Gaffurius Woodcuts of Pythagoras; Ratio, Logos and the Music of the Spheres

A page from Gaffurius’ Theorica musicae (1492).
Woodcut showing Pythagoras with bells, a kind of glass harmonica, a monochord with weighted strings and (organ?) pipes,
all in Pythagorean tuning.

The story of how Pythagoras discovered the ratios that make up a musical scale while passing by a blacksmith’s, and how that led to the grand theme of “The Music of the Spheres” is legendary. In fact, we must take nearly everything we know about Pythagoras as legend because students in his brotherhood were sworn to secrecy. Nevertheless, Pythagoras, sometimes called “the first pure mathematician” made his ideas known far and wide. His influence on Aristotle, Plato and the Neo-Platonic philosophers was heralded across the Mediterranean, the Middle East, and continues to this day.

The woodcut (1492) depicts Pythagoras (570-495 BC) performing various experiments concerning the mathematical ratios of musical intervals. According to legend, Pythagoras discovered the foundations of musical tuning by listening to the sounds of four blacksmith’s hammers, which produced consonance and dissonance when they were struck simultaneously. Pythagoras rushed into the blacksmith shop to discover why certain hammers produced consonant tones while other hammers did not. It is said that he found that the explanation was in the weight ratios. The hammers weighed 12, 9, 8, and 6 pounds respectively. Hammers A and D were in a ratio of 2:1, which is the ratio of the octave. Hammers B and C weighed 9 and 8 pounds. Their ratios with hammer A were (12:9 = 4:3 = perfect fourth) and (12:8 = 3:2 = perfect fifth). The space between B and C is a ratio of 9:8, which is equal to the musical whole tone, or whole step interval. [1]

The legend is, at least with respect to the hammers, demonstrably false. It is probably a Middle Eastern folk tale.[2] These proportions are indeed relevant to string length (e.g. that of a monochord) — using these founding intervals, it is possible to construct the chromatic scale and the basic seven-tone diatonic scale used in modern music, and Pythagoras might well have been influential in the discovery of these proportions (hence, sometimes referred to as Pythagorean tuning) — but the proportions do not have the same relationship to hammer weight and the tones produced by them.[3][4]

Earlier sources mention Pythagoras’ interest in harmony and ratio. Xenocrates (4th century BC), while not as far as we know mentioning the blacksmith story, described Pythagoras’ interest in general terms:

“Pythagoras discovered also that the intervals in music do not come into being apart from number; for they are an interrelation of quantity with quantity. So he set out to investigate under what conditions concordant intervals come about, and discordant ones, and everything well-attuned and ill-tuned.”[5]

Whatever the details of the discovery of the relationship between music and ratio, it is regarded[6] as historically the first empirically secure mathematical description of a physical fact. As such, it is symbolic of, and perhaps leads to, the Pythagorean conception of mathematics as nature’s modus operandi.[7] As Aristotle was later to write,

the Pythagoreans construct the whole universe out of numbers.[8]

The lower left quadrant of the woodcut pictures Pythagoras plucking a six-stringed “monochord” using weights to tighten the strings in place of gears or tuning pegs such as we would find today on guitars or other stringed instruments. Again, the ratios of the weights are arranged in a similar fashion as the length of the pipes, or the volume of water in the glasses, or the sizes of the bells pictured in the other quadrants of the woodcut.

But other accounts describe his monochord quite differently. It consisted of a single stretched gut-string attached at either end of a rectangular box with a movable bridge-piece, which enables one to demonstrate the harmonic laws by shortening or lengthening the part of the string to be plucked by moving the bridge.

(In this modern version you can see a tuning peg at the left end in place of the weights tied to the end of the strings in Gaffurius’ woodcut). If you put the bridge exactly in the middle, you will have mathematical ratio of 1:2 (that is, the ratio of the half to the whole); when plucked, the monochord will sound a perfect octave. If you set the movable bridge at three-fifths of the length of the string, creating a ratio of 3:2, the tone produced will be a perfect fifth – and so on. Pythagoras placed a sacred significance on the ratios that he would use to define his harmonic scale, believing these same ratios to be the foundation underlying all of creation – the harmony of the spheres. Nevermind that centuries later astronomer Johannes Kepler would puzzle over why his painstakingly accurate measurments of the planetary orbits did not follow these “divine” perfect circles, that the orbits were in fact elliptical! Pythagoras did not know yet about the Sun’s gravitational effect on orbital mechanics, as Kepler would learn from Newton. He also had not yet come to understand that the Sun was at the center of our solar system, not “one and one-half tones from Venus”!

Even though Pythagoras was the first to teach that the Earth is spherical, astronomy was still in its infancy. The heliocentric theory of the solar system would not come to be generally understood until Copernicus in the 16th century. Aristarchus of Samos (310-230 BC) , did present such a model that put the Sun at the center of the solar system, but it would be several hundred years before Copernicus would credit the work of Aristarchus and provide further proof. Aristotle adopted Ptolemy’s model of the geocentric solar system placing the Earth at the center and it worked so well that sailors navigated successfully by it for centuries, irregardless of it’s central tenet being wrong!

To understand why Pythagorean ideas carried such weight, especially in light of so many cases where the numbers just don’t add up, it is important to realize that while he may have been the “first pure mathematician“, the mathematics of philosophers in those times involved an inner dimension of special significance. In his Metaphysics (350 BC) Aristotle seems to take issue with this Pythagorean fascination with numbers.

“The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.”

Now it is true that Pythagoras made substantial contributions to mathematics, music and astronomy. His Pythagorean Theorem is standard fare in math texts.

The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).

Pythagoras’s biographers state that he also was the first to identify the five regular solids and that he was the first to discover the Theory of Proportions.

There is some debate among historians as to whether Pythagoras was the first to think of these, or merely the first to bring these teachings from earlier Babylonian and Indian sources to Greece. Irregardless, he understood the significance of these ideas and through his work these concepts found many applications and continue to be important to this day.

But according to Aristotle, the Pythagoreans used mathematics for solely mystical reasons, devoid of practical application.[Burkert, 1972] They believed that all things were made of numbers.[Kahn 2001] The number one (the monad) represented the origin of all things[Reidweg, 2005] and the number two (the dyad) represented matter. The number three was an “ideal number” because it had a beginning, middle, and end[Joost-Gaugier, 2006] and was the smallest number of points that could be used to define a plane triangle, which they revered as a symbol of the god Apollo. The number four signified the four seasons and the four elements. The number seven was also sacred because it was the number of planets and the number of strings on a lyre, and because Apollo’s birthday was celebrated on the seventh day of each month. They believed that odd numbers were masculine,[Burkert, 1972] that even numbers were feminine, and that the number five represented marriage, because it was the sum of two and three.

Ten was regarded as the “perfect number”[Burkert, 1972] and the Pythagoreans honored it by never gathering in groups larger than ten. Pythagoras was credited with devising the tetractys, the triangular figure of four rows which add up to the perfect number, ten. (See my post on Tetraktys in Seed of Life.) The Pythagoreans regarded the tetractys as a symbol of utmost mystical importance. Iamblichus, in his Life of Pythagoras, states that the tetractys was “so admirable, and so divinised by those who understood [it],” that Pythagoras’s students would swear oaths by it.

In Raphael’s fresco The School of Athens, Pythagoras is shown writing in a book as a young man presents him with a tablet showing a diagrammatic representation of what may be a lyre above a drawing of the sacred tetractys.

It is difficult for us today to tease apart the mathematics from the numerology. But more importantly we might ask, why do we feel compelled to do so? C.S. Lewis hid astrological symbols throughout his Chronicles of Narnia because as a scholar of medieval literature and thought, he could see how modern society was losing an appreciation for the mystery of inner meaning and the power of metaphor. To the medieval mind, and to all who preceded them, one could find layers of meaning in all things. It would not be until the 17th century, the so-called Age of Enlightenment, when science and “Newton’s sleep” would begin to train us to turn a blind eye to such things as metaphor and meaning. [See Planet Narnia: The Seven Heavens in the Imagination of C.S. Lewis by Michael Ward, or the BBC documentary based on his book.]

Revivals of Pythagorean philosophy would emerge regularly; Plato, Eudorus and Philo of Alexandria. Apollonius of Tyana sought to emulate Pythagoras and live by his teachings, and mathematician/musician Nichomachus expanded on Pythagorean numerology and music theory. The theme of the music of the spheres never went away, it only deepened with time. However much the mathematics and astronomical discoveries altered the details, the grand theme persists even to the string theory, gravitational waves and multiverse theories of today.

The Greek word for ratio is logos, which also means word, thought, and reason. To the early Christian philosophers, several hundred years after Pythagoras, logos had a special significance in the light of the first verse of the Gospel of John, sometimes referred to as “Hymn to the Word [Logos]”:

“In the beginning was the Word [logos], and the Word was with God, and the Word was God.”

Clement of Alexandria (150-215 AD) and the other early church fathers who wrote on music could argue that Pythagoras’s identification of ratio, or logos, with the divine principle of universal order harmonized with the gospel’s identification of logos with God, of which Jesus was the manifestation. Jesus was the

“new song which composed the entire creation into melodious order, and tuned into concert the discord of the elements, that the whole universe might be in harmony with it.”

In his Exhortation to the Greeks, written in the late second century to reveal the errors of paganism and the perfect truth of Christianity, Clement ridicules the Greek myths concerning music, especially the legends about Orpheus, whom he calls the “Thracian wizard.” (- The Music of the Spheres, Music, Science and the Natural Order of the Universe by Jamie James, 1993).

But the beat goes on … The logos of the Gospel of John shows up in a different guise than the logos of Pythagorean ratios, but it is still Logos.

The woodcuts are by Franchinus Gaffurius (Franchino Gaffurio; 14 January 1451 – 25 June 1522) who was an Italian music theorist and composer of the Renaissance. He was an almost exact contemporary of Josquin des Prez and Leonardo da Vinci, both of whom were his personal friends. He was one of the most famous musicians in Italy in the late 15th and early 16th centuries.

Portrait of a Musician (possibly Gaffurius) by Leonardo da Vinci

Gaffurius was born in Lodi to an aristocratic family. Early in life he entered a Benedictine monastery, where he acquired his early musical training; later he became a priest. He then lived in Mantua and Verona before settling in Milan as the maestro di cappella at the cathedral there, a position which he accepted in January 1484.

Gaffurius retained the post at the cathedral for the rest of his life, and it was in Milan that he knew both Josquin des Prez and Leonardo da Vinci.

Gaffurius was widely read, and showed a strong humanist bent. In addition to having a thorough understanding of contemporary musical practice, he met composers from all over Europe, since he had the good fortune to be living and working at one of the centers of activity for the incoming Netherlanders. His books have a pedagogical intent, and provide a young composer with all the techniques necessary to learn his art.

The major treatises of his years in Milan are three: Theorica musicae (1492), Practica musicae (1496), and De harmonia musicorum instrumentorum opus (1518). The second of these, the Practica musicae, is the most thorough, proceeding through subjects as diverse as ancient Greek notation, plainchant, mensuration, counterpoint, and tempo. One of his most famous comments is that the tactus, the tempo of a semibreve, is equal to the pulse of a man who is breathing quietly—presumably about 72 beats per minute. [This is useful information for Music Practitioners providing live acoustic music at the bedside].

The information on Gaffurius is from "Franchinus Gaffurius," "Leonardo da Vinci" in The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie.  20 vol.  London, Macmillan Publishers Ltd., 1980.   
Some of the information on Pythagoras' hammers and his other experiments involving musical ratios are from the following sources:

1.  Weiss, Piero, and Richard Taruskin, eds. Music in the Western World: A History in Documents. 2nd ed. N.p.: Thomson Schirmer, 1984.
2.  Kenneth Sylvan Guthrie, David R. Fideler (1987). The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings which Relate to Pythagoras and Pythagorean Philosophy, p.24. Red Wheel/Weiser. 
3. Christensen, Thomas, ed. The Cambridge history of Western music theory. Cambridge: Cambridge University Press, 2002. 143.
4. Burkert, Walter (1972). Lore and Science in Ancient Pythagoreanism, p.375.  
5. Barker (2004).  Andrew (ed.). Greek musical writings (1st pbk. ed.). Cambridge: Cambridge University Press. p. 30. 
6. Lucas N.H. Bunt; Phillip S. Jones; Jack D. Bedient (1988). The historical roots of elementary mathematics (Reprint ed.). New York: Dover Publications. p. 72. 
7. Christian, James. Philosophy An Introduction to the Art of Wondering. Wadsworth Pub Co. p. 517. 
8. Waterfield, transl. with commentary by Robin (2000). The first philosophers : the Presocratics and Sophists (1. publ. as an Oxford world's classics paperback ed.). Oxford: Oxford Univ. Press. p. 103. 

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