti 1585, pp. 277–283.



6.7 The Rebirth of Aristoxenus, or Vincenzio Galilei



309



he remained linked to Pythagorean ratios, only enriched by the Venetian Zarlino,

without taking into consideration the alternative of Aristoxenus. And yet somebody

else was to do so: a musician or a natural philosopher?

Musicorum et cantorum magna est distantia

Isti dicunt, illi sciunt, quae componit musica.

Nam, qui facit quod non sapit, diffinitur bestia.

[The distance is great between musicians and singers]

The latter say, the former know, the things that music composes.

For he who does what he does not know is defined as a beast.

Guido D’Arezzo



6.7 The Rebirth of Aristoxenus, or Vincenzio Galilei

Zarlino was criticised by the Florentine noble, Vincenzio Galilei (1520–1591) in

his Dialogo della musica antica et della moderna [Dialogue between ancient and

modern music]. A lute-player, composer and connoisseur of rival theories (he had

even been a pupil of Zarlino), this musician from Florence immediately denounced

“: : : la poca fede d’alcuni stampatori di Venezia : : :” [the little faith of some printers

in Venice] who are said to have boycotted him “: : : per compiacere ad alcuno il

quale o tratto da invidia impediva che queste mie fatiche uscissero fuore : : :” [to

please someone who, moved by envy, prevented these efforts of mine from coming

out]. As a result of these editorial intrigues, he consequently printed the book at

Florence, and in Italian instead of Latin, under the patronage of Count Giovanni

Bardi. Actually, Vincenzio Galilei was a member of the famous Camerata de Bardi,

which was renewing musical style, moving away from polyphonic complications in

search of a monodic, melodic simplicity, suitable to let the meaning of the poetic

verses in music be understood: “recitar cantando” [reciting in song]. Our Florentine

musician expounded his theory in the form of a Platonic dialogue between two

characters, one of whom was his patron, the musician Giovanni Bardi.99 It can

already be seen, from the way the subject under discussion was presented, that

even though he dealt with the numerical ratios for notes in detail, Vincenzio Galilei

evaluated the theory on the basis of the requirements of musicians. Definitely,

beside his calculations for the ratios, he always put a stave with the relative notes

(Fig. 6.7).

To convince the reader, whether theoretician or musician, of his arguments, he

described how to construct an instrument on which they could be verified. “Tirinsi

sopra una piana superficie due corde all’unisono, d’un’istessa lunghezza, grossezza,

& bontà; dividasi poi col compasso una di esse : : : & chi volesse ancora udire qual

si voglia intervallo in una sola corda, : : :”. [Let two strings be extended over a plane



99

Vincenzio Galilei 1581, p. [iii]. As Fabio Fano wrote in the “Prefazione”, the charge against

Zarlino of having prevented the publication of the book at Venice became explicit in the Discorso

intorno all’opere di Messer Gioseffo Zarlino of 1589, p. 8. Cf. Massera 1977, pp. 140–148.



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6 With the Latin Alphabet, Above All



Fig. 6.7 How Vincenzio Galilei assigned the relative ratios to the notes on a stave (Vincenzio

Galilei 1581, p. 20)



surface in unison, with the same length, breadth, and good quality: then let one of

them be divided with a compass : : : and anyone who wants to hear any interval

on a single string, : : :].100 “: : : in reprovare l’opinione” [in reproving the opinion]

of “Reverendo M. Gioseffo Zarlino”, the Florentine composer went back over the

current theory, which had modified the Pythagorean order into Ptolemy’s “syntone”,

and contained the (already seen above) new consonances of the third and the sixth

in the new ratios of 5:4, 6:5, 8:5, 5:3. But, this other Galilei noted, “(: : : contro

l’opinione del prattico)” [in contrast with the opinion of the practical musician], in

the new scale, not all the minor thirds have the same ratio, because this depends on

where they begin. On the contrary, the musician would like to play on the keys (or

on the lute) in the same way the intervals indicated at the same distance on the stave.

The problem derived from the difference between the Pythagorean major tone 9:8

and the minor one 10:9, called the syntonic comma 81:80, already seen in Benedetti.

Or again, Zarlino’s minor third, together with the major tone, exceeded the correct

Pythagorean fourth by a comma. And so on, the book multiplied examples that put

Zarlino in contrast with musicians, the “prattici moderni contrappuntisti” [practical

modern contrapuntists], with the numerical ratios calculated by theory, and lastly

artificial theory with “Nature”, including and excluding those syntonic commas,

which others cheerfully ignored.

Also Galilei father wrote of how many of these commas were contained in major

and minor semitones and tones, though without offering any calculations. And as

regards Pythagorean commas, instead, he trusted Boethius, who was honoured with

a “very well”, which we now know he did not deserve.101 He represented Zarlino’s

senarius in a figure which contained its intervals (Fig. 6.8).

He insisted on pointing out that all this was not new at all, but had been

taken from Ptolemy’s Harmonics; in his Quadripartite [Tetrabiblos], Ptolemy had

even compared “gli aspetti de’ pianeti alle forme degli intervalli musici : : :” [the



100

101



Vincenzio Galilei 1581, p. 3.

Vincenzio Galilei 1581, pp. 9–10.



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