Ad Quadratum Construction and Study of the Regular Polyhedra

Ad Quadratum and Music
Music and World View:
128
Pythagoras is generally credited for introducing mathematics into music, i.e., associating
intervals between pitches with ratios of numbers representing frequencies or lengths of
resonating devices such as taut strings or wind pipes.
However, the idea is, in all probabilities, much more ancient. Sumerian, Babylonian,
Egyptian, Chinese civilizations show evidence that music, very early, was related to the
concept of number 34 . By Plato’s time music had evolved into two distinct and virtually
separate branches – one practical, having to do with musical performance, the other
theoretical, consisting of the study of the mathematical relationships between tones, or
more generally proportion.
The relationship between numbers and acoustical phenomena seems to have given the
ancients the notion that all other phenomena were also governed by numbers 35 . Music
being the most advanced science of the time, most other aspects of human experience
were cast according to its model, as in our own age, physics has been used as a paradigm
for most other branches of knowledge, a function that biology now seems poised to take
over.
It is the existence of such paradigms, which provides a degree of unity and coherence to
the world views that dominate historical periods. For instance, the dominance of musical
models 36 from antiquity to well into the 17 th Century in the mentalities of philosophers,
theologians, astronomers and astrologers as well as mathematicians, architects and
physicians made plausible the concept of universal harmony. Boethius (c. 480-524) is
among the first to have explicitly presented this idea through his division of music in
three types: musica mundana, or music of the universe, describing the movement of the
celestial spheres; musica humana, or music of the human being, expressing the
correspondence between soul and body; and musica instrumentalis, instrumental (and
vocal) music. Only the last is concerned with acoustic phenomena while the other two
relate to proportions and intellectual harmony.
The human mind being so designed that it seems perpetually in search of a unifying
viewpoint to conceptualize its experience, found in the musical model an enduring and
34
McClain, Ernest G. The Pythagorean Plato. Nicholas-Hays, York Beach (Maine) 1978.
35
Dae- Am Yi. Musical Analogy in Gothic and Renaissance Architecture. Unpublished Ph.D. thesis, University of
Sydney, 1991.
36
“The eternal universe, originally without order and in all that pertained to it formless and devoid of all things that
are most clearly distinguished according to the categories of quality and quantity and the rest, was rendered discrete
by number as the most sovereign and artistic form, and its elements were given a most distinctive organization and it
came to partake of harmonious variation and perfect congruity in accordance with its affinity for and imitation of the
unique properties of music.” De Falco 44.7, [Iamblichi] Theologoumena Arithmeticae. Pp. 33-34, Leipzig, 1932. To
Robert Fludd (Utriusque Cosmi, 1619) the monochord is “the most exact symbol of the nature of the world and the
figure of truth itself” (Amman: The Musical Theory and Philosophy of Robert Fludd, pp. 223-224) and to Kepler it is
all part of a Universal harmony (Harmonices Mundi).