Esther M. Zimmer Lederberg
The Views of Rudolf Wittkower1
François Carter2
In the Greek view (Plato), there are five elements: fire
(tetrahedron), air (octohedron), earth (cube), water
(icosahedron), and the quintessence (dodecahedron: the
quintessence was an element that kept heavenly bodies
moving in their geometric figure). One should not overlook
the circle, which is so important it represents god or the
cosmos, as the circle is "perfect: no beginning, no end and
one can traverse the circle an infinite number of times"
[Sir Francis Bacon felt that science and religion should be
separated and constantly attacked Plato and Mathematics].3
Furthermore, according to the Pythagoreans, the
numbers 1, 2, 3, 4 are very special, as: "1" determines a
point, "2" points determine a line, "3" points determine a
(flat) surface, and "4" points determine a 3-dimensional
solid, and 1+2+3+4 = 10 (a special number). Thus numbers
like "12" are almost magical, as they are associated with:
a) the quintessence [dodecahedron], b) the number of signs
of the zodiac, c) the number of apostles (later followers
of Plato), etc. Thus we find special relationships in masques:
Magic number "12":
Ballet de Monsieur de Vendôme: used 12 masquer dancers plus other dancers
Balet - Comique: by Beaujoyeulx, used 12 masquer dancers
Vision of Twelve Goddesses: by Samuel Daniel, used 12 masquer dancers
Masque of Queens: by Ben Jonson, used 12 masquer dancers
Pleasure Reconciled to Virtue: by Ben Jonson, used 12 masquer dancers
Magic shape "circle":
Hymenaei: by Ben Jonson, used a circle dance
Oberon: by Ben Jonson, used a circle dance
Vision of Twelve Goddesses: by Ben Jonson, used a circle dance
In addition to the above "special number" relationships,
these relationships repeatedly appear to assert that
they are special, not accidental. Thus in "Ballet de
Monsieur de Vendôm", each figure is danced using
a different step, and there are 12 "figures" (each one for
one of the figures of the Zodiac), danced by 12 masquers.
These figures are composed of the number "12" (quintessence)
click here.
We shall see below that the views of Plato that the cosmos is
structured upon arithmetic and geometry was also employed by
the Pythagoreans to assert harmonic ratios (1:2, 2:3, 3:4) to
underlie the cosmos. During the Renaissance, harmonic ratios
were the basis of dance, music and architecture.4
Rudolf Wittkower's "Architectural Principles in the Age of Humanism"
analyzes architecture from the viewpoint of musical harmony. Given the
vibrations of a string, the lengths found are in the ratios 1:2:3:4.
1:2:4 are octaves, 1:2:3 produce perfect fifths. From these ratios,
we find:
2n: 1, 2, 4, 8, 16,... (designated as "female")
3n: 1, 3, 9, 27,... (designated as "male")
These sequences are found in Plato's "Timaeus", interpreted as
evidence that the entire universe of god has its origin (male
and female souls) in mathematical sequences. Plato further expounds
a general cosmology based upon an animistic theory of the Microcosm
in which celestial bodies are seen as living beings '...endowed with
the fairest of bodies and the happiest and best of souls.' These
heavenly beings are formed mainly of the element five ("Quintessence")
and demonstrate their superior intelligence by moving in a procession
that is fixed, regular, and completely uniform; whereas the earthly
beings, including mankind, are made predominantly of the grosser
element "Earth" and their movements are much less orderly and
intelligent. Lacking true wisdom and understanding, etc., etc.
"It now emerges that Plato's interpretation of the entire visible
world depends absolutely on the concept of measured dance, that is,
the mathematics of cosmic choreography."
5.
Relationships between these "female" and "male" sequences were
central to the philosophy of the Pythagoreans, as well. Thus the
view entertained by Renaissance architects and artists Ficino, Georgi,
Alberti, Palladio, Barbaro, and Serli 6
that musical harmony of the universe, as well as the "perfect"
proportions of god's greatest creation, man, have a basis in
mathematics. Thus harmony and the proportions of man, must be
used in art, architecture, dance, and music, all interrelated
and bound together using linear perspective, proper "proportions",
and rhetoric tacens.
It has been pointed out that the basis of our modern [musical]
notational system is a chain of decreasing note-values, each of
which is divided into two parts. A whole-note is divided into
two half-notes, a half-note into two quarter-notes, a quarter-note
into two eighth-notes, and so on.
The mensural system is also based on the idea of a chain of
decreasing note-values; the longest is the maxima, then come the
longa, breve, semibrev, and minim. But in contrast to the modern
notational system, the mensural system allows a division of the
notes into either two or three parts. In 'inperfect' mensurations,
the notes are divided into two parts, in 'perfect' into three.
Any note can be divided into two or three parts except the minim,
which is always binary. The fact that the mensural system allows
for ternary and binary divisions is perhaps the most fundamental
difference between our modern notational system and the mensural
system.
See mensuration7.
Thus, as in architecture, two sequences are possible in music:
(1/2)n: 1, 1⁄2,
1⁄4, 1⁄8,
1⁄16,... (designated as "female")
(1/3)n: 1, 1⁄3,
1⁄9, 1⁄27,
1⁄81,... (designated as "male")
Ficino, a neo-Platonist (Plotinus) mystic, held the Christian
view that man was the image of god embodied in the harmonies
of the universe, and was also influenced by Vitruvius' idea of
a figure inscribed in a square, the square inscribed in a circle,
symbolizing the mathematical relationship of a microcosm within
a macrocosm. Thus the harmonies and proportions of man were
thought to apply to the entire universe.
7
1
"Architectural Principles in the Age of Humanism",
by Rudolf Wittkower, W. W. Norton, New York, 1971, p. 104
2
"Number Symbolism and Renaissance Choreography"; Carter,
François; Dance Research: The Journal of the
Society for Dance Research, Vol. 10, No. 1, 1992, 21-39,
see pp. 27-32
3
Ibid., "Mathematicians, he [Sir Francis Bacon] complains,
have a preconceived notion of order and symmetry which
they impose upon the universe in their desire to control
the natural world, with the result that nature is distorted
to fit their systems.", p. 34
"... he [Sir Francis Bacon] rejects the Pythagorean notion
of the mystical decade." (ie:1+2+3+4=10). p. 35
4
"Mensuration and Proportion Signs: Origins and Evolution",
by Anna Maria Busse Berger, Oxford Univ. Press, 2000, p. 1
5
"Dance, Spectacle, and the Body Politick, 1250-1750",
Nevile, Jennifer, Indiana Univ. Press, 2008, p. 274
6
"Mensuration and Proportion Signs: Origins and Evolution",
by Anna Maria Busse Berger, Oxford Univ. Press, pp. 38-43.
We find that doing fractions in arithmetic using Roman numerals
was not an easy task. A primitive system of positional notation
slowly developed. Thus 4326 was written as: IV(M) III(C) XXVI.
More consistently, it should have been IV(M)
III(C) II(X) VI. However, fractions were
expressed in base 12 (dodecahedron of quintessence), as the
Roman unit for money was in base 12. Thus 5/9 = x/12, or x = 20/3 ≈
7. One can begin to appreciate the accomplishment of
astronomers needing trigonometric tables!
7
"Architectural Principles in the Age of Humanism",
by Rudolf Wittkower, W. W. Norton, New York, 1971, p. 16
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