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The Birth of
By Stuart Isacoff
At left, from Gregor Reisch’s Arithmetica (1503), Boethius and
Pythagoras in an imaginary competition using Arabic numerals
and an abacus. Above left, Albrecht Altdorfer’s Birth of the Virgin
(1525) and above, Masaccio’s fresco The Trinity (c.1428). Opposite
page, Piero della Francesca’s The Flagellation (c.1460).
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Winter 2005 Early Music America
Temperament
and Perspective
In the cradle of the Renaissance, new ideas about
proportion affected the development of painting and music
A
often
reflect dramatic shifts in a
society’s take on reality. Indeed, when
aesthetic issues generate high emotions,
it’s a good bet that ideological or religious sensibilities are also at play. These
become clear only by examining a multitude of cultural forces – going
beyond narrow musical, literary, or
painterly issues to develop a broader
context.
When I set out to learn about historical tunings, it was easy enough to
locate material about the mathematics
of temperaments but more difficult to
discover why people were at each
other’s throats over the matter. Why,
for example, did Giovanni Battista
Doni attempt to discredit Girolamo
Frescobaldi in 1640 with charges of
drunkenness, stupidity, and worse when
they disagreed about the tuning of an
organ in Rome? And what of Gioseffo
Zarlino’s assertion, a generation earlier,
that Vincenzo Galilei’s rejection of
sanctioned tuning theory was the equivalent of committing an immoral act?
These stories carry the weight of philosophical and societal currents beyond
matters of mere personal taste or
artistic inclination.
The particular bone of contention
in these cases, of course, was just how
far musicians could deviate from
music’s pure number harmonic ratios.
The parameters had been established
long before. Pythagoras, in the sixth
century B.C.E., was said to have discovered in a blacksmith shop, as the
smithies hurled their hammers against
anvils, that a pure octave is created by
RTISTIC DEVELOPMENTS
two tones, the higher of which is
vibrating twice as fast as the lower, that
a pure fifth is produced by two tones
vibrating in the ratio 3 to 2, and that a
pure fourth by two tones vibrating in
the ratio 4 to 3. These musical concordances sounded so right, so
harmonious, that they were
believed in Ancient Greece
to be nothing less than
signposts for the eternal
order of the universe. By
the second century, Clement of Alexandria would
gather music’s proportions
into the Christian fold by
proclaiming that Christ
himself had decreed them.
Yet another “divine”
ratio gained prominence in
the 13th century. This was
the pure third, beloved particularly by the Renaissance-era
English, and constructed, in its major
form, by two tones vibrating in the
proportion 5 to 4.
The simple-ratio musical concords
are beautiful – one might say magical.
And yet they embody a distressing paradox. Tune a fixed-pitch instrument so
that from any note you can find a perfect octave, and you will be unable to
do the same for perfect fifths. Octaves
are based on multiples of 2 (formed by
the proportion 2 to 1) while fifths are
based on multiples of 3 (formed by the
proportion 3 to 2). Both 2 and 3 are
prime numbers; and powers of prime
numbers can never be equal. Thus their
progeny are incommensurate. Indeed,
pure octaves, fifths, and thirds, when
multiplied across a keyboard, all
produce differing versions of the scale
tones. Therefore, except in extremely
limited circumstances, the harmonic
building blocks they form are not very
practicable.
Cultural influences
In the 15th century, grappling with
such intricate problems involving proportion spurred techniques that would
radically alter the course of all Western
arts. In painting, the outcome was
artistic perspective. In music, it was
temperament. Though it has been little
noticed, the ideas that gave birth to
each emerged from the same cultural
wellspring.
A number of factors made the time
ripe for these developments. Michael
Baxandall points out in his book
Painting and Experience in Fifteenth Century
Italy (Oxford University Press, second
edition, 1988) that one of the most
important skills taught in Florence
schools was that of gauging. “It is an
important fact of art history that commodities have come regularly in standard-sized containers only since the
nineteenth century,” he writes; “previously a container – the barrel, sack or
bale – was unique, and calculating its
volume quickly and accurately was a
condition of business.” Indeed, in this
respect painters and merchants had
much in common; to the commercial
man, “a pile of grain [could be] reduced
to a cone, the barrel to a cylinder.”
Piero della Francesca, a highly gifted
perspectivist painter of the era, actually
wrote a mathematical handbook for
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merchants entitled De abaco. So the
conscious exploration of proportion
was ingrained in everyday life to an
extent we can little appreciate today.
The tool used most often in commerce was called the Rule of Three
(also known as the Golden Rule and
the Merchant’s Key). Piero della
Francesca described it in this way: “The
Rule of Three says that one has to
multiply the thing one wants to know
about by the thing that is dissimilar to
it, and one divides the product by the
remaining thing.” To demonstrate, he
applied it to the following problem: if
seven bracci of cloth are worth nine lire,
how much will five bracci be worth?
Renaissance merchants used this
rule in dealing with questions of pasturage, discounts, the adulteration of
commodities, and currency exchange;
even with it, things could get horribly
complex, because each city had not
only its own currency but also its own
weights and measures. In any case,
proportion was in the air. As Baxandall
states, “Piero della Francesca had the
same equipment for a barter deal as for
the subtle play of intervals in his pictures.” Indeed, Leonardo da Vinci used
the Rule of Three for a problem about
weight in a balance and came up with
the terms 6, 8, 9, 12 – the Pythagorean
harmonic scale as discussed in 15thcentury musical and architectural theory. (The numbers 6 and 12 formed an
octave, the interval between 6 and 9
and between 8 and 12 generated a fifth,
and so on.) Pietro Cannuzio’s Rules of
Music’s Flowers placed precisely this
mathematical notation at the top of its
title page: an invitation, says Baxandall,
“to the mercantile eye.”
Other aspects of this culture also
influenced the direction art would now
take. The Aristotelian vision of a universe immutable in form, with all
things striving to fulfill their singular
purpose in a great chain of being,
began to crumble. Renaissance minds
shifted away from an abstract sense of
formal order and toward the lessons of
concrete experience. Thus Filippo
Brunelleschi’s radical experiment in
perspective art – through which he created the illusion of three-dimensional
space on a flat surface – arose because
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Winter 2005 Early Music America
How orthagonals appear to converge
at a point in the distance, from Leon
Battista Alberti’s De pictura (1435).
the personally “subjective” had gained
philosophical legitimacy. Leon Battista
Alberti, who canonized the technique
in the first published method utilizing
the “vanishing point” – that imaginary
spot in a painting where receding parallel lines beginning at the front (called
orthogonals) appear to converge in the
“distance” – explained that the effect
depended entirely on the idea of seeing
things through a “window,” that is,
from a particular (not universal) point
of view.
An event that also helped push
things in this direction was the arrival
in Italy, shortly after 1400, of copies of
Ptolemy’s long-forgotten Geographia,
which had been written around 150
C.E. Brought from Constantinople in
the form of Greek manuscripts, it provided a lesson in how to depict a
It was easy enough to
locate material about
the mathematics of
temperaments but more
difficult to discover why
people were at each other’s
throats over the matter.
curved map on a flat surface. An
important printing of the work, with
additional maps, was issued in Florence
in 1482. Ptolemy’s maps helped spur a
cartography craze that would continue
through the 16th century.
Altering proportions
There were moral and religious
implications to it all. Plato had condemned the beginnings of perspectival
art in his own day for distorting the
true proportions of things, replacing
reality and nomos (law) with subjective
appearance and a resulting arbitrariness.
(The proportions of music’s simple
concordances were, of course, similarly
regarded as eternal law; St. Augustine
even declared that they should be used
as a template in the construction of
churches.) Now, however, Renaissance
artist Lorenzo Ghiberti would include
in his Commentaries translations of
Alhazen, the Islamic natural philosopher whose optics sought a balance
between mathematical certainty and
direct experience. Perspective was being
transformed into an emblem of spiritual truth. Peter of Limoges’s Treatise on
the Moral Eye, which appeared in an
Italian translation in 1496, declared, “a
consideration of the eye and of such
things as appertain to it is a very useful
means of knowing more fully about
divine wisdom.”
The most significant and useful
aspect of what Brunelleschi and Alberti
accomplished was that they described
the subjective world according to a
rational and repeatable procedure.
Erwin Panofsky, in his Perspective as
Symbolic Form (Zone Books, 1997), calls
it the “objectification of the subjective,” or the “carrying over of artistic
objectivity into the domain of the phenomenal.” Quoting from Pomponius
Guaricus’s De sculptura, Panofsky writes
that there was now a mathematically
justifiable rule to determine “how far
two things ought to stand from another, or how closely they ought to cohere,
in order that the intelligibility of the
subject matter is neither confused by
crowding nor impaired by sparseness.”
Of course, as Panofsky also reminds
us, “it is essential to ask of artistic periods and regions not only whether they
have perspective, but also which perspective they have.” Once the genie
was out of the bottle, it took on a life
of its own. Visual artists like Leonardo
and Piero, not content to depict the
world “realistically” (in their subjective
fashion), experimented with proportion
and placement to create a myriad of
illusionistic effects. The results could
be awe-inspiring: Masaccio’s fresco The
Trinity conveyed depth so realistically
that viewers believed they were viewing
real holes in the image.
The analogy to temperament is
striking. Here, too, the inviolable universal proportions of music – what
earlier musicians might have labeled
“reality” and “law” – were altered
through a rational, repeatable procedure, governing how far pitches ought
to stand from one another or how
closely “they ought to cohere” – for
purely subjective reasons. The first text
that mentioned the process was by
Leonardo’s friend, Franchinus
Gaffurius. In his Practica musicae (completed in 1483), he noted that organists
were slightly diminishing their fifths.
The reason is clear: if an instrument is
tuned in the normal Pythagorean practice, stacking pure fifths to arrive at the
pitches to be used, the resulting thirds
will be unpleasantly wider than pure.
By adjusting the fifths – diminishing
each by one fourth of the amount by
which those thirds would be too wide –
those narrower stacked fifths will now
yield a third that is pure. Gaffurius,
though a strong advocate of Pythagorean tuning, was nonetheless reporting on the emergence of early
meantone temperament: the musical
equivalent of perspective.
By tampering with proportion, temperament (among its meanings are “to
tamper with,” “to mix ingredients proportionately,” and “to regulate”) creates
an aural “window” through which
musical relationships can be shaped. In
many temperaments (and this is
notable to a high degree in the later
non-equal circulating kind), melodies
and harmonies can, through the formulaic alteration of pitch placement, project shades and contours to the ear corresponding to the way foreshortening
and other techniques project a very
particular visual “angle” to the eye.
(For a fascinating account of Bach’s
Brandenburg Concertos No. 2 and No.
5 performed in Werckmeister III, a
classic non-equal circulating temperament, listen to the CD Early on the
Pitch label, produced by the American
Festival of Microtonal Music under
the direction of Johnny Reinhard
(available at www.afmm.org.)
Deciding the temperament that is
most appropriate depends, of course,
on many factors, including the musical
material at hand and the medium to be
used (in the case of harpsichords, for
example, this involves such issues as
string material and scaling – the length
of each string in relation to its pitch).
Certainly no single approach fits all
cases, either in painting or in music.
For instance, Piero della Francesca’s
The Flagellation (c.1460) uses conflicting
sets of proportions: the floor tile pattern in the area that features Jesus is
based on incommensurable ratios,
while the floor pattern beyond is based
on a simple arithmetical division.
Albrecht Altdorfer, in the Munich
Birth of the Virgin (1525), writes
Panofsky, created an “absolute oblique
space” – that is, a space in which there
are no discernable orthogonals or
frontals. This corresponds in time
roughly to Adrian Willaert’s Quid non
ebrietas of 1519, which Edward
Lowinsky demonstrated was a theoretical argument in favor of equal temperament (“Adrian Willaert’s Chromatic
‘Duo’ Re-Examined,” Tijdschrift voor
Muziekwetenschap 18). And equal temperament, in which the octave is divided into 12 equal parts, contains, like
Piero’s floor under Jesus, irrational proportions, and like Altdorfer’s Birth of the
Virgin, is often described as presenting
an oblique or non-directional frame.
There are advantages to oblique
space and to equal-tempered instruments, however (provided the instruments can accommodate the roughness
or “grit” inherent in this tuning – and
not all can). As Jean-Philippe Rameau
suggested in arguing for its adoption,
equal temperament achieves its incredible utility by allowing the imagination
of the listener to create the proper
musical hierarchy intended by the composer. Rather than being, as it is usually
described, a tuning of all “grays” – that
is, of no real perspective – equal temperament actually provides a ground
for simultaneous multiple perspectives.
The listener can discern from the musical context what role a particular pitch
is meant to play, and he or she can do
this no matter what the key or musical
texture. Thus, as music, like all art, continued to stretch free of its old moorings, this tuning – though admittedly
hard to achieve and vilified by many –
provided an unobtrusive canvas on
which composers were free to play.
From the Theorica Musicae (1492) of
Franchinus Gaffurius: Pythagoras
discovering the harmonic ratios in music.
Stuart Isacoff is the author of Temperament:
How Music Became a Battleground for the
Great Minds of Western Civilization (Vintage,
2003) and editor of Piano Today magazine.
In Practica musicae, Gaffurius
noted that organists were
slightly diminishing their
fifths. Though a strong
advocate of Pythagorean
tuning, he was nonetheless
reporting on the emergence
of early meantone
temperament: the musical
equivalent of perspective.
,
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