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:
  1. Magic number "12":
    1. Ballet de Monsieur de Vendôme: used 12 masquer dancers plus other dancers
    2. Balet - Comique: by Beaujoyeulx, used 12 masquer dancers
    3. Vision of Twelve Goddesses: by Samuel Daniel, used 12 masquer dancers
    4. Masque of Queens: by Ben Jonson, used 12 masquer dancers
    5. Pleasure Reconciled to Virtue: by Ben Jonson, used 12 masquer dancers
  2. Magic shape "circle":
    1. Hymenaei: by Ben Jonson, used a circle dance
    2. Oberon: by Ben Jonson, used a circle dance
    3. 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")

See F. Giordi's Harmonic Ratios

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 mensuration 7.

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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