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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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.