Rudolf Wittkower's "Architectureal 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")
Similar "female" and "male" sequences arise in the
problem of "mensuration" and "misura" in music (dance).
Examine the section dealing with dance, choreography
and music for further details, and the relation to
"corporeal rhetoric".
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. Relationships
between these 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 1
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 and architecture, all interrelated and bound
together using linear perspective and rhetoric tacens.
We focus on musical harmony. Given a triple of numbers found
in a ratio: <a:b:c> where the ratio at interest is a:c,
then "b" is the mean. Three different means may be found:
Thus
<2:3:4> has an arithmetic mean, as 3-2 = 4-3 = 1.
<4:6:9> has a geometric mean, as 4:6 = 6:9 (both are 2:3)
<6:8:12> has a harmonic mean, as (8-6)/6 = (12-8)/12 = ⅓
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.2
Using these ideas, Palladio and Alberti, as well as others
reformulated architecture. Palladio recommends seven rooms that
would have musical harmony (in this, he departs from the views
of Alberti, as Alberti didn't discuss circular rooms):
Circular 3
Square = <x, x>
Rectangular = <x√2, x>
Rectangular = <3x, 4x> (harmonic ratio = 4:3)
Rectangular = <2x, 3x> (harmonic ratio = 3:2)
Rectangular = <3x, 5x> (harmonic ratio = 5:3)
Rectangular = <1x, 2x> (harmonic ratio = 2:1)
The height of a room could be one of the three means:
arithmetic, geometric or harmonic. 4
Thus <6,9,12> has room height of 9 (arithmetic mean), as
<6,12> has room dimensions of <3x, 4x>, but could also have
a room height of 8 as <6,12> = <1:2>.
Thus "...the numbers by which the agreement of sounds
affects our ears with delight, are the very same which
please our eyes and our minds,...".
5
"We shall therefore borrow all our rules for harmonic
relations from musicians to whom this kind of numbers
is extremely well known, and from those particular
things wherein Nature shows herself most excellent and
complete."5
"For Alberti, harmonic ratios inherent in nature are
revealed in music. The architect who relies on those
harmonies is not translating those musical ratios into
architecture, but is making use of a universal harmony
apparent in music..." 5
"...when Palladio wants churches to be built in such
a manner and with such proportions that all the parts
together may convey a sweet harmony to the eyes of the
beholders'..." and "The proportions of the voices are
harmonies for the ears; those of the measurements are
harmonies for the eyes."6
"...for both, music and painting, convey harmonies;
music does it by chords and paintings by its proportions.
Musical intervals and linear perspective are subject to
the same numerical ratios, for objects of equal size
placed so as to recede at regular intervals diminish in
'harmonic' progression." Quote from Leonardo da Vinci.7
1
"Architectural Principles in the Age of Humanism",
by Rudolf Wittkower, W. W. Norton, New York, 1971, p. 104
2
Ibid., p. 16
3
It is not clear that non-circular rooms (square or rectangular)
exhausted all possibilities. Thus a room that is a regular hexagon
with one length "x" might be included, as well as other regular
shapes such as pentagons, etc.
4
"Architectureal Principles in the Age of Humanism",
by Rudolf Wittkower, W. W. Norton, New York, 1971, p. 108, 109
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